SystematicOptions Trading

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SystematicOptionsTradingEvaluating, Analyzing, and Profiting from

Mispriced Option Opportunities

Sergey Izraylevich and Vadim Tsudikman

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Printed in the United States of America

First Printing September 2010

ISBN-10: 0-13-708549-4

ISBN-13: 978-0-13-708549-1

Pearson Education LTD.Pearson Education Australia PTY, Limited.Pearson Education Singapore, Pte. Ltd.Pearson Education North Asia, Ltd.Pearson Education Canada, Ltd.Pearson Educatión de Mexico, S.A. de C.V.Pearson Education—JapanPearson Education Malaysia, Pte. Ltd.Library of Congress Cataloging-in-Publication Data

Izraylevich, Sergey, 1966-Systematic options trading : evaluating, analyzing, and profiting from mispriced option opportunities / Sergey Izraylevich and Vadim Tsudikman.

p. cm.ISBN-13: 978-0-13-708549-1 (hardback : alk. paper)ISBN-10: 0-13-708549-4

1. Stock options. 2. Options (Finance) 3. Investment analysis. 4. Portfolio management. I. Tsudikman, Vadim, 1965- II. Title. HG6042.I97 2011332.63’2283—dc22

2010006153

This book is dedicated to Professor Uri Gerson, Hebrew University of Jerusalem.

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

PART I Criteria as the Basis of a Systematic Approach

chapter 1 General Presentation and Review of Criteria Properties . . . 3

1.1 The Main Tool for Solving the Selection Problem . . . 3

1.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Philosophy of Criteria Creation . . . . . . . . . . . . . . . . . 5

1.4 Mission Fulfilled by Criteria . . . . . . . . . . . . . . . . . . . . 6

1.5 Forecast as a Key Element of the Criterion . . . . . . . . . 8

1.6 Classification of Criteria . . . . . . . . . . . . . . . . . . . . . . . 9

1.6.1 Universal Criteria . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6.2 Specific (Nonuniversal) Criteria . . . . . . . . . . . . 11

chapter 2 Review of the Main Criteria . . . . . . . . . . . . . . . . . . . . . 132.1 Criteria Based on Lognormal Distribution . . . . . . . . 13

2.1.1 Description of Lognormal Distribution . . . . . . 13

2.1.2 Expected Profit on the Basis of LognormalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Profit Probability on the Basis of LognormalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Criteria Based on Empirical Distribution . . . . . . . . . 22

2.2.1 Description of Empirical Distribution . . . . . . . . 22

2.2.2 Expected Profit on the Basis of EmpiricalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Profit Probability on the Basis of EmpiricalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.4 Simplified Calculation Algorithm . . . . . . . . . . . 28

2.2.5 Modifications of Empirical Distribution . . . . . . 31

2.3 Criteria Based on the Ratio of ExpectedProfit to Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Basic Concept and Criteria CalculationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.2 Criteria Calculation Example . . . . . . . . . . . . . . 36

2.4 Criteria Based on Expert Distribution . . . . . . . . . . . 38

2.4.1 Basic Concept and Criteria CalculationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.2 Set of Standard Distributions . . . . . . . . . . . . . . 39

2.4.3 Combining Separate Standard Distributionsinto a Unified Probability Density Function . . . . . . 45

2.4.4 Criteria Calculation on the Basis of the UnifiedProbability Density Function . . . . . . . . . . . . . . . . . 47

2.4.5 Construction and Valuation of ComplexStrategies Based on the Unified ProbabilityDensity Function . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5 Specific (Nonuniversal) Criteria . . . . . . . . . . . . . . . . 50

2.5.1 Break-Even Range . . . . . . . . . . . . . . . . . . . . . . . 50

2.5.2 IV/HV Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5.3 Relative Frequency Criterion . . . . . . . . . . . . . . 57

2.5.4 The Ratio of Normalized Time Value to theCoefficient of Absolute Price ChangesDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

chapter 3 Evaluation of Criteria Effectiveness . . . . . . . . . . . . . . . . 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Methods of Criteria Effectiveness Evaluation . . . . . . 64

3.2.1 Correlation Between a Criterion andProfit as the Main Effectiveness Indicator . . . . . . . 64

3.2.2 Transformation of the Criteria EffectivenessIndicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.3 The Dynamics of Transformed EffectivenessIndicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.4. Selection of the Averaging Period . . . . . . . . . . . 72

viii Systematic Options Trading

3.3 Peculiarities of Criteria Effectiveness Evaluation . . . 75

3.3.1 Number of Combinations Used inthe Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Expressing Profit . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.3 Expressing Effectiveness Indicators . . . . . . . . . 80

3.4 Review of Criteria Effectiveness Indicators . . . . . . . . 84

3.4.1 Correlation Between Criterion andProfit Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4.2 Correlation Between a Criterion andProfit Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4.3 Correlation Between the Sharpe Ratios ofCriterion and Profit . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4.4 Areas Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4.5 Other Effectiveness Indicators . . . . . . . . . . . . . . 96

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

PART II The Main Areas of Criteria Application

chapter 4 Selection of Option Combinations . . . . . . . . . . . . . . . . 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 Analysis of Criteria Effectiveness in the Selectionof Option Combinations . . . . . . . . . . . . . . . . . . . . . 106

4.3 Factors That Affect Option CombinationsSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3.1 Absolute Values of the Criterion . . . . . . . . . . . 110

4.3.2 Strategy and Underlying Assets . . . . . . . . . . . . 112

4.3.3 Simultaneous Analysis of Factors AffectingCombinations Selection . . . . . . . . . . . . . . . . . . . . 113

4.4 Multistrategy, Long-Term Evaluation of CriteriaEffectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

chapter 5 Selection of Option Strategies . . . . . . . . . . . . . . . . . . 1215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Evaluation of Criterion Effectiveness by RankingAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Contents ix

5.2.1 Methods of Ranking Analysis . . . . . . . . . . . . . 123

5.2.2 Results of Ranking Analysis . . . . . . . . . . . . . . . 131

5.2.3 Generalized Ranking Analysis andIntroduction of the Threshold Parameter . . . . . . . 139

5.2.4 Results of Generalized Ranking Analysis . . . . . 140

5.2.5 Maximum Obtainable Values of theCriterion Effectiveness Coefficient . . . . . . . . . . . . 142

5.3 Traditional Methods of Evaluating the CriterionEffectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.4 Synthetic Approach to Criterion EffectivenessAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.5 The Model for Optimizing the ThresholdParameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

chapter 6 Selection of Underlying Assets . . . . . . . . . . . . . . . . . . 1616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.2 Analysis of Criteria Effectiveness in Selectionof Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.3 Multistrategy, Long-Term Evaluation ofCriteria Effectiveness . . . . . . . . . . . . . . . . . . . . . . . 166

6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.4 The Optimization Model for the Number ofUnderlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.4.1 Utility Indicators . . . . . . . . . . . . . . . . . . . . . . . 169

6.4.2 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . 171

6.4.3 Convolution of Utility Functions and DerivingOptima for Different Strategies and Criteria . . . . 173

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

x Systematic Options Trading

PART III Multicriteria Analysis

chapter 7 Basic Concepts of Multicriteria Selection as

Applied to Options . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 The Pareto Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

7.2.1 The Algorithm of Forming the Pareto Set . . . 183

7.2.2 Widening the Pareto Set and the “Layer”Notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.4 Comparative Analysis of Multicriteria andMonocriterion Selection Effectiveness . . . . . . . . . . . 191

7.5 Comparative Analysis of Two MulticriteriaSelection Methods: Pareto Versus Convolution . . . . 197

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

chapter 8 The Impact of Criteria Correlation on Multicriteria

Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.2 Evaluation of Criteria Interrelationship . . . . . . . . . 206

8.3 Criteria Correlation and Profitability of ParetoSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.4 Criteria Correlation and Profitability ofSelection Using the Convolution Method . . . . . . . . . 212

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

appendix Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Contents xi

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xiii

Acknowledgments

The authors would like to express their gratitude to the team at High Technology Invest Inc.Special thanks are due to Arsen Balishyan, Ph.D., and Vladislav Leonov, Ph.D., for theirskillful assistance in research and manuscript preparation. We are also indebted to MikhailKolkovsky, Sergey Anikeev, and Eugen Masherov, Ph.D., for their useful comments andcontinued help at all stages of this project.

About the Authors

xiv Systematic Options Trading

Sergey Izraylevich, Ph.D., chairman of the board of High Technology Invest Inc., beganhis career as a lecturer at The Hebrew University of Jerusalem and Tel-Hay AcademicCollege. He received numerous awards for academic excellence, including Golda Meir’sprize and the Max Shlomiok honor award of distinction. Sergey has traded options formore than 10 years and engages in creating automated systems for the algorithmic tradingof options. He is the author of numerous articles published in highly rated, peer-reviewedscientific journals. Sergey is a columnist for Futures magazine.

Vadim Tsudikman, president of High Technology Invest Inc., is a financial consultant andinvestment advisor specializing in derivatives valuation, hedging, and capital allocation inextreme market environments. With 12 years of options trading experience, he developscomplex trading systems based on multicriteria analyses and genetic optimizationalgorithms. In co-authorship with Sergey Izraylevich, Vadim contributes articles toFutures magazine on the cutting-edge issues related to options pricing, volatility, and riskmanagement.

Introduction

xv

What the Book Is About and Who Should Read It

This book discusses the procedures of multidimensional search, selection, and utilization ofpotential trading opportunities existing in the options market. It contains no magic rulespromising quick and guaranteed enrichment. Instead, you find comprehensive research aimedat discovery and practical application of statistical regularities and probabilistic characteristicsof option trading. The aim of our systematic approach is not the creation of ever-winningstrategies. Rather we strive to implement a realistic idea—developing a system of algorithmsand rules that provide you with statistical advantage over the average market participant. Thetrading system based on consistent application of the principles discussed in this book enablesyou to create and maintain positions with high (higher than the market average) expectedprofits and lower forecast risks.

The substantial part of this book is devoted to the problem of selection. Statistical edge andprobabilistic advantages depend on our ability to select the best variants from a great numberof available alternatives. The options market is incredibly broad and diverse, whereaspromising trading opportunities are rare and hard to identify. To avoid missing the chance todiscover these scarce “pearls,” an ample quantity of alternatives should be thoroughlyestimated and analyzed. Continuous analyses of large data sets covering the entire optionsmarket is the only way to identify sparse trading opportunities that can be described as “thebest of the available ones.” Therefore, the issues related to the development, optimization,and practical application of selection criteria are discussed broadly and examined in depththroughout the book.

This book is intended for you—traders, investors, portfolio managers, theoreticians, andeconomists—with different grounding level in options and mathematics.

If you are familiar with the basics of statistics and probability theory and have masteredthe fundamentals of option trading, you can now proceed to reading this book. For those ofyou who have no previous experience with options but are familiar with the first two disci-plines, we recommend you to start with the appendix in which we list the main definitionsand explain the notions and terms that are necessary to understand the contents of the book.

Those who are not familiar with probability theory and statistics have two options. Youcan start reading without delving into proofs and arguments, focusing rather on patterns andregularities described in the text and on conclusions resulting from them. In this case you mustrely on the results presented by the authors and fully trust the validity of their judgments andconclusions. An alternative way, which is to dig into the basics of statistics and probability

theory, can enable you to examine the material of the book critically. Even superficialknowledge of the basics of these subjects provides an opportunity to form your own opinionon many important issues of option trading. The first way can take less of your time andeffort, whereas the second one allows for getting the most out of this book.

xvi Systematic Options Trading

IntroductionHistory of options dates back thousands of years. Social and commercial relations governedby rules similar to option terms came into existence at the dawn of human society. Variousrecords are found in ancient documents and archeological sources dating back to the ages ofPentateuch. In Genesis Jacob purchased an option to marry Laban’s daughter Rachel inexchange for 7 years of labor. His prospective father-in-law, however, reneged, perhapsmaking this the first precedent of option default. Laban required Jacob to marry his olderdaughter Leah. Jacob obeyed his will, but because he loved Rachel, he purchased anotheroption requiring 7 more years of labor. He exercised the second option on the expirationdate and finally married his sweetheart.

Before the early 1970s the options market was poorly organized. Most transactions wereexecuted over the counter, often through the mediation of banks or other financial institu-tions. Essential terms of trade were not standardized, and in each case they were establishedthrough negotiations of the parties concerned. There was no formal and objective pricingmechanism that could be used as the starting point to determine the option premium. Thewatershed point happened in 1973, when two events brought about a fundamental changein the financial world. This was the year when Fischer Black and Myron Scholes publishedtheir famous option pricing model (Black & Scholes, 1973) and the Chicago Board OptionsExchange (CBOE) began trading standardized option contracts. The first event providedtraders with a formalized algorithm of option pricing. Despite numerous drawbacks, thispricing model had one indisputable advantage: It enabled the comparison of market priceswith a benchmark value. The second event initiated the development of an organizedoptions market. This process is still underway today involving a growing number of investorsand financial flows in option trading.

At the dawn of the new millennium, an important milestone in the development of theworld derivatives market was passed. For the first time the volume of exchange tradedoptions (having less than 30 years of history) exceeded the volume of futures traded since1848. Since then, options have been continuously dominating among other derivatives.

The undisputed leader in option trading is the U.S. market that absorbs more than two-thirds of the world trading volume. An important peculiarity of the U.S. options market isthe competition of many exchanges offering the same product (that is, options for the sameunderlying asset). Although CBOE surpasses other exchanges with regard to the volume oftraded contracts (approximately 30% of the total option trading volume), none of them

controls more than one-third of the market. Such a competitive environmentcontributes to liquidity growth, spread shortening, and commission declining thatattracts new market participants.

The prospects of options market development are beyond any doubt. Every yearbrings in additional financial flows; new trading strategies evolve; and option-basedadvanced structured products become more and more popular. As time goes by, theinfluence of large institutional investors will strengthen. In the last several years hedgefunds became one of the dominating market drivers, and analysts forecast furtherinflow of their capital into the option trading. At the same time activity of individualinvestors on the derivatives markets is expected to become more intense.

The area of options application is extremely wide. Mutual funds, banks, andinvestment companies use options as an instrument to regulate their investment risks.Buying Put options prevents financial institutions from liquidating long positionswhen they anticipate the underlying asset plunging. On the other hand, when marketgrowth is forecast, buying Call options limits potential losses (if the forecast fails) tothe premium paid for options. Buying options also creates considerable leverageadding to the investment potential and increasing the effectiveness of assetmanagement.

Producers of various goods and consumers of raw materials use options to hedgethe risks of market price fluctuations. For example, by purchasing a Put option, an oilproducer ensures that its future output will be sold at a price not lower than the optionstrike price. This is the way the company can be secured against a possible fall in theprice of its production. On the other hand, an oil-refining company can buy a Calloption for oil, thereby ensuring that its raw material will be purchased at a price nothigher than the option strike price. Thus, the oil consumer can be secured against theprice growth. International companies can hedge currency risks of their export/importoperations by purchasing corresponding currency options.

Use of options to manage risks is called hedging. Another area of applying options,often opposed to hedging, includes a class of speculative strategies aimed at earningprofits by creating various structures composed of long and short options.

Speculative option strategies give investors broad opportunities incomparable withpossibilities provided by other financial instruments in respect to their flexibility andpotential profitability. The main feature of options distinguishing them from themajority of other financial instruments is the nonlinearity of their payoff function(which is the relationship between profit and the future underlying asset price). Thisfeature enables the creation of option combinations possessing almost any desiredprofit profile that makes options an indispensable instrument in achieving variousgoals for many financial market participants.

This book is intended for investors who strive to make profits using speculativeoption trading. The principles described here can be applied to all option strategies.

Introduction xvii

Although some of them are frequently used to demonstrate the techniques of discovering thetrading opportunities, whereas other strategies are not even mentioned, this selectivity ismerely due to our wish to keep the text within reasonable limits.

The systematic approach presented in this book is based on universal principles that canbe applied to options on any type of underlying assets: stocks, futures, currencies, interestrates, and commodities. The same is true regarding different markets: Despite certainnational specificities in legislation and regulation terms, options markets of all countries aresuited for implementing the systematic approach.

To illustrate the different aspects of the systematic approach and to demonstrate itspotential effectiveness in exploring the opportunities of option trading, we use historicaldata from U.S. exchanges. The research described in this book is based on a databasecontaining 7 years of price history of 2,500 stocks and their options.

xviii Systematic Options Trading

Options: What Is Known and What Is NotIn the 1980s, when the first option exchange and the first pricing model emerged, optionsmarkets began to develop so fast that the existing theoretical background could not satisfyincreasing practical needs. The facilities required to store and to process informationincoming from trading floors were not yet established. As a result, statistical data processingand theoretical developments could not satisfy growing demands of market professionals.

However, as time goes by the stream of information grows and the scope of theoreticalresearch widens. Every year brings more and more professional publications on the subject.Options are thoroughly studied at universities, becoming one of the most popular topics ofeconomical, mathematical, and interdisciplinary research. Option exchanges arrangeseminars popularizing basic knowledge among beginners and organize advanced-levelcourses intended for market professionals.

A significant bulk of knowledge on options has been accumulated. These attainments aresystematized and published in popular and professional sources. The literature on optionscan be divided into two main categories.

The first category includes theoretical research on the basis of financial mathematics. Asubstantial number of scientific articles and books are devoted to the development ofadvanced option pricing models. They apply probability theory and discuss various compli-cated issues, including volatility abnormalities, nonlinearities, and interrelationshipsbetween parameters.

A strict and extensive mathematical background of option theory is given by Peter James(James, 2003). This book represents the basis for researchers entering the world of options,though its complexity makes it comprehensible only to specialists with deep knowledge ofmathematics. The basics of derivatives theory are perfectly described by John Hull (Hull,

2008). This is a textbook that covers all essential issues from basic terminology to compli-cated problems of financial mathematics. Option pricing is widely discussed in Espen Haug’sbook (Haug, 2006). It can be used as a universal handbook covering up-to-date progress inprice modeling (see also Achdou and Pironneau, 2005, Rouah and Vainberg, 2007). Mathe-matical fundamentals of derivatives theory (not only options) are widely covered by SalihNeftci (Neftci, 2000). Although the majority of theoretical works have not yet been imple-mented, some of the mathematical models are widely used by option exchanges, brokers,market-makers, and traders. The ability to apply theoretical attainments becomes increas-ingly essential and publications dedicated to this issue gain considerable practical value(Reehl, 2007).

Various aspects of volatility modeling and their implications on derivatives pricing werereviewed by Jim Gatheral and Nassim Taleb (Gatheral and Taleb, 2006). The authorsexamine all main properties of stochastic, local, and implied volatilities and describe manyclassical and advanced mathematical models. A special emphasis is placed on the dynamicproperties of the volatility surface and its relationship to options valuation. The discussionof volatility derivatives, barrier end exotic options is of particular interest. Besides this work,the theoretical problems of volatility modeling and forecasting were comprehensivelytreated by Ser-Huang Poon and Riccardo Rebonato (Poon, 2005, and Rebonato, 2004).

The second category of publications is based on practical option trading and summarizesthe experience accumulated by market practitioners. It discusses strategies based oncombining different options and describes methods of building desirable profit profiles onthe basis of option positions structuring (Banks and Siegel, 2007, Cohen, 2005, Cohen,2009, Courtney, 2008, Vine, 2005). Strong emphasis is placed on methods of derivingarbitrage profit.

Lawrence McMillan is a widely known author of popular books on options. His publi-cations (McMillan, 2002; Lehman and McMillan, 2003) include a detailed description ofdifferent option strategies and are extremely useful. The author highlights a multitude ofversatile techniques indispensable for any option trader. Plentiful examples based on realmarket data, simple language, and broad coverage—those are the distinguishing features ofhis books. You can find not only an encyclopedic review of option strategies in McMillan’sbooks, but also a comprehensive description of delta-neutral hedging, arbitrage, and otherspecific techniques.

An excellent example of a handbook covering most aspects of option trading is the workby Michael Thomsett (Thomsett, 2009). An option trader can find there a detaileddescription of many useful strategies. The problems of return calculation and risk evaluationare also discussed in detail. The author gives much attention to technical aspects of optiontrading—information sources, taxation, accounting for dividends, and so on.

Books in which authors do not limit themselves to mere review of option strategies butdiscuss serious theoretical and practical issues without involving complicated mathematicsare also helpful.

Introduction xix

Sheldon Natenberg (Natenberg, 1994, 2007) describes the key elements of option theoryin a popular and yet precise language. He discusses the peculiarities of implied volatilitybehavior and investigates the characteristics of the Greeks and specifics of their applicationas the instruments of risk analysis. Without superfluous mathematics, the author investigatessuch important phenomena as volatility smiles and skews. Comprehension of complicatedtheoretical issues is facilitated by intelligible charts and tables.

The book by Allen Baird (Baird, 1992) targets the same audience. Being a fairly compre-hensive introduction to option theory and practice, it spares the reader the wilds of compli-cated mathematics. Accurate description of risk management basics is among the mainvantages of the book. The section devoted to the most typical mistakes made by tradingbeginners also deserves a special mention.

Certain books are dedicated to specific option strategies. For example, the idea ofvolatility trading is popularly described in the work of Kevin Connolly (Connolly, 1997).Without resorting to complicated mathematics, the author applies dynamic hedging tocombinations consisting of options and their underlying assets.

The book by Nassim Taleb (Taleb, 1997) also discusses various aspects of dynamichedging and peculiarities of delta-neutral volatility trading strategies. This is the workwritten by a professional with years of experience in risk management. Although containingsome inevitable portion of mathematics, it is still comprehensible to the majority of readers.In most cases the author uses diagrams and tables instead of formulae.

The book by Leonard Yates (Yates, 2003) belongs to the same category. The authordiscusses interesting ideas and gives ground for original trading strategies based, inparticular, on negative correlation between VIX and S&P indices. The strategy is testedusing historical data and the results indicate its potential applicability.

Many particular features of options trading were recently covered in impressive depth.These include pricing and risks associated with exotic options (De Weert, 2008), applicationof foreign exchange (Wystup, 2007), and commodity options (Garner & Brittain, 2007),trading at expiration (Augen, 2009), intraday trading (Augen, 2009), protective strategiesbased on Put options, and so on.

Our knowledge on options goes beyond the literature dedicated to this narrow topic.Theory and practice of option trading apply various elaborations originating from differentareas of finance, statistics, probability theory, and applied mathematics. For example,creating their classical option pricing model, Black and Scholes used the well-knownlognormal distribution that was widely discussed and cited in statistical and mathematicalliterature. Later other authors created their own pricing models using other known distributions.

The potential benefit of adopting ideas from adjacent scientific fields is far from beingexhausted. For example, in classical option pricing theory the assumption of randomness ofunderlying asset price changes is the most questionable issue. Basically, it follows fromapplying lognormal distribution and means that the underlying asset price moves according

xx Systematic Options Trading

to geometrical Brownian motion laws. The work of Edgar Peters (Peters, 1996) representsan interesting example of a more sophisticated approach to the description of pricebehavior. It applies chaos mathematics, fractal theory, and nonlinear dynamics to accountfor asset price fluctuations. Peters claims that these models describe price behavior moreaccurately than standard probability distribution functions. Therefore, their applicationopens the gates for more accurate option price modeling. There is a lot of work to be donehere, and new research of physicists and mathematicians will surely contribute to elaboratingoption theory.

The up-to-date achievements in the sphere of options theory can be summarized asfollows. There is an adequate, albeit with certain drawbacks, option pricing model.Numerous versions of the basic model, eliminating some of its drawbacks and making theestimations more accurate, are also available. The basic principles of creating option pricingmodels, based on assumptions about the main underlying asset characteristics, are reliablyestablished. Basic option risk indicators (“the Greeks”) are grasped. We know their features,interrelationships, and applicability in different situations. Various aspects of impliedvolatility behavior, including its dynamics, specific relationships with different parameters,and numerous anomalies, are profoundly investigated. We also possess an extensive set ofadvanced option strategies allowing construction of almost any desired payoff profile.

Despite this impressive progress, some important aspects still remain beyond theoreticaland practical studies. Next we summarize issues still requiring additional investigation.

The main topic of theoretical research (though directly related to investment practice) isthe determination of the fair option value. The term fair value stands for the price thatimplies zero profit for both option sellers and buyers. This requires creation of realisticoption pricing models (Katz and McCormick, 2005). It is common knowledge that apartfrom parameters that are objectively defined (current underlying asset price, strike, risk-freeinterest rate, and so on), the option price is determined by the forecast of underlying assetprice dynamics. In the classical model this forecast is expressed by a probability densityfunction of lognormal distribution that is specified by two parameters: variance derivedfrom historical volatility and mean value that is usually considered to be equal to the currentprice. This form of forecast has a number of drawbacks, though attempts to use other proba-bility distributions gave only local improvements and added new drawbacks. Hence the maingap in option theory can be defined as the absence of alternative methods for creating proba-bility forecasts of the future underlying asset price.

If the price is assumed to be a continuous value, then the forecast can take the form ofprobability density function. The construction of such functions should be the principaltopic of future research. We consider attempts to create one universal function for all casesto be unproductive. It should rather be a set of rules and algorithms for generating a wholeclass of density functions, each of which will be appropriate in certain conditions. The devel-opment of effective algorithms generating appropriate probability density functions willminimize the difference between modeled prices and fair values of options.

Introduction xxi

Apart from developing high-quality probabilistic forecasts, further research should targetthe development of optimization algorithms for parameters used in option pricing models.Even in the Black-Scholes model—which is relatively simple and contains only a few param-eters—the outcome strongly depends on the variance value. Historical volatility, which isusually used to derive variance, depends on the length of the historical period used for itscalculation. The value of this parameter can change the modeled option price considerably.As models become more complicated, the number of parameters increases and theircombined influence becomes more pronounced.

Another essential drawback of option theory consists in the insufficient development ofspecific risk indicators. (Some alternative indicators are described in Izraylevich andTsudikman, 2009d, 2010.) The majority of works on this issue are based on calculating theGreeks that are derivatives of the option price with respect to the underlying asset price(delta), volatility (vega), time (theta), and the interest rate (rho). (Derivatives of higherorders are also used.) Derivatives are calculated analytically using formulae of option pricingmodels. This implies that risk indicators obtained in this way inherit all the drawbacks ofinitial models. Such an approach to expressing option risks seems to be rather lopsided. Justas options market prices rarely match with the modeled ones, the Greeks calculated analyt-ically almost never coincide with real changes in option values. We believe that futureresearch of option risk management should focus on three main issues.

The first one relates directly to the problem of improvements in pricing models. Themodeling formulae should be modified to include not only high-quality probabilisticforecasts (previously mentioned) but also to enable calculation of useful indicators (deriva-tives or any other coefficients) that accurately reflect corresponding risks.

The second issue represents the empirical study of option price increments in responseto changes of underlying asset price, volatility, time to expiration, and risk-free interest rate.The patterns established in the course of these investigations can then be used (i) asindependent risk indicators, (ii) for adjustment of the Greeks derived analytically, and (iii)to calibrate option pricing models.

The third issue corresponds to the estimation of risks of an option portfolio as a wholeentity. Some risk indicators, such as theta and rho, are additive. Hence the dependence ofthe portfolio on time decay or interest rate change can be easily expressed as the sum ofthetas or rhos of all options included in the portfolio. On the contrary, delta and vega arenonadditive. Therefore, if the portfolio consists of options on several underlying assets,summing separate deltas and vegas is meaningless. One of the possible ways to solve thisproblem is to present the delta of each option as a derivative with respect to some index(such as S&P500 or NASDAQ) rather than with respect to the price of a correspondingunderlying asset. (This issue is discussed in Izraylevich and Tsudikman, 2009b.) In the sameway vegas of separate options can be expressed as derivatives with respect to volatility index(such as VIX or VXN) rather than with respect to volatility of a separate underlying asset.

xxii Systematic Options Trading

These procedures produce additive deltas and vegas that enable calculation of risk indicators(by summation of additive deltas and vegas) characterizing the whole portfolio. Other waysto estimate risks of a complex portfolio should also be examined. Research in this field willcertainly bring useful practical results.

In this review we outlined what is already known about options and how much still liesahead of us. We defined the main lines of future research that are, in our point of view, ofspecial interest. Some of the gaps in option knowledge are partially filled in this book.

Introduction xxiii

The Concept of the Systematic Approach

This section introduces the basic concept of the systematic approach including itsphilosophy, objectives, and methodology. Here we overview the essence of operationsrequired for consecutive execution of valuation, comparative analyses, and selection proce-dures. We strongly recommend you get acquainted with this material as it represents an all-embracing description of the general framework for systematic options trading.

The Goals and ObjectivesOne of the main issues in the option trading is the problem of selecting the best variantsamong many available alternatives. The choice is wide and the objects to examine and assessare compound structures. Although continuous functioning in such complicatedenvironment hampers the investment process significantly, it provides at the same time abroad spectrum of promising trading opportunities.

In the literature and in multiple services offered by brokerage firms and Internet sources,the problem of choice is generally solved through application of different market scannersand rankers. A typical scanner screens the market for underlying assets that currently haveextreme characteristics, such as divergence between historical and implied volatilities, dailyvolatility fluctuations, changes in trading volume, and so on. Afterward a ranker ordersunderlying assets according to the suitability for a particular option strategy. Then suitablecombinations should be designed for all chosen underlying assets. Because a great numberof combinations can be constructed for a given underlying asset within a given strategy, it isusually advised to use combinations’ payoff charts (the functional relationship between theprice of the underlying asset and a combination’s profit estimated for a certain future date)as a basis for their comparison and decision making. However, in most cases visual analysisis quite unfeasible if a large quantity of option strategies and underlying assets have to becompared simultaneously.

We regard the choice of suitable underlying assets for the a priori defined strategy as adifferential approach. It is a forced measure resulting from the imperfection of analyticaltools limited to simple scanning and visual analysis of payoff functions. Differential selection

deprives the investor of the potential to utilize the whole spectrum of various trading oppor-tunities provided by the market completely and effectively.

What we oppose to a differential approach is an integral systematic approach based onthe strictly formalized assessment criteria, universal procedures of multicriteria analysis, andwell-structured selection algorithms. The systematic approach enables simultaneousprocessing of a considerable number of option strategies and underlying assets. Withoutsuch an integral system, the investor has little or even no chance to make prompt selectiondecisions and to adapt successfully to changing market conditions.

The main goal of the systematic approach is to create a complex portfolio containing apotentially unlimited number of option combinations corresponding to a variety ofstrategies and underlying assets. Its application ensures that all trading opportunitiesappearing at any particular moment will be thoroughly estimated and none of the variantsworth considering will be omitted. The systematic approach is absolutely indispensable forturning option trading into a long-term continuous process of income generation withcontrollable parameters of risk and profitability.

Valuation, Comparative Analysis, and SelectionThe systematic approach is realized through consecutive execution of the following proce-dures: valuation, comparative analyses, and selection. These procedures are applied to themultitude of option combinations. The combination represents a complex structureconsisting of any number of long and/or short individual options corresponding to certainunderlying assets. Each option combination can be characterized by the shape of its payofffunction. When referring to the option trading strategy, we will imply a certain definitiveshape of the payoff function that is inherent to all combinations belonging to the samestrategy and that is qualitatively distinguishable from payoff functions of combinations notbelonging to this strategy. The set of option combinations available at any given moment intime for valuation, analyses, and selection of promising trading opportunities will bereferred to as the initial set.

ValuationOption combinations are valuated through the application of strictly formalized criteriadeveloped specially for this purpose and expressing potential profitability and risk of theassessed variants in different ways. Criteria represent mathematical constructions withdifferent degrees of complexity and one or many parameters. Optimization of parameters isperformed either by means of statistical analyses of historical time series or by expertforecasts. Because parameters optimized on historical data are inclined to suffer from the

xxiv Systematic Options Trading

disadvantage of curve fitting, close attention should be paid to the validity of statisticalpatterns used to determine their optimal values. Expert forecasts also have significantdrawbacks because they reflect opinions of particular specialists and thus represent rathersubjective estimates. A systematic approach, applying both statistical analyses and expertforecasts, allows diminishing their drawbacks while amplifying advantages of these twoparameterization methods.

Development of sophisticated criteria capable of valuating option combinationsadequately, and optimization of their parameters, are the crucial issues that determine thepractical success of systematic approach. The first part of this book discusses the basicprinciples of criteria construction and parameterization; the main criteria are described andanalyzed in detail.

After being valuated by criteria, every combination receives a numerical characteristicreflecting its investment attractiveness. Option combinations can be valuated by one orseveral criteria. In the latter case the number of characteristics attributed to each combi-nation is equal to the number of criteria.

Comparative AnalysisFollowing the completion of the valuation stage, the characteristics attributed to combina-tions need to be analyzed. During the analysis every combination is compared with all theothers according to their characteristics. As a result, all variants constituting the initial set areordered according to their quality indicators.

If the valuation was based on several criteria, then the analysis generates severalorderings. In this case the same combination can have different positions in differentorderings. For example, a combination can be the best one according to its expected profit,but at the same time it can be at the end of the list in an ordering obtained by the applicationof some risk-related criterion. Subsequently all orderings can be either used separately orcombined into a unified one.

The unified ordering can be either complete or partial. Usually partial ordering appearswhen a complete one is unachievable. This may happen if some items turn out to be incom-parable by certain criteria or if they are valued differently according to different criteria. Insuch cases the entire set of alternatives is divided into groups, and these groups are conse-quently ordered as joint entities. Different methods appropriate for execution of such proce-dures are discussed in Chapter 7, “Basic Concepts of Multicriteria Selection as Applied toOption Combinations.”

Introduction xxv

SelectionAt the next stage the results of the comparative analysis are used to select a limited numberof combinations possessing superior quality characteristics. This procedure needs to bearranged thoroughly because it leads to the irreversible decision as to which combinationswill enter the portfolio and which ones will be rejected.

You need to consider three main principles when choosing combinations suitable forinclusion into the portfolio.

● The number of combinations selected should be large enough to maintaindiversification of the portfolio above some minimum level. Like in the classic portfoliotheory, it minimizes specific risks related to individual underlying assets.

● Criteria values of selected combinations should exceed the values of the rejected ones.The minimal threshold for this excess should be established for each particularsituation.

● The relative superiority of some combinations over others (resulting from thecomparison of their criteria values) should not be considered as the sufficient reasonfor selection. The absolute criteria values must also be taken into account. Forexample, between two combinations with an expected return of –$2 and –$10, the firstvariant is preferable and, in principle, can be selected as the one with relatively bettercharacteristics. However, the absolute value of the expected return corresponding tothe first combination is negative and hence this combination, just as the second one,cannot be selected to enter the portfolio.

In practice, however, these principles contradict one another. Thus, following the secondand the third principles an investor endeavors to decrease the number of combinationsselected. At the same time the principle of portfolio diversification induces the oppositetendency—to increase this number. Thereby the structure of the resulting portfolio repre-sents a compromise (trade-off) between all three principles.

The selection procedure represents a set of rules determining how to draw the lineseparating potentially profitable combinations from those that lack such potential. Considera simple situation: The initial set consists of N combinations ordered according to the valuesof a certain criterion; the investor must select N’ best variants out of N alternatives. Thisproblem may be solved by creating one or several utility functions. The argument of suchfunction is the number of selected combinations (numerical value of the place occupied bythe last selected combination in the ordering). The value of the utility function is an indicatorreflecting the measure of utility arising from the selection of this particular number of combi-nations. In other words, the utility function may be defined as the relationship between anaverage return (the maximum drawdown, the Sharpe ratio, or any other characteristicreflecting the investor’s satisfaction) and the number of combinations selected.

xxvi Systematic Options Trading

Analytical methods are not applicable to the majority of utility functions because noformulae establish the relations between the value and the argument of these functions.Hence the values of utility functions are usually derived empirically from historical timeseries using different statistical techniques.

If several utility functions are used simultaneously, they need to be combined into oneunified function. Such unification is possible because all utility functions have the sameargument (the number of selected combinations). The main requirement for the methodsused to combine different utility functions is the unambiguity of the outcome that must beconsistently interpretable. It means that the resultant function should be unimodal with asingle evident maximum corresponding to the optimal number of combinations to beselected. Statistics offers several methods to combine empirical functions; the most popularare multiplicative and additive convolutions. We have developed an additional method—aminimax convolution (see Chapter 5, “Selection of Option Strategies”) that in most casesbrings more reliable and unambiguous results.

Sequence of Operations, Notion of a “Matrix” and Its ReductionThe initial set consists of a huge number of option combinations that must be processedduring the execution of valuation, analyses, and selection procedures. Suppose that at anytime moment there are mi options traded for every underlying asset i. Assuming that anyoption can either be absent or present in the combination (in the latter case it can be eitherlong or short) and that the proportion of different options is the same in all combinations,the number of possible combinations for one underlying asset is determined as 3m

i. Accord-ingly, the total number of combinations for n underlying assets can be estimated as follows:

Even if only 1,000 underlying assets are available for trading, and on average only 20different options are traded for every underlying asset, then the procedures of valuation,analyses, and selection will cover more than 300 billion combinations! Moreover, the possi-bility to use unequal proportions of different options within one combination—which isquite realistic—generates a truly enormous number of variants to process. This number is sogigantic that computational procedures become unrealizable even for the most advancedcomputer hardware. Therefore, the initial set needs to be decreased to some reasonablequantities that are possible to work with on personal computers. You can achieve thisdecrease through the creation of combination-generating algorithms that produce only

combinations m

i

ni=

=∑ 3

1

Introduction xxvii

potentially appropriate combinations instead of generating all possible variants. (Theirappropriateness is determined by specific requirements and limitations of the particularinvestor.)

First, the investor must decide what strategies to use and then generate combinationscorresponding exclusively to these strategies. This can significantly decrease the number ofvariants in the initial set. Then additional reasonable limitations should be applied withinevery strategy. For example, the following limits can be used for the short strangle strategy:The strike of the Call option must be greater than the strike of the Put option; the differencebetween Call and Put strikes must not exceed 25% of the underlying asset’s price; the ratioof Put options to Call options must be between 0.8 and 1.2. Such limits, on the one hand,are well founded and, on the other hand, they do not prevent an investor from taking fulladvantage of the majority of opportunities appearing in the options market. At the sametime, these limitations reduce the initial set to such an extent that makes it processible forpersonal computers.

Further facilitation of computational procedures can be achieved if selection is realizedas a series of consecutive subselections. We propose to adhere strictly to the followingsequence of operations. Before initiating any selection procedure, a range of potentiallysuitable underlying assets and trading strategies should be determined. At the same time thealgorithms used to generate option combinations must be established. After that the initialset of variants available for trading can be represented as a three-dimensional space ({under-lying assets × strategies × combinations}) on which the consecutive subselection proceduresare executed.

If the algorithms used to generate option combinations allow creating only one combi-nation for every underlying asset within every strategy (that is, one single combination corre-sponds to each {underlying assets × strategies}), then the three-dimensional space of theinitial set turns into a two-dimensional space. The two-dimensional initial set can bevisualized as a table with lines corresponding to underlying assets and columns—tostrategies. Each cell of this table contains one option combination relating to a given under-lying asset and to a certain strategy. Such a table can further be referred to as a two-dimen-sional matrix.

If several option combinations are created for every underlying asset within everystrategy, then each cell in the table will contain more than one combination for any {under-lying assets × strategies}. In this case the two-dimensional matrix becomes a three-dimen-sional one. (Its elements form the initial set.)

The procedure of selection can be viewed as a reduction of the three-dimensional matrix.We propose to realize it as a sequence of three consecutive operations (each of which can beregarded as a subselection).

The first operation represents selection of one or several best option combinations corre-sponding to a specific underlying asset and a given strategy. Suppose that the initial set

xxviii Systematic Options Trading

consists of 30,000 elements (1,000 underlying assets, 5 strategies, and 6 combinations forevery {underlying assets × strategies}). If during the first subselection procedure only onecombination is chosen out of the 6 possibilities, then the initial set of 30,000 items decreasesto 5,000 and the three-dimensional matrix becomes two-dimensional. Chapter 4, “Selectionof Option Combinations,” discusses the methodology of this operation.

The second operation consists of choosing one or several superior option tradingstrategies for every underlying asset. In our example this leads to a further decrease of theinitial set. If only one strategy is chosen, the initial set declines to 1,000 combinations. Theresult of this operation is the ultimate reduction of the matrix because after the execution ofthis subselection a unique list of underlying assets corresponds to every strategy. Therefore,the remaining part of the initial set cannot anymore be presented as an entire table withoutgaps. This operation is discussed in Chapter 5.

The third operation is intended to select the best variants from the lists of underlyingassets corresponding to every strategy. If this procedure selects approximately 10% ofcombinations out of those that were chosen during two previous operations (we assume thispercentage as an average estimate though in practice it can vary substantially), then the initialset is finally reduced to just 100 variants. Chapter 6, “Selection of Underlying Assets,”describes this operation in detail.

This sequence of operations represents only one possible way to perform the proceduresof valuation, analyses, and selection. Some other, more complicated approaches to thereduction of the initial set matrix can be developed. However, the scope of this book islimited to the preceding scheme because even such a relatively simple algorithm has morethan enough particular features and specific peculiarities.

Introduction xxix

Overview of Trading Opportunities

This book covers various aspects of dealing with trading opportunities provided by options:from their detection and investigation to selection and deriving profits. This statement apriori assumes the existence of trading opportunities. Although this is quite obvious to theauthors, it would be more appropriate to demonstrate the permanent presence of varioustrading opportunities in the options market. Besides, it is also useful to investigate theirdynamics and structure.

The aim of this section is to perform the statistical investigation of trading opportunitiesexisting at different moments in time for various underlying assets. An overview of tradingopportunities is also expedient because it provides you with analytical tools for evaluatingthe potential profitability of different options markets and underlying asset types.

What Is a Trading Opportunity?A trading opportunity is the deviation of the market price of an option (or any option combi-nation) from its fair value. The fair value is the price that implies zero profit for both theseller and the buyer of the option. This interpretation is related to the efficient markethypothesis stating that any new information is immediately priced in and hence all tradedassets are fairly valued, and extracting profit is impossible neither for sellers nor for buyers.It is common knowledge that the efficient market concept is an idealization unachievable atpresent days. Financial markets are ineffective; asset prices constantly fluctuate and deviatefrom their fair value thereby creating various trading opportunities.

Because we define the trading opportunity as the difference between the market priceand the fair value, we need to establish the algorithms to estimate these variables.

At each moment the market price is characterized by three indicators: last, bid, and askprices. The first indicator is of little importance because it deals with the past whereas weare interested in current trading opportunities. To discover them it is preferable to use bidand ask prices. If the investor prefers to be on the conservative side, the worst of these twoprices should be used (that is, ask price—when buying options, and bid price—when sellingthem). This approach decreases the probability of a mistake but reduces the number oftrading opportunities considerably. The less conservative investor can use the combinationof bid and ask prices—their simple average or weighted average with different weights forthe best and the worst price. In this case the probability of erroneous inclusion of the optioninto the category of “trading opportunities” is higher, but the sample is more representative.

Estimating the fair value of any asset is an extremely difficult task, and options are notan exception. Their distinctive feature is that, in contrast to other financial instruments,options have expiration dates. This enables us to evaluate the accuracy of the fair valueestimate in a reasonably short time period. The common way to estimate the fair value ofoptions is the Black-Scholes formula and other similar models. However, they have anumber of significant drawbacks and cannot be used to obtain the fair values suitable for theestimation of trading opportunity. That is why we use another method to get more accuratefair value estimates.

Method for the Evaluation of Trading Opportunities The quantitative expression of trading opportunities can be obtained by subtracting the fairvalue of an option from its current market price. For comparability of results the differenceshould be normalized by the strike price (allowing us to express the differences between fairand market prices in percentage).

xxx Systematic Options Trading

We assume the market price of an option to be equal to the average between the bid andthe ask prices. The option fair value can be calculated using the method proposed by RalphVince (Vince, 1992):

Option fair value = ,

where pi is the probability of outcome i,ai is the profit or loss of outcome i, andN is the number of possible outcomes.

To obtain the most accurate estimate of the fair value, Vince permits looking into thefuture. Knowing the underlying asset price at the expiration date, we can find out the exactfair value of any option. In this case we have the only outcome (N=1) with probability pi =1. Accordingly, the fair value of the option is ai.

For the Call option ai is equal to the difference between the underlying asset price (UAP)at the expiration date and the strike price (SP) if UAP > SP, otherwise ai = 0. For the Putoption ai is equal to the difference between the SP and the UAP at the expiration date if UAP< SP, otherwise ai = 0. To be fully accurate, we need to discount this fair value by the risk-free interest rate normalized by the time to expiration. However, within the framework ofcurrent research, this correction can be neglected.

Zero (or close to zero) difference between the market price and the fair value indicatesabsence of trading opportunities. The positive difference indicates that the option isovervalued and there is a trading opportunity to sell it. Similarly, the negative differenceindicates that the option is undervalued and there is a trading strike price opportunity to buy it.

To avoid zero fair values we evaluated trading opportunities of simple option combina-tions (straddles) rather than of separate options. The market prices and the fair value ofcombinations have been calculated for each of the 2,500 stocks and each date of the periodfrom January 2, 2001, to August 16, 2007. Straddles were created using contracts with thenearest expiration date and strike prices closest to the current underlying price. Thus weobtained a table of 2,500 lines (according to the number of stocks) and 1,564 columns (thenumber of dates). Each cell of this table contains the value of the difference between themarket price and the fair value of the combination corresponding to a certain date and to agiven stock. In total, we calculated 3,910,000 values characterizing the presence or absenceof trading opportunities.

Structure and Dynamics of Trading OpportunitiesTo demonstrate the intraday structure of strike price trading opportunities existing in theoptions market, we arbitrarily chose several dates (May 1, June 1, July 2, 2007) and analyzedthe deviations between the market and the fair values on these days. The quantity and quality

( )p ai ii

N

=∑

1

Introduction xxxi

of trading opportunities are vividly depicted in Figure I.1 with the market price of a combi-nation on one axis and its fair value on another. Each point on the figure relates to thecombination corresponding to a certain underlying asset.

On the whole we can observe a firm relation between the two indicators (R2 = 0.32).Points at the line relate to combinations with no trading opportunities. Points to the northof the line represent overvalued combinations. Undervalued combinations are representedby points under the line. The scattering pattern of points in Figure I.1 indicates the presenceof a considerable number of trading opportunities. Both overvalued and undervalued combi-nations are observed in large quantities. However, the extent to which they are over- orundervalued varies widely. Although many points are not situated exactly on the separatingline, they are still very close to it, meaning that the trading opportunities in these cases arenegligible.

xxxii Systematic Options Trading

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

Market price

Fai

r va

lue

1 May 2007

1 June 2007

2 July 2007

Figure I.1 Relation between market price and fair value of straddles observed as of May 1, June1, and July 2, 2007, for 2,500 stocks. The line separates overvalued (points above it) and under-valued (points below it) combinations. All values are expressed as the percentage of the strikeprice.

We propose the following heuristic rule to separate combinations with trading opportu-nities from others that do not possess any trading potential or only have an insignificant one.Combinations with a difference between the market price and the fair value of no more thanone percent (that is, within the range of –1% to 1%) are considered as lacking trading oppor-tunities. Combinations with a difference that is outside this range are considered as havingtrading opportunities. Combinations with a difference of >1% are overvalued; combina-tions with a difference of <–1% are undervalued. Based on this classification we can analyzedata presented in Figure I.1. (Only one date is featured in the following discussion becausethere is no significant difference between different days.)

The best way to represent the structure of trading opportunities is to build a frequencydistribution of the differences between market and fair prices (see Figure I.2). The distri-bution of trading opportunities existing as of July 2, 2007, is characterized by the skewtoward positive values. This indicates that overvalued combinations prevailed over under-valued ones. Only 19% of combinations fall into the –1% to 1% interval that we considerto be the range with negligible trading opportunities. Fifty-two percent of combinations areovervalued and 29% are undervalued. This means that more than 80% of stocks had thepotential of realizing either short- or long-straddle strategies. Short positions could beopened for more than half of the combinations, whereas long straddles turned out to beprofitable in slightly less than one-third of all cases.

The frequency distribution of differences between market and fair values deviates fromnormal distribution considerably (see Figure I.2). Small differences corresponding to theabsence of trading opportunities are more frequent than it is expected under the normaldistribution. Medium differences are observed less frequently than under normal distri-bution. Comparison of two distributions reveals asymmetry in distribution of trading oppor-tunities. Moderately overvalued combinations are more frequent than under normaldistribution whereas moderately undervalued combinations are less frequent. However, thesituation with the distribution of big differences is the contrary—highly undervalued combi-nations (left tail of the distribution) are more frequent than highly overvalued ones (right tailof the distribution) (see Figure I.2).

Introduction xxxiii

xxxiv Systematic Options Trading

0

2

4

6

8

10

12

14

-20 -15 -10 -5 0 5 10 15 20

Difference between market price and fair value

Fre

qu

ency

, %

normal distribution empirical distribution

Figure I.2 Two distributions of differences between market prices and fair values of straddlesobserved on July 2, 2007, for 2,500 stocks. Prices and differences are expressed as thepercentage of the strike price. Positive differences correspond to overvalued straddles, negativedifferences—to undervalued straddles.

So far we have been analyzing the distribution of trading opportunities between differentunderlying assets (to be more exact, between combinations corresponding to these assets)within one day. At the next stage the time dynamics of trading opportunities will beconsidered for separate underlyings. We begin with one stock (AAPL will be used as anexample) and calculate the difference between market prices and fair values of straddles forall dates within the period from January 2001 until August 2007. Figure I.3 shows thesedifferences plotted against the corresponding dates. Visual analysis of these data reveals thatundervalued periods alternate with overvalued periods. In general, we can say that thedynamics of this process is characterized by quasiperiodical cycles. Although at first sightthese cycles have similar periodicity, their detailed investigation suggests that trading oppor-tunities can hardly be forecasted on their basis.

Introduction xxxv

-15

-10

-5

0

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10

02.01.2001

01.06.2001

13.11.2001

23.04.2002

03.10.2002

14.03.2003

22.09.2003

08.03.2004

26.08.2004

01.03.2005

31.08.2005

07.03.2006

05.09.2006

05.03.2007

08.08.2007

Diff

eren

ce b

etw

een

mar

ket

pri

ce a

nd

fai

r va

lue

Figure I.3 Dynamics of the differences between the market price and fair value of straddles forAAPL stock. Prices and differences are expressed as the percentage of the strike price.

Using AAPL as an example we illustrated the dynamics of trading opportunities for justone underlying asset (Figure I.3). However, the analysis of other stocks (not presented here)gives similar results. The overwhelming majority of underlying assets shows similarbehavior—more or less regular fluctuations between overvalued and undervalued areas. Asin the AAPL case, there are periods when trading opportunities are negligibly small.

Previously we mentioned that the intensity of trading opportunities can vary widely.Although we agreed to consider the difference between the market and the fair values ofmore than 1% as indicating the presence of a trading opportunity, the profit potential maybe quite low if the differences exceed 1% by just a slight margin. On the other hand, thedifference of 5% and over has a strong profit potential. As it follows from Figure I.3 (andother research not presented here), medium trading opportunities (with a profit potentialof approximately 2% to 4%) are prevailing in the market. However, it should not have anegative impact on our evaluation of trading opportunities because these medium devia-tions of market prices from their fair values occur quite frequently.

The dynamics of trading opportunities analyzed by the example of AAPL indicates theapproximate equality of periods when the options of a certain underlying asset areovervalued and when they are undervalued. Does such uniform distribution of tradingopportunities (between over- and undervalued periods) reside in all stocks? Are thereunderlying assets with options that are undervalued or overvalued most of the time? Toanswer these questions we calculated the number of days when options were overvaluedand undervalued for each of the 2,500 stocks. We divided the obtained values by 1,564 (thetotal number of days) to express them as the percentage fraction of time. This data was usedto build two distributions of time fractions: when options were overvalued and when theywere undervalued (see Figure I.4).

Two distributions are shifted relative to each other: overvalued—toward the longerfractions of time, undervalued—toward the shorter fractions of time (see Figure I.4). Thismeans that options are more often overvalued than undervalued. In general, we canconclude that most of the options are undervalued for no more than 25% to 40% of the timeand are overvalued for 45% to 60% of the time (see Figure I.4). Moreover, options relatingto approximately 5% of stocks are overvalued for more than 70% of the time. On the otherhand, options relating to only 3% of stocks are undervalued for just 50 % to 60% of thetime. This means that options of some stocks are permanently overvalued, whereasconstantly undervalued options are relatively rare.

xxxvi Systematic Options Trading

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As shown in Figure I.1, a multitude of trading opportunities consisting of both under-valued and overvalued options exists in the market simultaneously. To get a detailed notionof their dynamic structure, we can analyze the proportions of combinations possessingtrading potential and those lacking it. This investigation demonstrates how the ratio ofovervalued, undervalued, and fairly valued combinations changes in time. To clarify theseissues we use 2,500 straddles (one for every underlying asset) for each of the 1,564 dates.For every combination we calculate the difference between the market price and its fairvalue. Based on this indicator, the straddles are classified into three categories in accordancewith the heuristic rule previously proposed. Combinations with a difference of >1% areconsidered to be overvalued; those with a difference of <–1% are undervalued. Combina-tions with a difference between –1% and 1% are considered to be fairly valued (that is,lacking any trading opportunities). For each date we calculate the proportion of combina-tions belonging to each of the three categories and observe the dynamics of their changes intime.

The proportion of fairly valued combinations is relatively stable in time—it fluctuatesslightly within the range from 10% to 20% (see Figure I.5). Overvalued combinationsprevail over undervalued ones. The former constitute approximately 50% to 60%throughout most of the time, whereas there are only 30% to 40% of undervalued combina-tions. (This corresponds with the conclusions drawn from the analysis in Figure I.5.) At thesame time there are periods when the proportion of undervalued combinations rises sharply(see Figure I.5).

Introduction xxxvii

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The following conclusions can be drawn from this statistical investigation. Considerabletrading opportunities consisting of overvalued and undervalued options are constantlyavailable in the market. However, their relative ratios have complicated time dynamics thatcan hardly be forecasted by discovering persistent cycles with regular periodicity. Conse-quently, an accurate prediction of future trading opportunities—whether most of theoptions will be overvalued or undervalued—seems to be unfeasible. Therefore, at everymoment in time the investor should determine the trading potential of each separate combi-nation and create corresponding strategies based on selling overvalued, buying undervalued,and excluding fairly valued options and their combinations.

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chapter 3

Evaluation of Criteria Effectiveness

3.1 Introduction

Evaluation of criteria effectiveness requires addressing the following question: To whatextent is the potential effect of their exploitation consistent with our expectations? Inprevious chapters we described the philosophy, methodology, and main approaches used tocreate criteria. While developing and constructing criteria we pursue quite concrete anddefinite goals. Hence, upon completion of this comprehensive creative process, it would benatural to check if the result is satisfactory and whether the goals have been achieved.

Before judging on the suitability of a particular criterion, it has to be tested thoroughly.To carry out such a test, we have to establish a set of numerical characteristics that wouldreflect the performance of a criterion. Hereafter such characteristics will be referred to as“indicators of criteria effectiveness.” (Using other terms, that is criteria quality, suitability,and such, we will imply the same indicators.) To maximize the test reliability, it is desirablenot to be restricted by one indicator but to use as many of them as possible (within reasonablelimits, of course). Next we consider a set of effectiveness indicators. Despite the fact that eachof them has strengths and weaknesses, their joint use allows you to evaluate the criteria effec-tiveness adequately.

The criteria effectiveness strongly depends on the type of the task it is intended to dealwith. This should be taken into account while selecting indicators and developing evaluationprocedures. Talking about types of task, we mean different areas of criteria application suchas selecting option combinations, strategies, and underlying assets (see Chapters 4 through6). In Chapter 1, “General Presentation and Review of Criteria Properties,” we distinguishedbetween two basic groups of criteria: universal and specific ones. As follows from the name,

the former can be applied to resolve almost any kind of problems whereas the latter are notas multifunctional. However, even universal criteria show heterogeneous suitability fordealing with different tasks.

The criterion often shows high effectiveness in certain situations but may become uselessin other circumstances. Hence, in this chapter we concentrate on evaluation of criteriaquality when they are used to accomplish one general task: sorting option combinations bytheir potential profitability. We consider this task to be general because its realization isrequired in all areas of criteria application. The methods intended to evaluate the effec-tiveness of criteria when they are used to fulfill specific tasks are described in the second partof this book dedicated to the main areas of criteria application.

The criteria effectiveness varies depending on the type of option strategy. A certaincriterion may successfully detect profitable variants within the set of combinations createdunder one strategy, but it may be ineffective in selecting combinations created under anotherstrategy. This feature is described in detail here, and it must also be taken into account whenevaluating criteria effectiveness.

Four option strategies are used in this chapter: short strangle/straddle, longstrangle/straddle, short calendar spread, and long calendar spread. (Here, strangles andstraddles are considered as combinations relating to one strategy.) A short calendar spreadis a combination consisting of a short Call (Put) with the nearest expiration date and a longCall (Put) with the same strike price but with the next expiration date. (Thus, this combi-nation is short because its payoff function is similar to short positions.) Accordingly, a longcalendar spread is a combination consisting of a long Call (Put) with the nearest expirationdate and a short Call (Put) with the same strike price but with the next expiration date. (Itspayoff function is similar to long positions.) Note that these definitions differ slightly fromthe common ones.

These particular four strategies were chosen because they represent a wide range ofpayoff functions that are opposite to each other and neutral to an underlying asset market(see the appendix). In this chapter we develop a set of effectiveness indicators intended toevaluate criteria described in Chapter 2, “Review of the Main Criteria.” The quality of eachcriterion will be evaluated using many indicators within different option strategies.

64 Systematic Options Trading

3.2 Methods of Criteria Effectiveness Evaluation

3.2.1 Correlation Between a Criterion and Profit as the MainEffectiveness IndicatorYou cannot gain full insight into the effectiveness of the criterion relying solely on a singleindicator. Nevertheless, there is one fundamental characteristic—the correlation between thecriterion value and the realized profit—which for several reasons should be considered in thefirst place. Firstly, the correlation between the outcome and the forecast (most criteria doinclude a forecast in one form or another) is the most direct and intuitively understandable

indicator of the criterion’s suitability for determining trading opportunities. Secondly,this indicator is universal in the sense that it is suitable for evaluation of any criterionwithin all option strategies. Finally, it is easy to compute and does not require anycomplicated data processing except building a simple regression model.

The evaluation of criterion effectiveness by means of the correlation is based onthe relationship between criteria values and realized profit. Applying a regressionmodel to this relationship allows an estimation of the fraction of the profit varianceexplained by the criterion variation and of the fraction of the profit variance causedby random factors. The square of the correlation coefficient (also referred to as“coefficient of determination”) shows how well the regression equation (whichreflects the linear relationship between the profit and the criterion) fits actual data. Inother words, it reflects the dispersion of data around the regression line. The coeffi-cient of determination ranges from zero (when the profit does not depend on thecriteria) to one (profit changes are fully determined by criterion changes).

One particular issue should be taken into account to perform a high-quality evalu-ation of criterion effectiveness using the correlation method. The value of the corre-lation coefficient expressing the {criterion × profit} relationship is usually small andvariable in time even when the criterion is correctly parameterized and possesses goodforecasting properties. This holds true even when the predictive power of the criterionis known in advance and was verified both empirically and in back-testing. Thisfeature is not specific to this particular indicator, but is rather common for mostindicators that will be discussed next. However, this should not upset us or make usthink that it is impossible to develop adequate methods for evaluation of criteria effec-tiveness. In section 3.2.2 we show how to overcome these difficulties by applying asimple transformation procedure.

The previously mentioned issue can be illustrated by the following examples. Fivehundred U.S. stocks were chosen according to the liquidity of their options. From thebeginning of 2001 until September 2007, for each stock and each date we createcombinations within short and long strangle/straddle strategies. The combinations areconstructed using four strikes that are the closest to the current stock price (two strikesbelow the current price and two strikes above it). Thus, 10 combinations are createdfor each stock: 4 straddles and 6 strangles (provided that Call strikes in a strangle arealways higher than Put strikes). Combinations are made of contracts with the nearestexpiration date and with a Call-to-Put ratio of 1:1. These 10 combinations areevaluated using a certain criterion, and three best variants are selected. In this way1,500 combinations are produced for 500 stocks for each date. After all these combi-nations are evaluated by the corresponding criterion; their profit (or loss) values arerecorded as of the expiration date. (Values of both the criterion and the profit areexpressed in percentage of margin requirements.) Relationships between criterion andprofit values are used to obtain correlation coefficients. The whole procedure isrepeated for each criterion under examination.

Evaluation of Criteria Effectiveness3 65

We begin with considering the correlation between criterion and profit values for severalrandomly selected dates. As an example we chose the “expected profit on the basis oflognormal distribution” criterion and the short strangle/straddle strategy. Two dates wererandomly selected: August 27, 2007 and September 10, 2007. The regression model for{criterion × profit} relationship is shown in the top-left corner of Figure 3.2.1. Each pointhere represents a certain combination, and all points of the same color correspond to combi-nations built within one date.

66 Systematic Options Trading

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Figure 3.2.1 Regression model for the relationship between the “expected profit on the basis oflognormal distribution” criterion (horizontal axis) and the profit of option combinations (verticalaxis). Option strategy: short strangle/straddle. The black points correspond to August 27, 2007, thegray points to September 10, 2007. The legend shows the values of determination coefficients.Each chart corresponds to a certain amount of averaging days (from 1 to 10).

As we already mentioned, the correlations between the criterion value and the realizedprofit are very low. (Determination coefficients are negligibly small: R2 = 0.043 and R2 =0.001, see the top-left chart of Figure 3.2.1.) However, only one strategy, one criterion, andjust two dates (out of almost 7 years) were used in this analysis. Therefore, it is necessary toverify if the obtained results are representative. To extend this study we estimated the effec-tiveness of many criteria, for both long and short strategies and considered their dynamicsduring the 7-year period.

Regressions similar to those shown on the top-left chart of Figure 3.2.1 were constructedfor each {date × strategy × criterion} conjunction, and their determination coefficients werecalculated. We found again that in the course of all 7 years the correlations between criteriavalues and profits of option combinations were extremely low (Figure 3.2.2). The coeffi-cients of determination chaotically fluctuated within the range from almost 0 to 0.2 withoutany definite cycles. Such relationships were observed for all the 11 criteria regardless of thestrategy. This extensive dynamic analysis confirms the validity of our prior conclusions basedon two random samples.

Evaluation of Criteria Effectiveness3 67

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Figure 3.2.2 Dynamics of determination coefficients corresponding to {criterion × profit}regressions for long and short option strategies. Each line corresponds to one of 11 criteria.

Does it mean that our criteria are ineffective and do not possess real forecasting abilities?Not necessarily! It only shows that this particular method in its present form is unsuitablefor the analysis of criteria effectiveness. In the next section we show that applying a simpletransformation technique allows you to increase the significance and productivity of thisevaluating procedure substantially.

3.2.2 Transformation of the Criteria Effectiveness IndicatorLet us have another look at the top-left chart of Figure 3.2.1. It is quite reasonable tosuppose that the weak relationship between the criteria and the profit values is caused by theinfluence of external factors that add noise to the system and reduce correlation. We supposethat at least some of these factors have a chaotic nature, and their impact on profit canchange a vector in the course of time. If this assumption is true, these factors can beneutralized by combining several days in the analysis.

Combining does not mean mixing data that belongs to several days in one analysis (likemerging black and gray points on the top-left chart of Figure 3.2.1). This could possiblyincrease the correlation coefficient but just due to the increase in the sample size. Wepropose the following procedure of data processing. Two successive dates are selected andcombinations corresponding to each of them are sorted by criterion values. Then wecombine these two orderings by calculating the average of two criterion values for combi-nations occupying the two first positions. The average profit value for these two combina-tions is calculated in a similar way. Then we go on to the second positions in both orderingsand calculate the averages for the criterion and the profit. The same procedure is applied forall positions up to the 1,500th.

This transformation provides us with a set of data consisting of averaged criterion andprofit values. Their interrelationship is presented on the top-right chart of Figure 3.2.1.Compare it with the similar relationship before averaging (the top-left chart of Figure 3.2.1).The averaging of data corresponding to August 27, 2007 and the adjacent date induced theincrease of the determination coefficient from 0.043 to 0.075 (black points on the figure).In another case, when averaging included September 10, 2007 and the adjacent date, theincrease was from 0.001 to 0.004. Although such augmentations seem to be negligible, itcould be due to the fact that only two dates have been combined so far. To obtain a morepronounced effect, we should proceed further.

Averaging 4-days data (beginning from the same dates as in the previous example)resulted in further increase of correlations. In one case R2 doubled in comparison with 2-days averaging, in another case it increased almost tenfold (Figure 3.2.1). The more dateswe include in the averaging, the higher the determination coefficients are. In the case of 10days the strong correlation between criterion and profit values becomes obvious (Figure3.2.1) (coefficients of determination are R2 = 0.24 for August 27, 2007 and R2 = 0.16 forSeptember 10, 2007).

68 Systematic Options Trading

It is interesting to trace the evolution of interrelationships between the criterion and theprofit as the averaging period increases. Going from the top-left corner of Figure 3.2.1 to itsbottom right corner, we can easily observe the gradual increase in correlation.

The data presented so far proves the effectiveness of the single criterion when applied toone strategy. Simple transformation was sufficient to acquire a notion of it. At the same timemany questions still remain unconsidered. Examining the averaging period, we stopped atthe tenth day. It is necessary to continue expanding this period to obtain the optimal valueof this parameter. Furthermore, we have to find out how the averaging procedure influencesthe evaluation of other criteria. The applicability of this procedure to other option strategiesshould also be ascertained. The influence of the transformation procedure on the dynamicsof the correlation coefficients should be analyzed as well.

3.2.3 The Dynamics of Transformed Effectiveness IndicatorsTo investigate the influence exerted by the transformation procedure on the dynamics of{criterion × profit} relationships, we analyze three averaging periods: 30, 60, and 100 days.Determination coefficients will be calculated for each date through the entire 7-year periodfor both long and short strategies and for all the 11 criteria.

Figure 3.2.3 shows the dynamics of the effectiveness indicator for three averagingperiods and two opposite-directed strategies. Comparison of diagrams with increasingaveraging periods reveals that the transformation procedure allows detecting cycles in thedynamics of the criteria effectiveness. Although the dynamics is absolutely chaotic in theabsence of transformation (Figure 3.2.2), 30-day averaging favors the appearance of someregularities in fluctuations of the effectiveness indicator (Figure 3.2.3). Increasing the periodup to 60 days makes the regularities even more observable, and the 100-day period revealsobvious cycles in the dynamics of the effectiveness indicator. These regularities are moreapparent for the short strangle/straddle strategy. Although the long strategy shows a similardynamic, the cycles here are less obvious and harder to recognize (Figure 3.2.2).

The data presented in Figure 3.2.3 also throws light upon the individual features ofcertain criteria. In general, most of the criteria show similar cycling within the short strategy.Nevertheless, they do not always coincide in terms of phase, amplitude, and duration ofcycles. In the case of the long strategy, such discrepancies are even more apparent. Suchproperty of the criteria to follow individual patterns in their behavior is quite a useful featurein view of their practical use. When several criteria are used simultaneously to evaluateoption combinations (see Chapters 7 and 8), low effectiveness of some criteria is balancedwith higher effectiveness of the others, which do not coincide with them in time phase. Thisincreases the combined effectiveness of the whole set of criteria.

Evaluation of Criteria Effectiveness3 69

70 Systematic Options Trading

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Figure 3.2.3 Dynamics of the determination coefficient corresponding to {criterion × profit}regressions for long and short strategies and for three averaging periods (30, 60, and 100 days).Each line corresponds to one of 11 criteria.

Evaluation of Criteria Effectiveness3 71

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The cycling depicted on Figure 3.2.3 means that periods of high criteria effectiveness aresucceeded by periods of their lower effectiveness and vice versa. If that is the case, howshould we derive a unique numerical indicator that reflects the general effectiveness of acertain criterion regardless of its current position in the time cycling? This is necessary inorder to (1) compare the effectiveness of different criteria as applied to a given optionstrategy; (2) compare the effectiveness of a certain criterion as applied to different strategies;and (3) solve other problems that require selecting the most suitable criteria. This issue isdiscussed in the next section.

3.2.4. Selection of the Averaging PeriodIn view of the dynamic patterns displayed by the determination coefficient and the influenceof the averaging period on this indicator, we proceed to the issue that is essential for evalu-ation of criteria effectiveness. To conduct a thorough analysis of different criteria we needto determine the optimal averaging periods for all of them. These periods can be eitherindividual for each criteria (and even change depending on the option strategy used) or bequite universal. Given the goals of the current investigation, we argue in favor of univer-sality. (The arguments will be given next.) At the same time an individual approach could bepreferred for practical purposes.

In the previous section we saw that despite the averaging procedure, the determinationcoefficient is quite variable in time and sometimes drops to rather low values. To obtain thegeneralized characteristic of criteria effectiveness, we need to express the determinationcoefficient in the form that describes the criterion regardless of the phase of its dynamiccycle. Now we propose the procedure that generalizes the effectiveness indicator.

Consider Figure 3.2.3 (short strategy, 30 days) again. Averaging all determination coeffi-cients related, for example, to the EPLN criterion, gives R2 = 0.16. This indicator describesthe generalized effectiveness of the EPLN criterion given the 30-day averaging period. Appli-cation of the 60-day period gives R2 = 0.21 and application of the 100-day period gives R2

= 0.25. Reproducing such calculations for other periods (the range from 2 to 900 days issufficient) we obtain the relationship between the generalized coefficient of determinationand the averaging period. Having such type of analysis performed for all criteria (for longand short strategies separately), we obtain a set of relationships that serve as the basis forsolving the problem of selecting the optimal averaging period.

The results of the short strategy analysis are displayed in Figure 3.2.4. For all the criteriathe increase of the averaging period causes an increase of the generalized determinationcoefficient (Figure 3.2.4). The relationships are nonlinear. As the averaging period increases,the correlations first grow rapidly. Further increase of the averaging period induces furthergrowth of the effectiveness indicator but at a slower rate. This pattern can be viewed as anapproximate description matching the behavior of most criteria. However, despite the

72 Systematic Options Trading

Evaluation of Criteria Effectiveness3 73

Short strangle/straddle

0

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0 100 200 300 400 500 600 700 800 900Averaging period

R2

IV/HV, PPLN, PPEM, RPLE BEVR

EPES

EPLN

RPLSRPLL

EPEM PPES

similarity, some criteria demonstrate their own specificity. For example EPLN and EPEShave the highest determination coefficients almost at all averaging periods. The formerdominates the latter when the averaging period is small (less than 200 days) and vice versawhen the period is large (more than 200 days). Other criteria show weaker effectivenessalthough some of them are better than EPEM when the averaging period is short.

Figure 3.2.4 Relationship between the determination coefficient and the averaging period for theshort strategy. Criteria abbreviations: EPLN, EPEM, EPES—expected profit on the basis oflognormal, empirical, and symmetrized empirical distributions; PPLN, PPEM, PPES—profitprobability on the basis of the same distributions; RPLL, RPLE, RPLS—the ratio of expected profitto loss based on these distributions; IV/HV—the ratio of implied to historical volatility; BEVR—break-even range.

In the case of the long strategy, the relationships between the generalized determinationcoefficient and the averaging period are similar to those observed in the case of the shortstrategy; the correlation rises as the period increases (Figure 3.2.5). However, the nonlin-earity here is much stronger. Starting from the 50th day, the increase in the averaging periodresults in the steep fall of the correlation growth rate. When the number of days exceeds100, the correlation growth completely stops. Such trend is typical for most but not allcriteria. For example, EPEM and EPLS reach their maximums earlier than other criteria.BEVR and IV/HV do not have any evident points of saturation. To summarize, for the longstrategy the generalized indicators of criteria effectiveness are lower, and the saturations arereached earlier than in the case of the short strategy (compare Figures 3.2.4 and 3.2.5).

74 Systematic Options Trading

Long strangle/straddle

IV/HV

BEVR

EPLNEPES

PPLN

PPEMPPES

RPLL

EPEM RPLSRPLE

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R2

0 100 200 300 400 500 600 700 800 900

Figure 3.2.5 Relationship between the determination coefficient and the averaging period forthe long strategy. Criteria abbreviations are the same as in Figure 3.2.4.

What averaging period should be used to evaluate criteria effectiveness? Should thisperiod be universal for all criteria or chosen on an individual basis? These questions areessential not only to the indicator considered up to now ({criterion × profit} correlation)but also regarding other criterion effectiveness indicators that have not been discussed yet.Considering these questions, we should not forget that averaging is just an auxiliaryprocedure intended to increase the significance of criterion effectiveness evaluation. Theprocedure of evaluation is used not merely to judge about the quality of a certain criterionbut also to compare it with other criteria. Therefore, it is preferable to use the sameaveraging period for all criteria used in the same research to ensure their comparability.Hence, the most acceptable solution would be to determine the universal (that is, suitablefor all the criteria) value for the averaging period.

How should we define the value for the universal averaging period? The simplestdecision to use the maximum period (of 900 or more days) seems to be erroneous. Althoughin most cases the correlation reaches its maximum under the longest period, such approachdoes not comply with one of the fundamental principles of trading systems development—to refrain from using excessively long historical intervals to avoid over-optimization andother adverse effects of data fitting. Therefore, we suggest using a relatively small thresholdperiod corresponding to the point at which correlation growth reaches saturation. In otherwords, the optimal period value is the moment when the growth rate of the determinationcoefficient starts dropping.

Adopting this approach and analyzing the data in Figures 3.2.4 and 3.2.5, we face diffi-culties with fixing an averaging period that could be considered as common for all thecriteria. The figures clearly show that the optimal period value, as it was previously defined,

is individual for each criterion and can vary depending on the option strategy. Nevertheless,we need to fix a compromise solution that will be at least to some extent suitable fordifferent situations that we discuss next.

Bearing in mind all these arguments and considerations, we decided that the mostacceptable value for the averaging period is 100 days. This value will be used further toevaluate the criteria effectiveness. However, it is not supposed to be regarded as aninvariable, permanent, and the most optimal solution for criteria quality evaluation. This isjust a convenient compromise allowing us to demonstrate different aspects of criteria effec-tiveness evaluation. (This averaging period is equally suitable for several option strategies,many criteria, and numerous effectiveness indicators.) On the contrary, when selecting theaveraging period for practical use, such global universality is not required. It would beenough to adhere to it only within the local system of compared criteria and strategies.Hence, when creating a real trading system, we can use any number of averaging periodsdepending on the criteria and strategies used by the system.

Evaluation of Criteria Effectiveness3 75

3.3 Peculiarities of Criteria Effectiveness Evaluation

3.3.1 Number of Combinations Used in the AnalysisAssessing the criterion effectiveness through the correlation between criterion and profitvalues requires the calculation of the determination coefficient on the basis of a certainsample. We decided that this sample was going to include 1,500 combinations: three foreach of the 500 stocks. However, if we used only one combination instead of the three bestones, then the sample would consist of just 500 elements. Likewise, using five combinationsfor each stock would expand the sample to include up to 2,500 elements. Moreover, thenumber of stocks itself was determined arbitrarily. We could restrict the number of under-lying assets to 100 or, on the contrary, we could use several thousand of them.

In fact, the sample size represents an additional parameter influencing the evaluation ofthe criterion effectiveness (another parameter, the averaging period, was previouslydiscussed). We call this parameter “the number of combinations used in the analysis.” Herewe examine the sensitivity of the effectiveness indicator to changes in the number of combi-nations and determine the optimal value for this parameter.

Careful examination of Figure 3.2.1 reveals that the distribution of points on thecoordinate space deviates significantly from the normal cigar-shaped pattern. (You canclearly see it on the charts with high averaging periods: 8 and 10 days.) It is rather pear-shaped, linearly stretched in the area of high criterion values and with a considerabledispersion of points, which seems to be random, in the area of low criterion values. Visualanalysis suggests that if the sample consists of exclusively combinations with high criterionvalues, the coefficient of determination would be much higher than if the sample includes

elements with both high and low criterion values. In other words, adding combinations withlow criterion values into the sample weakens the correlation.

Now we present the market data providing formal grounds for these visual observations.Using the data from Figure 3.2.1 (August 27, 2007) we begin with the 10-day averagingperiod. All combinations are sorted according to criterion values and the determinationcoefficient is calculated for the first 100 combinations. Then we add the 101st combinationand compute the effectiveness indicator value again. We continue adding combinations oneafter another right up to the last one (1,500th), every time obtaining the coefficient of deter-mination. The result is depicted by the top line in Figure 3.3.1.

76 Systematic Options Trading

0.1

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100 300 500 700 900 1100 1300 1500

Number of combinations

10 days

4 days

6 days

8 days

R2

Figure 3.3.1 Relationship between the coefficient of determination and the number of combina-tions used in the regression analysis. Criterion: expected profit on the basis of lognormal distri-bution; strategy: short strangle/straddle; date: August 27, 2007. Four averaging periods are shown.

As was expected, the maximum value of the determination coefficient corresponds to theminimal sample of 100 combinations. The gradual increase of the sample size to 400 combi-nations entailed a sharp drop in the criterion effectiveness indicator. Afterward, adding newelements up to the 700th element did not change the correlation value. Further increase inthe number of combinations led to the decrease of the determination coefficient down to itsminimum reached upon inclusion of the 1,500th combination (Figure 3.3.1, the top line).The same trend was detected for the 8-day averaging period (Figure 3.3.1, the second linefrom the top). At the same time, lower averaging periods did not show any definite dynamics(Figure 3.3.1, the two bottom lines). This once again confirms the usefulness of the averagingprocedure.

The analysis presented in Figure 3.3.1 indicates the negative influence of the sample sizeon the criterion effectiveness indicator. Beside this, we can draw another important

conclusion: The criterion used in this investigation more accurately (that is, producing fewererrors) identifies combinations as “good” rather than as “bad” ones. This means thatalthough unprofitable combinations would not be mistakenly included in the portfolio,many potentially profitable combinations could be undervalued and wrongly excluded fromthe potential investment universe.

Nevertheless, we refrain from making any generalizations regarding the shape and thedegree of the relationship between the effectiveness indicator and the sample size. After all,we presented data corresponding to only one criterion, one strategy, the only randomlyselected date, and one effectiveness indicator. Hence, we recommend treating these resultsmerely as an indication that the sample size actually has influence upon the value of thecriterion effectiveness indicator.

We do not overcharge this chapter with illustrations of all possible relationships betweenthe effectiveness indicator and the number of combinations used in the analysis. However,it is worth mentioning that in other cases these relationships could become positive ratherthan negative. In fact, positive relationships are quite natural because, all other things beingequal, adding data to the sample normally causes an increase of the correlation coefficient.Yet this phenomenon could be further amplified by more efficient identification of “bad”combinations accompanied by less effective recognition of “good” ones.

Deciding on the optimal number of combinations is a difficult task, and the solutiondepends on a considerable number of interrelated factors such as criterion, strategy, effec-tiveness indicator, and so on. Just as in the case of parameterization of the averaging period,it is preferable to select optimal values for different situations individually. Nevertheless, forthe purpose of reviewing various effectiveness indicators (see section 3.4) we need to set oneuniversal value for the number of combinations used in the analysis. It is essential for thecomparability of the results obtained in different studies.

The optimal parameter value is chosen based on our experience and a large number ofpreliminary investigations. In the following research we continue to use 500 stocks as under-lying assets, but the number of combinations selected for each of them will be reduced fromthree to one. Thus, all samples will consist of 500 combinations. This compromise valueseems to be quite optimal if it has to be universal.

3.3.2 Expressing ProfitEvaluation of criterion effectiveness is based on two variables: criterion and profit. Theirvalues must be estimated for all the combinations used in the analysis. In the examplesdiscussed we expressed the values of the EPLN criterion and the values of profit in U.S.dollars normalized by the margin requirements. Thus, both the criterion and the profit wereexpressed in the same units; they were similarly scaled and, in fact, represented variableswith the same meaning.

Evaluation of Criteria Effectiveness3 77

When evaluating the effectiveness of criteria forecasting other values (rather thanexpected profit), we are obliged to choose between two alternatives. One possibility is toexpress realized profits in normalized U.S. dollars regardless of what the criterion forecasts.Another alternative is to express profit in the same way as it is forecasted by the criterion.

Unfortunately, it is not always possible to express profit exactly in the same way asforecasted by the criterion. Still, even in these cases it can at least be expressed in the sameunits as criterion values. For example, when evaluating the effectiveness of the criterionforecasting profit probability, it is impossible to express the realized profit of a single combi-nation as a probability. However, if we consider a group of combinations (grouping is basedon the positions occupied by combinations in the orderings), we can calculate the proportionof profitable combinations (ratio of profitable combinations to the total number of combi-nations). This value is easily comparable with profit probability (that is, with the criterionvalue averaged for the group) and is expressed in the same units.

In most cases the second alternative—to express profit as it is forecasted by thecriterion—is preferable. The following examples will support this statement.

We continue using the determination coefficient of the {criterion × profit} relationshipas the indicator of criterion effectiveness. It will be applied to analyze two criteria: expectedprofit and profit probability based on the lognormal distribution (EPLN and PPLN, respec-tively)—within the short strangle/straddle strategy. The parameters are fixed as follows: the100-day averaging period, 500 combinations used in the analysis. Every group of combina-tions is formed according to its position in the orderings. (The first group corresponds to thefirst positions in the orderings, the second group to the second positions, and so on.) Becausewe use a 100-day averaging period, there will be 100 elements in each group. The totalnumber of groups is 500 (according to the number of combinations used in the analysis). Therealized profit values (calculated separately for every group) are expressed in two differentways: (i) in U.S. dollars normalized by margin (which corresponds to the EPLN criterion)and (ii) as proportion of profitable combinations (which corresponds to the PPLN criterion).

Here we analyze four {criterion × profit} relationships:

● Criterion in U.S. dollars (EPLN)—profit in U.S. dollars

● Criterion in U.S. dollars (EPLN)—profit as proportion

● Criterion as proportion (PPLN)—profit as proportion

● Criterion as proportion (PPLN)—profit in U.S. dollars

The first and third alternatives correspond to situations when the criterion and profit areexpressed in the same units (Figure 3.3.2(a) and 3.3.2(c)). The second and fourth alterna-tives depict the situations when the criterion and profit are expressed in different units(Figure 3.3.2(b) and 3.3.2(d)).

The analysis of four regression models leads to the following conclusion: When profitand the criterion are measured in the same units, the effectiveness indicator is significantly

78 Systematic Options Trading

higher. For example, the determination coefficient of EPLN, when both the criterion andprofit are expressed in US dollars, is R2 = 0.66 (Figure 3.3.2[a]). However, expressing profitas the proportion of profitable combinations leads to a substantial drop in the effectivenessindicator (R2 = 0.36, Figure 3.3.2[b]). Similarly, the correlation of the PPLN criterion andthe profit is very high, when both values are expressed as proportions (R2 = 0.92, Figure3.3.2[c]). Still, expressing profit in U.S. dollars decreases the effectiveness indicator to R2 =0.37 (Figure 3.3.2[d]).

Evaluation of Criteria Effectiveness3 79

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Figure 3.3.2 Relationships between values of two criteria (EPLN and PPLN) and profit expressedin two ways (in U.S. dollars and as proportion of profitable combinations) for the shortstrangle/straddle strategy. (a) and (c)—criterion and profit are expressed in the same units. (b)and (d)—criterion and profit are expressed in different units.

The results of this investigation suggest that expressing profit in the same units as forecastby criterion substantially increases the effectiveness indicator. Besides, such approach seemsto be more correct from the methodological point of view and increases the reliability of thewhole process of criteria effectiveness evaluation.

We propose the following approach to profit expression when evaluating the effec-tiveness of the main criteria described in Chapter 2.

● For criteria forecasting future profit (expected profit on the basis of differentdistributions), we recommend to express profit in U.S. dollars with the samenormalization as for criteria values.

● For criteria forecasting profit probability on the basis of different distributions, it isconvenient to express profit as a proportion of profitable combinations. The sameapproach can be used for the BEVR criterion. Although profit cannot be expressed inthe same units as forecast by this criterion, the notion of the “positive profits area” isclose to probability and can be approximated by the proportion of profitablecombinations.

● For criteria based on the ratio of expected profit to loss, we recommend to expressprofit as the ratio of the difference between profitable and unprofitable combinationsto the total number of combinations.

● The IV/HV ratio criterion does not forecast profit. Therefore, to evaluate itseffectiveness we can calculate (instead of estimating realized profits) the ratio ofhistorical volatility realized on a certain date in the future (the forecast date) tohistorical volatility used to obtain the criterion value. Likewise, for all other specific(not universal) criteria, we should establish special forms of profit expression thatcorrespond to the criterion nature as much as possible.

3.3.3 Expressing Effectiveness Indicators In the previous sections we evaluated the criterion effectiveness using a regression modelbetween criterion and realized profit values. The determination coefficient that reflects thevariability of data around the regression line was accepted as the effectiveness indicator.Though the determination coefficient adequately reflects the degree of interdependencebetween the criterion and the profit, it says nothing about the shape of their relationship.Consider the following hypothetical situation. Suppose that we have to evaluate threedifferent samples, each consisting of ten combinations created within a certain optionstrategy. We have to assess the criterion effectiveness in selecting combinations for eachstrategy.

We purposely generated data for this simulation in such a way that in each case the co-efficient of determination is very high and equal for all three strategies. It seems that the criterion effectiveness (as it was defined in the previous sections) is the same for each strategy. However, the regression chart points at the prematurity of this conclusion(Figure 3.3.3).

80 Systematic Options Trading

Evaluation of Criteria Effectiveness3 81

y = -0.41x + 4.51

R 2 = 0.98

y = 0.53x - 0.21

R 2 = 0.98

y = 0.10x + 2.98

R 2 = 0.98

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In one case the combination profit grows steeply as the criterion value increases. In thesecond case the growth is more gradual. And finally, in the third case the increase in thecriterion value results in the drop in the combination profit. Given these forms of {criterion× profit} relationships, we should conclude that this criterion is more effective for the firststrategy, less effective when applied to the second one, and unsuitable for the third strategy.However, our indicator (the coefficient of determination) indicates equal effectiveness ofthis criterion in all three situations!

This simulation experiment indicates the necessity to establish an additional indicator ofcriterion effectiveness, which takes into account not the {criterion × profit} correlationstrength but rather the shape of this relationship. The shape can be described by theregression equation with the regression line slope being the most important coefficient forour purposes. In the models presented in Figure 3.3.3, the greatest slope is attributed to thefirst strategy (0.53); it is lower for the second one (0.1) and negative for the third strategy(–0.41).

Figure 3.3.3 Hypothetical model simulating three different types of relationships betweencriterion and profit values.

It seems reasonable to express the criterion effectiveness indicator in two supplementalforms: as the determination coefficient (which reflects the correlation strength between thecriterion and the profit) and as the slope coefficient (which reflects the shape of the

relationship). At the same time we should mention that simulating the hypothetical situationpresented in Figure 3.3.3 we intentionally generated data in the way that made determi-nation coefficients equal and independent of regression slopes. Therefore, it would be usefulto analyze the real relationship existing between these two criterion effectiveness indicatorsusing the market data.

Consider the {criterion × profit} regression models for three criteria (expected profit onthe basis of lognormal (EPLN), empirical (EPEM), and symmetrized empirical (EPES) distri-butions) applied to two opposite option strategies (short and long strangle/straddle). Theparameter values are equal to those fixed in the previous studies: a 100-day averaging periodand 500 combinations used in the analysis. For the EPLN criterion the effectivenessindicator expressed as a determination coefficient is maximal for both short and longstrategies (Figure 3.3.4). If criterion effectiveness is measured by regression slope, the resultis exactly the same—for both strategies the highest coefficient is detected for the EPLNcriterion (Figure 3.3.4). EPEM and EPES criteria take the second and the third positions forboth the long and the short strategies. In these cases two forms of the effectiveness indicatoragain produced consistent valuations—both the determination coefficient and the slopewere higher for EPEM relative to EPES (Figure 3.3.4).

Summarizing the aforesaid, we conclude that usually the determination coefficient ispositively correlated with the slope coefficient. This suggests that both forms of the effec-tiveness indicator evaluate the forecasting abilities of criteria quite similarly. To providemore ground for this suggestion, we build regression models analogous to those presentedin Figure 3.3.3 for two more option strategies: short and long calendar spread. The valuesof both coefficients were computed for each regression, and the relationship between deter-mination and slope coefficients were analyzed.

Our expectation proved to be true. There is a strong correlation between two forms ofthis effectiveness indicator (Figure 3.3.5). Hence, evaluation of the criterion effectivenessusing the {criterion × profit} regression can be limited to only one of them. At the same timeone should remember that, when evaluating other criteria and using them with otherstrategies, we can face other shapes of the relationship between different forms of a certaineffectiveness indicator. (There may even be no relationship at all.) In such cases it would bereasonable to use several forms of the effectiveness indicator because information containedin each of them could be supplemental rather than overlapping.

82 Systematic Options Trading

Evaluation of Criteria Effectiveness3 83

short strangle/straddle

y = 0.22x - 0.46 y = 0.17x - 0.55y = 0.39x - 1.25R 2 = 0.53 R 2 = 0.40R 2 = 0.66

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long strangle/straddle

y = 0.39x - 23.42y = 0.42x - 22.27y = 0.91x + 3.00R 2 = 0.53R 2 = 0.58R 2 = 0.89

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Figure 3.3.4 Regression analysis of {criterion × profit} relationships for three criteria and twooption strategies. Criteria abbreviations are the same as in Figure 3.2.4.

In this section we review the main effectiveness indicators intended to evaluate the criteriaforecasting abilities. The indicators described here do not constitute a complete set of criteriaeffectiveness analysis tools. The development of new effectiveness indicators and elabo-ration of existing ones is a continuous process progressing in parallel with creating andupgrading of criteria as such.

Properties of different effectiveness indicators are illustrated by examples based on realmarket data. Using September 10, 2007, as the starting point, we calculate indicators on thebasis of a 100-day averaging period. (The grounds for this choice were discussed in section3.2.5.) The number of combinations used in the analysis will be 500. (This choice wasdiscussed in section 3.3.1.)

84 Systematic Options Trading

3.4 Review of Criteria Effectiveness Indicators

0.30

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1.00

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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R2

Figure 3.3.5 Relationship between the determination and the slope coefficients of the {criterion× profit} regression models estimated for three criteria and four option strategies. The EPLNcriterion is in black, EPEM is in gray and EPES is denoted by contour signs. Diamonds stand forthe short strangle/straddle strategy, circles for the long strangle/straddle strategy, triangles forthe short calendar spread, and squares for the long calendar spread.

Evaluation of Criteria Effectiveness3 85

Four criteria (EPLN, EPEM, PPLN, EPLL) and four option strategies (long and shortstrangle/straddle, long and short calendar spread) serve as the examples for each effec-tiveness indicator. Wherever it is possible, profit is expressed in the same form as criterion(as described in section 3.3.2).

3.4.1 Correlation Between Criterion and Profit ValuesThis effectiveness indicator was previously discussed in detail. Hence, we will not return tothe calculation method and will proceed directly to its application to different criteria andoption strategies.

Obviously, this indicator evaluates the qualities of different criteria differently (Figure3.4.1). The values of the criteria, which forecast profit probability (PPLN) and the profit toloss ratio (RPLL), demonstrate a much stronger correlation with realized profit than thecriteria forecasting expected profit (EPLN and EPEM). This is true for all strategies exceptlong calendar spread where the highest determination coefficient was detected for the EPLNcriterion (R2 = 0,96).

Concerning the differences between strategies, the following conclusions can be drawn.The PPLN and RPLL criteria show quite similar effectiveness for all four option strategies(Figure 3.4.1). The only exception is the short calendar spread for which both criteriashowed lower effectiveness than for the other strategies. In contrast to PPLN and RPLL, theeffectiveness of the other two criteria turned out to be heterogeneous with respect todifferent strategies (Figure 3.4.1). EPLN and EPEM were the most effective when appliedto the long calendar spread strategy. EPLN was the least effective for the short calendarspread and EPEM for the short strangle/straddle strategy.

Summarizing the analysis of data presented in Figure 3.4.1, we conclude that this effec-tiveness indicator allows ranking of criteria according to their appropriateness for selectingpotentially profitable option combinations. It is noteworthy that a certain criterion is highlyeffective when applied to one strategy yet displays inferior effectiveness with others.Similarly, a certain criterion may effectively evaluate option combinations created within agiven strategy while other criteria fail in accomplishing this task.

This means that the quality of every criterion should be thoroughly evaluated with regardto different option strategies. This evaluation results in classifying of criteria and optionstrategies according to their mutual compatibility. Such approach establishes a correspon-dence between strategies and their most suitable criteria.

86 Systematic Options Trading

EPLN

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long calendar spread

short strangle/straddle

short strangle/straddlelong strangle/straddle

long strangle/straddle

short calendar spread

short calendar spread

long calendar spread

long calendar spread

short calendar spread

Figure 3.4.1 Regression analysis of {criterion × profit} relationships for four option strategiesand four criteria.

3.4.2 Correlation Between a Criterion and Profit IndexesThis indicator is similar to the previous one in its nature and calculation methodology. Thedifference is that instead of measuring the correlation between the absolute values ofcriterion and profit we analyze their relative values. All combinations are sorted by criterionvalues and get indexes corresponding to their positions in the ordering. The first combi-nation, having the highest criterion value, is assigned index 1; the second combinationreceives index 2, the last combination, having the lowest criterion value, is given index 500.Then all combinations are sorted once again according to their profit values and receive

Evaluation of Criteria Effectiveness3 87

second indexes according to their positions in this ordering. As a result, each combinationpossesses two indexes corresponding to its criterion and profit values. The criterion effec-tiveness is determined by the regression of two indexes, and the determination coefficient isused as an indicator of effectiveness.

The advantage of this effectiveness indicator over the previous one (section 3.4.1) is insmoothing extreme criterion and profit values. Apart from that, the criterion and the profitvariables always have the same scales ranging from 1 to 500. This allows distributing dataevenly within a two-dimensional coordinate system that makes visualization easier (compareFigures 3.4.1 and 3.4.2).

Visual observation in Figure 3.4.2 reveals that the EPLN criterion gives better results forthe long strategies than for the short ones. This criterion is the least effective for the shortstrangle/straddle strategy as evidenced by the fuzzy cloud of points broadly scattered on theregression plane. The same conclusion relates to the EPEM criterion. Moreover, thiscriterion is characterized by greater fuzziness of points in the bottom-left corner of the graphin comparison with the top-right corner (Figure 3.4.2). It means that EPEM erroneouslymore frequently identifies combinations as “good” (the bottom-left corner of the chartcorresponding to high criterion values) than identifying them as “bad” (the top-right cornercorresponding to low criterion values).

With regard to all strategies, PPLN and RPLL are more effective than the other twocriteria. (It complies with the results obtained on the basis of another effectiveness indicatordescribed in section 3.4.1.) Charts corresponding to PPLN and RPLL (Figure 3.4.2) clearlyshow that when applied to the short calendar spread strategy these criteria effectively revealbad combinations but often make mistakes identifying good ones. This feature is proved bythe fuzziness of points in the bottom-left corners of these charts and their concentrationalongside the diagonals of top-right corners. Interestingly, the situation for the shortstrangle/straddle strategy is the opposite: Points are fuzzy in top-right corners and are closeto the diagonal in the bottom-left corners of the charts (Figure 3.4.2).

The effectiveness indicator based on the correlation between criterion and profitindexes is similar to the one we discussed in section 3.4.1. Furthermore, the numericalvalues of both indicators (determination coefficients) are so close that the two evaluatingmethods could be considered as identical. However, their visual characteristics differ. Theregression of indexes effectively detects distinctive peculiarities of the relationship betweenthe criterion and the profit that remain hidden if absolute values are analyzed. In particular,index analysis effectively displays the heterogeneity of forecasting abilities of the criterionwithin different ranges of its values. Some criteria can be highly effective in the area of highvalues and rather ineffective in the area of low values (or vice versa). This information,being readily obtainable by the visual analysis of index regression, is important for criteriaevaluation.

88 Systematic Options Trading

EPLN

0

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fit

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fit

short strangle/straddle long strangle/straddle

short calendar spread long calendar spread

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short strangle/straddle long strangle/straddle

short calendar spread long calendar spread

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0 50 100 150 200 250 300 350 400 450 500

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short strangle/straddle long strangle/straddle

short calendar spread long calendar spread

Figure 3.4.2 Regression analysis of criterion and profit indexes for four option strategies andfour criteria.

3.4.3 Correlation Between the Sharpe Ratios of Criterion and ProfitThis method of criterion effectiveness evaluation is also based on regression analysis. Incontrast to the previous methods, here coordinates of points on the regression plane aredetermined by the Sharpe ratios of the criterion and profit and not by their raw values.

Because we set a 100-day averaging period, there is a group of 100 elements for eachposition (ranging from 1 to 500) in 100 orderings of combinations according to the criterionvalues. These groups are formed in the following way. For a certain date the combinationwith the maximum criterion value is selected. Similarly, the combination with the maximumcriterion value is chosen for the previous date and so on up to 100 preceding days. As a result,we have a group of 100 combinations each of which was the best in one of 100 days. A group

of 100 combinations that were at the second positions for each day is created in an analogousway. Finally, we obtain 500 groups (because we use 500 combinations for each date).

Thus, we have 100 combinations with the maximum criterion values, 100 combinationsoccupying the second positions in the orderings by criterion values, 100 combinationssituated at the third positions, and so on up to the 500th positions. For each of these groupswe can compute not only the average criterion and profit value (we used it in sections 3.4.1and 3.4.2 for the EPLN and EPEM criteria) and the ratio of profitable and unprofitablecombinations (we used it in sections 3.4.1 and 3.4.2 for the PPLN and RPLL criteria) butalso the indicator showing the variability of criterion and profit values within each of the500 groups.

We express the variability by calculating a standard error that is the ratio of a standarddeviation to the square root of the number of elements in the sample (in this case, thenumber of elements is 100). For each group the Sharpe ratios of the criterion and the profitare computed by dividing the average by the standard error. (Traditionally, the Sharpe ratiois defined as the ratio of the excess return [difference between the average profit and the risk-free rate] to standard deviation. In our version of the Sharpe ratio, we assume the risk-freerate to be zero and use standard error instead of standard deviation.) Next, we applyregression analysis to these data. Point coordinates are the Sharpe ratios of the criterion, andthe profit and each point corresponds to one of the 500 groups. The determination coeffi-cient of this regression reflects the strength of the relationship between the profit Sharperatio and the criterion Sharpe ratio. This method allows evaluating the {criterion × profit}relationship taking into account not only the averages of these indicators but of theirvariability as well.

Regression analysis of the Sharpe ratios shows that EPLN is more effective than EPEMfor the majority of option strategies (Figure 3.4.3). The only exception is the short calendarspread strategy for which the first criterion is slightly less effective than the second one (R2

= 0.50 and R2 = 0.53 correspondingly). Unfortunately, this method is inapplicable forevaluation of the effectiveness of the other two criteria: PPLN and RPLL. Because values ofthese criteria correspond to probabilities and ratios, the profits should be expressed asproportions and ratios of profitable combinations within a certain group (see section 3.2.2).Therefore, for each group only one profit indicator is obtainable (in contrast to other criteriain which each of the 100 combinations in a group possessed its own profit value). Hence,we cannot estimate any intra-group variability measure.

For the long strangle/straddle strategy, the effectiveness of both criteria is extremely low(Figure 3.4.3; R2 = 0.13 for EPLN, R2 = 0.01 for EPEM). This is quite surprising knowingthat the determination coefficients obtained using the other two methods were rather high(regression based on relationships between criterion and profit: R2 = 0.89 for EPLN, R2 =0.58 for EPEM; regression based on criterion and profit index: R2 = 0.89 for EPLN, R2 =0.58 for EPEM). Apparently, the strong correlation between the average values of thecriterion and the profit coincides in this case with the high variability of these indicatorshereby leading to the low correlation of the Sharpe ratios.

Evaluation of Criteria Effectiveness3 89

Figure 3.4.3 Regressionanalysis of the criterion andprofit Sharpe ratios for fouroption strategies and twocriteria.

90 Systematic Options Trading

EPEM

-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1012345

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Criterion Sharpe coefficient

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

Criterion Sharpe coefficient

Pro

fit

Sh

arp

e co

effi

cien

t

short strangle/straddle long strangle/straddleshort calendar spread long calendar spread

short strangle/straddle long strangle/straddleshort calendar spread long calendar spread

EPLN

-15-14-13-12-11-10-9-8-7-6-5-4-3-2-1012345

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Evaluation of Criteria Effectiveness3 91

The latter example illustrates the interesting possibility of opposite evaluations whileestimating a certain criterion using different effectiveness indicators. The awareness aboutsuch inconsistency is crucial for comprehensive criteria quality analysis. We consider theeffectiveness indicator based on the regression of the Sharpe ratios to be an essentialinstrument of criterion forecasting abilities evaluation. It contains important informationthat is not readily disclosed by other indicators.

3.4.4 Areas RatioSorting combinations by criteria values, we assume that each subsequent combination in theordering is inferior to the previous ones in terms of profit potential or other profit-relatedcharacteristics. The effectiveness indicator based on the areas ratio is intended to test thisassumption. Unlike all criteria effectiveness evaluation techniques described in the previoussections, this method is not based on regression analysis. The peculiarities of its calculationmake it applicable only to criteria forecasting the expected profit of combinations. Next wedescribe the procedure for calculating this effectiveness indicator using the example of theshort strangle/straddle strategy.

Imagine the relationship between the profit of the portfolio consisting of combinationssituated at n top positions in the ordering and the number of combinations included in theportfolio (that is, n that ranges from 1 to 500). If the criterion can indeed identify the bestcombinations effectively, the ordering would consist of profitable combinations situatedmostly at top positions and unprofitable ones positioned mainly at lower positions. In thiscase, the relationship would have a shape of an ascending convex curve that reaches itsmaximum at the point corresponding to the combination with a zero profit. Then the curvedescends down to its minimum.

The bold curves in Figure 3.4.4 illustrate these relationships for three criteria forecastingexpected profit on the basis of different distributions (EPLN, EPEM, and EPES). The shapeof these curves corresponds to the theoretical scheme previously described. This suggeststhat the criteria indeed can sort combinations by their potential profit. An additional confir-mation of the criteria effectiveness is the shape of the thin curves in Figure 3.4.4. They areobtained in the same manner except that the combinations were sorted not by the criteria (asit was done in the cases of the bold lines) but randomly. One can easily see that in the caseof sorting combinations by the criteria portfolio profit at first increases (because profitablecombinations are situated mostly at the first positions in the ordering) and then decreases.On the other hand, random ordering resulted in a decrease of profits from the beginning(Figure 3.4.4).

92 Systematic Options Trading

Figure 3.4.4 Relationship between the portfolio profit and the number of combinations includedin the portfolio (n). The bold curves correspond to portfolios based on combinations sorted bycriteria values. (Combinations were sorted by criteria, and top n ones were included in theportfolio.) Thin curves correspond to portfolios of combinations sorted randomly. Criteria abbre-viations are the same as in Figure 3.2.4.

Now we proceed from the visual analysis of the data shown in Figure 3.4.4 to the formal-ization and numerical expression of the criteria’s effectiveness indicator. An ideal ordering,which serves as the reference point, should be created for this purpose. Assume that thecriterion under consideration is perfect. This means that sorting combinations by thecriterion values completely coincides with sorting by realized profit. Our effectivenessindicator is based on measuring the degree of coincidence between the two rankings—ordering by the real criterion and ordering by an idealized criterion (that is, by profit).

Idealizations of EPLN, EPEM, and EPES criteria are shown in Figure 3.4.5 by thin curves(obtained by sorting combinations according to their profit values). Obviously, idealizationsare better than real orderings obtained by sorting combinations by the criteria values (thebold curves). The degree of this superiority has to be expressed numerically to produce anew criterion effectiveness indicator.

The basic idea of this effectiveness indicator can be easily appreciated by graphical repre-sentation. Consider again the relationship between portfolio profit and the number ofcombinations forming the portfolio (n) in two cases—sorting by values of the EPLN criterionand sorting by future profit (idealization). Let us connect the first and the last points of thetwo graphs with a straight line (Figure 3.4.6). The starting points of both graphs alwayscoincide; they represent “empty” portfolios (without any combinations) with zero profits.

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n

EPLN EPEM EPES

Evaluation of Criteria Effectiveness3 93

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EPLN EPEM EPES

Figure 3.4.5 Relationship between the portfolio profit and the number of combinations includedin the portfolio (n). Bold curves correspond to portfolios based on combinations sorted by criteriavalues. (Combinations were sorted by criteria and the top n ones were included in the portfolio.)Thin curves correspond to portfolios created by idealized sortings (that is, by future profit). Criteriaabbreviations are the same as in Figure 3.2.4.

(They are needed for standardization and comparability of the two orderings.) The lastpoints of both graphs also coincide and correspond to the portfolios consisting of all thecombinations (500 in this case). Therefore, regardless of the sorting method, its profit isequal to the total profit of all combinations used in the analysis.

Connecting the first and the last points with a straight line produces two plane figureswith a common base (Figure 3.4.6). The first figure is bounded with a straight line frombelow and with an idealized curve representing sorting by profit from above. The secondfigure is bounded with the same straight line from below and with the curve representingsorting by the criterion from above. The area of the first figure is always larger or equal tothe area of the second one. The closer the area of the second figure to the area of the firstone, the higher the quality of the criterion. In the extreme case the two areas coincide, andthis means that the criterion possesses maximum effectiveness because there were nomistakes in its ordering.

The ratio of areas of these two figures stands for numerical expression of the criterioneffectiveness indicator. It ranges from 0 to 1 and can be computed in the following way.

Consider an ordering of combinations sorted by descending profit. The positionoccupied by a combination in the ordering is denoted as k, k = 1,…,N. In our case N = 500.Let p(k) be the profit value of combination k. In accordance with the ordering, the follow-ing inequality holds: p(n) ≤ p(m) for all n and m so that n > m. Evaluation of these combinations using a certain criterion results in their descending ordering by its values. We

94 Systematic Options Trading

denote the position of a combination situated at the k-th position as ik. Sums and are profits of two portfolios consisting of n first combi-P n p ii

kk

n( ) ( )=

=∑

1

P n p kk

n( ) ( )=

=∑

1

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nFigure 3.4.6 Graphical representation of two figures used for computing the effectivenessindicator based on the areas ratio. (This example is based on the EPLN criterion.) Both figuresare bounded with the straight line from below. From above each figure is bounded with a curvecorresponding to the relationship between the portfolio profit and the number of combinationsincluded in the portfolio (n). The bold curve represents the portfolios based on the sorting ofcombinations by criteria values. The thin curve represents the portfolios based on the sorting ofcombinations by future profit.

nations: P(n) stands for the sorting by profit and Pi(n) stands for the sorting by criterionvalues. One can easily see that for any n from 1 to N holds P(n) ≥ Pi(n), which explains thefact that for any n the curve for the profit of the portfolio corresponding to sorting by thecriterion is lower than the idealized curve. We denote the total profit of the portfolioconsisting of N combinations as . Then the criterion effectivenessindicator representing the ratio of areas of the two figures described is

P P N P NN i= =( ) ( )

Evaluation of Criteria Effectiveness3 95

,or after a simple transformation:

(3.4.1).

The indicator of criterion effectiveness computed for the data presented in Figure 3.4.6using formula 3.4.1 equals I = 0.72. Note that the areas ratio is quite sensitive to the totalnumber of combinations used in the analysis. (The similar effect was described in section3.3.1 for another effectiveness indicator.) The degree of this sensitivity can be estimated byconstructing figures based on different numbers of combinations.

Figure 3.4.7 shows pairs of figures corresponding to 100, 200, 300, 400 and 500 combi-nations used in the analysis of the EPLN criterion effectiveness. As the number of combina-tions decreases, the ratio of areas changes. It is I = 0.59 for 400 and 300 combinations, I =0.63 for 200 combinations and I = 0.65 for 100 combinations. Although these changes arenot very significant, they can become rather substantial in other circumstances (over othertime periods, for other criteria and option strategies). Our statistical investigations andresults of real trading point to about 200—300 as the optimal number of combinations tobe used in the criterion effectiveness analysis based on the areas ratio.

Ip i

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Figure 3.4.7Graphical repre-sentation of figuresused for calcu-lating the effec-tiveness indicatorbased on the areasratio for the EPLNcriterion. Each thincurve coupled witha thin line corre-sponds to a pair offigures with aspecific number ofcombinations usedin the analysis.

3.4.5 Other Effectiveness IndicatorsThe described indicators do not constitute a complete list of evaluation techniques intendedfor assessment of criteria effectiveness. Analytical methods based on similar or fundamen-tally different ideas are constantly developed. Different versions of effectiveness indicatorscan be produced on the basis of the same principles described in sections 3.4.1–3.4.4. Suchindicators, evaluating the forecasting qualities of criteria in a slightly different way, areuseful supplemental tools that are indispensable for a comprehensive comparative analysisof many criteria.

Below we present several examples illustrating such additional versions of criteria effec-tiveness indicators. (All examples are based on the EPLN criterion and the shortstrangle/straddle strategy.)

The relationship between the profit and the position of a combination in the ordering.This evaluation method represents a mixture between index regression (section 3.4.2) and{criterion × profit} regression (section 3.4.1). The difference is that here we estimate therelationship between profit expressed in U.S. dollars and the criterion value presented as anindex. What is the purpose of such hybridization? It turned out that presenting data in thisway may reveal the nonlinear nature of a profit-criterion relationship (Figure 3.4.8). Theincrease of the combination’s position in the ordering from 1 to 100 leads to a sharp fall inprofit. Further promotion from the 100th up to 400th position does not have much impacton the returns but further increase of the position from the 400th to 500th position againentails an abrupt profit decline. This nonlinearity (which is hardly detectable by means ofthe other effectiveness indicators) is extremely important for the clear perception of differ-entiated criteria effectiveness within its different value ranges.

Deviations of profits forecasted by the criterion from realized values. Criteria based onmathematical expectations forecast future profits in the most direct way. Therefore, it wouldbe useful to analyze deviations of forecasted values from their realizations and possiblerelationship between these deviations and forecasted values themselves. Figure 3.4.9(a)shows that in most cases forecast values are overestimated. However, it is even moresurprising that there is a clear negative relationship between deviations and criterion values.The higher the criterion values, the more overvalued are the forecasts. On the contrary,many negative forecasts turned out to be undervalued because combinations with negativeforecast values actually performed better than expected.

96 Systematic Options Trading

Figure 3.4.8 Relationship between profit of combinations and their position in the ordering.EPLN criterion; short strangle/straddle strategy.

The relationship between deviations and the positions of combinations in the ordering isshown in Figure 3.4.9(b). Significant overvaluing observed for combinations situated at thefirst positions in the ordering sharply diminishes as the position number increases. However,this relationship is nonlinear and, therefore, the rate of the overvaluing disappearance slowsdown after the 100th position. Although after the 200th position a few overvalued forecastsare still observed, after the 300th position profits of most combinations are on averageforecasted correctly.

Certainly, deviations of profit from the forecast values are inevitable. The average valueand dispersion of these deviations should be analyzed by statistical methods and must betaken into account in practical application of criteria. It is particularly important not to losesight of nonlinearity of relationship between deviation and position of combinations in theordering. As shown in Figure 3.4.9, combinations with the highest criterion values are mostlikely to be overestimated. Notwithstanding their ultimate profitability, the realized gainvalue is frequently lower than it was predicted. Therefore, when estimating future return ofan option portfolio, we have to consider this feature and try to offset it by introducing coeffi-cients correcting the absolute profit values forecasted by criteria.

Evaluation of Criteria Effectiveness3 97

-6

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Position in the ordering

Figure 3.4.9 Difference between profit and criterion values and its relationship with (a) thecriterion value and (b) the position of combinations in the ordering. EPLN criterion; shortstrangle/straddle strategy.

98 Systematic Options Trading

(b)

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When developing correction coefficients and their application methods, you should takeinto account specific forms of the relationship between deviations and position of combina-tions in the ordering. In some cases the relationship may be hardly detectable or even absent;in other cases it may have a different shape not similar to that shown in Figure 3.4.9. Thefollowing example is intended to illustrate this. Sticking to the same strategy (shortstrangle/straddle) we replace the criterion—instead of the criterion forecasting profit value(EPLN) we apply another criterion forecasting profit probability (PPLN). Such substitutionleads to a different form of relationship (Figure 3.4.10). First, the relationship between thedifference and the criterion value changed its shape from linear (Figure 3.4.9(a)) tononlinear (Figure 3.4.10(a)). Secondly, the relationship between the differences and theposition of combinations in the ordering—albeit remained nonlinear—changed its shapefrom convex (Figure 3.4.9(b)) to concave (Figure 3.4.10(b)). Finally, the degree of these tworelationships is also different. There is no need for computing correlation coefficients fornonlinear regressions to see that data dispersion is much higher for the PPLN than for theEPLN criterion. (Compare Figures 3.4.9 and 3.4.10.) The relationships between deviationsand values of other criteria can be even more specific, especially if these criteria are used toevaluate combinations within other option strategies. All peculiarities of such relationshipsmust be taken into account while developing and applying correction coefficients.

Evaluation of Criteria Effectiveness3 99

Figure 3.4.10 Difference between profit and criterion values and its relationship with (a) thecriterion value and (b) the position of the combination in the ordering. PPLN criterion; shortstrangle/straddle strategy.

100 Systematic Options Trading

(a)

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Analysis of the Sharpe ratios. In section 3.4.3 we already used the Sharpe ratios tocalculate the effectiveness indicator using the regression model. (This was a regressionreflecting the relationship between the Sharpe ratios based on profit values and the Sharperatios based on criterion values.) This kind of analysis can easily be expanded. The Sharperatio is an extremely important indicator containing information not only on theprofitability of combinations but on their variability as well. Thus, it should be appliedextensively to obtain a versatile and comprehensive evaluation of criteria forecastingabilities.

The following example demonstrates extraction of additional information about criteriaeffectiveness by application of the Sharpe ratio analysis. Figure 3.4.11 illustrates therelationship between the profit Sharpe ratio and the position of the combination in theordering by criterion values. As we have already noticed in the earlier examples, relation-ships of this kind are often nonlinear. At first, the increase in the position number leads to asharp fall in the Sharpe ratio. After the 100th position the decrease of the coefficient slowsdown and resumes its pace only after the 400th position. The nonlinearity of such relation-ships is extremely important for estimating the heterogeneity of criteria effectiveness at itsdifferent value ranges.

Evaluation of Criteria Effectiveness3 101

-0.3

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Sh

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Figure 3.4.11 Relationship between the Sharpe ratio and the position of the combination in theordering.

The criterion effectiveness analysis is a complex procedure requiring consideration of theexhaustive number of factors having both direct and indirect influence on the evaluationprocess.

First, the analysis is complicated by the time-to-time variability of all effectivenessindicators regardless of the option strategies and criteria. In this chapter we illustrated thedynamics of only one effectiveness indicator (to avoid overloading the text with monot-onous figures). Nevertheless, our exhaustive research suggests that all effectivenessindicators without exception are to a greater or lesser extent variable in time. We proposedthe way to master this problem and described the methods allowing mitigation of the effectof indicators’ variability. However, there is no universal solution suitable for all circum-stances. Hence, each situation requires individual and comprehensive analysis.

Furthermore, the evaluation results depend on the averaging procedure necessary toincrease the reliability of criteria effectiveness estimation. (This procedure requiresoptimization of the “averaging period” parameter.) Besides, there are many additionalfactors each of which should be thoroughly considered and parameterized. We discussedonly one of them (the number of combinations involved in the analysis) because its influenceis inevitable for almost all criteria, option strategies, and effectiveness indicators. However,the influence of other specific factors is not so evident and does not always become apparent.

Another difficulty arises from the necessity to express profit in the same form and in thesame units as forecast by the criterion. Unfortunately, it is not always possible, and one hasto resort to various tricks to bring these values together.

Our research suggests that to ensure the successful evaluation of criterion effectiveness itis not enough to select a correct analytical procedure. Because one procedure can generateseveral numerical indicators (for example, the determination coefficient and the slope coeffi-cient in the regression analysis), we must select those that reflect the criterion characteristicsbetter.

As to the number of effectiveness indicators, we can develop as many of them as ourimagination and computer capacities allow us to. We described only basic methods ofcriteria effectiveness evaluation choosing the most universal, understandable, and easilyinterpreted ones. Undoubtedly, a creative approach allows developing a lot of additionalindicators based on various evaluation principles. This clears the way for a most compre-hensive criteria effectiveness analysis.

Applying the techniques described in this chapter in practice requires (1) developmentand adaptation of transformation procedures, (2) optimization of parameters, and (3)adjustment of evaluation algorithms to the current market conditions and to the optionstrategies used by an investor.

102 Systematic Options Trading

3.5 Summary

235

A

absolute price changes distribution,58-61

absolute values of criterion, 110-112adding combinations, 76additive convolution, 190algorithms

of combination selection, 121-122

empirical distribution, 28-31optimization, xxiiPareto sets, 183-189

allocation of capital, 172-173analysis

comparative, xxiv-xxvcriteria effectiveness, 106-110,

150-155dynamic, 67factors affecting combination

selection, 113-114multicriteria, xxiv

number of combinations usedin, 75-77

rankinggeneralized results,

140-142methods of, 123-131results of, 131-138

regression, 83selection of underlying assets,

162-165Sharpe ratios, 101visual, xxxiv

applyingconvolution methods, 212-214lognormal distributions, 42regression models to profit and

value relationships, 65area ratios, 91-95assets

fair value, xxxprice variability, 57

Index

underlying, 112analysis of criteria

effectiveness, 162-165long-term evaluation of

criteria effectiveness,166-168

optimization, 168-176selection of, 161-162

assignment of distribution of probabilities, 5

averagingfour (4)-days data, 68selection of periods, 72-73, 75

B

Baird, Allen, xxbanks, options, xvii. See also optionsbear spreads, 233-234behavior, prices, xxiblack symbols, 149Black, Fischer, xviBlack-Scholes model, xxx, 6, 57break-even range, 50-53Brownian motion, 42bull spreads, 233-234buy low, sell high strategy, 7butterfly strategy, 37

C

calculationsdetermination coefficients, 69empirical distribution, 28-31lognormal distribution, 15-22ratio of expected profit and loss,

34-38relative changes, 14returns, xix

TimeValue, 60unified probability density

functions, 47calendar spreads, 231-233

comparison of, 151calls

options, xviispreads, 12strike prices, 3

capital allocation, 172-173Chicago Board Options Exchange

(CBOE), xviclassification of criteria, 9-12coefficients

criterion effectiveness, 133, 142-145

determination, 65-69combinations

optionsabsolute values, 110-112analysis of criteria

effectiveness, 106-110long-term evaluation of

criteria effectiveness,115-118

selection of, 105simultaneous analysis of

factors affectingselection, 113-114

underlying assets, 112selection of algorithms, 121-122several days in analysis, 68

common uncertainty, 6comparative analysis, xxiv-xxv

of multicriteria, 191-202comparisons

of calendar spreads, 151of heterogeneous strategies, 122

236 Index

Index 237

Connolly, Kevin, xxconstruction, unified probability

density functions, 48-49convolution, 157-159, 190-191

applying, 212, 214minimax, 156Pareto sets, 197-202utility functions, 173-176

correlationcoefficients, 65between criterion and profit

indexes, 86-87between criterion and profit

values, 85multicriteria selection

applying convolutionmethods, 212-214

evaluation of criteria inter-relationships, 206-208

impact on, 205selection profitability

of Pareto selections,208-212

between Sharpe ratios ofcriterion and profit, 88-91

criteriaclassification, 9-12effectiveness, 63-64

analysis of, 106-110correlation between profit

and, 64-68expressing indicators,

80-82expressing profit, 77-80long-term evaluation of,

115-118number of combinations

used in analysis, 75-77

review of indicators, 84-89,92-99, 101

selection of averagingperiods, 72-75

selection of underlyingassets, 162-168

transformation ofindicators, 68-69

values of coefficients, 142-145

expected profit on basis of empirical

distribution, 25-28on basis of lognormal

distribution, 15-22expert distribution, 9, 38-39

set of standard distributions, 39-45

unified probability densityfunctions, 45-49

forecasting, 8-9, 155inefficient, 126based on lognormal distribution,

13-15mission fulfilled by, 6modifications of empirical

distribution, 31-34multicriteria selection, 181-182

applying convolutionmethods, 212-214

comparative analysis, 191-202

convolution, 190-191correlation, 205evaluation of criteria inter-

relationships, 206-208Pareto sets, 183-185

profitability of Pareto selections, 208-212

widening Pareto sets, 186-189

nonexpert, 39nonuniversal, 50

break-even range, 50-53IV/HV range, 53-57ratio of normalized time

value, 58-61relative frequency criterion,

57-58perfect, 124philosophy of creation, 5probability on basis of empirical

distribution, 28profit

correlation betweenindexes, 86-87

correlation between Sharperatios and, 88-91

correlation between values, 85

ratio of expected profit to loss,34-38

simplified calculation algorithms,28, 31

specific (nonuniversal), 11-12cycles, quasiperiodical, xxxiv

D

days, combining several days inanalysis, 68

definitions, systematic approach, 4-5degree of symmetry (DSYM), 115-117delta, xxii

derivatives, xxiitheory, xviii

determination coefficients, 67-69development of possible future

scenarios, 9deviations in measurements, 144distribution

absolute price changes, 58-61empirical

criteria based on, 22-24expected profit on basis,

25-28modifications of, 31-34probability on basis of, 28simplified calculation

algorithms, 28, 31expert

criteria based on, 38-39set of standard

distributions, 39-45unified probability density

functions, 45-49exponential, 43frequency, xxxiii, 131, 136Gaussian, 14lognormal, xx, 41-42

description of, 13-15expected profit on basis of,

15-22normal, 14, 132probabilities, 5, 19standard, 8of time fractions, xxxviuniform, 40

domination of certain strategies overothers, 122

238 Index

Index 239

dynamicsanalysis, 67of trading opportunities, xxxi,

xxxiii, xxxv, xxxviiof transformed effectiveness

indicators, 69-71

E

effectivenesscriteria, 63-64

analysis, 106-110correlation between profit

and, 64-68expressing indicators,

80-82expressing profit, 77-80long-term evaluation of,

115-118number of combinations

used in analysis, 75-77review of indicators, 84-89,

92-101selection of averaging

periods, 72-75selection of underlying

assets, 162-168transformation of

indicators, 68-69values of coefficients,

142-145monocriterion selection,

191-196empirical distribution

criteria based on, 22-24expected profit on basis, 25-28modifications of, 31-34probability on basis of, 28simplified calculation algorithms,

28-31

estimation of risk, xxiievaluation

of criteria effectiveness, 63-64correlation between profit

and, 64-68expressing indicators,

80-82expressing profit, 77-80long-term, 115-116, 118number of combinations

used in analysis, 75-77review of indicators, 84-89,

92-101selection of averaging

periods, 72-75transformation of

indicators, 68-69of criteria interrelationships,

206-208of risk, xixof trading opportunities, xxx

events, frequency, 8expectation, 41expected profit

on the basis of empirical distribution (EPEM), 25-28,115-117

on the basis of lognormal distribution (EPLN), 15-22,115-117, 133-137

Pareto sets, 185-187expert distribution

criteria based on, 38-39set of standard distributions,

39-45unified probability density

functions, 45-49expert forecasts, 42

exponential distribution, 43expressing

effectiveness indicators, 80-82profit, 77-80

F

F-tests, 132fair value, xxi, xxxforecasting, xxii

correlation between outcomeand, 64

criteria, 155expert, 42future profit, 80horizons, 27as key elements of criterion, 8-9profit probability, 80universal criteria, 10

formal definition of systematicapproach, 4-5

forming Pareto sets, 183-185formulas

Black-Scholes, xxxlognormal distribution, 14

frequencydistribution, xxxiii, 131, 136past events, 8relative frequency criterion,

57-58functions

payoff, 15, 47, 226option combinations,

228-234separate options, 226-227

probability density, 19, 28underlying assets, 171-176

unified probability density, 45-49

utility, 155-159weight, 32

future scenarios, 9

G

Gatheral, Jim, xixGaussian distribution, 14generalized ranking analysis results,

140-142geometrical Brownian motion, 42goals of systematic approaches,

xxiii-xxivgray symbols, 149Greeks, xxii, 11. See also risk

H

Haug, Espen, xixhedging, xviiheterogeneous strategies,

comparisons, 122High FrIV(B), 58history

of options, xvi-xviivolatility, xxii, 16, 80

holding periods, 31horizon of history, 27

building empirical distributioninfluences, 26

empirical distribution, 28weight functions as substitute, 32

Hull, John, xviii

I

implied volatility, 57in-the-money options, 58

240 Index

Index 241

indexes, correlation between criterionand profit, 86-87

indicatorscriteria effectiveness, 68-69effectiveness

expressing, 80-82review of, 84-89, 92-101

profit, 64-68utility, 169

individual uncertainty, 6inefficient criterion, 126interactive probability scenario

builder, 39, 43interrelationships, evaluation of

criteria, 206-208intraday structure, xxxiinvestment firm options, xvii. See also

optionsIV/HV

Pareto sets, 185-187ranges, 53-57ratios, 80

J–K–L

James, Peter, xviii

kurtosis, 8

layers, Pareto sets, 186-189, 210lognormal distribution, xx, 41

applying, 42description of, 13-15expected profit on basis of,

15-17, 19-20, 22long calendar spreads, 64, 84,

127-128, 231, 233long history horizons, 27long straddle strategies, 3, 84, 115,

127-128, 133, 137, 143, 228-229

long strangle strategies, xxxiii, 3, 84,115, 229-230

long strategiesaveraging periods, 73-74dynamics of transformed

effectiveness indicators, 71long-term evaluation, 1156-118,

166-168losses, criteria based on ratio of

expected profit and loss, 34-38

M

management, risk, xviimarkets

development of options, xviiiprice, xxxi

matrices, xxvii-xxixmaximum drawdowns, 169McMillan, Lawrence, xixmean, 8, 17MeanEmpiric(B,S), 31MeanSymEmpiric(B,S), 34measurements, deviations in, 144methods

convolution, 212-214of criterion effectiveness,

145-150of long-term evaluation of

criteria effectiveness, 115-116of ranking analysis, 123-131underlying assets, 163, 166

minimax convolution, 156, 173, 190mission fulfilled by criteria, 6models

historical volatility, 17optimization, 168-176price, xxi

regression, 65-66standard price distribution, 14threshold parameters, 155-159

modes, 39modification of empirical distribution,

31-34monocriterion selection effectiveness,

191-192, 195-196multicriteria

analysis, xxivselection, 181-182

comparative analysis, 191-202

convolution, 190-191correlation, 205-214Pareto sets, 183-189

multiplicative convolution, 190mutual fund options, xvii. See also

options

N

narrow trade ranges, 31Natenberg, Sheldon, xxNeftci, Salih, xixnonexpert criteria, 39nonforecasting universal criteria, 11nonformalized analysis, 9nonuniversal criteria, 50

break-even range, 50-53IV/HV ranges, 53-57ratio of normalized time value,

58-61relative frequency criterion,

57-58normal distribution, 14, 132number of combinations used in

analysis, 75-77

O

Objectives of systematic approach,xxiii-xxiv

Operations, sequences of, xxvii-xxixopportunities, trading, xxix-xxx

evaluation, xxxstructure and dynamics of, xxxi,

xxxiii, xxxv, xxxviioptimization

algorithms, xxiithresholds

parameters, 155-156, 159values, 145

underlying assets, 168-176options

combinations, 228absolute values, 110, 112analysis of criteria effec-

tiveness, 106-110bear spreads, 233-234bull spreads, 233-234calendar spreads, 231-233long-term evaluation of

criteria effectiveness,115-118

selection of, 105simultaneous analysis of

factors affectingselection, 113-114

straddles, 228-229strangles, 229-230underlying assets, 112

criteria based on ratio ofexpected profit to loss, 34-38

empirical distributionexpected profit on basis of,

25-28

242 Index

Index 243

modifications of, 31-34probability on basis of, 28simplified calculation

algorithms, 28-31expert distribution

criteria based on, 38-39set of standard

distributions, 39-45unified probability density

functions, 45-49history of, xvi-xviilognormal distribution

description of, 13-15expected profit on basis of,

15-22markets, development of, xviiimulticriteria selection, 181-182

applying convolutionmethods, 212-214

correlation, 205evaluation of criteria

interrelationships, 206-208

Pareto sets, 183-185profitability of Pareto

selections, 208-212widening Pareto sets,

186-189prices, 6selection problem

classification of criteria, 9-12

definition of, 4-5forecasting, as key elements

of criterion, 8-9mission fulfilled by

criteria, 7

philosophy of criteriacreation, 5-6

tools for solving, 3-4separate, payoff functions,

226-227strategies, xix

analysis of criterion effec-tiveness, 150-151, 155

generalized ranking analysisresults, 140-142

methods of criterion effectiveness, 145-150

methods of rankinganalysis, 123-131

results of ranking analysis,131-138

selection of, 121-122threshold parameters,

139-140trading, xxvivalues of criterion

effectiveness coefficients,142-145

orderingunderlying assets, 171unified, xxv

out-of-the-money options, 58outcome, forecasts, 64overlapping of information, 155overoptimization, 26

P

parametersempirical distribution, 26horizon of history, 32lognormal distribution, 18

thresholds, 139-140optimizing, 155-156, 159values, 150

variance, 44Pareto sets, 183-185

convolution, comparisons, 197-202

profits, criteria correlation and,208-212

widening, 186-189past events frequency, 8patterns, scattering, xxxiipayoff function, 15, 47, 226

option combinations, 228bear spreads, 233-234bull spreads, 233-234calendar spreads, 231-233straddles, 228-229strangles, 229-230

separate options, 226-227peculiarities, 7Pentateuch, xviperfect criterion, 124periods, averaging, 72-75Peters, Edgar, xxiphilosophy of criteria creation, 5Poon, Ser-Huang, xixposition holding periods, 31premium, risk, 6prices

behavior, xximarket, xxximodels, xxioptions, 6standard price distribution

models, 14strike, xxx, 3

probabilitiesdensity functions, 19distribution of, 5, 19sum of, 8empirical distribution, 28theory, xv

ProbEmpiric(B,S), 31problems, selection

classification of criteria, 9-12definition of, 4-5forecasting, as key elements of

criterion, 8-9mission fulfilled by criteria, 7philosophy of criteria creation,

5-6tools for solving, 3-4

ProbSymEmpiric(B,S), 34profit

criteria correlationbetween effectiveness

indicators, 64-68between indexes, 86-87based on ratio of expected

profit and loss, 34-38between Sharpe ratios and,

88-91between values, 85

expectedon basis of empirical

distribution, 25-28on basis of lognormal

distribution, 15-22expressing, 77-80probability

on the basis of empiricaldistribution, 28

on the basis of lognormaldistribution, 115-117

244 Index

Index 245

Pareto setscriteria correlation and,

208-212selection methods, 193

proportional long strangle, 17-19puts

to call ratios, 3options, xviishort options, 28spreads, 12strike prices, 3

Q–R

quasiperiodical cycles, xxxiv

rangesbreak-even, 50-53IV/HV, 53-57

ranking analysismethods of, 123-131results of, 131-138

ratiosareas, 91-95of expected profit to loss, 34-38of implied to historical volatility

(IV/HV), 80, 115-117of normalized time value, 58-61Put to Call, 3Sharpe, 11, 88-89, 91, 150-165

analysis, 101Pareto sets, 188underlying assets, 171

underlying assets, 169Rebonato, Riccardo, xixreduction of matrices, xxvii-xxix

regressionanalysis, 83models, 65-66relationships, 135

regularities, observation of, 69relationships

between criterion and profitvalues, 81

between determination and theslope coefficients, 84

lognormal distribution, 18between profit of combinations

and positions, 97regression, 135

relative changes, calculating, 14. Seealso yields

relative frequency criterion, 57-58results

of long-term evaluation ofcriteria effectiveness, 116, 118

of ranking analysis, 131-142underlying assets, 167-168

returns, calculating, xixreview of effectiveness indicators,

84-89, 92-101rho, xxiirisk

characteristics, 11estimation of, xxiievaluation, xixmanagement, xviipremium, 6

S

scales, thresholds, 153scattering patterns, xxxiischematic representations, 123-125

Scholes, Myron, xviselection, xxiv, xxvi-xxvii

of averaging periods, 72-75combinations of algorithms of,

121-122multicriteria, 181-182

applying convolutionmethods, 212-214

comparative analysis, 191-202

convolution, 190-191correlation, 205evaluation of criteria

interrelationships, 206-208

Pareto sets, 183-185profitability of Pareto

selections, 208-212widening Pareto sets,

188-189of option combinations, 105

absolute values, 110-112analysis of criteria

effectiveness, 106-110long-term evaluation of

criteria effectiveness,115-118

simultaneous analysis of factors affecting, 113-114

underlying assets, 112of option strategies, 121-122

analysis of criterion effectiveness, 150-155

generalized ranking analysisresults, 140-142

methods of criterion effectiveness, 145-150

methods of rankinganalysis, 123-131

results of ranking analysis,131-138

threshold parameters, 139-140

values of criterion effectiveness coefficients,142-145

problemsclassification of criteria,

9-12forecasting as key elements

of criterion, 8-9formal definition of, 4-5mission fulfilled by

criteria, 6philosophy of criteria

creation, 5tools for solving, 3-4

of underlying assets, 161-162analysis of criteria

effectiveness, 162-165long-term evaluation of

criteria effectiveness,166-168

optimizing, 166-176selective convolution, 190selling straddles, 11separate options, payoff functions,

226-227sequences of operations, xxvii-xxixsets

Pareto, 183-185comparisons to

convolution, 197-202

246 Index

Index 247

criteria correlation andprofitability of, 208-212

widening, 186-189of standard distributions, 39-45

Sharp ratio, 11Sharpe ratios, 11, 88-91, 150,

153, 165analysis, 101Pareto sets, 188underlying assets, 171

short calendar spreads, 64, 127-128,231-233

short Put options, 28short straddle strategies, xxxiii, 11,

127-128, 133, 137, 143short strangle strategies, xxviii, 3,

229-230short strategies

averaging periods, 73dynamics of transformed

effectiveness indicators, 71simplified calculation algorithms,

28-31simultaneous analysis of factors

affecting combinations selection,113-114

skewness, 8specific criteria, 50

break-even range, 50-53IV/HV range, 53-57nonuniversal, 11-12ratio of normalized time value,

58-61relative frequency criterion,

57-58speculative option strategies, xviispreads, 12

square of the correlation coefficients, 65

standard distribution, 8Standard Error (SE), 132standard price distribution models, 14statistics, xv, 31

testing, 127straddles, 3, 228-229strangles, 229-230strategies

heterogeneous, comparisons of, 122

long-straddle, xxxiiioptions, xix

analysis of criterion effec-tiveness, 150-151, 155

generalized ranking analysisresults, 140-142

methods of criterion effectiveness, 145-150

methods of rankinganalysis, 123-131

results of ranking analysis,131-138

selections of, 121-122threshold parameters,

139-140trading, xxivvalues of criterion

effectiveness coefficients,142-145

short-straddle, xxxiiishort-strangle, xxviiispeculative option, xviiand underlying assets, 112,

116-118strike prices, xxx, 3

strong trends, 31structure of trading opportunities,

xxxi, xxxiii, xxxv, xxxviisum of probabilities, 8supply and demand, 7symmetrization of empirical

distribution, 33-34synthetic approach to criterion

effectiveness analysis, 150-151, 155systematic approach

formal definition of, 4-5goals and objectives of,

xxiii-xxivoverview of, xxiii

T

Taleb, Nassim, xix-xxtesting

F-tests, 132statistics, 127

theories, xviiitheta, xxiiThomsett, Michael, xixthree-dimensional matrices, xxviiithresholds

parameters, 139-140, 155-159scales, 153values, 140-144

optimizing, 145parameters, 150

timefractions, distributions of, xxxvivalue, 6

TimeValue, 60tools, solving selection problems, 3-4

tradingopportunities, xxix-xxx

evaluation, xxxstructure and dynamic of,

xxxi, xxxiii, xxxv,xxxvii

option strategies, xxivselection problems

classification of criteria, 9-12

definition of, 4-5forecasting, as key elements

of criterion, 8-9mission fulfilled by

criteria, 7philosophy of criteria

creation, 5-6tools for solving, 3-4

transformation of criteria effectivenessindicators, 68-69

trends, 31two-dimensional matrices, xxviiitypes

of convolution, 157-159, 190of uncertainty, 6

U

uncertaintycommon, 6individual, 6

underlying assets, 112, 116-118analysis of criteria effectiveness,

162-165long-term evaluation of criteria

effectiveness, 166-168optimizing, 168-176selection of, 161-162

248 Index

Index 249

unified ordering, xxvunified probability density functions,

45-49uniform distribution, 40universal criteria, 9-10

forecasting, 10nonforecasting, 11

utility functions, 155-159, 169-176

V

validation of unified probabilitydensity functions, 48-49

valuation, xxiv-xxvvalues

absolute of criterion, 110, 112of criterion effectiveness

coefficients, 142-145fair, xxxmean, 17profit, correlation between

criterion and, 85thresholds, 140-144

optimizing, 145parameters, 150

time, 6variance, 8vega, xxiiVince, Ralph, xxxivisual analysis, xxxivvolatility, xxii, 41

historical, 16, 80

W–Z

weight functions, 32widening Pareto sets, 186-189

Yates, Leonard, xxyields, 14

AutomatedOption Trading

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AutomatedOption Trading

Create, Optimize, and

Test Automated Trading Systems

Sergey Izraylevich, Ph.D., and Vadim Tsudikman

Vice President, Publisher: Tim MooreAssociate Publisher and Director of Marketing: Amy NeidlingerExecutive Editor: Jim BoydEditorial Assistant: Pamela BolandOperations Specialist: Jodi KemperSenior Marketing Manager: Julie PhiferAssistant Marketing Manager: Megan GraueCover Designer: Alan ClementsManaging Editor: Kristy HartProject Editor: Betsy HarrisCopy Editor: Cheri ClarkProofreader: Debbie WilliamsSenior Indexer: Cheryl LenserSenior Compositor: Gloria SchurickManufacturing Buyer: Dan Uhrig

© 2012 by Pearson Education, Inc.Publishing as FT PressUpper Saddle River, New Jersey 07458

This book is sold with the understanding that neither the author nor the publisher is engaged in rendering legal, accounting, or otherprofessional services or advice by publishing this book. Each individual situation is unique. Thus, if legal or financial advice or otherexpert assistance is required in a specific situation, the services of a competent professional should be sought to ensure that the situationhas been evaluated carefully and appropriately. The author and the publisher disclaim any liability, loss, or risk resulting directly orindirectly, from the use or application of any of the contents of this book.

FT Press offers excellent discounts on this book when ordered in quantity for bulk purchases or special sales. For more information, please contact U.S. Corporateand Government Sales, 1-800-382-3419, corpsales@pearsontechgroup.com. For sales outside the U.S., please contact International Sales atinternational@pearson.com.

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All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Printed in the United States of America

First Printing March 2012

ISBN-10: 0-13-247866-8ISBN-13: 978-0-13-247866-3

Pearson Education LTD.Pearson Education Australia PTY, LimitedPearson Education Singapore, Pte. Ltd.Pearson Education Asia, Ltd.Pearson Education Canada, Ltd.Pearson Educatión de Mexico, S.A. de C.V.Pearson Education—JapanPearson Education Malaysia, Pte. Ltd.Library of Congress Cataloging-in-Publication DataIzraylevich, Sergey, 1966-

Automated option trading : create, optimize, and test automated trading systems / Sergey Izraylevich, Vadim Tsudikman.p. cm.

Includes bibliographical references and index.ISBN 978-0-13-247866-3 (hardcover : alk. paper)

1. Stock options. 2. Options (Finance) 3. Electronic trading of securities. 4. Investment analysis. 5. Portfolio management. I. Tsudikman, Vadim, 1965- II.Title.

HG6042.I968 2012332.64’53—dc23

2011048038

This book is dedicated to our parents,

Izraylevich Olga, Izraylevich Vladimir,

Tsudikman Rachel, Tsudikman Jacob.

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xv

Chapter 1 Development of Trading Strategies . . . . . . . . . . . . . . . . .11.1 Distinctive Features of Option Trading Strategies . . . . . .1

1.1.1 Nonlinearity and Options Evaluation . . . . . . . . . .1

1.1.2 Limited Period of Options Life . . . . . . . . . . . . . . . .2

1.1.3 Diversity of Options . . . . . . . . . . . . . . . . . . . . . . . . .3

1.2 Market-Neutral Option Trading Strategies . . . . . . . . . . . .4

1.2.1 Basic Market-Neutral Strategy . . . . . . . . . . . . . . . . .4

1.2.2 Points and Boundaries of Delta-Neutrality . . . . . .6

1.2.3 Analysis of Delta-Neutrality Boundaries . . . . . . .10

1.2.4 Quantitative Characteristics of Delta-Neutrality

Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

1.2.5 Analysis of the Portfolio Structure . . . . . . . . . . . .21

1.3 Partially Directional Strategies . . . . . . . . . . . . . . . . . . .34

1.3.1 Specific Features of Partially Directional

Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

1.3.2 Embedding the Forecast into the Strategy

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

1.3.3 The Call-to-Put Ratio at the Portfolio Level . . . . .40

1.3.4 Basic Partially Directional Strategy . . . . . . . . . . .42

1.3.5 Factors Influencing the Call-to-Put Ratio in an

Options Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . .44

1.3.6 The Concept of Delta-Neutrality as Applied

to a Partially Directional Strategy . . . . . . . . . . . .49

1.3.7 Analysis of the Portfolio Structure . . . . . . . . . . . .57

1.4 Delta-Neutral Portfolio as a Basis for the Option

Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61

1.4.1 Structure and Properties of Portfolios Situated

at Delta-Neutrality Boundaries . . . . . . . . . . . . . .62

1.4.2 Selection of an Optimal Delta-Neutral

Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

Chapter 2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .732.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

2.1.1 Parametric Optimization . . . . . . . . . . . . . . . . . . .73

2.1.2 Optimization Space . . . . . . . . . . . . . . . . . . . . . . . .75

2.1.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . .78

2.2 Optimization Space of the Delta-Neutral Strategy . . . .79

2.2.1 Dimensionality of Optimization . . . . . . . . . . . . .80

2.2.2 Acceptable Range of Parameter Values . . . . . . . .85

2.2.3 Optimization Step . . . . . . . . . . . . . . . . . . . . . . . . . .87

2.3 Objective Functions and Their Application . . . . . . . . .88

2.3.1 Optimization Spaces of Different Objective

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89

2.3.2 Interrelationships of Objective Functions . . . . . .91

2.4 Multicriteria Optimization . . . . . . . . . . . . . . . . . . . . . .96

2.4.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97

2.4.2 Optimization Using the Pareto Method . . . . . . . .99

2.5 Selection of the Optimal Solution on the Basis of

Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

2.5.1 Averaging the Adjacent Cells . . . . . . . . . . . . . . . .103

2.5.2 Ratio of Mean to Standard Error . . . . . . . . . . . .104

2.5.3 Surface Geometry . . . . . . . . . . . . . . . . . . . . . . . . .106

2.6 Steadiness of Optimization Space . . . . . . . . . . . . . . . .108

2.6.1 Steadiness Relative to Fixed Parameters . . . . . .109

2.6.2 Steadiness Relative to Structural Changes . . . .110

2.6.3 Steadiness Relative to the Optimization

Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

viii Automated Option Trading

2.7 Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . .114

2.7.1 A Review of the Key Direct Search Methods . . .116

2.7.2 Comparison of the Effectiveness of Direct

Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .127

2.7.3 Random Search . . . . . . . . . . . . . . . . . . . . . . . . . .131

2.8 Establishing the Optimization Framework:

Challenges and Compromises . . . . . . . . . . . . . . . . . . .134

Chapter 3 Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . .1353.1 Payoff Function and Specifics of Risk Evaluation . . . .135

3.1.1 Linear Financial Instruments . . . . . . . . . . . . . .136

3.1.2 Options as Nonlinear Financial

Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138

3.2 Risk Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139

3.2.1 Value at Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . .140

3.2.2 Index Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141

3.2.3 Asymmetry Coefficient . . . . . . . . . . . . . . . . . . . . .157

3.2.4 Loss Probability . . . . . . . . . . . . . . . . . . . . . . . . . .159

3.3 Interrelationships Between Risk Indicators . . . . . . . .161

3.3.1 Method for Testing the Interrelationships . . . . .161

3.3.2 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . .162

3.4 Establishing the Risk Management System:

Challenges and Compromises . . . . . . . . . . . . . . . . . . .165

Chapter 4 Capital Allocation and Portfolio Construction . . . . . . . . .1674.1 Classical Portfolio Theory and Its Applicability

to Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167

4.1.1 Classical Approach to Portfolio Construction . .168

4.1.2 Specific Features of Option Portfolios . . . . . . . .169

4.2 Principles of Option Portfolio Construction . . . . . . .170

4.2.1 Dimensionality of the Evaluation System . . . .170

4.2.2 Evaluation Level . . . . . . . . . . . . . . . . . . . . . . . . . .173

4.3 Indicators Used for Capital Allocation . . . . . . . . . . . .174

4.3.1 Indicators Unrelated to Return and Risk

Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174

4.3.2 Indicators Related to Return and Risk

Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178

Contents ix

4.4 One-Dimensional System of Capital Allocation . . . . . .183

4.4.1 Factors Influencing Capital Allocation . . . . . . .183

4.4.2 Measuring the Capital Concentration in the

Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192

4.4.3 Transformation of the Weight Function . . . . . . .196

4.5 Multidimensional Capital Allocation System . . . . . . . .204

4.5.1 Method of Multidimensional System

Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204

4.5.2 Comparison of Multidimensional and

One-Dimensional Systems . . . . . . . . . . . . . . . . . .206

4.6 Portfolio System of Capital Allocation . . . . . . . . . . . . .209

4.6.1 Specific Features of the Portfolio System . . . . . .209

4.6.2 Comparison of Portfolio and Elemental

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211

4.7 Establishing the Capital Allocation System:

Challenges and Compromises . . . . . . . . . . . . . . . . . . .214

Chapter 5 Backtesting of Option Trading Strategies . . . . . . . . . . . .2175.1 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217

5.1.1 Data Vendors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218

5.1.2 Database Structure . . . . . . . . . . . . . . . . . . . . . . .219

5.1.3 Data Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220

5.1.4 Recurrent Calculations . . . . . . . . . . . . . . . . . . . .221

5.1.5 Checking Data Reliability and Validity . . . . . . .222

5.2 Position Opening and Closing Signals . . . . . . . . . . . .225

5.2.1 Signals Generation Principles . . . . . . . . . . . . . .225

5.2.2 Development and Evaluation of

Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226

5.2.3 Filtration of Signals . . . . . . . . . . . . . . . . . . . . . . .227

5.3 Modeling of Order Execution . . . . . . . . . . . . . . . . . . .228

5.3.1 Volume Modeling . . . . . . . . . . . . . . . . . . . . . . . . .229

5.3.2 Price Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .230

5.3.3 Commissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231

x Automated Option Trading

5.4 Backtesting Framework . . . . . . . . . . . . . . . . . . . . . . .232

5.4.1 In-Sample Optimization and Out-of-Sample

Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232

5.4.2 Adaptive Optimization . . . . . . . . . . . . . . . . . . . .233

5.4.3 Overfitting Problem . . . . . . . . . . . . . . . . . . . . . . .234

5.5 Evaluation of Performance . . . . . . . . . . . . . . . . . . . . .236

5.5.1 Single Event and Unit of Time Frame . . . . . . . .236

5.5.2 Review of Strategy Performance Indicators . . .237

5.5.3 The Example of Option Strategy Backtesting . .242

5.6 Establishing the Backtesting System: Challenges and

Compromises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .246

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .247

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .251

Payoff Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254

Separate Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254

Option Combinations . . . . . . . . . . . . . . . . . . . . . . . . . .255

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261

Contents xi

The authors would like to express their gratitude to the team of High Technology Invest

Inc. Special thanks are due to Arsen Balishyan, Ph.D., CFA, and Vladislav Leonov, Ph.D.,

and Eugen Masherov, Ph.D., for their continued help at all stages of this project.

Acknowledgments

Sergey Izraylevich, Ph.D., Chairman of the Board of High Technology

Invest Inc., has traded options for well over a decade and currently creates

automated systems for algorithmic option trading. A Futures magazine

columnist,he has authored numerous articles for highly rated,peer-reviewed

scientific journals. He began his career as a lecturer at The Hebrew

University of Jerusalem and Tel-Hay Academic College, receiving numerous

awards for academic excellence, including Golda Meir’s Prize and the Max

Shlomiok honor award of distinction.

Vadim Tsudikman, President of High Technology Invest Inc., is a financial

consultant and investment advisor specializing in derivatives valuation,

hedging, and capital allocation in extreme market environments.With more

than 15 years of option trading experience, he develops complex trading

systems based on multicriteria analysis and genetic optimization algorithms.

Izraylevich and Tsudikman coauthored Systematic Options Trading (FT

Press) and regularly coauthor Futures magazine articles on cutting-edge

issues related to option pricing, volatility, and risk management.

About the Authors

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IntroductionThis book presents a concept of developing an automated system tailored specifically for

options trading. It was written to provide a framework for transforming investment ideas into

properly defined and formalized algorithms allowing consistent and disciplined realization of

testable trading strategies.

Extensive literature has been published in the past decades regarding systematic,

algorithmic, automated, and mechanical trading. In the Bibliography of this book, we list some

of the comprehensive works that deserve special attention. However, all books dedicated to

the creation of automated trading systems deal with traditional investment tools, such as

stocks, futures, or currencies. Although the development of options-oriented systems requires

accounting for numerous specific features peculiar to these instruments, automated trading of

options remains beyond the scope of professional literature.The philosophy, logic, and quanti-

tative procedures used in the creation of automated systems for options trading are completely

different from those used in conventional trading algorithms. In fact, all the components of a

system intended for automated trading of options (strategy development,optimization,capital

allocation, risk management, backtesting, performance measurement) differ significantly from

their analogs in the systems intended for trading of plain assets.This book describes consecu-

tively the key stages of creating automated systems intended specifically for options trading.

Automated trading of options represents a continuous process of valuation,structuring,and

long-term management of investment portfolios (rather than individual instruments). Due to

the nonlinearity of options, the expected returns and risks of their complex portfolios cannot

be estimated by simple summation of characteristics corresponding to individual options.

Special approaches are required to evaluate portfolios containing options (and their combina-

tions) related to different underlying assets. In this book we discuss such approaches,describe

systematically the core properties of option portfolios, and consider the specific features of

automated options trading at the portfolio level.

The Book StructureAn automated trading system represents a package of software modules performing the

functions of developing, formalizing, setting up, and testing trading strategies.

Chapter 1, “Development of Trading Strategies,” discusses the development and formal-

ization of option strategies. Since there is a huge multitude of trading strategies somehow

related to options, we limit our discussion to market-neutral strategies. The reason for

selecting this particular type of option strategies relates to its wide popularity among private

and institutional investors.

Strategy setup includes optimization of its parameters, capital allocation between

portfolio elements, and risk management. Chapter 2, “Optimization,” deals with various

optimization aspects. In this chapter we discuss various properties of optimization spaces,

different types of objective functions and their interrelationships, several methods of multi-

criteria optimization,and problems of optimization steadiness relative to small changes in the

parameters and strategy structure. Special attention is given to the application of traditional

methods of parametric optimization to complex option portfolios.

In Chapter 3,“Risk Management,” we discuss a set of option-specific risk indicators that

can be used for developing a multicriteria risk management system. We investigate the

influence of different factors on the effectiveness of the risk indicator and on the number of

indicators needed for effective risk measuring.

In Chapter 4,“Capital Allocation and Portfolio Construction,”we consider various aspects

of capital allocation among the elements of an option portfolio. Capital can be allocated on

the basis of different indicators not necessarily expressing return and risk. This chapter

describes the step-by-step process of constructing a complex portfolio out of separate option

combinations.

The testing of option strategies using historical data is discussed in Chapter 5,“Backtesting

of Option Trading Strategies.” In this chapter we stress the particularities of creating and

maintaining option databases and provide methods to verify data accuracy and reliability.The

problem of overfitting and the main approaches to solving it are also discussed.Performance

evaluation of option strategies is also the topic of this chapter.

xvi Automated Option Trading

Strategies Considered in This BookThe nature of options makes it possible to create a considerable number of speculative

trading strategies.Those can be based on different approaches encompassing the variety of

principles and valuation techniques.

In many strategies options are used as auxiliary instruments to hedge main positions. In

this book we are not going to delve into this field of options application since hedging repre-

sents only one constituent part of such trading strategies, but not their backbone.

Options may be used to create synthetic underlying positions. In this case the investor

aims for the payoff profiles of an option combination and its underlying asset to coincide.

This can increase trading leverage significantly. However, apart from leverage, automated

trading of synthetic assets is no different from trading in underlying assets (besides the

certain specificity regarding execution of trading orders,higher brokerage commissions, and

the necessity to roll over positions).Thus, we will not dwell on such strategies either.

Most trading strategies dealing with plain assets (not options) are based on the forecast

of the direction of their price movement (we will call them directional strategies). Options

can also be used in such strategies. For example, different kinds of option combina-

tions, commonly referred to as spreads, benefit from the increase in the underlying

price (bull spreads), or from its decline (bear spread). Despite the fact that trading

strategies based on such option combinations possess many features distinguishing

them from plain assets strategies, the main determinant of their performance is the

accuracy of price forecasts.This quality makes such strategies quite similar to common

directional strategies, and therefore we will not consider them in this book.

The focus of this book is on strategies that exploit the specific features of options.

One of the key differences of options from other investment assets is the nonlinearity

of their payoff functions. In the trading of stocks, commodities, currencies, and other

linear assets, all profits and losses are directly proportional to their prices. In the case

of options, however, position profitability depends not only on the direction of the

price movement, but on many other factors as well. Combining different options on

the same underlying asset can bring about almost any form of the payoff function.

This feature of options permits the creation of positions that depend not only on

the direction and the amplitude of price fluctuations, but also on many other param-

eters, including volatility, time left until the expiration, and so forth.The main subject

of our consideration is a special type of trading strategies sharing one common

property referred to as market-neutrality. With regard to options, market-neutrality

means that (1) small changes in the underlying price do not lead to a significant change

in the position value,and (2) given larger price movements, the position value changes

by approximately the same amount regardless of the direction of the underlying price

movement. In reality these rules do not always hold, but they serve as a general

guideline for a trader striving for market-neutrality. The main analytical instrument

used to create market-neutral positions is delta.The position is market-neutral if the

sum of the deltas of all its components (options and underlying assets) is equal to or

close to zero. Such positions are referred to as delta-neutral.

Another type of trading strategy that will be considered in this book is a set of

market-neutral strategies whose algorithms contain certain directional elements.

Although in this case positions are created while taking into consideration the value of

delta, its reduction to zero is not an obligatory requirement. Forecasts of the direc-

tions of future price movements represent an integral part of such a strategy.These

forecasts can be incorporated into the strategy structure in the form of biased proba-

bility distributions or asymmetrical option combinations, or by application of

technical and fundamental indicators.We will call such strategies partially directional.

Generally, automated strategies are designed to trade one or just a few financial

instruments (mainly futures on a given underlying asset). Even if several instruments

are traded simultaneously, in most cases positions are opened, closed, and analyzed

independently. Options are no exception. Most traders develop systems oriented

solely at trading OEX (options on S&P 100 futures) or options on oil futures. In this

Introduction xvii

book we will consider strategies intended to trade an unlimited number of options relating

to many underlying assets. All positions created within one trading strategy will be

evaluated and analyzed jointly as a whole portfolio.

xviii Automated Option Trading

Scientific and Empirical Approaches to DevelopingAutomated Trading Strategies

There are two main approaches to the development of automated trading strategies.The first

approach is based on the principles and concepts defined by a strategy developer. All the

elements composing such a strategy originate from economic knowledge, fundamental

estimates, expert opinions, and so forth. Formalization of such knowledge, estimates, and

assumptions in the form of algorithmic rules provides a basis for creating an automated

trading strategy. Following the example of Robert Pardo, we will call this a scientific

approach.

At its extreme, the scientific approach provides for a total rejection of optimization

procedures. All the rules and parameters of a trading system are determined solely on the

basis of knowledge and forecasts of the developer. Apparently, the likelihood of creating a

profitable strategy, while avoiding engagement in optimization procedures, is extremely

low. Scientific approach in its pure form is hardly applicable in real trading.

The alternative approach is based on the complete denial of any a priori established

theories, models, and principles while developing automated trading strategies. This

approach requires extensive use of computer technologies to search for profitable trading

rules. All algorithms can be tested for this task (with no concern for any economically sound

reasons standing behind their application). Candidate algorithms can be selected from a

number of ready-made alternatives available or actually constructed by the system developer.

The method of algorithm creation is not determined by preliminary assumptions and is not

limited by any exogenous reasoning. Trading rules are selected solely on the basis of

their testing using historical data. The resulting strategy is devoid of any behavioral logic

or economic sense. Following Robert Pardo, we will call it an empirical approach.

At its extreme,the empirical approach is a purposeful quest for algorithms and parameters

that maximize simulated profit (minimize loss or satisfy any other utility function). This

approach is based exclusively on optimization. Nowadays there is a wide choice of high-

technology software that facilitates fast development of effective algorithms and provides for

establishment of optimal parameter sets. For example, neuron networks and genetic

methods represent powerful tools that enable relatively fast finding of optimal solutions

through the creation of self-learning systems.

Usually trading strategies constructed on the basis of the empirical approach show

remarkable results when tested on historical time series, but demonstrate failure in real

trading.The reason for this is overfitting. Even walk-forward analysis does not eliminate this

threat because the significant number of degrees of freedom (which is not unusual under the

empirical approach) allows choosing such a set of trading rules and parameters that would

generate satisfactory results not only during the optimization period, but in the walk-

forward analysis as well (we will examine this in detail in Chapter 5).Thus, practical use of

the empirical approach exclusively is risky and hardly applicable in real trading.

Introduction xix

Rational Approach to Developing Automated TradingStrategies

Most developers of automated trading strategies combine scientific methods and empirical

approaches. On the one hand, strategies resulting from such combining are based on strong

economic grounds. On the other hand, they benefit from the numerous advantages of

optimization and from the impressive progress in computer intelligence.We will call this the

rational approach.

Under the rational approach a set of rules,determining the general structure of a trading

strategy, is formed at the initial stage of strategy creation.These rules are based on the prior

knowledge and assumptions about market behavior. The results of statistical research,

either received by the strategy developer or obtained from scientific publications and

private sources, can also be used to shape the general framework. Obviously, patterns estab-

lished during such research introduce certain logic into the strategy under development. At

the same time, statistical research may result in the discovery of inexplicable relationships

lacking any economic sense behind them. Such relationships should be treated with special

care since they may either be random in nature or result from data mining.

The initial stage of strategy creation is based mainly on the elements of scientific

approach. At this stage the following must be determined:

● Principles of generating the signals for opening and closing trading positions

● Indicators used to generate open and close signals

● A universe of investment assets that are both available and suitable for trading

● Requirements to the portfolio and restrictions imposed on it

● Capital allocation among different portfolio elements

● Methods and instruments of risk management

At the next stage of developing a trading strategy, the rules laid down on the basis of

scientific approach are formalized in the form of computable algorithms. This stage is

congested with elements of the empirical approach.These are the essential steps:

● Defining specific parameters. All rules formulated on the basis of the scientific

approach should be formalized using a certain number of parameters.

● Specifying the algorithms for parameter calculation. Different algorithms may be

invented for calculating the same parameter.

● Establishing the procedures for the selection of parameter values.This requires

adopting a certain optimization technique.

Usually the decision on the number of parameters and selection of methods for their

optimization does not depend on the economic considerations of the developer, but follows

from the specific requirements to the strategy and from its technical constraints. These

requirements and constraints are developed with regard to the reliability, stability, and other

strategy features, among which the capability to avoid the overfitting is one of the most

important properties.

In this book we will follow the principles of rational approach to the creation of trading

strategies.The main task of the developer is to combine methods attributed to the scientific

and the empirical approaches in a reasonable and balanced manner. In order to accomplish

this task successfully, all basic components of a trading strategy should be clearly identified

as belonging either to components that are set on the basis of well-founded reasoning or to

an alternative category of components that are formed primarily by applying various

optimization methods.

xx Automated Option Trading

As of today, there is no universal definition that would embrace all aspects of such a complex

and versatile concept as “risk.”Actually, we must admit that millions of people—journalists,

businessmen,scientists,professional investors,and financial services consumers who use this

term daily—are still unable to give a strict and universal definition of the matter in question.

This is especially striking considering that the notion of risk is the cornerstone of economics,

finance, and many other related disciplines. Moreover, the same situation is observed in two

other, undoubtedly fundamental, areas of scientific cognition of the material world: biology

and physics. In biology, there is no universal definition of “species,” while this notion is the

basis for the whole macroevolution theory—the signal achievement of this science.Physicists

also did not arrive at a common opinion regarding a unique and comprehensive definition of

“energy.”Without precise understanding of this key element, neither the development of the

quantum theory at the microcosmic level nor the creation of the “final theory of everything,”

pretending to describe the origin, evolution, and future fate of the universe, is possible.

Isn’t it strange that the three basic areas of human knowledge—biology, economics, and

physics—erect their theories on the basis of core elements lacking strict and unambiguous

scientific definitions? We leave this question unanswered since even the slightest attempt to

straighten out this confusing situation will distract us not only from the main topic, but also

from the system of rational reasoning that we rigorously adhere to in this book.

chapter 3

Risk Management

3.1 Payoff Function and Specifics of Risk EvaluationAll financial instruments can be classified as linear or nonlinear according to their payoff

function. The former category includes stocks, commodities, currencies, and other assets

whose profits and losses are directly proportional to their price. Nonlinear assets include

derivative financial instruments with prices depending on the prices of their underlying

assets.The relationships between the profits and losses of these instruments and the under-

lying asset prices are not linear.Options represent one of the main categories of the nonlinear

instruments.The approaches used to evaluate the risks of linear and nonlinear instruments

differ fundamentally.

3.1.1 Linear Financial InstrumentsThe grounds for risk management theory were established when derivatives had not yet

begun their full-scale advance on the financial markets. As a result,all classical risk evaluation

models were developed for linear instruments.The general concept stating that “the risk of a

specific asset is proportional to the variability of its price”was accepted as the basic paradigm

for risk quantification.

The objective estimation of price variability can be derived only from the past price

fluctuations (other estimates, for example, those based on expert opinions, cannot be

completely objective). Such an approach has a significant drawback, since it requires extrap-

olation of historical data and is based on the assumption that probabilities of future outcomes

are proportional to the frequency of similar occurrences that have been realized in the past.

Although in many areas (for example, in car insurance) this method is widely accepted, it was

repeatedly proved that with regard to financial markets it is, to say the least, imperfect.Never-

theless,despite all the drawbacks and inaccuracies that arise when risks are estimated on the

basis of historical data, this approach is widely applied because there are no better alterna-

tives as of today.

It was suggested to use the standard deviation of asset returns (or their dispersion) as the

measure of price variability, which is usually denoted by the term “historical volatility.”

Usually, historical volatility is calculated as annualized mean-square deviation of daily

logarithm price returns.The length of the historical period used to calculate volatility is a key

parameter in measuring the risk of linear assets. If time series are too long,the estimate of the

current risk is based on outdated data having no direct relation to the actual market

dynamics. Such a valuation cannot be reliable. On the other hand, using time series that are

too short brings about unstable risk estimates reflecting only momentary market trends.

Thus, the choice of the historical horizon length is a product of compromise and should be

defined within the context of the whole risk management system (while taking into account

the specific features of the trading strategy under development).

Although standard deviation by itself reflects the risk, it is also used to calculate more

complex indicators that express risks in a form that is more convenient for practical appli-

cation.The well-known example of such an indicator is Value at Risk (VaR), which estimates

an amount of loss that will not be exceeded with a given probability during a certain period.

136 Automated Option Trading

The history of emergence and broad acceptance of this indicator date back to the

stock market crash of 1987,when the failure of existing risk management mechanisms

became evident. The search for new approaches to risk measurement led to quick

development and an extensive application of the innovative technology, expressing

the risk through estimation of capital that may be lost with a given probability, rather

than by calculation of some statistical indicators (like standard deviation). In the 1990s

VaR became the universally recognized standard of risk measurement, and in 1999 it

obtained the official international status set in the Basel Agreements.With the lapse of

time,VaR became the compulsory indicator appearing in the accounting reports of the

majority of financial institutions.

Although from a practical point of view, VaR is more informative than standard

deviation, these indicators do not differ from each other conceptually. This can be

demonstrated by considering the VaR calculation method. There are three main

approaches to VaR calculation: analytical, historical, and the Monte-Carlo method. The

analytical method is based on using the parameters of specific return distribution.

Despite its numerous drawbacks, lognormal distribution is used most often. Since the

main parameter of this distribution is the standard deviation (the second parameter,

expected price, is usually set to be equal to the current asset price),one can claim that

VaR is just a secondary indicator derived from the standard deviation. The historical

method is based on using asset price changes that were observed during a predeter-

mined period in the past. Since the standard deviation is calculated on the basis of the

same data, both indicators strongly correlate with each other and, in fact, express the

same concept. The Monte-Carlo method generates a multitude of random price (or

return) outcomes.The algorithm of price generation utilizes the probability density

function of a certain distribution. And again, the lognormal distribution (with the

standard deviation as its main parameter) is used most frequently.Thus, the emergence

of VaR added more convenience for its users, but did not lead to the development of

new risk evaluation principles.

There is nothing particularly complicated in risk evaluation for portfolios

consisting of linear assets. In most cases the standard deviation and VaR of such

portfolios can be calculated using simple analytical methods.The minimal input data

required for these calculations include the standard deviation of each instrument, its

weight in the portfolio, and the covariance matrix.The matrix is necessary to account

for the effect of diversification (a decrease in portfolio risk as a result of including low-

correlated assets or assets with negative correlations in it). Conversely, the risk for a

portfolio containing at least some nonlinear instruments cannot be calculated analyti-

cally. In this case we need to apply different simulation methods, of which the Monte-

Carlo is the most common.

Risk Management 1373

3.1.2 Options as Nonlinear Financial InstrumentsTraditional risk evaluation methods that are commonly used for linear assets are inappro-

priate for financial instruments with a nonlinear payoff function.The reason is that the return

distribution of nonlinear assets is not normal. For example, a call option has an unlimited

profit potential which implies that the right tail of its return distribution is unlimited. At the

same time, its maximum possible loss cannot exceed the premium paid at the position

opening. Hence, the left tail of the distribution is limited to this amount, which means that

the returns distribution is asymmetric and cannot even roughly be considered as normal.The

nonnormality is so pronounced that the application of logarithmic transformation does not

solve the problem, as this is the case for linear assets. (Although logarithmic transformation

brings the return distribution of linear assets closer to normal distribution, there are many

evidences of deviation of logarithm return distribution from normality. Nevertheless, in this

case we speak only about deviations,while for nonlinear assets the distribution does not even

get close to the normality.)

Despite the aforesaid, methods developed for risk measurement of linear assets can be

applied to some nonlinear instruments, provided that certain conditions are observed. For

example, although the VaR of an option cannot be calculated analytically, it can be estimated

using the Monte-Carlo method.However, these methods can be used only as auxiliary instru-

ments for the risk evaluation of automated option trading strategies.Dedicated methods with

due regard to the specifics of nonlinear assets in general and options in particular should be

given priority.

The common tools for risk evaluation of separate options are the “Greeks”that express the

change in option price given the small change in a specific variable.These indicators can be

interpreted as the sensitivity of an option to fluctuations in a variable value.The variables

include an underlying asset price,volatility, time,and the interest rate.These variables may be

seen as risk factors that give rise to instability of option prices.

The Greeks are calculated analytically as partial derivatives of the option price with

respect to the given variable. One of the option pricing models (for example, the Black-

Scholes formula) is used to calculate these derivatives. Delta is the derivative of the option

price with respect to the underlying asset price.The derivative with respect to volatility is

called vega, with respect to time—theta—and with respect to the interest rate—rho. Deriva-

tives of the second and higher orders can also be used (for example, gamma—the second

order derivative with respect to price).

The Greeks represent a convenient and rather adequate instrument of options risk evalu-

ation.The risk of combination, consisting of options related to one underlying asset, can be

estimated by summing up the corresponding Greeks. However, problems do arise in

measuring the risks of complex structures rather than separate options. Inclusion of

option combinations, related to different underlying assets, in the portfolio makes it impos-

sible to evaluate certain risks by mere summation. Some Greeks, such as theta and rho, are

138 Automated Option Trading

additive for options pertaining to different underlying assets. Thus, the sensitivity of the

complex portfolio to time decay or the interest rate change is easily calculated as the sum of

thetas and rhos of separate options. Things get more complicated with non-additive risk

indicators, such as delta and vega (these two characteristics are the most important for risk

management). If the portfolio consists of options relating to several underlying assets, the

summation of separate deltas and vegas makes no sense.These characteristics are not appro-

priate for complex portfolios since they express changes in the option price depending on

the changes in price or volatility of one specific underlying asset. Differentiated analysis of

positions in separate underlying assets is not efficient since the whole portfolio must be

evaluated as a single structure.

The problem of Greeks non-additivity can be solved by expressing the deltas of all options

as the derivatives with respect to some common index rather than to their corresponding

underlying assets.Similarly,vegas of different options can be calculated as the derivative with

respect to the volatility of the same index rather than to the volatilities of separate underlying

assets. Such procedures impart additivity properties to delta and vega, which allows for the

calculation of risk indicators for the whole portfolio as a single entity. S&P 500 or any other

index constructed by the system developer (for example,an index based on the prices of only

those stocks that are underlying assets for combinations included in the portfolio) can be

used as a common index.The selection or construction of the common index represents a

separate complex task; its solution depends on the relationships between the portfolio

elements and the chosen index and on many risk management parameters.

Index delta is undoubtedly one of the most important instruments for measuring the risk

of option portfolios. However, apart from this indicator, risks of automated options trading

strategies should and must be evaluated using additional complementary characteristics (in

this chapter, apart from index delta, we discuss three other indicators:VaR, loss probability,

and asymmetry coefficient).This enables us to put together a complete picture of different

risk aspects that are peculiar to the strategy under development.

Risk Management 139

3.2 Risk IndicatorsThe risks of option portfolios may be estimated using traditional indicators, such as VaR, and

specific characteristics developed deliberately for this purpose. Although the risk evaluation

of option portfolios by means of indicators that are usually used for linear assets is technically

possible, one should be extremely careful in interpreting the obtained results. These

indicators can be used only as auxiliary instruments of risk management.The main indicators

should be developed subject to options peculiarities.To get a thorough comprehension of

different risks of the portfolio under construction,several indicators should be applied simul-

taneously. Moreover, these indicators should correlate with each other as little as possible.

This will allow us to form a set of unique risk indicators,each of which will complement,but

not duplicate, the information contained in other indicators.

3

Four risk indicators are discussed in this section.We begin with VaR and explain how this

indicator can be calculated for the option portfolio. Then, we describe three indicators

developed specifically for risk evaluation of complex option portfolios.Particular attention is

focused on the index delta.

3.2.1 Value at Risk (VaR)As defined earlier,VaR is an estimate of a loss that will not be exceeded with a given proba-

bility over a certain period (in further examples we will use the 95% probability). For a

portfolio consisting of nonlinear assets,the Monte-Carlo simulation has to be applied for a VaR

calculation. A multitude of underlying asset price outcomes is generated using the prob-

ability density function of lognormal (or another) distribution (constructed on the basis of

standard deviation derived from historical data).Then, for each price outcome the option

payoff is estimated. Finally, these values are used to calculate VaR.The same method (with an

allowance for correlations between different underlying assets) is used to calculate the VaR

of the option portfolio.

Despite all its advantages and practical utility,VaR has a number of significant drawbacks

(Tsudikman, Izraylevich, Balishyan, 2011), which become especially apparent when it is

applied to assess the risks of nonlinear assets.The following are the main drawbacks:

● Lack of reliable technology to derive robust parameters of return distribution from

historical data. Consequently, the probability density functions used for VaR estimation

fail to forecast severe economic shocks. For complex portfolios that include options

relating to different underlying assets, the possibility of obtaining reliable and robust VaR

estimates is even more questionable.

● Besides underestimating the magnitude and the frequency of extreme outcomes,VaR

may also overestimate both unsystematic (diversifiable) and systematic risks.

● VaR is a point estimate and thus does not reflect the whole range of potential outcomes.

Since it is calculated for only one or several probability values, the general properties of

the distribution left tail (containing all adverse outcomes) remain vague.

● An estimated VaR value can be easily manipulated by changing the length of the

historical price data.This issue becomes especially problematic if the portfolio includes

less liquid securities (like many options) lacking a verified price history.

● Like many other indicators developed for estimating the risk of linear assets,VaR relies,

to a certain extent, on effective market hypothesis and, hence, inherits its numerous

drawbacks.This becomes especially apparent in times of extreme market fluctuations.

Until recently it was widely believed that VaR adequately describes the risk of all

investment portfolios regardless of their composition and structure. However, the global

140 Automated Option Trading

financial crisis that erupted in 2007 vividly demonstrated the mismatch of forecasts based on

VaR and realized losses.The reason for the divergence between forecasts and reality was that

in the past 20 years financial markets underwent cardinal changes, caused by the devel-

opment of complex financial technologies and the shifting of emphasis from simple linear

assets to derivatives (many of which are nonlinear).The share of nonlinear instruments (of

which options are not the last) in the asset structure of large financial institutions grew

steadily.This impressive advance notwithstanding, risk evaluation mechanisms remained the

same or lagged behind significantly.

In the development of option-based trading strategies, the use of a risk forecasting system

based solely on VaR or similar dispersion-related indicators is inadmissible. A fully fledged risk

management system should include the whole package of evaluation algorithms based on

various principles and take specific option properties into account.

3.2.2 Index DeltaThe index delta characterizes the sensitivity of an option portfolio to broad market fluctua-

tions.This indicator expresses quantitatively the change in the portfolio value under a small

index change. The index delta can be used to evaluate and manage the risk of complex

portfolios in the same way as an ordinary delta does (when applied for portfolios consisting

of options related to a single underlying asset). Besides, the index delta can be applied to

create delta-neutral portfolios (as described in Chapter 1, “Development of Trading

Strategies”) and to restructure existing positions in order to maintain delta-neutrality over the

whole period of portfolio existence.

The AlgorithmThe algorithm for calculation of the index delta can be presented as a sequence of the

following steps:

1. Build regression models for the relationships between the prices of all underlying assets

(options on which are included in the portfolio) and the index.To build these models,

the length of the historical horizon has to be specified.

2. Using the regression models created at step 1, calculate the prices of all underlying

assets as implied by the one-point change in the index value (or by a certain percentage

index change).

3. Determine the fair value of each option in the portfolio given that its underlying asset

price is equal to the value obtained at step 2.This procedure is executed by substituting

the price obtained at step 2 (rather than the current price) into the selected option

pricing model.

4. For each option in the portfolio, calculate the price increment that would take place if

its underlying asset price changes by one point (or by the other value defined at step 2).

Risk Management 1413

The price increment is the difference between the value calculated at step 3 and the

current option market price.

5. Calculate the value of the index delta by summing up all the increments obtained at

step 4.

Analytical Method of Calculating Index DeltaConsider a portfolio consisting of options related to different underlying assets. Let

be a set of options included in the portfolio,while the underlying asset ofO O O ON1 2 3, , ,...,{ }

142 Automated Option Trading

option is . If different options relate to the same underlying asset (that is, assets AjAiOi

and coincide under ), these options form a structure that may correspond to onej k≠Ak

,∆ii

i

O

A=

∂∂

⎛⎝⎜

⎞⎠⎟

where and denote small changes in the price of the option and its underlying asset,

respectively.This expression interprets delta as the speed of the option price change relative

to price changes of its underlying asset.

∂Ai∂Oi

Computing delta of a separate option is a rather trivial task. It is realized in many

software programs based on the Black-Scholes and other, more sophisticated, models.The

delta of the portfolio consisting of options on different underlying assets cannot be

computed as the sum of all deltas since they are derivatives of premiums with respect to

different variables (prices of different underlying assets). As noted previously, this problem is

solved by calculating the sensitivity of the option price relative to the changes of index rather

than to changes in the prices of separate underlying assets. Following this approach, we

define index delta as a derivative of the option price with respect to the index value:

∆i

.

This index delta of a separate option can be expressed as follows:

IDO

Iii=

∂∂

⎛⎝⎜

⎞⎠⎟

(3.2.1).IDO

I

O

A

A

Iii i

i

i=∂∂

⎛⎝⎜

⎞⎠⎟=

∂∂

⎛⎝⎜

⎞⎠⎟⋅

∂∂

⎛⎝⎜

⎞⎠⎟

of the standard option combinations (strangle, straddle, calendar spread, and so on) or may

be of any arbitrary composition. Formally, the combination may consist of an unlimited

number of different options relating to one underlying asset, and the portfolio may include

an unlimited number of combinations.

For any option included in the portfolio, its delta can be expressed asOi

In this equation the value expressed as represents the change in the price of the

underlying asset in response to the index change. In this respect it is similar to the concept

of beta.The difference between these two characteristics is that the index delta is dimen-

sional, whereas beta is a nondimensional characteristic (expressed as the ratio of relative

price changes to a relative index change).Beta is commonly used to evaluate the relationship

between the changes in the index and a separate asset. For our purposes, it would be

convenient to express it as the following ratio:

∂∂

⎛⎝⎜

⎞⎠⎟

A

Ii

After simple transformations we obtain

βii iA A

I I=

∂∂

/

/.

.

By substituting this expression into equation 3.2.1, we get the formula for calculating the

index delta of a separate option,

∂∂

⎛⎝⎜

⎞⎠⎟=

∂∂

⎛⎝⎜

⎞⎠⎟⋅ = ⋅

A

I

A A

I I

A

I

A

Ii i i i

ii/

/β

IDA

Ii i ii= ⋅ ⋅∆ β , (3.2.2)

where is the ordinary delta of option . Let us denote the quantity of option in theOiOi

∆i

portfolio as .To calculate the index delta of the whole portfolio , the index deltas

of separate options included in this portfolio should be summed considering their quantities:

IDPortfolioxi

The index delta calculated using equation 3.2.3 can be interpreted as the change in the

portfolio value in response to the index change by one point. It might be more convenient to

express the portfolio value change in response to the percentage index change.This allows

evaluating portfolio sensitivity to relative index changes. For example, a simple transfor-

mation of equation 3.2.3 allows calculating the index delta for a 1% index change:

ID A x IPortfolio i i i ii

= ⋅ ⋅ ⋅∑∆ β . (3.2.3)

ID A xPortfolio i i i ii

% = ⋅ ⋅ ⋅∑1

100∆ β . (3.2.4)

Risk Management 1433

Example of Index Delta CalculationLet us consider an example of calculating the index delta for a small portfolio, consisting of

options on seven stocks.The S&P 500 will be used in the calculations, though other indexes

could be equally appropriate here.Table 3.2.1 shows the portfolio consisting of seven short

straddles.The quantities of these combinations are approximately inversely proportional to

the prices of the corresponding underlying stocks.The portfolio was created on January 2,

2009, using options with the nearest expiration date (January 16, 2009).The current index

value on January 2, 2009, was 931.8. Beta coefficients of the stocks were calculated on the

basis of daily closing prices using the 120-day historical horizon. Ordinary deltas of separate

options were derived using the Black-Scholes model with a risk-free interest rate of 3.3% and

the volatility estimated at the same horizon of price history.

144 Automated Option Trading

Table 3.2.1 Data required to calculate the index delta (relative to the S&P 500 index) of the portfolio consistingof seven short straddles.

Stock Stock Stock Option Delta of One Index Delta of Index Delta of Ticker Price Beta Strike Type Quantity Option One Option the Position

VLO 23.24 1.58 22.5 Call –400 0.63 0.0248 –9.93

Put –400 –0.37 –0.0146 5.83

PPG 43.55 1.01 45 Call –200 0.39 0.0184 –3.68

Put –200 –0.60 –0.0283 5.66

VNO 58.29 1.62 60 Call –100 0.46 0.0466 –4.66

Put –100 –0.55 –0.0557 5.57

SYMC 14.80 0.91 15 Call –600 0.46 0.0066 –3.99

Put –600 –0.53 –0.0077 4.60

CBE 30.67 1.02 30 Call –300 0.61 0.0205 –6.14

Put –300 –0.40 –0.0134 4.03

NKE 53.06 0.89 55 Call –100 0.35 0.0177 –1.77

Put –100 –0.64 –0.0324 3.24

MMM 59.19 0.70 60 Call –100 0.44 0.0196 –1.96

Put –100 –0.55 –0.0245 2.45

Total (index delta of the portfolio) –0.61

Risk M

anagement

1453

The next-to-last column of Table 3.2.1 shows index deltas of separate options calculated

by equation 3.2.2. For example, the index delta of one call option related to VLO stock is

calculated as The product of index delta of one

option and its quantity in the portfolio gives the index delta of the position corresponding to

this contract (the last column of the table). Thus, the index delta of the position corre-

sponding to the call of VLO stock is . Summing the index deltas

of all positions (that is, all values shown in the last column of the table) gives the index delta

of the portfolio, which corresponds to equation 3.2.3. In this example .

Using equation 3.2.4, the portfolio index delta can easily be expressed in percentage terms:

, which means that the portfolio value will decrease

by 5.69% if the S&P 500 changes by 1%.

Analysis of Index Delta Effectiveness in Risk EvaluationTo estimate the effectiveness of the index delta,we used a database containing eight years of

the price history of options and their underlying assets. The index deltas were estimated

relative to the S&P 500 index. Accordingly, stocks composing this index were used as under-

lying assets for options included in the portfolio.

To estimate the influence of portfolio creation timing on the index delta effectiveness,we

constructed a series of portfolios for each expiration date from the beginning of 2001 until

the beginning of 2009.These portfolios differed from each other in the number of trading

days between the time of portfolio creation and the expiration date. For a given expiration

date, the most distant of the portfolios was constructed 60 days before the expiration; the

next one, 59 days before the expiration; and so on, right up to the last portfolio that was

created only 1 day before the expiration.Thus, 59 portfolios (different from each other in

terms of time left until options expiration) were constructed for each expiration date.In total,

from 30 (for 60 days) to 90 (for 1 day) portfolios were created for each value of the “number

of days left until options expiration”parameter.

Each portfolio consisted of short straddles for all 500 stocks included in the index.The

straddles were created using the strikes closest to the current stock price.The quantity of

options corresponding to each stock was determined as , where 10000 repre-

sents the equivalent of the capital allocated to each straddle (see Chapter 4, “Capital

Allocation and Portfolio Construction,” for details), and A is the price of the stock i.The beta

of each stock, deltas of separate options, and index deltas were calculated using the same

techniques and parameters as in the preceding example (see Table 3.2.1). In addition, the

following characteristics were estimated for each portfolio:

x Ai i= 10000 /

IDPortfolio ( 0.61 931.8) 100 5.69% = − × = −

IDPortfolio = −0.61

IDi 400 0.0248 9.93= − × = −

IDi (23.24 1.58 0.63) 931.8=0.0248.= × ×

146 Automated Option Trading

● Percent change of the index from the time of portfolio creation until the expiration

date:

where is the index value at the time t of portfolio creation, and is the index valueIeIt

I I I It t%

e( )= ⋅ −100 ,

Risk Management 1473

at the expiration date.

● Percent change of the portfolio value from the time of its creation until the

expiration date:

where is the market value of the portfolio at the time t, and is the value of thePePt

P P P Prealized t t%

e( )= ⋅ −100 ,

portfolio at the expiration date.This characteristic reflects the realized portfolio risk.

● Expected percent change of the portfolio value:

This characteristic represents the estimate of portfolio risk given the index change.

● The difference between the actual and the expected change of portfolio value:

.

The closer this characteristic is to zero (that is, the lesser the difference between the

actual and the expected changes), the more accurate the index delta is in forecasting

portfolio value fluctuations.

Figure 3.2.1 shows the average differences between the realized changes in portfolio

values and the changes that were expected given the portfolio risk estimated by the index

delta. Differences of the portfolios created long before the expiration (30 to 60 days) were

close to zero, though their variability (shown on the chart as standard deviations) was

substantial.Therefore, these portfolios allowed for the most accurate estimation of the risk

with relatively high dispersion of individual outcomes. For portfolios created shortly before

the expiration (up to 20 days), the difference between the actual and the expected value

changes were high, positive, and rather stable (low dispersion).This implies that the index

delta underestimates the risk of such portfolios and the magnitude of this underestimation is

stable.

Difference P Prealized expected= −% %

P I ID Pexpected Portfolio t% %( ) .= ⋅ ⋅100 %

Figure 3.2.1 Relationship between the effectiveness of the index delta (expressed through thedifference of the actual and the expected portfolio value changes) and the number of days left untiloptions expiration at the moment of portfolio creation. Points denote average values; horizontalbars, standard deviations.

Positive differences imply that actual changes in the portfolio value

were higher than expected.This means that the index delta underestimated the risk of these

portfolios (since the loss of short positions is incurred when options values increase

. Correspondingly, negative differences imply that actual

changes in the portfolio value were smaller than expected (risk was overestimated). Bars

depicting the variability of outcomes are situated in both the negative and the positive areas

for portfolios created far from their expiration dates (see Figure 3.2.1).This indicates that

some of these portfolios were overestimated and some were underestimated.

Overall, we can conclude that at the moment of portfolio creation, the more days that

remain until the expiration date, the more accurate the average risk estimate is. At the same

time, the probability that a given estimate will turn out to be inaccurate increases as well.For

portfolios created shortly before the expiration date, the risk is underestimated, though the

magnitude of this underestimation is relatively stable (the standard deviation of the average

difference is rather low).Therefore, for portfolios created a long time before the expiration,

the effectiveness of the index delta can be increased by applying additional risk indicators,

while for portfolios created just before the expiration date, the introduction of adjusting

coefficients seems to be sufficient (since the variability of results is low).

P Prealized expected% %<( )( )%Prealized > 0

P Prealized expected% %>( )

148 Automated Option Trading

-8

-4

0

4

8

12

0 10 20 30 40 50 60

Days to expiration

Dif

fere

nce

ove

rest

imat

ion

RIS

Ku

nd

eres

tim

atio

n

Delta is a local instrument indicating the change in the option price under a small change

in its underlying asset price.This means that the higher the actual underlying price change,

the less accurate the forecast of the option price change based on the ordinary delta is.

Hence, the next issue addressed in our study is to test whether and by how much the effec-

tiveness of the forecast based on the index delta deteriorates when index changes are consid-

erable.To answer this question, we will examine the relationship between the effectiveness

of the index delta (expressed through differences between actual and expected changes in

the portfolio value) and the magnitude of the index change.

As it follows from Figure 3.2.2, big index changes indeed induce considerable differences

between realized and expected changes in portfolio values.The relationships are statistically

significant in both cases when the index grows and when it falls (in the former case the corre-

lation was lower than in the latter case).The most interesting fact in the context of evaluation

of the index delta effectiveness is that considerable index changes (regardless of whether it

increases or decreases) correspond to positive differences, and small index changes corre-

spond to negative ones (see Figure 3.2.2).This means that under significant index fluctua-

tions, the index delta tends to underestimate the risk, whereas under small and moderate

index moves, the risk turns out to be overestimated.The index delta demonstrates the highest

effectiveness when the amplitude of market fluctuations is within the limits of 3% to 5%.

R 2 = 0.36

R 2 = 0.51

-60

-30

0

30

60

90

120

150

-30 -20 -10 0 10 20

Index change, %

Dif

fere

nce

Figure 3.2.2 Relationship between the effectiveness of the index delta (expressed through thedifference of the actual and the expected portfolio value changes) and the percentage indexchange. Empty points correspond to positive index changes; filled points, to negative indexchanges.

Risk Management 1493

Analysis of Index Delta Effectiveness at Different Time HorizonsThe research presented in the preceding section was limited to the cases when risk is

estimated only once—at the time of portfolio creation.The accuracy of this estimate was also

verified only once—at the expiration date of options included in the portfolio (for

simplicity’s sake we assume that all options expire simultaneously). In this section we will

study the situations in which risk is estimated repeatedly during the portfolio life span and

the effectiveness of this estimate is tested at different time intervals.

Contrary to the previous study (in which 59 portfolios were formed for each expiration

date), in this study a single portfolio was constructed for each expiration date.The time of

each portfolio creation was 50 trading days away from the given expiration date.The number

of portfolios totaled 90. All other parameters of the trading strategy and the algorithm used

for portfolio creation were the same as in the previous study.

The effectiveness of the index delta in forecasting portfolio risk was evaluated using the

method that was applied in the preceding section (the difference between risk estimated on

the basis of the index delta and its realized value).To assess the quality of risk forecast at

different stages of portfolio existence, we had to (1) calculate values of the index delta daily

through the whole life span of the portfolio, and (2) estimate the changes in the portfolio

value during different time periods (we will refer to these periods as “testing horizons”). All

periods, from the shortest (1 day) to the longest ones (49 days) had to be tested.This data

allows us to perform a detailed evaluation of the differences between forecasts and reality.

Table 3.2.2 presents an example of one of the portfolios and shows the evolution of its

characteristics. At the time of portfolio creation (August 7,2008; first row of the table), there

were 50 trading days until the options expiration date (October 17,2008).The second row of

the table shows characteristics of this portfolio the day after its creation. Accordingly, the

third row shows its characteristics after three days, and so on, until the expiration date (only

20 first days of portfolio existence are included in Table 3.2.2).The table shows the values of

the following characteristics that are necessary to estimate the effectiveness of the index

delta at different time horizons:

● d—The number of days from the valuation moment until the expiration date.This

characteristic is used for indexing the moments of valuation and testing.

● —The index delta calculated at the moment of valuation using equation 3.2.3.IDd

150 Automated Option Trading

● —The percentage of index delta (expresses the change in the portfolio value if the

index changes by 1%) calculated at the moment of valuation using equation 3.2.4.

IDd%

● —The percentage of index change calculated as whereI I I Id j d d% ( )= ⋅ −−100 ,I %

is the index value at the moment of valuation, is the index value at the

moment of testing, and j is the testing horizon (number of days between the valuation

date and the testing date).

Id j−Id

● —The percentage of change of the portfolio value calculated asPrealized%

where is the market value of the portfolio at thePdP P P Prealized d j d d

% ( )= ⋅ −−100 ,

moment of valuation d, and is the value of the portfolio at the moment of testing.Pd j−

time horizon.

● —The expected percentage change of portfolio value calculated asPexpected%

This characteristic estimates the portfolio risk under theP I ID Pexpected d d% %( ) .= ⋅ ⋅100 %

Risk Management 1513

This characteristic reflects the change in the portfolio value that occurs during j days.

Therefore, represents the risk of the portfolio that was realized at the specifiedPrealized%

condition that during j days the index changes by .

● Difference—A reflection of the divergence between the actual and the expected change

in the portfolio value . A positive difference means that the index delta

underestimates the risk, whereas a negative difference points to the risk overestimation.

The closer the difference is to zero, the more accurate the index delta is in forecasting

the portfolio risk.

P Prealized expected% %−( )

I %

152Autom

ated Option Trading

Table 3.2.2 Characteristics of the option portfolio measured during 20 days from the time of its creation (August 7, 2008). Valuespresented in the table are used to evaluate the effectiveness of the index delta at different time horizons.

Testing Horizon j = 1 Day Testing Horizon j = 5 DaysDate d

Difference Difference

7 Aug 08 50 1,224 –0.59 –7.5 1,266

8 Aug 08 49 1,237 –1.38 –17.8 1,296 2.4 1.1 –1.5 2.5

11 Aug 08 48 1,267 –1.74 –22.8 1,305 0.7 2.4 –1.0 3.4

12 Aug 08 47 1,243 –1.32 –17.0 1,290 –1.2 –1.9 2.2 –4.1

13 Aug 08 46 1,225 –1.16 –14.9 1,286 –0.3 –1.4 0.4 –1.8

14 Aug 08 45 1,235 –1.48 –19.1 1,293 0.6 0.8 –0.7 1.4 2.1 0.9 –1.3 2.2

15 Aug 08 44 1,235 –1.68 –21.8 1,298 0.4 0.0 –0.6 0.6 0.1 –0.2 –0.2 0.0

18 Aug 08 43 1,183 –1.13 –14.5 1,279 –1.5 –4.2 2.7 –6.9 –2.0 –6.7 3.7 –10.3

19 Aug 08 42 1,159 –0.73 –9.2 1,267 –0.9 –2.0 1.1 –3.2 –1.8 –6.7 2.4 –9.2

20 Aug 08 41 1,151 –0.82 –10.5 1,275 0.6 –0.7 –0.5 –0.2 –0.9 –6.1 1.1 –7.2

21 Aug 08 40 1,148 –0.87 –11.1 1,278 0.2 –0.3 –0.2 –0.1 –1.2 –7.1 1.8 –8.9

22 Aug 08 39 1,126 –1.27 –16.5 1,292 1.1 –1.9 –1.1 –0.8 –0.5 –8.8 0.8 –9.6

25 Aug 08 38 1,100 –0.58 –7.3 1,267 –2.0 –2.3 2.9 –5.2 –0.9 –7.0 1.1 –8.1

26 Aug 08 37 1,087 –0.66 –8.4 1,272 0.4 –1.3 –0.2 –1.0 0.4 –6.3 –0.3 –6.0

27 Aug 08 36 1,074 –0.98 –12.5 1,282 0.8 –1.1 –0.6 –0.5 0.6 –6.7 –0.5 –6.2

28 Aug 08 35 1,071 –1.51 –19.7 1,301 1.5 –0.3 –1.7 1.4 1.8 –6.7 –1.7 –5.0

29 Aug 08 34 1,060 –1.08 –13.8 1,283 –1.4 –1.0 2.5 –3.5 –0.7 –5.8 1.1 –6.9

2 Sep 08 33 1,062 –0.98 –12.5 1,278 –0.4 0.1 0.5 –0.4 0.8 –3.5 –0.6 –3.0

3 Sep 08 32 1,078 –0.96 –12.2 1,275 –0.2 1.6 0.2 1.3 0.3 –0.8 –0.2 –0.6

4 Sep 08 31 1,088 0.08 1.0 1,237 –3.0 0.9 3.4 –2.5 –3.5 1.2 4.1 –2.9

Pexpected%

Prealized%

I %Pexpected%

Prealized%

I %

IdDd%DdPd

The characteristics shown in Table 3.2.2 correspond to two testing horizons,one and five

days.Let us consider as an example the 41st day to options expiration (d = 41,highlighted in

gray in the table).The following is a step-by-step description of the calculations necessary to

estimate the difference between realized and expected risks.The index delta corresponding

to this date is equal to

.

For the one-day testing horizon (j = 1) the percentage index change is estimated as

The percentage change of the portfolio value is calculated as

,

and the expected percentage change of the portfolio value is

The difference between the actual and the expected changes in the portfolio value is equal

to , which indicates that on the 41st day the risk was

slightly overestimated (when the estimate is tested just one day after it has been made). For

a five-day testing horizon (j = 5), the percentage index change is

The percentage change of the portfolio value is equal to

,

and the expected portfolio value change is

The difference between the actual and the expected changes in the portfolio value is much

higher: , which implies that when the risk is tested five

days after the forecast, it becomes highly overestimated.

Characteristics of all portfolios created from the beginning of 2001 until the beginning of

2009 were calculated using the same methodology.The index delta of each portfolio was

estimated daily during the whole period of portfolio existence.The effectiveness of these

estimates was tested at all time horizons (from 1 to 49 days).

Graphical presentation of the relationship between the realized and the forecast changes

in the portfolio value provides a visual insight into the effectiveness of the index delta.Figure

3.2.3 shows such relationships for the initial stage of the portfolio life (from the 50th to the

40th days until options expiration). The highest correlation was detected for the one-day

testing horizon. In this case the cloud of points (each of which represents an individual

portfolio) is blurred to the smallest extent (the highest determination coefficient R2 = 0.29

was obtained in this case).The extension of the testing horizon to 5, 10, and 20 days leads to

a gradual deterioration in the predicting quality of the index delta. An analysis of the data

presented in Figure 3.2.3 suggests that the higher the testing horizon, the weaker the

Difference = − − − = −6 7 0 5 6 2. ( . ) . %

P I ID Pexpected Portfolio% %( ) ( .= ⋅ ⋅ = ⋅ −100 100 0 6 1041

% .. ) . %.5 1151 0 5= −

P P P Prealized% ( ) ( )= ⋅ − = ⋅ −100 100 1074 1151 115136 41 41 == −6 7. %

I I I I% ( ) ( ) .= ⋅ − = ⋅ − =100 100 1282 1275 1275 0 636 41 41 %.

Difference = − − − = −0 3 0 2 0 1. ( . ) . %

P I ID Pexpected Portfolio% %( ) ( .= ⋅ ⋅ = ⋅ ⋅ −100 100 0 2 141

% 00 5 1151 0 2. ) . %.= −

P P P Prealized% ( ) ( )= ⋅ − = ⋅ −100 100 1148 1151 115140 41 41 == −0 3. %

I I I I% ( ) ( ) .= ⋅ − = ⋅ − =100 100 1278 1275 1275 0 240 41 41 %.

ID I ID41 41 41 100 1275 0 82 100 10 5% ( ) ( . ) / .= ⋅ = ⋅ − = −

Risk Management 1533

relationship between the risk forecast and its actual realization, right up to its full absence.

This conclusion follows both from a comparison of correlation coefficients (R2 = 0.04 for 20

days) and from the patterns of points scattering over the regression plane. Besides, the

extension of the testing horizon results in the decrease of the slope coefficients of corre-

sponding regression lines.This also supports our observation that the forecasting qualities of

the index delta weaken at longer time horizons.

154 Automated Option Trading

R 2 = 0.29R 2 = 0.13R 2 = 0.11R 2 = 0.04

-20

-15

-10

-5

0

5

10

15

20

-30 -20 -10 0 10 20 30 40 50

Expected change in portfolio value, %

Rea

lized

ch

ang

e in

po

rtfo

lio v

alu

e, %

20 days 10 days 5 days 1 day

Figure 3.2.3 The relationships between realized and expected (on the basis of index delta)changes in portfolio value. Four testing horizons are shown.

Regression analysis, similar to that presented in Figure 3.2.3, is a simple and intuitively

comprehensible tool for qualitative evaluation of the index delta. At the same time, it does

not provide a strict quantitative characteristic for measuring the effectiveness of this risk

indicator.For that purpose, it would be preferable to use the difference between the realized

and the expected changes in the portfolio value.Since short option positions generate losses

when their values increase , positive differences ( ) indicate that

the risk is underestimated by the index delta. Accordingly,negative differences imply that the

risk is overestimated.

P Prealized expected% %>( )%Prealized > 0

Figure 3.2.4 shows the average differences and standard errors (expressing the extent of

results variability) for different testing horizons. For the shortest testing horizon (one day),

the deviations of the realized changes in the portfolio values from expected ones were close

to zero through the whole period, right from portfolio creation up to about 20 days until

options expiration. After the 20th day, the closer the portfolios approached the expiration,

the more the risk became underestimated.This means that the index delta can forecast risk

rather accurately,but for a short period and only at the early stages of the portfolio life (during

30 days from its setup). A 5-day testing horizon gives similar results with the only difference

that in this case the underestimation of risk begins earlier (from the 30th day until expiration)

and reaches higher values. Further increases in the testing horizon make these tendencies

even more pronounced—the underestimation of risk begins earlier and reaches greater

extents (see Figure 3.2.4). Besides, it would be worth noting that for short testing horizons,

the relationships between the difference and the number of days left until options expiration

are nonlinear. At the same time, for the longer testing horizons these relationships gradually

become more linear and steeper. This steepness characterizes the speed of the forecast

quality deterioration occurring due to the passage of time (approaching the expiration date).

Risk Management 1553

Testing horizon:

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50

Days to expiration

Dif

fere

nce

1 day

5 days

10 days

20 days

30 days

40 days

49 days

Figure 3.2.4 The dependence of the difference between the realized and the expected changesin the portfolio value on the number of days left until options expiration. Seven testing horizonsare shown. Points denote average values; horizontal lines, standard errors.

Complete information on the effectiveness of the index delta and its dependence on the

two parameters under examination—the valuation moment (relative to the expiration date)

and the testing horizon—can be gathered by presenting the data in the form of a topographic

map. In the two-dimensional coordinate system we will plot the number of days left to

options expiration (which corresponds to different moments of risk valuation) on the

horizontal axis, and the testing horizon on the vertical axis. The height marks of this

topographic surface reflect the average difference between the realized and the expected

changes in the portfolio value.The topography of such a surface is shown in Figure 3.2.5.The

area corresponding to the highest efficiency of the index delta is situated at the bottom-right

corner of the surface (highlighted by the broken line on the chart). In general, we can

conclude that (1) during the 30 days from the moment of portfolio creation (which covers

the period from portfolio creation up to the day when only 20 days are left until the

expiration date), the index delta can estimate the risk quite effectively, and (2) the precision

of these estimates holds during approximately 25 days from the time when the estimates

were made.The triangular shape of the area where the index delta demonstrates its highest

efficiency indicates that forecasting the risk for longer periods is possible only at an early

stage of the portfolio existence. As the expiration approaches,the forecast horizon should be

shortened.

156 Automated Option Trading

Helvetica Regular 8/10

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

Days to expiration

Tes

tin

g h

ori

zon

-15-0 0-15

15-30 30-45

45-60 60-75

Figure 3.2.5 Dependence of difference between the realized and the expectedchanges in the portfolio value on the number of days left until options expiration andon the testing horizon. The broken line highlights the area where the risk wasestimated by the index delta efficiently.

3.2.3 Asymmetry CoefficientThis indicator expresses the skewness of the payoff function of the option portfolio.The idea

underlying this concept consists in the fact that most strategies based on selling naked

options are built on the market-neutrality principle. If the portfolio is really market-neutral,

its payoff function should be sufficiently symmetrical relative to the current value of an index

reflecting the broad market. Such symmetry implies that the value of the portfolio will

change roughly equally regardless of the market direction. If market-neutrality is violated, the

payoff function is biased and the asymmetry coefficient can measure this bias.

Since the portfolio value P is the sum of options values that it includes, the relationship

between P and changes of the index can be expressed as

P I O A I xii

i i( , ) ( ( , )) ,δ δ= ⋅∑ (3.2.5)

I

Risk Management 1573

Applicability of Index DeltaThe index delta represents a convenient and easily calculable instrument that can be applied

to different types of option portfolios. However, its effectiveness may vary in quite a wide

range depending on several factors. In particular, the predicting power of this risk indicator

can be affected by the time left to options expiration.When a portfolio is constructed using

options with a relatively long expiration date,risk forecasts obtained on the basis of the index

delta are quite reliable. Conversely, if a portfolio consists of options that expire shortly,

the applicability of the index delta is limited (the risk may turn out to be underestimated

significantly).

Besides, we have detected that the effectiveness of forecasting the risk by the instrumen-

tality of the index delta depends on the magnitude of expected market fluctuations. This

indicator demonstrates high forecasting abilities during calm and moderately volatile periods.

However, when extreme conditions are prevailing over the markets, the index delta is

unsuitable for assessing the risk of option portfolios.

Other things being equal, applying the index delta for risk measurement is quite effective

during the initial stage of a portfolio’s existence.However, its effectiveness deteriorates as the

portfolio ages and the expiration date approaches.Besides, the index delta is able to produce

a reliable risk forecast only for relatively short time intervals.There is a direct relationship

between the number of days left until options expiration and the forecast horizon—the

closer the expiration is, the shorter the forecast should be. Otherwise, the risk may turn out

to be underestimated significantly.

where is the value of the option, is the underlying asset, is the quantity of optionxiAiOi

158 Automated Option Trading

i in the portfolio, and is the index change (for example, means that the index0.07δ =δrose by 7%).The value of function can be established using beta , estimated as a

slope coefficient in a linear regression between the returns of the underlying asset and the

index. If we know beta, we can roughly estimate the value of the underlying asset given that

the index changes by a specified amount:

βiA Ii ( , )δ

By applying the Black-Scholes model, we can estimate the values of all options included

in the portfolio under the assumption that prices of their underlying assets are equal to the

A I Ai i i( , ) ( ).δ β δ≈ ⋅ + ⋅1 (3.2.6)

values obtained using equation 3.2.6.Summing all values,we get the portfolio value corre-

sponding to equation 3.2.5.Two P values have to be calculated in order to estimate the degree

Oi

of portfolio skewness—for the case of index growth by and for the case of its declineδ × I

by the same value.These values will be denoted as and , respec-

tively. Knowing these two values, the portfolio asymmetry coefficient can be calculated as

P A Ii( ( , ))δ −P A Ii( ( , ))δ +

If we present the portfolio payoff function by plotting the index values on the horizontal

axis and the portfolio values on the vertical axis, then Asym can be visualized as the slope of

AsymP A I P A I

Ii i=

−⋅ ⋅

+ −( ( , )) ( ( , )).

δ δδ2

(3.2.7)

the line connecting the two points with abscissas and and

ordinates corresponding to the payoff function.The higher the absolute value of the slope,

the more asymmetric the payoff function (if the slope is zero, the payoff function is perfectly

symmetric).

Table 3.2.3 contains the interim data required to calculate the asymmetry coefficient for

the portfolio consisting of ten short straddles.This portfolio was created on July 21,2009 (all

options expire on August 21, 2009; the risk-free rate is 1%; the quantity of each option is

X I= ⋅ −( )1 δX I= ⋅ +( )1 δ

). For example, the price of ED stock, provided that and that the index risesβ = 0 23.xi = 1

by 10% ( ), is . Substituting this value into the Black-

Scholes formula (instead of the current stock price) gives us the prices of call and put options

($1.89 and $3.06, respectively). Having computed the values of all options in a similar way,

37.92 (1+0.23 0.1)=$38.79⋅ ⋅δ = 0 1.

we sum them up to obtain and . The table shows that if the

index rises, the portfolio will be worth $46.66, and if it falls, $31.46. Substituting this

data into equation 3.2.7 and taking into account that on the date of the portfolio crea-

tion the S&P 500 was 954.58, we can calculate the asymmetry coefficient:

P A Ii( ( , ))δ −P A Ii( ( , ))δ +

.Asym = − ⋅ ⋅ =( . . ) ( . . ) .46 66 31 46 2 954 58 0 1 0 08

Table 3.2.3 Data required to calculate the asymmetry coefficient for the option portfolio.

δ = 0.1 δ = –0.1Stock Stock Stock Option Stock Option Stock Option Ticker Price Volatility Beta Strike Type Price Price Price Price

Call 38.79 1.89 37.05 1.20ED 37.92 0.49 0.23 40 Put 38.79 3.06 37.05 4.11

Call 33.90 3.94 29.06 0.22EIX 31.48 0.15 0.77 30 Put 33.90 0.00 29.06 1.13

Call 56.90 6.95 49.76 0.60EXC 53.33 0.11 0.67 50 Put 56.90 0.00 49.76 0.79

Call 43.26 4.78 39.06 2.26FE 41.16 0.54 0.51 40 Put 43.26 1.48 39.06 3.16

Call 63.07 5.56 54.37 1.42FPL 58.72 0.49 0.74 60 Put 63.07 2.44 54.37 6.99

Call 13.76 3.78 11.68 1.76NI 12.72 0.40 0.82 10 Put 13.76 0.00 11.68 0.07

Call 39.69 0.68 37.31 0.08PCG 38.50 0.16 0.31 40 Put 39.69 0.94 37.31 2.73

Call 34.38 4.52 30.42 1.42PEG 32.40 0.31 0.61 30 Put 34.38 0.12 30.42 0.97

Call 33.11 3.15 30.75 1.05SO 31.93 0.15 0.37 30 Put 33.11 0.01 30.75 0.27

Call 18.33 3.35 14.03 0.14WMB 16.18 0.26 1.33 15 Put 18.33 0.00 14.03 1.10

Portfolio value P 46.66 31.46

3.2.4 Loss ProbabilityAs it follows from the name, the loss probability indicator reflects the probability that the

portfolio will yield a loss.For a portfolio that contains options,the probability of this negative

outcome can be estimated only by a simulation similar to the Monte-Carlo (as described

earlier for VaR calculations).To apply this method, a random price should be generated for

each underlying asset for a predefined future moment (for example, at the expiration date).

The prices are generated on the basis of probability density functions (selected by the system

developer) with parameters derived from historical data or predetermined by the system

Risk Management 1593

developer according to his personal consideration (scientific method). Then profits and

losses of options are calculated for generated stock prices.The sum of these values represents

an estimation of the portfolio profit or loss.This cycle represents one iteration.By repeatedly

performing many iterations,we can obtain a reliable estimate of the portfolio loss probability.

Table 3.2.4 shows two iterations performed for the portfolio used in the preceding

example (see Table 3.2.3).The stock prices were generated using the lognormal distribution

with historical volatility estimated on the basis of a 120-day period (correlations of stock

prices were taken into account). The first iteration generated the price of $31.04 for EIX

stock,which implies a profit of 30+1.74–31.04 = $0.70,while the second iteration performed

for the same stock incurred a loss in the amount of $1.28. For the whole portfolio, the first

iteration yielded a loss of $2.74, while the second one turned out to be profitable by $5.87.

Table 3.2.4 An example of two iterations performed to estimate the profit or the loss of the optionsportfolio.

160 Automated Option Trading

Iteration 1 Iteration 2Stock Straddle Ticker Strike Price Stock Price Profit Stock Price Profit

ED 40 5.08 37.80 2.88 36.03 1.11

EIX 30 1.74 31.04 0.7 33.02 –1.28

EXC 50 3.42 55.38 –1.96 53.22 0.2

FE 40 5.60 31.36 –3.04 35.39 0.99

FPL 60 7.39 68.73 –1.34 67.47 –0.08

NI 10 2.76 11.33 1.43 12.35 0.41

PCG 40 1.99 35.93 –2.08 40.38 1.61

PEG 30 3.17 31.93 1.24 31.30 1.87

SO 30 2.08 32.85 –0.77 31.37 0.71

WMB 15 1.44 16.24 0.2 16.11 0.33

Total –2.74 5.87

A full set of iterations performed for a given portfolio represents a simulation (in the corre-

lation analysis considered in the next section, we generated 20,000 iterations for each

simulation).The estimate of the portfolio loss probability is obtained by dividing the number

of unprofitable iterations by their total quantity.For example, if 7,420 out of 20,000 iterations

used are unprofitable, the loss probability is 0.37 (7,420 / 20,000).

An effective risk management system should involve a multitude of alternative indicators

based on different principles. Their application area may include initial portfolio

construction, assessment, and restructuring of the existing portfolio and generation of stop-

loss signals. Different risk indicators should be unique and, as far as possible, interdependent

(that is, they should be uncorrelated).This would ensure that each of them supplements the

information contained in the other indicators instead of duplicating it. In this section we

examine the interrelationships between the four risk indicators—VaR (the commonly used

indicator, applied primarily to linear assets) and three indicators that were developed specif-

ically for assessing the risk of an options portfolio (index delta, asymmetry coefficient, and

loss probability).

3.3.1 Method for Testing the InterrelationshipsTesting the extent of interrelationships between the four risk indicators is based on the

presumption that if different indicators are unique (not duplicating each other), the perform-

ances of portfolios created on their basis should be uncorrelated. To test the correlation

between the risk indicators, we used the database containing prices of options and their

underlying assets from January 2003 until August 2009. For each expiration date, we created

60 series of portfolios (each series was composed of 1,000 portfolios) corresponding to

different time intervals left until the expiration.The most distant series was set up 60 days

before the expiration, the next one 59 days before it, and so on, right up to the last series.

Thus, each number-of-days-to-expiration was represented by 1,000 portfolios, which gives

60,000 portfolios for each expiration day.

Each portfolio consisted of ten short straddles relating to stocks that were randomly

selected from the S&P 500 index. Each straddle was constructed using the strike closest to

the current stock price.The quantity of options corresponding to each stock was determined

as , where 10000 represents the equivalent of the capital allocated to each

straddle, and Ai is the stock price.

Within each series the values of the four risk indicators were calculated for each of the

1,000 portfolios. Subsequently, the best portfolio was selected for each indicator (thereby, 4

portfolios were chosen from every series). The returns of the selected portfolios were

recorded on the expiration date. The returns were expressed in percentage terms and

normalized by the time spent in a position (from portfolio creation until the expiration date).

x Ai i= 10000 /

3.3 Interrelationships Between Risk Indicators

Risk Management 1613

3.3.2 Correlation AnalysisAs expected, the returns of portfolios selected on the basis of the four risk indicators were

interrelated to a certain extent. However, apart from a single exception, the correlations

turned out to be relatively low—in five of the six cases the determination coefficient

(squared correlation coefficient) was within the range of 0.3 to 0.4 (see Figure 3.3.1).This

implies that information contained in our risk indicators is duplicated by only 30% to 40%.

Therefore, the introduction of an additional indicator to the risk evaluation system that is

based on a single criterion can enrich this system with about 60% to 70% of new information.

The exception was represented by one pair of indicators—VaR and loss probability.Their

high correlation can be interpreted as an indication of the similarity of the basic ideas under-

lying these indicators.Therefore,their simultaneous application is inexpedient because it will

not add a sufficient volume of new information to the whole risk management system.

Let us consider the following issue.Can the degree of correlation between risk indicators

vary depending on certain factors? Two factors should be analyzed in the first place: the time

interval from the moment of portfolio creation until the options expiration date and the

market volatility at the moment of portfolio creation.

Within one day, the extent of the interdependence between the risk indicators can be

measured by the variance of returns of the four portfolios selected by these indicators on that

day.The higher the variance, the lower the interrelationship between the risk indicators. If

the indicators are perfectly correlated, each of them chooses the same portfolio, in which

case the variance is zero.

162 Automated Option Trading

VaR &index delta

R 2 = 0.36

-2

-1

0

1

2

-2 -1 0 1 2

VaR & loss probability

R 2 = 0.76

-2

-1

0

1

2

-2 -1 0 1 2

VaR &asymmetry

R 2 = 0.38

-2

-1

0

1

2

-2 -1 0 1 2

Index delta &loss probability

R 2 = 0.32

-2

-1

0

1

2

-2 -1 0 1 2

Index delta &asymmetry

R 2 = 0.40

-2

-1

0

1

2

-2 -1 0 1 2

Asymmetry &loss probability

R 2 = 0.35

-2

-1

0

1

2

-2 -1 0 1 2

Figure 3.3.1 Correlation of returns of portfolios selected on the basis of the four risk indicators.Each chart shows the pair of indicators and the corresponding determination coefficient.

Figure 3.3.2 shows the inverse non-linear relationship between variance and the time

interval left until the expiration.This means that close to the expiration date, risk indicators

are weakly correlated and,hence, all of them (or, at least, some of them) carry a considerable

load of additional information. On the other hand, the variance of returns of the portfolios

created long before the expiration is rather low which means that risk indicators are strongly

interrelated during this period (hence,the information contained in these indicators overlaps

significantly).

Risk Management 1633

Figure 3.3.2 Relationship between the variance of returns of portfolios selected on the basis ofthe four risk indicators and the number of days left to options expiration. Higher variance points tolower correlation of risk indicators. Dots denote separate variance values; diamonds denoteaverage values.

Market volatility also influences the interrelationships between the risk indicators,though

this effect is much stronger shortly before the expiration than further away from this date.

Thus, for portfolios created two days before the expiration, the correlation coefficient

between variance and volatility was r = 0.62 for historical volatility and r = 0.68 for implied

volatility (see Figure 3.3.3).Yet, for portfolios created 60 days before expiration, the corre-

lation coefficients were much lower (r = 0.24 and r = 0.28, respectively). These results

indicate that during volatile periods, risk indicators contain additional information, provided

that the portfolio is formed of options with a close expiration date.On the other hand,under

calm market conditions, different risk indicators carry less nonduplicating information

(regardless of the time left to options expiration).

164 Automated Option Trading

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60

Days to expiration

Var

ian

ce

Figure 3.3.3 Relationship between the variance of returns of portfolios selected on the basis offour risk indicators and market volatility (historical and implied). The figure shows only portfolioscreated two days before options expiration.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60

Volatility

Var

ian

ce

Historical volatility

Implied volatility

3.4 Establishing the Risk Management System: Challengesand Compromises

The set of risk indicators described in this chapter represents an example of an evaluation

tool that can be used to develop a multicriteria risk management system. These four

indicators by no means exhaust potential opportunities for creating additional risk

forecasting instruments.The efforts in this direction are continuing by both theoreticians and

practitioners, which will undoubtedly lead to the development of many useful instruments

allowing for a versatile analysis of potential risks threatening option portfolios.The devel-

opers of automated trading systems participate actively in this creative process of

constructing new and elaborating existing risk management tools. The ever-widening

spectrum of available valuation algorithms turns the selection of appropriate indicators that

correspond to specific features of the trading strategy under development into a difficult

challenge.

Risk Management 1653

This task is further complicated by the fact that the effectiveness of any particular risk

indicator may vary depending on many factors.This feature was clearly demonstrated by an

example of the index delta, one of the most universal indicators that is suitable for assessing

the risk of most options portfolios.The predicting power of the index delta was shown to

change depending on the timing of the risk evaluation (relative to the expiration date),on the

length of the forecasting horizon, and on the magnitude of expected market fluctuations.

Therefore, a wise approach to forming the set of risk indicators for a particular trading

strategy consists of including a multitude of them into a risk management system and devel-

oping a switching mechanism that allows them to be turned on and off (depending on the

favorability of different factors prevailing at that specific moment).

With all that in mind, there is still a necessity to avoid the inclusion of redundant items in

the risk evaluation model.This is an important issue, for which a compromise settlement is

needed. An effective risk management system should not involve too many indicators.

Otherwise, this would overburden calculations and lower the efficiency of the evaluation

procedures. While the introduction of any additional indicator into the existing risk

management system is being considered, their uniqueness should be thoroughly estimated.

All potential candidates must carry really new information not contained in other gauges.Two

of the four indicators analyzed in this chapter turned out to duplicate each other:VaR and loss

probability.Hence,when creating a trading strategy,founded on the principles that are similar

to those used in our examples, the developer should include only one of them in the multi-

criteria risk management system.

The results of our study suggest that the number of indicators needed for effective risk

measuring may change depending on the timing of portfolio creation and on the prevailing

market conditions. In particular, the multicriteria approach to risk evaluation could be the

most appropriate one when portfolios are formed close to the expiration date and when the

market volatility is relatively high.Under these conditions,evaluation tools are less correlated

and, hence, each of them introduces a substantial amount of additional information into the

complex risk management system. Undoubtedly, further research will reveal additional

factors that influence the extent of interrelationships between different risk indicators and

determine the necessity for introducing additional gauges.

166 Automated Option Trading

A

acceptable values, determining rangeof, 76, 85-87

access in historical database (inbacktesting systems), 220-221

adaptive optimization, 233-234

additive convolution, 97, 172

adjacent cells, average of, 103-104

alternating-variable ascentoptimization method, 116-118

comparison with random searchmethod, 132

effectiveness of, 128

American option, defined, 251

amoeba search optimization method,123-127

analysis of variance (ANOVA) in one-dimensional capital allocationsystem, 190-191

analytical method (index delta calculation), 142-143

analytical method (VaR calculation), 137

arbitrage situations, tests for, 223

assets

in delta-neutral strategy, 5

in portfolio

analysis of delta-neutralitystrategy, 24-25

analysis of partially direc-tional strategies, 58

price forecasts, 35

call-to-put ratio inportfolio, 40-49

delta-neutrality appliedto, 49-57

embedding in strategystructure, 36-40

Index

asymmetry coefficient, 157-159,180-181

asymmetry of portfolio

analysis of delta-neutralitystrategy, 29-30

analysis of partially directionalstrategies, 60

at-the-money, defined, 252

attainability of delta-neutrality, 14,19-20, 51, 54-55

automated trading systems

defined, xv

empirical approach to development, xviii-xix

rational approach to development, xix-xx

scientific approach to development, xviii

average of adjacent cells, 103-104

B

backtesting systems

challenges and compromises, 246

framework for, 232

adaptive optimization,233-234

in-sample optimization/out-of-sample testing,232-233

overfitting problem,234-236

historical database, 217

data access, 220-221

data reliability/validity,222-224

data vendors for, 218

recurrent calculations,221-222

structure of, 219-220

order execution simulation, 228

commissions, 231

price modeling, 230-231

volume modeling, 229

performance evaluationindicators, 236

backtesting example,242-245

characteristics of return,237-238

consistency, 241

maximum drawdown,238-239

profit/loss factor, 240-241

Sharpe coefficient, 239-240

single events, 236

unit of time frame, 236

signals generation, 225

filtration of signals, 227-228

functionals development/evaluation, 226-227

principles of, 225-226

bear spreads, payoff functions, 258-259

boundaries of delta-neutrality, 6-10

in calm versus volatile markets,10-11, 13

optimal portfolio selection, 67-72

in partially directional strategies,49-55, 57

portfolio structure andproperties at, 62-65

quantitative characteristics of,14-21

262 Index

Index 263

broker commissions (in backtestingsystems), 231

bull spreads, payoff functions, 258-259

C

calendar optimization, defined, 73

calendar spreads, payoff functions,257-258

call options

defined, 251

payoff functions, 254-255

call-to-put ratio in portfolio, 40-42

factors affecting, 44-49

calm markets

delta-neutrality boundaries in,10-13, 51-52

portfolio structure analysis

long and short combinations, 26-27

loss probability, 31-33

number of combinations inportfolio, 22-24

number of underlying assetsin portfolio, 24-25

portfolio asymmetry, 29-30

straddles and strangles,28-29

VaR, 33-34

capital allocation

challenges and compromises,214-216

classical portfolio theory, 168-169

option portfolios, featuresof, 169-170

in delta-neutral strategy, 5

indicators

asymmetry coefficient,180-181

delta, 180

expected profit, 179

inversely-to-the-premium,175-176

inversely-to-the-premiumversus stock-equivalency,176-178

profit probability, 179

stock-equivalency, 174-175

VaR, 181-183

weight function forreturn/risk evaluation,178-179

multidimensional system, 172,204-205

one-dimensional systemversus, 206-209

one-dimensional system, 170, 172

analysis of variance in,190-191

factors affecting, 183-186

historical volatility in,186-187

measuring capital concen-tration, 192-195

multidimensional systemversus, 206-209

number of days toexpiration in, 187-188

number of underlying assetsin, 188-190

weight function transformation, 196-204

in partially directional strategies, 43

portfolio system, 209-211

elemental approach versus,173-174, 211-214

capital concentration

concave versus convex weightfunction comparison, 202-204

measuring, 192-195

one-dimensional versus multi-dimensional capital allocationsystems, 208-209

portfolio versus elemental capitalallocation systems, 213-214

capital management systems

first level of, 167

in partially directional strategies, 43

second level of, 167

characteristics of return performanceindicator, 237-238

classical portfolio theory, 168-169

option portfolios, features of,169-170

closing signals

in delta-neutral strategy, 4

generating (in backtestingsystems), 225

filtration of signals, 227-228

functionals development/evaluation, 226-227

principles of, 225-226

in partially directional strategies, 42

combination of options, 3

combinations

defined, 252

factors affecting call-to-put ratio,44-49

long and short combinations,analysis of delta-neutralitystrategy, 26-27

in partially directional strategies, 43

payoff functions for, 255

bull/bear spreads, 258-259

calendar spreads, 257-258

straddles, 256

strangles, 256-257

in portfolio

analysis of delta-neutralitystrategy, 22-24

analysis of partially directional strategies,57-59

commissions (in backtesting systems), 231

computation, defined, 74

concave weight function, 196, 198

convex weight function compared

by capital concentration,202-204

by profit, 200-202

conditional optimization, 74

consistency performance indicator, 241

constraints in conditionaloptimization, 74

264 Index

Index 265

convex weight function, 196-198

concave weight function compared

by capital concentration,202-204

by profit, 200-202

convolution

of indicators, 172

in multicriteria optimization,97-98

correlation analysis

of objective functions, 91-96

risk indicator interrelationships,162-165

correlation coefficient in objectivefunction relationships, 91

criterion parameters in partially direc-tional strategies, 42

criterion threshold index, 14-17, 51-52

point of delta-neutrality,determining, 7-8

D

data access in historical database (inbacktesting systems), 220-221

data reliability in historical database(in backtesting systems), 222-224

data validity in historical database (inbacktesting systems), 222-224

data vendors for historical database (inbacktesting systems), 218

database (in backtesting systems), 217

data access, 220-221

data reliability/validity, 222-224

data vendors for, 218

recurrent calculations, 221-222

structure of, 219-220

deformable polyhedron optimizationmethod, 123-127

delta, 138-139, 169, 180, 253. See alsoindex delta

delta-neutral strategy, xvii, 4

basic form of, 4-5

optimal portfolio selection, 67-72

optimization space of, 79-80

acceptable range ofparameter values, 85-87

optimization dimension-ality, 80-85

optimization step, 87-88

partially directional strategiesversus, 34

points and boundaries of, 6-10

in calm versus volatilemarkets, 10-11, 13

quantitative characteristicsof, 14-21

portfolio structure analysis, 21-34

long and short combinations, 26-27

loss probability, 31-33

number of combinations inportfolio, 22-24

number of underlying assetsin portfolio, 24-25

portfolio asymmetry, 29-30

straddles and strangles,28-29

VaR, 33-34

portfolio structure andproperties at boundaries,62-65

price forecasts versus, 35

delta-neutrality

applied to partially directionalstrategies, 49-55, 57

attainability, 14, 19-20, 51, 54-55

derivatives, Greeks as, 138

determination coefficient in objectivefunction relationships, 91

dimensionality of optimization, 80-85

one-dimensional optimization,80-82

two-dimensional optimization,83-85

direct filtration method, 227

direct methods, defined, 78

direct search optimization methods, 115

alternating-variable ascentmethod, 116-118

comparison of effectiveness,127-130

drawbacks to, 115-116

Hook-Jeeves method, 118-120

Nelder-Mead method, 123-127

Rosenbrock method, 120-123

directional strategies, xvi

diversification, underlying assets inportfolio, 24

diversity of options available, 3

domination in Pareto method, 99

E

elemental capital allocation approach,173-174

portfolio system versus, 211-214

embedding price forecasts in strategystructure, 36-40

empirical approach to automatedtrading system development,xviii-xix

equity curve in backtesting results,242-244

European option, defined, 251

evaluation

of option pricing, 1-2

of performance (in backtestingsystems), 236

backtesting example,242-245

characteristics of return,237-238

consistency, 241

maximum drawdown,238-239

profit/loss factor, 240-241

Sharpe coefficient, 239-240

single events, 236

unit of time frame, 236

exhaustive search optimizationmethod, 114-115

expected profit (capital allocationindicator), 179

expiration date, 169

defined, 251

in delta-neutral strategy, 8-13

266 Index

Index 267

effect on call-to-put ratio, 48-49

in one-dimensional capitalallocation system, 187-188

exponential annual return, 237

F

fair value pricing, determining, 1-2

filtration of signals, 227-228

first level of capital managementsystems, 167

fixed parameters, steadiness ofoptimization space, 109-110

forecasts of underlying asset prices, 35

call-to-put ratio in portfolio, 40-42

factors affecting, 44-49

delta-neutrality applied to, 49-57

embedding in strategy structure,36-40

full optimization cycle, defined, 74

functionals, development and evaluation of, 226-227

fundamental analysis for priceforecasts, 35

G

gamma, 138, 253

generating signals in delta-neutralstrategy, 4

genetic algorithms, 115

global maximum, defined, 75

Greeks, 169

defined, 253

in risk evaluation, 138-139

H

hedging strategies, xvi

historical database (in backtestingsystems), 217

data access, 220-221

data reliability/validity, 222-224

data vendors for, 218

recurrent calculations, 221-222

structure of, 219-220

historical method (VaR calculation), 137

historical optimization period,steadiness of optimization space,112-114

historical volatility

defined, 252

in delta-neutral strategy, 80

in one-dimensional capitalallocation system, 186-187

recurrent calculations applied,221-222

in risk evaluation, 136

Hook-Jeeves optimization method,118-120

comparison with random searchmethod, 132

effectiveness of, 128

I–J–K

ideal strategy, 241

implied volatility

defined, 252

estimations, 224

in-sample optimization, 232-233

in-the-money, defined, 252

index delta, 139, 141, 169

analysis of effectiveness

at different time horizons,150-156

in risk evaluation, 146-149

analytical method of calculation,142-143

applicability of, 156-157

calculation algorithm, 141-142

example of calculation, 144-146

indirect filtration method, 227

interrelationships

between risk indicators, 161

correlation analysis, 162-165

testing, 161

of objective functions, 91-96

intrinsic value, defined, 251

inversely-to-the-premium (capitalallocation indicator), 175-176

stock-equivalency versus,176-178

investment assets

in delta-neutral strategy, 5

in portfolio

analysis of delta-neutralitystrategy, 24-25

analysis of partially directional strategies, 58

price forecasts, 35

call-to-put ratio in portfolio,40-49

delta-neutrality applied to,49-57

embedding in strategystructure, 36-40

isolines in delta-neutrality boundaries, 9

L

length of delta-neutrality boundary, 14,17-19, 51-54

life span of options, limited nature of, 2-3

linear annual return, 237

linear assets, options versus, xvii

linear financial instruments

nonlinear instruments versus, 135

risk evaluation, 136-137

local maximum, defined, 75

long calendar spreads, 258

long combinations

analysis of delta-neutralitystrategy, 26-27

factors affecting call-to-put ratio,44-49

in portfolio, analysis of partiallydirectional strategies, 58-59

long options, payoff functions, 254-255

long positions, limitations on, 111

long straddles, 256

long strangles, 256

loss probability, 159-160

analysis of delta-neutralitystrategy, 31-33

in portfolio, analysis of partiallydirectional strategies, 60

268 Index

Index 269

M

margin requirements, 170, 252

market impact, 230

market volatility, effect on call-to-putratio, 46-47

market-neutral strategies, 258. See alsodelta-neutral strategy

market-neutrality, xvii

Markowitz, Harry, 168

maximum drawdown, 238-239

as objective function

effect on optimizationspace, 90

relationship withpercentage of profitabletrades, 93

relationship with profit, 92

mean, ratio to standard error, 104-105

minimax convolution, 97

modeling (in backtesting systems), 228

commissions, 231

price modeling, 230-231

volume modeling, 229

money management in delta-neutralstrategy, 5

Monte-Carlo method (VaR calculation), 137

moving averages, compared withaverage of adjacent cells, 103

multicriteria optimization, 79

convolution, 97-98

nontransitivity problem, 96-97

Pareto method, 99-102

robustness of optimal solution, 102

averaging adjacent cells,103-104

ratio of mean to standarderror, 104-105

surface geometry, 106-108

steadiness of optimization space,108-109

relative to fixed parameters,109-110

relative to historicaloptimization period,112-114

relative to structuralchanges, 110-111

multidimensional capital allocationsystem, 172, 204-205

one-dimensional system versus,206-209

multiple regression analysis in one-dimensional capital allocationsystem, 190-191

multiplicative convolution, 97, 172

N

Nelder-Mead optimization method,123-129

neuronets, 115

nodes, defined, 74

nonlinear financial instruments

linear instruments versus, 135

risk evaluation, 138-139

risk indicators, 139

asymmetry coefficient,157-159

index delta, 141-157

interrelationships between,161-165

loss probability, 159-160

VaR (Value at Risk), 140-141

nonlinearity, options evaluation and, 1-2

nonmodal optimization, 76

nonmodal optimization space, 82

nontransitivity in multicriteriaoptimization, 96-97

normalization, 184

O

objective function

defined, 74

effect on optimization space,89-91

explained, 78-79

interrelationships of, 91-96

usage of, 88

one-dimensional capital allocationsystem, 170-172

analysis of variance in, 190-191

factors affecting, 183-186

historical volatility in, 186-187

measuring capital concentration,192-195

multidimensional system versus,206-209

number of days to expiration in,187-188

number of underlying assets in,188-190

weight function transformation,196-204

one-dimensional optimization, 80-82

opening signals

in delta-neutral strategy, 4

generating (in backtestingsystems), 225

filtration of signals, 227-228

functionals development/evaluation, 226-227

principles of, 225-226

in partially directional strategies, 42

optimal area, defined, 75

optimal delta-neutral portfolioselection, 67-72

optimal solution

defined, 75

robustness of, 82, 102

averaging adjacent cells,103-104

ratio of mean to standarderror, 104-105

surface geometry, 106-108

optimization

adaptive optimization, 233-234

challenges and compromises in, 134

defined, 73

dimensionality of, 80-85

one-dimensionaloptimization, 80-82

two-dimensionaloptimization, 83-85

270 Index

Index 271

in-sample optimization, 232-233

multicriteria optimization

convolution, 97-98

nontransitivity problem,96-97

Pareto method, 99-102

robustness of optimalsolution, 102-108

steadiness of optimizationspace, 108-114

objective function

effect on optimizationspace, 89-91

explained, 78-79

interrelationships of, 91-96

usage of, 88

parametric optimization,explained, 73-75

terminology, 74-75

optimization methods

direct search methods, 115

alternating-variable ascentmethod, 116-118

comparison of effectiveness, 127-130

drawbacks to, 115-116

Hook-Jeeves method,118-120

Nelder-Mead method,123-127

Rosenbrock method,120-123

exhaustive search, 114-115

random search, 131-133

optimization space

defined, 74

of delta-neutral strategy, 79-80

acceptable range ofparameter values, 85-87

optimization dimensionality,80-85

optimization step, 87-88

effect of objective functions on,89-91

explained, 75-77

steadiness of, 108-109

relative to fixed parameters,109-110

relative to historicaloptimization period,112-114

relative to structuralchanges, 110-111

optimization step, 76, 87-88

option combinations

defined, 252

factors affecting call-to-put ratio,44-49

long and short combinations,analysis of delta-neutralitystrategy, 26-27

in partially directional strategies, 43

payoff functions for, 255

bull/bear spreads, 258-259

calendar spreads, 257-258

straddles, 256

strangles, 256-257

in portfolio

analysis of delta-neutralitystrategy, 22-24

analysis of partially direc-tional strategies, 57-59

option portfolios

capital allocation indicators

asymmetry coefficient,180-181

delta, 180

expected profit, 179

inversely-to-the-premium,175-176

inversely-to-the-premiumversus stock-equivalency,176-178

profit probability, 179

stock-equivalency, 174-175

VaR, 181-183

weight function forreturn/risk evaluation,178-179

capital allocation systems,challenges and compromises,214-216

features of, 169-170

multidimensional capitalallocation system, 172, 204-205

one-dimensional systemversus, 206-209

one-dimensional capitalallocation system, 170-172

analysis of variance in,190-191

factors affecting, 183-186

historical volatility in,186-187

measuring capital concentration, 192-195

multidimensional capitalallocation system versus,206-209

number of days toexpiration in, 187-188

number of underlying assetsin, 188-190

weight function transformation, 196-204

portfolio capital allocationsystem, 209, 211

elemental system versus,173-174, 211-214

option strategies, payoff functions for, 255

bull/bear spreads, 258-259

calendar spreads, 257-258

straddles, 256

strangles, 256-257

option trading strategies

limited options life span, 2-3

nonlinearity and options evaluation, 1-2

option diversity, 3

options, linear assets versus, xvii

order execution simulation (inbacktesting systems), 228

commissions, 231

price modeling, 230-231

volume modeling, 229

272 Index

Index 273

out-of-sample testing, 232-233

out-of-the-money, defined, 252

overfitting problem, 234-236

P

parameter values, determiningacceptable range of, 76, 85-87

parametric optimization, 73-75. Seealso optimization

Pareto method, 172

in multicriteria optimization,99-102

partially directional strategies, xvii

basic form of, 42-43

call-to-put ratio, 40-42

factors affecting, 44-49

delta-neutrality applied to, 49-57

delta-neutrality strategy versus, 34

embedding forecast in strategystructure, 36-40

features of, 35

portfolio structure analysis, 57-61

payoff functions

call-to-put ratio in portfolio, 40-42

factors affecting, 44-49

defined, 253

for option combinations, 255

bull/bear spreads, 258-259

calendar spreads, 257-258

straddles, 256

strangles, 256-257

in option portfolios, 169

for separate put/call options,254-255

percentage of profitable trades asobjective function

effect on optimization space, 90

relationship with maximumdrawdown, 93

relationship with profit, 92

performance evaluation indicators (inbacktesting systems), 236

backtesting example, 242-245

characteristics of return, 237-238

consistency, 241

maximum drawdown, 238-239

profit/loss factor, 240-241

Sharpe coefficient, 239-240

single events, 236

unit of time frame, 236

points of delta-neutrality, 6-10

in calm versus volatile markets,10-11, 13

quantitative characteristics of,14-21

polymodal optimization, 76

polymodal optimization space, 82

portfolio asymmetry, analysis ofpartially directional strategies, 60

portfolio capital allocation approach,173-174, 209-211

elemental system versus, 211-214

portfolio construction

capital allocation indicators

asymmetry coefficient,180-181

delta, 180

expected profit, 179

inversely-to-the-premium,175-176

inversely-to-the-premiumversus stock-equivalency,176-178

profit probability, 179

stock-equivalency, 174-175

VaR, 181-183

weight function forreturn/risk evaluation,178-179

capital allocation systems,challenges and compromises,214-216

classical portfolio theory, 168-169

option portfolios, featuresof, 169-170

multidimensional capitalallocation system, 204-205

one-dimensional systemversus, 206-209

one-dimensional capitalallocation system

analysis of variance in,190-191

factors affecting, 183-186

historical volatility in,186-187

measuring capital concentration, 192-195

multidimensional systemversus, 206-209

number of days toexpiration in, 187-188

number of underlying assetsin, 188-190

weight function transformation, 196-204

option portfolios

multidimensional capitalallocation system, 172

one-dimensional capitalallocation system,170-172

portfolio versus elementalapproach to capitalallocation, 173-174

portfolio capital allocationsystem, 209-211

elemental system versus,211-214

portfolio structure at delta-neutralityboundaries, 62-65

portfolio structure analysis

of delta-neutrality strategy, 21-34

long and short combinations, 26-27

loss probability, 31-33

number of combinations inportfolio, 22-24

number of underlying assetsin portfolio, 24-25

portfolio asymmetry, 29-30

straddles and strangles,28-29

VaR, 33-34

of partially directional strategies,57-61

274 Index

Index 275

position closing signals

in delta-neutral strategy, 4

in partially directional strategies, 42

position opening signals

in delta-neutral strategy, 4

in partially directional strategies, 42

position opening/closing signals,generating (in backtesting systems), 225

filtration of signals, 227-228

functionals development/evaluation, 226-227

principles of, 225-226

premium

defined, 251

inversely-to-the-premium (capitalallocation indicator), 175-176

stock-equivalency versus,176-178

price forecasts, 35

call-to-put ratio in portfolio, 40-42

factors affecting, 44-49

delta-neutrality applied to, 49-57

embedding in strategy structure,36-40

price modeling (in backtestingsystems), 230-231

profit

concave versus convex weightfunction comparison, 200-202

as objective function

effect on optimizationspace, 89

relationship with maximumdrawdown, 92

relationship withpercentage of profitabletrades, 92

relationship with Sharpecoefficient, 91

one-dimensional versus multi-dimensional capital allocationsystems, 206-208

portfolio versus elemental capitalallocation systems, 211-213

profit probability (capital allocationindicator), 179

profit/loss factor, 240-241

put options

defined, 251

payoff functions, 254-255

Q–R

quantitative characteristics of delta-neutrality boundaries, 14-21

quantitative characteristics analysis,244-245

random search optimization method,131-133

range of acceptable values,determining, 76, 85-87

rational approach to automatedtrading system development, xix-xx

recurrent calculations in historicaldatabase (in backtesting systems),221-222

regression analysis, 154

reliability of data in historical database(in backtesting systems), 222-224

requirements in delta-neutral strategy, 5

restrictions in delta-neutral strategy, 5

rho, 138, 253

risk, lack of definition for, 135

risk evaluation

effectiveness of index delta,146-149

linear financial instruments,136-137

nonlinear financial instruments,138-139

risk indicators, 139

asymmetry coefficient, 157-159

establishing risk managementsystems, 165-166

index delta, 141

analysis of effectiveness atdifferent time horizons,150-156

analysis of effectiveness inrisk evaluation, 146-149

analytical method of calculation, 142-143

applicability of, 156-157

calculation algorithm,141-142

example of calculation,144-146

interrelationships between, 161

correlation analysis, 162-165

testing, 161

loss probability, 159-160

VaR (Value at Risk), 140-141

risk management

in delta-neutral strategy, 5

establishing risk indicators,165-166

the Greeks, 169

in partially directional strategies, 43

robustness of optimal solution, 82, 102

averaging adjacent cells, 103-104

defined, 75

ratio of mean to standard error,104-105

surface geometry, 106-108

Rosenbrock optimization method,120-123, 129

rotating coordinates optimizationmethod, 120-123

S

scientific approach to automatedtrading system development, xviii

second level of capital managementsystems, 167

selecting optimal delta-neutralportfolio, 67-72

selective convolution, 97

Sharpe coefficient, 239-240

as objective function

effect on optimizationspace, 90

relationship with profit, 91

short calendar spreads, 257

short combinations

analysis of delta-neutralitystrategy, 26-27

276 Index

Index 277

factors affecting call-to-put ratio,44-49

in portfolio, analysis of partiallydirectional strategies, 58-59

short options, payoff functions,254-255

short straddles, 256

short strangles, 256

signal-generation indicators in delta-neutral strategy, 4

signals generation

in backtesting systems, 225

filtration of signals, 227-228

functionals development/evaluation, 226-227

principles of, 225-226

in delta-neutral strategy, 4

simplex search optimization method,123-127

simulation of order execution (inbacktesting systems), 228

commissions, 231

price modeling, 230-231

volume modeling, 229

single events in performance evaluation, 236

slippage, 230

smoothing, advantages of, 88

smoothness of optimization space, 77

spreads, xvii

standard deviation of asset returns inrisk evaluation, 136

standard error, ratio to mean, 104-105

steadiness of optimization space, 77,108-109

relative to fixed parameters,109-110

relative to historical optimizationperiod, 112-114

relative to structural changes,110-111

stock-equivalency (capital allocationindicator), 5, 174-175

inversely-to-the-premium versus,176-178

straddles

analysis of delta-neutralitystrategy, 28-29

payoff functions, 256

strangles

analysis of delta-neutralitystrategy, 28-29

payoff functions, 256-257

strike price, defined, 251

strikes range index, 14-17, 51-52

structural changes, steadiness ofoptimization space, 110-111

structural optimization, defined, 73

surface geometry, determiningrobustness of optimal solution,106-108

survival bias problem, 218

synchronization, 219

synthetic assets strategies, xvi

T

technical analysis for price forecasts, 35

testing risk indicator interrelation-ships, 161. See also backtestingsystems

theta, 138, 253

three-dimensional optimization, 77

time decay, defined, 252

time horizons, effectiveness of indexdelta, 150-156

time value, defined, 252

transformation of weight function,196-204

transitivity in multicriteriaoptimization, 96-97

two-dimensional optimization,77, 83-85

U

unconditional optimization, 74

underlying assets in one-dimensionalcapital allocation system, 188-190

unimodal optimization, 76

unimodal optimization space, 82

unit of time frame in performanceevaluation, 236

V

validity of data in historical database(in backtesting systems), 222-224

Value at Risk. See VaR

values, determining acceptable rangeof, 76, 85-87

VaR (Value at Risk), 181-183

analysis of delta-neutralitystrategy, 33-34

calculation methods, 137

drawbacks to, 140-141

in portfolio, analysis of partiallydirectional strategies, 61

in risk evaluation, 136

variation coefficients, 181, 183

vega, 138-139, 253

vendors for historical database (inbacktesting systems), 218

visual analysis of backtesting results,242-244

volatile markets

delta-neutrality boundaries in,10-13

delta-neutrality boundaries inpartially directional strategies,51-52

historical volatility in risk evaluation, 136

portfolio structure analysis

long and short combinations, 26-27

loss probability, 31-33

number of combinations inportfolio, 22-24

number of underlying assetsin portfolio, 24-25

278 Index

Index 279

portfolio asymmetry, 29-30

straddles and strangles,28-29

VaR, 33-34

volume modeling (in backtestingsystems), 229

W–Z

walk-forward analysis, 235

weight function

for return/risk evaluation,178-179

transformation of, 196-204