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PRE-PRINT ABSTRACTS

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Director: Dr Nick Webber Research Co-ordinator: Rhona Macdonald Financial Options Research Centre Telephone: +44 (0)24 76524118 Warwick Business School Fax: +44 (0)24 76524167 University of Warwick Email: [email protected] Coventry CV4 7AL – UK Website: www.warwick.ac.uk/go/forc

Financial Options Research Centre Preprint Abstracts

The Financial Options Research Centre produces its own working papers (known as Pre-prints), many of which are subsequently published in Journals. Abstracts for each Pre-print are outlined in this brochure including, where applicable, publication. This list was correct as at 12 November 2004.

Contents 04-145 The Greek Implied Volatility Index: Construction and Properties

04-144 Volatility Options: Hedging Effectiveness, Pricing and Model Error

04-143 A Parsimonious Continuous Time Model of Equity Index Returns (Inferred From High Frequency Data)

04-142 The Value of Storage Facility

04-141 The Logistic Function and Implied Volatility: Quadratic Approximation and Beyond

04-140 Valuing Discrete Barrier Options on a Dirichlet Lattice

04-139 Valuing Bermudan Options When Asset Returns are Lévy Processes

04-138 Correcting for Simulation Bias in Monte Carlo methods to Value Exotic Options in Models Driven by Lévy Processes

04-137 An EZI method to reduce the rank of a correlation matrix

04-136 Valuing Path Dependent Options in the Variance-Gamma Model by Monte Carlo with a Gamma Bridge

04-135 Valuing Continuous Barrier Options on a Lattice Solution for a Stochastic Dirichlet Problem

04-134 An Asset Based Model of Defaultable Convertible Bonds with Eondogenised Recovery

04-133 A Monte-Carlo Method for the Normal Inverse Gaussian Option Valuation Model using an Inverse Gaussian Bridge

04-132 A Two-Factor Model for Commodity Prices and Futures Valuation

04-131 A Contango-Constrained Model for Storable Commodities

04-130 Equilibrium Model for Commodity Prices: Competitive and Monopolistic Markets

04-129 Term Structure Models via Matrix Completion

03-128 Implied Kernel Models

03-127 A Note on Optimal Calibration of the Libor Market Model to the Correlations

03-126 Pricing of Implied Volatility Derivatives

03-125 The Favorite/Long-shot Bias in S&P 500 and FTSE 100 Index Futures Options: The Return to Bets and the Cost of Insurance

03-124 The Relation Between Implied and Realised Probability Density Functions

03-123 The Effect of Mis-Estimating Correlation on Value-at-Risk

03-122 Hedging Errors under Mis-specified Asset Price Processes

Financial Options Research Centre - Pre-print Abstracts

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02-121 The Valuation of Convertible Bonds: A Study of Alternative Pricing Models

02-120 A New Approach to Modelling the Dynamics of Implied Distributions: Theory and Evidence from the S&P 500 Options

01-119 Numerical Valuation of Discrete Barrier Options

01-118 A Reduced-form Model incorporating Fundamental Variables

01-117 Equilibrium Price Processes, Mean Reversion and Consumption Smoothing

01-116 Valuation of Claims on Non-Traded Assets using Utility Maximisation

01-115 Static Dynamic Hedging of Exotic Options: An Evaluation of Hedge Performance via Simulation

01-114 Pricing of Defaultable Coupon Bonds Under a Jump-Diffusion Process

01-113 A New Class of Commodity Hedging Strategies: A Passport Options Approach

00-112 Multivariate Distributional Tests Based on the Empirical Characteristic Function Approach: A Comparison

00-111 Simulating the Evolution of the Implied Distribution

00-110 Real Options with Constant Relative Risk Aversion

00-109 A Computational Framework for Contingent Claim Pricing and Hedging under Time Dependent Asset Processes

00-108 The Languages of Contingent Claim Contracts

00-107 Taxonomy of Algorithms

00-106 Stochastic Volatility Models with Jumps: Implications for Smiles in Foreign Exchange Markets

00-105 Stock Index Futures Markets: Stochastic Volatility Models and Smiles

00-104 Fixed Income Futures Markets: Stochastic Volatility Models and Smiles

00-103 The Sampling Properties of Volatility Cones

00-102 An Evaluation of Tests of Distributional Forecasts

00-101 Volatility Smile Consistent Option Models: A Survey

99-100 Pricing of Occupation Time Derivatives: Continuous and Discrete Monitoring

99-99 Corridor Options and Arc-Sine Law

99-98 Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: an Arrow-Debreu Pricing Approach

99-97 Conditional Gaussian Models of the Term Structure of Interest Rates

98-96 The Relationship between Stock Returns and Tobin’s q: Tobin’s q Effect

98-95 Implied Risk-Neutral Distribution: A Comparison of Estimation Methods

98-94 Rational Bounds on the Prices of Exotic Options

98-93 Implied Volatility Surfaces: Uncovering Regularities for Options On Financial Futures

98-92 The Theory of No Good Deal Pricing in Financial Markets


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98-91 The Extension Theorem and the Unified Approach to No-Arbitrage Pricing

98-90 Pricing By Arbitrage Under Arbitrary Information

98-89 Auditor Performance, Implicit Guarantees, and the Valuation of Legal Liability

98-88 A Generalization of the Sharpe Ratio and its Applications to Valuation Bounds and Risk Measures

98-87 Empirical Properties of Asset Price Processes

98-86 The Dynamics of Implied Volatility Surfaces

98-85 The Dynamics of Smiles

97-84 A Comparison of Alternative Methods for Hedging Exotics Options

97-83 Modelling Commodity Futures Spreads: An Empirical Study

97-82 Hedging Option Position Risk: An Empirical Examination

97-81 The Potential for Profitable Stock Market Manipulation in the Presence of Positive Feedback Trading

97-80 Kalman Filtering of Generalized Vasicek Term Structure Models

97-79 Can a Diffusion Recover a Lognormal Jump Process?

97-78 Numerical Procedures for Pricing Interest Rate Exotics Using Markovian Short Rate Models

97-77 American Featured Options

97-76 Efficient Pricing of Caps and Swaptions in a Multi-Factor Gaussian Interest Rate Model

97-75 Hedging Barrier Options in Incomplete Markets with Transactions Costs

97-74 Computational Aspects of Term Structure Models and Pricing Interest Rate Derivatives

97-73 Mathematical Programming and Risk Management of Derivative Securities

97-72 Simple Resettable Cap and Floor Pricing Formulae

96-71 Calibration of Kennedy and Multi-Factor Gaussian HJM to Caps and Swaptions Prices

96-70 Option Pricing and Smile Effect when Underlying Stock Prices are Driven by a Jump Process

96-69 Pricing by Arbitrage in Incomplete Markets

96-68 Optimal Delta-Hedging Under Transactions Costs

96-67 Monte Carlo Valuation of Interest Rate Derivatives Under Stochastic Volatility

96-66 A Note on the Efficiency of the Binomial Option Pricing Model

96-65 The Risk Premium in Trading Equilibria Which Support Black-Scholes Option Pricing

96-64 Equilibrium and the Role of Options in an Economy with Stochastic Volatility

95-63 Martingale Restriction Tests of Option Pricing Models and Their Interpretation

95-62 Non-Negative Affine Yield Models of the Term Structure

95-61 Term Structure Modelling Under Alternative Official Regimes

95-60 A Comparison of Alternative Covariance Matrices for Models with Over-Lapping


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Observations

95-59 Can Dividend Yields Forecast Returns?

95-58 An Econometric Analysis of Long Horizon Mean Reversion in UK Stock Prices

95-57 Arbitrage in a Fractional Brownian Motion Market

95-56 On the Simulation of Contingent Claims

94-55 Efficient and Flexible Bond Option Valuation in the Heath Jarrow and Morton Framework

94-54 Pricing Exotic Options in a Black-Scholes World

94-53 The Dynamics of Stochastic Volatility

94-52 Gamma Hedging in Incomplete Markets Under Transactions Costs

94-51 Multi-Period Minimax Hedging Strategies

94-50 Minimax Hedging Strategy

94-49 A Theory of the Term Structure with an Official Short Rate

94-48 A Model of UK Libor as a Jump-Diffusion Process

94-47 A Comparison of Models for Pricing Interest Rate Derivative Securities

94-46 A Comparison of Models of the Term Structure

94-45 Option Prices as Predictors of Stock Prices: Intraday Adjustments to Information Releases

94-44 On a Free Boundary Problem That Arises in Portfolio Management

93-43 Dynamic Asset Allocation: Insights From Theory

93-42 Computing the Fong and Vasicek Pure Discount Bond Price Formula

93-41 A Note on Parameter Estimation in the Two Factor Longstaff and Schwartz Interest Rate Model

93-40 The Magnitude of Implied Volatility Smiles: Theory and Empirical Evidence for Exchange Rates

93-39 Convergence to Efficiency of the Nikkei Put Warrant Market of 1989-1990

93-38 Contingent Claims Analysis

93-37 Interest Rate Volatility and the Term Structure of Interest Rates

93-36 Optimal Delta-Hedging Under Transactions Costs

92-35 A Review of Option Pricing with Stochastic Volatility

92-34 Recent Developments in Derivative Securities

92-33 Conditional Volatility and the Informational Efficiency of the PHLX Currency Options Market

92-32 The Term Structure of Volatility Implied by Foreign Exchange Options

92-31 The Characterization of Economic Equilibria Which Support Black-Scholes Option Pricing

92-30 Efficient Monte Carlo Valuation and Hedging of Contingent Claims

92-29 Current Research on Derivative Products


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92-20 The Term Structure of Spot Rate Volatility and the Behaviour of Interest Rate Processes

92-19 Do Derivative Instruments Increase Market Volatility?

91-28 A Note on the Breen Binomial Acceleration Technique

91-27 Interest Rate Swaps and Default-Free Bonds: A Joint Term Structure Model

91-26 Behaviour of the FTSE 100 Basis

91-25 The Delivery Options in Bond Futures Contracts

91-24 A Family of Ito Process Models for the Term Structure of Interest Rates

91-23 Ex-Post Evaluation of Dynamic Portfolio Strategies (or How to Tell Whether a Million Dollars has been Thrown Away)

91-22 An Introduction to Parallel Processing for Financial Valuation Problems

91-21 The Term Structure of Interest Rates and Associated Options; Equilibrium vs Evolutionary Models

91-18 Testing for Overreaction in Short Sterling Options

90-17 Interest-Rate Derivatives: Evolutionary Valuation and Hedging

90-11 Valuing Average Rate ('Asian') Options

90-10 Finite Difference Techniques for One and Two Dimension Option Valuation Problems

90-09 A Survey of Elementary Techniques for Pricing Options on Bonds and Interest Rates

89-08 Two Factor Models in Option Pricing

89-07 Optimal Replication of Contingent Claims Under Transactions Costs

89-06 Financial Engineering: New Approaches to Managing Risk Exposure

89-05 Valuing Interest Rate Options via a Primitive Theory of the Term Structure

89-04 A Primitive Theory of the Term Structure of Interest Rates

89-03 Expected Turnover in a Binomial Tree

89-02 Modelling the Dynamics of the Term Structure of Interest Rates

89-01 Estimating the Gilt-Edged Term Structure: Basis Splines and Confidence Intervals

88-05 The Ho and Lee Term Structure Theory: A Continuous Time Version

88-04 Financial Options: New Markets and New Valuation Techniques

88-03 Valuation and Hedging: Theoretical Issues

88-02 American Options: Theory and Numerical Analysis

88-01 Numerical Methods


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04-145 The Greek Implied Volatility Index: Construction and Properties There is a growing literature on implied volatility indices in developed markets.

However, no similar research has been conducted in the context of emerging markets. In this paper, an implied volatility index (GVIX) is constructed for the fast developing Greek derivatives market. Next, the properties of GVIX are explored. In line with earlier results, GVIX can be interpreted as a gauge of the investor’s sentiment. In addition, we find that the underlying stock market can forecast the future movements of GVIX. However, the reverse relationship does not hold. Finally, a contemporaneous spillover between GVIX and the US volatility indices VXO and VXN is detected. The results have implications for portfolio management.

04-144 Volatility Options: Hedging Effectiveness, Pricing and Model Error Motivated by the growing literature on volatility options and their imminent

introduction in major exchanges, this paper addresses two issues. First, we examine whether volatility options are superior to standard options in terms of hedging volatility risk. Second, we investigate the comparative pricing and hedging performance of various volatility option pricing models in the presence of model error. Monte Carlo simulations within a stochastic volatility setup are employed to address these questions. Alternative dynamic hedging schemes are compared, and various option‐pricing models are considered. The results have important implications for the use of volatility options as hedging instruments, and for the robustness of the volatility option pricing models.

04-143 A Parsimonious Continuous Time Model of Equity Index Returns (Inferred from High Frequency Data)

In this paper we propose a continuous time model capable of describing the dynamics of futures equity index returns at different time frequencies. Unlike several related works in the literature, we avoid specifying a model a priori and we attempt, instead, to infer it from the analysis of a data set of 5‐minute returns on the S&P500 futures contract. We start with a very general specification. First we model the seasonal pattern in intraday volatility. Once we correct for this component, we aggregate intraday data into a daily volatility measure to reduce the amount of noise and its distorting impact on the results. We then employ this measure to infer the structure of the stochastic volatility model and of the leverage component, as well as to obtain insights on the shape of the distribution of conditional returns. Our model is then refined at a high frequency level by means of a simple non‐linear filtering technique, which provides an intraday update of volatility and return density estimates on the basis of observed 5‐minute returns. The results from a Monte Carlo experiment indicate that a sample of returns simulated according to our model successfully replicates the main features observed in market returns.

04-142 The Value of Storage Facility The paper derives the value of a storage facility that is too small to affect an

exogenously defined price process of a storable good with seasonal and mean‐reverting components. It provides an elegant new continuous‐time model of storage under simple assumptions, which could be applied, for example, to natural gas. In the case without a seasonal component, closed form solutions are obtained as functions of the underlying price. A local time analysis provides an even simpler unconditional formula, which generalizes to the full model. The value of a storage facility under a model with a seasonal and a stochastic component is represented as a time integral which is easily evaluated numerically. The analysis provides a proper treatment of


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the true nature of the “option store” and insights into the value of each component. An interesting feature of the model is that all transactions (whether to buy or to sell) are triggered by a single critical price. The analysis enables us to compare the profitability of storing under alternative price process assumptions.

04-141 The Logistic Function and Implied Volatility: Quadratic Approximation and Beyond

We introduce a new methodology for estimating implied volatilities, and other option pricing parameters. Almost all valuation formulae are linear combinations of the special functions, whose arguments contain some or all of the parameters. We obtain our estimates by replacing these functions with a surrogate. Consequently, we obtain simple formulae, when options are not necessarily at‐the‐money. For the extended Black‐Scholes‐Merton formula, the logistic distribution replaces the cumulative normal distribution. These formulae, which are identical for both European pouts and calls, are at least quadratic approximations, and substantially extend and improve previous approximation validity ranges.

04-140 Valuing Discrete Barrier Options on a Dirichlet Lattice

Discrete barrier options can be valued by quadrature, on a lattice or by Monte Carlo integration. Prices found by an ordinary lattice method will have a large discretisation bias. A good Monte Carlo method will have less bias, but will face difficulties in pricing American style discrete barrier options. Quadrature methods are relatively slow for American barrier options. We provide a rigorous mathematical framework for valuing discrete barrier options. We show how the Dirichlet lattice of Kuan and Webber can be extended to remove discretisation bias in the lattice valuation of discrete barrier options. Unlike a plain lattice method, the lattice can value American barrier options by backwards or forwards induction and can price a wide range of complex barrier options, including those with multiple and non‐constant barrier levels. Numerical results are given. We conclude the lattice is a relatively simple method of obtaining accurate option values for a wide range of complex European and American discrete barrier options.

04-139 Valuing Bermudan Options When Asset Returns are Lévy Processes Evidence from the financial markets suggests that empirical returns distributions, both

historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Lévy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods that can be used when explicit solutions are not present. In particular, the valuation of the Bermudan options in problematical. This paper investigates a lattice method that can be used when the returns process is Lévy, based upon an approximation to the transition density function of the Lévy process. We find alternative derivations of the lattice, stemming from alternative representations of the Lévy process, that may be useful if the transition density function is unknown or intractable. We apply the lattice to models based on the variance‐gamma and NIG processes. We


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find that the lattice is able to price Bermudan style options to acceptable accuracy.

04-138 Correcting for Simulation Bias in Monte Carlo methods to Value Exotic Options in Models Driven by Lévy Processes

Lévy processes can be used to model asset return’s distributions. Monte Carlo methods must frequently be used to value path dependent options in these models, but Monte Carlo methods can be prone to considerable simulation bias when valuing options with continuous reset conditions. In this paper we show how to correct for this bias for a range of options by generating a sample from the extremes distribution of the Lévy process on subintervals. We work with the variance‐gamma and normal inverse Gaussian processes. We find the method gives considerable reductions in bias, so that it becomes feasible to apply variance reduction methods. The method seems to be a very fruitful approach in a framework in which many options do not have analytical solutions.

04-137 An EZI method to reduce the rank of a correlation matrix Reducing the number of factors in a model by reducing the rank of a correlation

matrix is a problem that can arise in many areas of finance, for instance pricing interest rate derivatives with libor market models. We introduce and describe a simple iterative algorithm for correlation rank reduction, the eigenvalue zeroing by iteration, EZI, algorithm. We investigate its convergence to an optimal solution and compare its performance with those of other methods. Several test data sets are used including an empirical forward Libor correlation matrix. The EZI algorithm is very fast even in computationally complex situations, and achieves a level of precision comparable to that of a computationally intensive optimization methods. From our results, the EZI algorithm has superior performance in practice to each of the two main methods in current use.

04-136 Valuing Path Dependent Options in the Variance‐Gamma Model by Monte Carlo with a Gamma Bridge

The Variance‐Gamma model has analytical formulae for the values of European calls and puts. These formulae have to be computed using numerical methods. In genera, option valuation may require the use of numerical methods including PDE methods, lattice methods, and Monte Carlo methods. We investigate the use of Monte Carlo methods, in the Variance‐Gamma model. We demonstrate how a gamma bridge process can be constructed. Using the bridge together with stratified sampling we obtain considerable speed improvements over a plain Monte Carlo method when pricing path‐dependent options. The method is illustrated by pricing lookback, coverage rate and barrier options in the Variance‐Gamma model. We find the method is up to around 400 times faster than plain Monte Carlo.


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04-135 Valuing Continuous Barrier Options on a Lattice solution for a Stochastic Dirichlet Problem

The stochastic Dirichlet problem computes values within a domain of certain functions with known values at the boundary of the domain. When applied to valuing barrier options, solutions are expressed as expected discounted payoffs achieved at hitting times to the boundary of the domain. We construct a lattice solution to the stochastic Dirichlet problem. In between time steps on the lattice, the lattice process is assumed to have the bridge distribution of the underlying stochastic process. We apply the Dirichlet lattice to valuing barrier options. A plain simple scheme converges very slowly. We find that the Dirichlet lattice is considerably faster than a plain lattice scheme, converging to 2 decimal places in only several hundred time steps. The Dirichlet lattice can directly value knock‐in barrier options, including knock‐in Bermudan barrier options which cannot normally be priced by a plain lattice method. It values Bermudan barrier options and barrier options with non‐linear barriers equally quickly. We present results demonstrating the superiority of the Dirichlet lattice over both a plain lattice method and a conditional Monte Carlo method.

04-134 An Asset Based Model of Defaultable Convertible Bonds with Endogenised Recovery

We describe a two factor valuation model for convertible bonds when the firm may default. We endogenize both default and the recovery value of a defaulted bond. A sophisticated numerical framework enables us to specify numerically and financially consistent boundary conditions and inequality constraints. We investigate the affect of changing the default, recovery and loss specification. The affect of introducing a stochastic interest rate is quantified, and asset and interest rate delta and gammas are found. The bond’s sensitivity to interest rate changes is about one tenth that of a corresponding defaultable straight bond, chiefly due to the presence of the conversion feature.

04-133 A Monte Carlo Method for the Normal Inverse Gaussian Option Valuation Model using an Inverse Gaussian Bridge

The normal inverse Gaussian process has been used to model both stock returns and interest rate processes. Although several numerical methods are available to compute, for instance, VaR and derivatives values, these are in a relatively undeveloped state compared to the techniques available in the Gaussian case. This paper shows how a Monte Carlo valuation method may be used with the NIG process, incorporating stratified sampling together with an inverse Gaussian bridge. The method is illustrated by pricing average rate options. We find the method is up to around 200 times faster than plain Monte Carlo. These efficiency gains are similar to those found in a relation paper, Riberio and Webber (02) [20], which develops an analogous method for the variance‐gamma process.

04-132 A Two‐Factor Model for Commodity Prices and Futures Valuation


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This paper develops a reduced form two‐factor model for commodity spot prices and futures valuation. This model extends the Gibson and Schwartz (1990)‐Schwartz (1997) two‐factor model by adding two new features. First the Ornstein‐Uhlenbeck process for the convenience yield is replaced by a Cox‐Ingersoll‐Ross (CIR) process. This ensures that our model is arbitrage‐free. Second, spot price volatility is proportional to the square root of the convenience yield level. We empirically test both models using weekly crude oil futures data from 17th of March 1999 to the 24th of December 2003. In both cases, we estimate the models parameters using the Kalman filter.

04-131 A Contango‐Constrained Model for Storable Commodities Diana R. Ribeiro and Stewart D. Hodges

In this article we develop a model for the commodity price dynamics under the risk‐neutral measure where the spot price switches between two distinct stochastic processes depending on whether or not inventory is being held. Specifically, when‐ever the drift of the spot price exceeds the cost of carrying inventory (interest rate plus storage costs) the inventory is being held. Conversely, whenever the drift of the spot price is less than the cost of carry, all the inventory is sold and the storage facility becomes empty. If inventory is being held, we assume that the spot price follows a geometric Brownian motion with drift equal to the cost of carrying inventory. Otherwise, the price follows a Ornstein‐Uhlenbeck stochastic process. This model verifies arbitrage‐free arguments since the commodity price process has a drift lees or equal to the cost of carry under the risk neutral measure. We illustrate and analyze the properties of the spot price and the forward curves implied by this model using numerical examples. The spot price sample paths and the corresponding forward curves are constructed by applying trinomial tree techniques. For comparison, we also provide the equivalent numerical examples for the single‐factor model provided by Schwartz (1997), which correspond to the unconstrained version of the spot price process in this model.

04-130 Equilibrium Model for Commodity Prices: Competitive and Monopolistic Markets Diana R. Ribeiro and Stewart D. Hodges

In this article, we develop an equilibrium model for storable commodity prices. The model is formulated as a stochastic dynamic control problem and considers two state variables – the exogenous supply and the inventory. The inventory is a fully controllable endogenous variable. We assume that the uncertainty arises from the supply, which evolves as a Ornstein‐Uhlenbeck stochastic process. This model is developed under a general framework which provides two distinct forms for the alternative economic scenarios of perfect competition and monopolistic storage. Since an analytical solution to the problem is not possible, we obtain a numerical solution that provides an optimal storage policy and generates the price dynamics. We also compute and analyse the equilibrium forward curves that result from the steady state optimal storage policy. The results are consistent with the theory of storages: the presence of storage in both economies stabilizes the natural prices. We also show that this effect is greater when the storage is competitive. The resulting forward curves take two fundamental shapes. If the initial spot price is greater than the long‐run natural price we observe backwardation; otherwise the market is in contango. Furthermore, the degree of contango is greater in the competitive market.

04-129 Term Structure Models via Matrix Completion Charles R. Johnson and Peter Weigel

For any type of term structure field model the essential input is the instantaneous


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correlation function. We suggest a construction of a strictly positive definite correlation function which is consistent with the observed sample correlation matrix, infinite-dimensional structure of the field, and monotonicity requirements. The construction is based on a solution of a constrained matrix completion problem.

03-128 Implied Kernel Models Peter Weigel

We develop a class of models within the pricing kernel framework. I.e., we model the pricing kernel directly, and not a particular interest rate or a set of rates. The construction of the kernel is explicitly linked to the calibrating set of instruments. Thus, once the kernel is constructed it will price correctly the chosen set instruments, and have a low‐dimensional Markow structure. We test our model on yield, at‐the‐money cap, caplet implied volatility surface, and swaption data. The quality of fit is very good.

03-127 A Note on Optimal Calibration of the Libor Market Model to the Correlations Peter Weigel

We develop a simple, fast, non‐parametric method for calibrating Libor market models to historical or implied correlation matrices. For a given symmetric matrix, the method utilises alternating projections to find the nearest correlation matrix of a lower rank.

03-126 Pricing of Implied Volatility Derivatives Emanuele Amerio, Gianluca Fusai and Antonio Vulcano

Starting from the description of the real mechanism on the basis of which implied volatilities are actually quoted by traders in option markets, we construct the risk neutral dynamics of the implied volatility to price implied volatility futures and forward starting compound options. These are exotic derivative contracts whose payoff depends on the future implied volatility. In particular, we obtain the risk neutral drift restriction that must be satisfied by each single stochastic implied volatility on the volatility surface invariant both to time to maturity and to (forward) moneyness. Also, we show that the instantaneous spot volatility is a point of this volatility surface, so we can apply our technique to determine its risk neutral dynamics.

03-125 The Favorite/Long‐shot Bias in S&P 500 and FTSE 100 Index Futures Options: The Return to Bets and the Cost of Insurance

Stewart D Hodges, Robert G Tompkins and William T Ziemba This paper examines whether the favourite/long‐shot bias that has been found in gambling markets (particularly horse racing) applies to options markets. We investigate this for the S&P 500 futures, the FTSE 100 futures and the British Pound/US Dollar futures for the seventeen plus years from March 1985 to September 2002. Calls on the FTSE 100 with three months to expiration display a relationship between probabilities and average returns that are very similar to the favourite/long‐shot bias in horse racing markets pointed out by Ali (1979), Snyder (1978) and Ziemba & Hausch (1986). There are slight profits from deep in‐the‐money calls on the S&P 500 futures and increasingly greater losses as the call options are out‐of‐the‐money. For 3 month calls on the FTSE 100 futures, the favourite bias is not found, but a significant long‐shot bias has existed for the deepest out of the money options. For call options in both markets, for the one month horizon, only a long‐shot bias is found. For the put options on both markets, and for both 3 month and 1 month horizons, we find evidence consistent with the hypothesis that investors tend to overpay for all put


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options as an expected cost of insurance. The patterns of average returns is analogous to the favourite/long‐shot bias in racing markets. For options on the British Pound/US Dollar, there does not appear to be any systematic favourite/long‐shot bias for either calls or puts.

03-124 The Relation Between Implied and Realised Probability Density Functions In this paper, we test the ability of risk‐neutral densities (RNDs) extracted from option

data to produce correct forecasts of the true densities of the underlying assets at expiry. Implied RNDs are estimated with both parametric and non‐parametric methods. A number of new testing procedures to assess the efficiency and unbiasedness of these density forecasts are presented. The forecasting performance of RNDs is also compared to that of return distributions simulated from GARCH‐type models. Our findings suggest that implied RNDs represent poor forecasts of the actual densities. However, their forecasting performance improves substantially after adjusting for the risk premium.

03-123 The Effect of Mis‐Estimating Correlation on Value‐At‐Risk Vasiliki D. Skintzi, George Skiadopoulos and Apostolos‐Paul N. Refenes

This paper examines the systematic relationship between correlation mis‐estimation and the corresponding Value‐at‐Risk (VaR) mis‐calculation. To this end, first a semi‐parametric approach, and then a parametric approach is developed. Various linear and non‐linear portfolios are considered, as well as variance‐covariance and Monte‐Carlo simulation methods are employed. We find that the VaR error increases significantly as the correlation error increases, particularly in the case of well‐diversified linear portfolios. In the case of option portfolios, this effect is more pronounced for short‐maturity, in‐the‐money options. The use of MC simulation to calculate VaR magnifies the correlation bias effect. Our results have important implications for measuring market risk accurately.

03-122 Derivatives Hedging and Volatility Errors Iliana Anagnou‐Basioudis and Stewart Hodges

This paper provides a general representation for the errors of delta‐hedging derivatives contracts under misspecified asset price processes. A new option Greek η, ‘eta’, non‐linear but easily computable for portfolios, is developed, which quantifies the dependence between the prospective hedging errors and the volatility forecast errors. The hedging errors are studied in more detail for a standard vanilla option, a geometric average rate option, and an up‐and‐out call option with a continuously monitored barrier. Two alternative approaches are provided for deriving the conditional and unconditional distribution of hedging errors: binominal tree and kernel estimation. The techniques developed enable us to quantify the absolute and relative difficulties of hedging different instruments or portfolio of instruments.

02-121 The Valuation of Convertible Bonds: A Study of Alternative Pricing Models Russell Grimwood and Stewart Hodges

Convertible debt represents 10% of all USA debt yet despite its ubiquity it still passes difficult modelling challenges. This paper investigates alternative convertible bond model specifications. The work reviews the literature on convertible debt valuation especially the methodologies adopted by practitioners. Inadequacies in the historical and current valuation methods are highlighted. The different features used in convertible bond contracts found on the International Security Markets Association database are catalogued for both the Japanese and USA markets. Fashions in the contracts that have changed through time are noted. Modal, average, maximum and


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minimum USA contract parameters for various features are used to establish realistic and representative convertible bond contracts. The motivation for analyzing the ISMA data is determine which contracts features are important before investigating model errors. The model errors themselves are a function of the contract in question and cannot therefore, be examined in abstract. The sensitivity of the modal convertible bond contract price to the method of modelling the spot interest rate and the intensity process is examined. The convertible bond price sensitivity to the input parameters reveals that accurately modelling the equity process and capturing the contract clauses in the numerical approximation appear crucial whereas the intensity rate and spot interest rate processes are of second order importance.

02-120 A New Approach to Modelling the Dynamics of Implied Distributions: Theory and Evidence from the S&P 500

Nikolaos Panigirtzoglou and George Skiadopoulos This paper presents and new approach to modelling the dynamics of implied distributions. First, we obtain a parsimonious description of the dynamics of the S&P 500 implied cumulative distribution functions (CDFs) by applying Principal Components Analysis. Subsequently, we develop new arbitrage-free Monte-Carlo simulation methods that model the evolution of the whole distribution through time as a diffusion process. Our approach generalizes the conventional approaches of modelling only the first two moments as diffusion processes, and it has important implications for “smile-consistent” option pricing and for risk management. The out-of-sample performance within a Value-at-Risk framework is examined.

JEL Classification: G11, G12, G13 Keywords: Implied cumulative distribution function, Monte Carlo Simulation, Option Pricing, Principal Components Analysis and Value-at-Risk

01-119 Numerical Valuation of Discrete Barrier Options Gianluca Fusai and Christina Recchioni

We propose a numerical method for barrier options valuation assuming that the trigger is checked at fixed times. Our method can deal with the interesting cases of time-varying barriers and non equally spaced monitoring dates. The convergence of the method is proved and its computational cost is found to be linear in the number of monitoring dates and quadratic in the spatial discretization. We also discuss extensively the empirical performance of the method, including the calculation of the delta and gamma coefficients, under the GBM and CEV stochastic specifications.

01-118 A Reduced-form Model incorporating Fundametal Variables Mark Wong and Stewart D Hodges

This paper proposes a reduced-form model of corporate debt, by taking into account stochastic interest rates, a firm’s equity value, and hazard rates of default. We innovatively introduce structural.

01-117 Equilibrium Price Processes, Mean Reversion and Consumption Smoothing Stewart D Hodges and Chien-Hui Liao Motivated by the empirical observation that there exists some degree of mean

reversion in asset prices, this paper investigates the time-varying behaviour of the price of risk in a partial equilibrium framework. The paper characterizes the equilibrium conditions under which the asset price processes must satisfy and obtain closed-form solutions in a time homogeneous economy. We construct a model where the consumption is relatively smooth and risk premium falls as wealth rises. The representative agent also becomes less risk averse as wealth rises.

01-116 Valuation of Claims on Non-Traded assets using Utility Maximisation


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Vicky Henderson A recent topical problem is how to deal with claims on ‘non-traded’ assets. A natural

approach is to choose another similar asset or index which is traded to use for hedging purposes. To model this situation, we introduce a second non-traded log Brownian asset into the well known Merton investment model with power-law utility. The investor has an option on units of the non-traded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete marke5s. Employing utility maximisation and duality methods we obtain a series approximation to the optimal hedge and reservation price. These are computed for some example options and the results compared to those using exponential utility.

01-115 Static versus Dynamic Hedging of Exotic Options: An Evaluation of Hedge Performance via Simulation

Robert Tompkins A number of pricing models have been derived for exotic options that rely upon many

of the perfect market assumptions made in the original derivation of European call options by Black and Scholes. A keystone of the Black and Scholes derivation is that a hedge can be constructed in continuous time that will allow risk neutral evaluation techniques to be employed. Due to product specific characteristics that some of these exotic options have, these assumptions may be inappropriate when dynamically hedging these products. This paper relaxes three of the perfect market assumptions including continuous rebalancing of the hedged portfolio, no transaction costs and constant (and known) volatility. Hedging simulations were run for a wide variety of exotic options with these assumptions relaxed and comparisons were made between the theoretical value of the option from the ‘perfect markets’ pricing model and the estimated cost of hedging the option. In almost all cases, it is shown hat the inclusion of transaction costs significantly increases the cost of hedging the option above the theoretical value and the introduction of stochastic volatility increases the variability in the hedge performance. Given that a number of static hedging approaches have been proposed in the literature for a number of these exotic options, we also examine the relative performance of these strategies for the hedging. Our findings suggest that neither dynamic nor static hedging strategies are superior for all exotic options. However, for the majority of the exotic options tested, the static hedging strategies seem to have less impacts from transaction costs and stochastic volatility.

00-114 Pricing of Defaultable Coupon Bonds Under a Jump-Diffusion Process Mark C Wong and Stewart D Hodges This paper employs a structural approach to analyse term structure of credit risk and

yield spreads for the corporate debt of firms when the value of underlying assets follows a jump diffusion process. We present a tractable, discrete time model for valuing general coupon bonds. We show several significant implications of the jump process for the term structure of credit spreads, when systematic jumps are present in the firm’s asset value. We also discuss the effects of diversifiability of jumps on corporate debt pricing. Other important factors include taxes and dividends. The main results are as follows. Firstly, the presence of jumps in asset values eliminates the undesirable qualitative feature of credit spreads decreasing to zero at the short end. The effects on credit spreads become more persistent when downward jumps are of higher volatility while the total variance of the firm’s asset value remains the same. Secondly, taxes do have significant effects on levels of credit spread. Interestingly, the model implies that a decrease in the federal tax rate may precipitate earlier default of low-grade bonds.

00-113 A New Class of Commodity Hedging Strategies: A Passport Options Approach Vicky Henderson, David Hobson and Glenn Kentwell We provide a new way of hedging a commodity exposure which eliminates downside


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risk without sacrificing upside potential. The tool used is a variant on the equity passport option and can be used with both futures and forwards contracts as the underlying hedge instrument. Results are given for popular commodity price models such as Gibson-Schwartze and Black with convenience yield. Two different scenarios are considered, one where the producer places his usual hedge and undertakes additional trading, and the other where the usual hedge is not held. In addition, a comparison result is derived, showing that one scenario is always more expensive than the other. The cost of these methods are compared to buying a put option on the commodity.

00-112 Multvariate Distributional Tests based on the Empirical Characteristic Function Approach: A Comparison

Mascia Bedendo and Stewart Hodges With the purpose of identifying appropriate testing procedures for multivariate

distributional forecasts, in this paper we compare the power of two versions of multivariate goodness-of-fit tests based on the Empirical Characteristic Function (ECF) in detecting deviations of the true distribution of the data from the forecast. Various Monte Carlo experiments carried out for dimensions up to 16 suggest the superiority of the continuous version of the test over the discrete one, in terms of both computational feasibility and statistical properties. The applicability of this testing procedure to the evaluation of density forecasts of financial asset returns generated in the context of risk management and Value at Risk models is carefully investigated.

00-111 Simulating the Evolution of the Implied Distribution Goerge Skiadopoulos and Stewart Hodges Motivated by the implied stochastic volatility literature (Bretten-Jones) and Neuberger

(1998), Derman and Kani (1997), Ledoit and Santa-Clara (1998), this paper proposes a new and general method for constructing smile-consistent stochastic volatility modes. The method is developed by recognising that option pricing and hedging can be accomplished via the simulation of the implied distribution, when the first two moments change over time. The algorithm can be implemented easily, and it is based on an economic interpretation of the concept of mixture of distributions. It can also be generalised to cases where more complicated forms for the mixture are assumed.

00-110 Real Options with Constant Relative Risk Aversion Vicky Henderson and David Hobson Real options problems have recently attracted much attention worldwide. One such

problem is how to deal with claims on ‘untraded’ assets. Often there is another traded asset which is correlated to the untraded asset, and this traded asset is used as a proxy for hedging purposes. We introduce a second (untraded) log Brownian asset into the well known Merton investment model with power-law utility. The investor has a claim on units of the untraded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete markets. We propose an approximation to the solution for the ‘optimal’ reservation price and hedge which is accurate when the position is small in comparison to wealth. The resulting loss when a suboptimal proxy strategy is followed is shown to be approximately quadratic in 1 – p.

00-109 A Computational Framework for Contingent Claim Pricing and Hedging under Time Dependent Asset Processes

Russell Grimwood and Les Clewlow This paper proposes a general computational framework for pricing exotic options in

the presence of volatility smiles. The paper extends and unifies Dupire’s (1994) work


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on implied trinomial trees with fully explicit, fully implicit and Crank Nicolson finite difference methods. We investigate the computational efficiency of this new framework for pricing exotic options. As American barrier option is priced as an example of applying the framework to complex exotic options.

00-108 The Languages of Contigent Claim Contracts Russell Grimwood and Stewart Hodges The work investigates the different languages used to characterise contingent claim

payoffs and cash flows. The mathematical operators used to mark up these payoffs and cash flows are formalised. A translation is provided between equivalent colloquial (jargon), legal and computational terms used. This part of the work should therefore be useful for those learning about derivatives who are unclear on the legal and mathematical terminology. The tensions and inadequacies inherent in using different languages to describe contingent claims are illustratd and thus the desirability of a legally binding machine readable language becomes clear. The progress towards a coherent computer language for characterisation of contracts (and their improved classification in terms of information monitoring) is reviewed.

00-107 Taxonomy of Algorithms Russell Grimwood and Stewart Hodges Valuation and risk management systems use a variety of algorithms to determine the

price and the hedge parameters (Greeks) for different types of contingent claims. The motivation for this study is to investigate the “optimal” (in some sense) algorithm for pricing a given contingent claim with given modeling assumptions. This work catalogues the properties of various contingent claim pricing and hedging algorithms. The study includes the nature of the required input parameters (for the instrument, the valuation mode, the numerical method and the parameters implicit in model calibration) and the number of stochastic processes each algorithm accommodates. Issues of calibration, model complexity and isomorphism are also discussed.

00-106 Stochastic Volatility Models with Jumps: Implications for Smiles in Foreign Exchange Markets

Robert Tompkins A convenient starting point for option pricing has been to assume that markets

conform to Geometric Brownian Motion (GBM). Extensive empirical evidence has shown that Foreign Exchange markets reject this assumption. In addition, the prices of options on these markets systematically diverge from prices assumed by the Black-Scholes (1973), Garman-Kohlhagen (1983) and Black (1976) models. Patterns of implied volatilities for these markets display the now familiar convex shape as a function of striking prices, which is commonly known as the smile. This paper examines whether alternative models that include jumps and stochastic volatility can account for the empirical dynamics of Foreign and Exchange Futures and explain the implied volatility smiles for options on Foreign Exchange markets. In this paper, we consider a rich class of stochastic volatility models that include jumps and correlated processes. This work extends Bates (1996b) by considering more markets over a longer period of analysis and alternative jump process models. Two alternative Jump models are considered which include jumps either in the underlying asset price or the stochastic volatility processes. For the four Foreign Exchange futures markets examined, either class of jump models


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can explain the empirical features of these markets. For both types of jump models, prices of European options are determined. Comparisons of simulated and actual options prices for these markets find substantial differences. These results concur with previous studies that the inclusion of a model consistent with the objective process alone is insufficient to explain the existence of smiles. This points to the existence of a risk premium when pricing options on Foreign Exchange. This research identifies the dynamics of this risk premium across strike price and time and finds consistencies of this risk premium for all four Foreign Exchange markets. Such consistency suggests market agents are assuming a similar functional form fore the risk premium.

00-105 Stock Index Futures Markets: Stochastic Volatility Models and Smiles Robert Tompkins This paper asks why implied volatility smiles exist for options on Stock Index Futures.

In the literature, two possible reasons have been proposed to explain this phenomenon: the existence of market imperfections and that the underlying price process differs from the lognormal diffusion process assumed by the Black Scholes (1973) model. This paper examines the latter hypothesis. Four stock index futures markets were considered. From the return and unconditional volatility series for these markets, seven key attributes were chosen capturing the empirical non-normal and non-IID characteristics of these markets. Using these attributes, comparisons were made with attributes consistent with the Heston (1993) stochastic volatility mode. Four variants of the stochastic volatility model were examined: two models assumed the underlying price series follows GBM and two models assumed a Normal Inverse Gaussian (NIG) distribution (as a surrogate for a skewed jump process). For both alternative price processes, uncorrelated or negatively correlated subordinated stochastic volatility processes were considered. The alternative models were evaluated using a simulated method of moments approach. Model comparisons were based upon minimisation of the sum of squared errors for the seven attributes (consistent with the model and observed empirically). For all four stock index futures markets, models including a negatively correlated stochastic volatility process with non-normal price innovations performed best within the total sample period. This result is time invariant, consistently observed when the data set is split into sub-periods. Price series were simulated from these optimal stochastic volatility models and Monte Carlo values of European options determined. These option prices were then expressed as implied volatility surfaces and compared to the actual implied volatility surfaces for options on these same four Stock Index Futures. Comparisons between simulated and actual implied volatility surfaces for the four established Stock Index options markets suggest that the alternative price process hypotheses is insufficient to explain the existence of smiles. However, consistent divergences between simulated and actual implied volatility surfaces were found. This may provide insights into the nature of market imperfections and risk premia.

00-104 Fixed Income Futures Markets: Stochastic Volatility Models and Smiles Robert Tompkins This paper examines whether the inclusion of an appropriae stochastic volatility

model that captures kay distrinbutional and volatility facets of Fixed Income Futures is sufficient to explain implied volatility smiles for options on these markets. Two variants of stochastic volatility models related ot the Heston (1993) care considered. Either a Gauss-Wiener or Jump process is assumed to drive the


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underlying price process with a subordinated (and correltated) stochastic volatility process. For three bond futures markets examined, models including a negatively correlated stochastic volatility process with jump processes perform best within the total sample period and for sub-periods. Using these optimal stochastic volatility models, prices of European options are determined numerically. Comparisons of simulated and actual options prices for these markets find substantial differences. This suggests that the inclusion of a stochastic volatility process consistent with the objective process alone is insufficient to explain the existence of smiles.

00-103 The Sampling Properties of Volatility Cones Stewart Hodges and Robert Tompkins In this research we extend the original work on volatility cones by Burghardt and

Lane (1990) to consider of the sampling properties of the variance of variance (and the standard deviation of volatility) under a rich class of models that includes stochastic volatility and conditionally fat-tailed distributions. Because the volatility cone examines volatility at quite long horizons, the estimation requires the use of overlapping data. This theory confirms the casual observation that the estimation of the variance of variance is downward biased when estimation is done on an overlapping basis. Our principal contribution is to identify what this bias is and derive an adjustment factor that approximates an unbiased estimate of the true variance of variance when overlapping data is used. Another contribution is the derivation of a formula that describes the variance of the quadratic variation over different time horizons. Using the theory presented, we tested the bias adjustments to the standard deviation of volatility using simulations. Two cases were examined: a GBM i.i.d. process and a non-i.i.d. process associated with the stochastic volatility model suggested by Heston (1993). In both cases, the bias introduced by estimation of volatility with overlapping data becomes insignificant after making the theoretical adjustments. These results are relevant to those who must sell options and must understand the nature of quadratic variation in asset prices. This will provide clearer insights into the nature of hedging errors when dynamically hedging options. This research also suggests a new method for the estimation of stochastic volatility models, where estimation over a long horizon is likely to provide robustness not associated with current methods.

00-102 An Evaluation of Testrs of Distributional Forecasts Pablo Noceti, Jeremy Smith and Stewart Hodges Traditionally, forecasters have concentrated on the point forecasts from their models.

This has been increasingly seen as deficient, as individuals are not indifferent to the uncertainty associated with these forecasts. Consequently, more recently attention has been focused on the distribution associated with forecasts. This paper investigates the size and power of a number of (distribution free) tests for distributional forecasts.

00-101 Pricing of Occupation Time Derivatives: Continuous and Discrete Monitoring Gianluca Fusai and A Tagliani The developing literature on “smile consistent” no-arbitrage models has emerged from

the need to price and hedge exotic options consistently with the prices of standard European options. This survey paper describes the steps through which this literature has evolved by providing a taxonomy of the various models. It highlights the main ideas


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behind the different models, and it outlines their advantages and limitations. Practical issues in implementing the models are also discussed.

99-100 Pricing of Occupation Time Derivatives: Continuous and Discrete Monitoring Gianluca Fusai and A Taglianiqqqq In the present work we use different numerical methods (multidimensional inverse

Laplace transform, numerical solution of a PDE by finite difference scheme, Montecarlo simulation) for pricing occupation time derivatives in order to examine the effect of continuous and discrete time monitoring of the underlying asset. In particular we treat the problem of the numerical inversion of a multidimensional Laplace transform and we show that it can be performed very fast and with great accuracy. We conduct also an analysis of the numerical method for the solution of the PDE with discrete monitoring and we show that the proposed method avoids unwanted oscillations in the solution arising near the monitoring date due to the updating of the occupation time.

99-99 Corridor Options and Arc-Sine Law Gianluca Fusai We study a generalization of the Arc-Sine Law. In particular we provide new results

about the distribution of the time spent by a BM with drift inside a band, giving the Laplace transform of the characteristic function. If one of the extremes of the band goes to infinity, our formula agrees with the results given in Akahori (1995) and Takàcs (1996). We apply these results to the pricing of exotic option contracts known as corridor derivatives. We then discuss the inversion problem comparing different numerical methods.

99-98 Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: an

Arrow-Debreu Pricing Approach Joao Pedro Vidal Nunes

Simple analytical pricing formulae have been derived, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas for derivatives under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions are derived only involving a single integral. Such approximations are shown to be both fast and accurate.

99-97 Conditional Gaussian Models of the Term Structure of Interest Rates Simon H Babbs We present a new family of yield curve models, termed “Conditional Gaussian”. It

provides both simplicity and extreme flexibility in constructing “market models”. Any conditional co-variance structure - including features designed to capture volatility “skews” and/or GARCH effects - can be used, and the model can be embedded into a continuos-time whole yield curve model consistent with general equilibrium. Conditionally Gaussian increments in log one-plus-interest-rates enable “vanilla” and path-dependent derivatives to be valued easily by Monte Carlo without discretization error, whether or not their payoffs depend solely on the particular market rates being modelled directly.

98-96 The Relationship between Stock Returns and Tobin’s q: Tobin’s q Effect. Ian Davidson and George Lededakis This paper examines the relationship between expected returns and Tobin’s q, B and

market value. Our basic idea is to use Tobin’s q ratio, the ratio of the market value of a firm to the replacement cost of its assets., as an additional variable to explain the expected common stock returns. The results support the single risk CAPM specifications in an unconditional form for stocks listed on the London Stock Exchange from July 1975 to June


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1996. The unbalanced sample consists of 1420 UK quoted non-financial firms. In contrast to Fama and French’s (1992) US study, for the sample of UK non-financial firms, B is able to explain cross-sectional differences of expected returns at the individual stock level. We find the relationship between B and the average monthly return is significant, and that the Toxin’s q ratio and market value are strongly significant. These results confirm the existence of a Tobin’s q effect,; stocks with a smaller Tobin’s q yield a higher average return.

98-95 Implied Risk-Neutral Distribution: A Comparison of Estimation Methods Silio David Aparicio and Stewart Hodges This paper examines two alternative approaches to recovering the risk-neutral density

function from contemporaneous option prices. First, we propose to recover the risk-neutral probabilities through a paramenterization of the equivalent martingale measure using the Generalised Beta distribution. Then, we use a non-parametric method to approximate the volatility smile using B-splines approximating functions and use the chain rule of differentiation to recover the implied distribution. We end the paper with a comparison of the two estimation methods using a sample of daily closing prices of the Chicago Mercantile Exchange options on the S&P 500 index future to assess the quality and stability of the implied distributions through statistical analysis.

98-94 Rational Bounds on the Prices of Exotic Options Anthony Neuberger and Stewart Hodges The paper provides a methodology for setting no arbitrage bounds on the price of exotic

options. This also provides a method for obtaining robust hedges. The question posed is given the prices of a set of reference assets (say, a stock and a set of European options on that stock) together with the possibility of entering into further forward positions at future dates, what restrictions can we place on the price of some exotic options, such as a barrier option or a bookback. This work can be viewed as a generalisation and extension of Merton’s seminal (1973) work for conventional options.

The general framework has a natural formulation as a linear program. We look for the

cheapest strategies which super-replicate the exotic under weak assumptions. The dual LP involves looking for extreme valuations over all (risk neutral) processes which satisfy the usual martingale restrictions. For some common exotics such as a digital barrier option and a lookback we are able to provide simple characterisations for the super-replicating hedge portfolio and the optimization reduces to a one-dimensional search. In principle we can obtain slightly tighter bounds if we entertain restrictions on the future prices of options, or in the dual if we restrict the dispersion of the process at future dates. Future numerical work will explore this issue, and also the problems of obtaining bounds numerically for slightly more complex securities.

98-93 “Implied Volatility Surfaces: Uncovering Regularities for Options On Financial

Futures” Robert G Tompkins While it is now genWerally accepted that implied volatilities of European options differ

across strike prices and time, what has not been examined in the literature is the characteristics of the strike price biases between different assets and asset classes and the variability of these surfaces over time. This research examines twelve options markets on financial futures (comprising three asset classes) and compares the strike price biases both for the same markets and across all markets.

When implied volatility surfaces are standardised, their patterns are smooth, well behaved

and display the asymmetrical and convex patterns previously described in the literature. Furthermore, there appears to be consistency in how the implied volatility surfaces evolve over time for individual markets and we uncover consistencies in the behaviours of


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implied volatility surfaces between markets. To test the significance of these empirical regularities, we develop a model based upon a polynomial expansion across strike price and time. The polynomial expansion approach allows us to examine each effect separately and include time dependent interactions. Coefficients from an OLS regression model allow direct comparisons.

Cross-sectional comparisons suggest that the first order strike price biases (skewness)

differs between asset classes. However, the first order strike price biases display similar time dependencies within the same asset class. Regarding the second order strike price effect (Kurtosis), consistencies exist between the three financial asset classes examined. These models are stable overtime and lose little explanatory power outside of sample. This suggests the paradigm used by option participants to adjust option prices away from Black-Scholes-Merton prices is consistent over time.

98-92 The Theory of No Good Deal Pricing in Financial Markets Ales Cerny and Stewart Hodges

The Term ‘no good deal pricing’ stands for any pricing technique based on the absence of attractive investment opportunities - good deals - in equilibrium. The theory presented here shows that any such technique can be seen, remarkably, as a generalization of no arbitrage pricing and that, with a little bit of care, it contains no arbitrage and risk-neutral equilibrium as the two opposite ends of a spectrum of possible equilibrium restrictions. We derive the Extension and Pricing Theorem in no good deal framework and establish some general results for good deals determined by von Neumann-Morgenstern preferences. We apply our results to the multiperiod model with no intermediate consumption.

98-91 The Extension Theorem and a Unified Approach to No-Arbitrage Pricing Ales Cerny and Stewart Hodges

The paper examines an important result in arbitrage pricing - the extension of the pricing rule from the marketed subspace to the whole market. We give a simple exposition of the Extension Theorem, showing that the absence of portfolios that cost nothing to purchase and converge to a strictly positive claim is a necessary and sufficient condition for the existence of a strictly positive and continuous extension of the pricing rule. As a consequence it is demonstrated that no arbitrage implies the existence of an equivalent martingale measure provided the right class of trading strategies is chosen. Finally, we show that the Extension Theorem unifies pricing between the incomplete and the complete market case.

98-90 Pricing By Arbitrage Under Arbitrary Information Simon Babbs and Michael J P Selby

A substantial applications literature on pricing by arbitrage has effectively restricted information to that arising solely from securities markets; return distributions are then governed solely by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, may influence required returns. We show that contingent claims depending solely on securities’ normalized price histories, are priced by arbitrage if, and only if, all risk-adjustment probabilities agree upon the law of primitive securities’ normalized prices. We show how existing diffusion-based results can be preserved, and resolve an issue relating to Merton’s (1973) stochastic interest rate model.

98-89 Auditor Performance, Implicit Guarantees, and the Valuation of Legal Liability Miles B Gietzmann, Mthuli Ncube and Michael J P Selby

Liability exposure is now such a major concern for auditors that any discussion of equilibrium audit fee structures needs to take account of the expected costs of liability exposure. We develop a model of audit litigation risk and then proceed to apply insights gained from the application of finance theory to show how liability exposure is related to guarantee provision. Given auditors wish to incorporate the expected costs of guarantees when planning and pricing services, we apply contingent claims analysis to derive


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valuation equations for expected (litigation) costs. We then proceed to consider wider regulatory issues. Whereas the above analysis assumes stable legal liability rules, we subsequently consider the auditor incentive effects of parametric variation in the rules. Since the quality of audits is unobservable, we consider how the rules could be set so as to ensure auditors are not tempted to provide a low degree of care and collude with management. To illustrate the applicability of this approach we also consider whether recent proposals to reform auditor liability to a proportional basis will always provide appropriate incentives.

98-88 A Generalization of the Sharp Ratio and its Applications to Valuation Bounds and Risk Measures

Stewart Hodges The purpose of this paper is threefold. First it provides a generalization of the Sharpe Ratio which solves these problems. Second, it shows how this new measure provides an appropriate framework for deriving valuation bounds for derivatives in incomplete markets. Finally, it provides some further characterizations of these bounds and discusses their possible role as measures of risk (i.e. related to Value at Risk concepts).

98-87 Empirical Properties of Asset Price Processes Pablo Noceti and Stewart Hodges

The Motivation for this work was to produce models that would satisfy the empirical most often found in financial data, but starting from the data, rather than from any particular model. Our approach is to find the key empirical regularities present in the data, and then find and test the models that have properties that are compatible with those regularities. Most practitioners still use the Normal distribution to describe financial asset returns, or consider a conditional fat tailed distribution like Student’s -t together with a GARCH-type volatility model enough to capture most of the empirical regularities of these returns. Our study shows that to develop a model that truly captures the empirical characteristics of financial asset returns, a symmetric fat tailed distribution is not enough, and we need jumps and stochastic volatility type-model.

98-86 The Dynamics of Implied Volatility Surfaces George Skiadopoulos, Stewart Hodges and Les Clewlow

Motivated by the papers of Dupire (1992) and Derman and Kani (1967), we want to investigate the number of shocks that move the whole implied volatility surface, their interpretation and their correlation with percentage changes in the underlying asset. This work differs from Skiadopoulos, Hodges and Clewlow (1998) in which they looked at the dynamics of smiles for a given maturity bucket. We look at daily changes in implied volatilities under two different metrics: the strike metric and the moneyness metric. Since we are dealing with a three dimensional problem, we fix ranges of days to maturity, we pool them together and we apply the Principal Components Analysis (PCA) to the changes in implied volatilities over time across both the strike (moneyness) metric and the pooled ranges of days to maturity. We find similar results for both metrics. Two shocks explain the movements of the volatility surface, the first shock being interpreted as a shift, while the second one has a Z-shape. The sign of the correlation of the first shock with percentage changes in the underlying asset depends on the metric that we look at, while the sign is positive under both metrics regarding the second shock. The results suggest that the number of shocks, their interpretation and the sign of their correlation with changes in the underlying asset is the same for the whole implied volatility surface as it is for the smile corresponding to a fixed maturity bucket.


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98-85 The Dynamics of Smiles George Skiadopoulos, Stewart Hodges and Les Clewlow

This empirical study is motivated by the models of Dupire (1992) and Derman and Kani (1997). We investigate the number and shape of shocks that move implied volatility smiles and subsequently we look at the correlation of changes in volatility with changes in the underlying asset. We achieve this by applying Principal Components. Analysis on the changes in implied volatilities over time, for fixed ranges of days to maturity and in two different metrics: the strike and the moneyness metric. In contrast to earlier papers in the interest rate literature, we decide on how many principal components explain the implied volatilities dynamics, not by using rules of thumb, but by using Velicer’s non-parametric criterion. Subsequently we use a “Procrustes” type rotation in order to interpret the retained components. The retained rotated principal components are used for the calculation of the correlation coefficients. We find similar results in both metrics regarding the number and shape of shocks. Two principal components explain the dynamics of smiles. After the rotation the first one is interpreted as shift and the second one has a Z-shape. The correlations for the first principal component depend on the metric, while for the second are positive under both metrics.

97-84 A Comparison of Alternative Methods for Hedging Exotics Options Silio David Aparicio and Les Clewlow

Under perfect market assumptions dynamic rebalanceing is costless and the underlying asset trades continuously. In reality, asset prices have jumps, volatility changes randomly, there are transactions cost and trading can only take place at discrete points in time. For standard options optimal and sub-optimal delta hedging strategies have been suggested in the literature, however for exotics options all these strategies could be very expensive to implement. There path dependent options are characterised by high and rapidly changing gamma and this implies a high frequency and therefore costly rebalancing that makes it unattractive to hedge dynamically even when optimal strategies are used. To overcome this problem we extend Clewlow and Hodges (1994) simulation approach to search for optimal methods of hedging based on heuristics which are consistent with myopic strategies. We use Monte Carlo simulations to study the efficiency of this mode, static replication portfolios and other delta hedging strategies under transaction costs and jumps in the underlying process. We also show that mathematical optimisation is a useful way of constructing replicating portfolios. It allows replication under general asset pricing models involving jumps and stochastic volatility, trading restrictions can be considered and the replicating portfolio can be constructed in terms of the strikes and maturities available in the market.

97-83 Modelling Commodity Futures Spreads: An Empirical Study Leticia Veruete and Stewart Hodges

Understanding the behaviour of commodity futures spreads is important for traders as well as consumers and producers. However, relatively little research has been done in modelling commodity futures spreads. On the one hand, a state dependent variance function (nonlinear) for commodity spread has been theoretically acknowledged to be important. On the other hand, commodity spreads


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are bounded by full carrying charges. However, there is no empirical study which considers simultaneously these two features. In this paper, a new model which incorporates the above features is fitted empirically. It is hypothesised that a model with a quadratic variance function on spread size and with lower and upper bounds in spreads can capture the observed behaviour of commodity futures spreads. It has the notable advantage of being consistent with no-arbitrage under upper and lower bound constraints. Empirical results, using commodity futures prices from different American exchanges for the period 1985 to 1995, suggest that the proposed empirical model fits well for most commodities when using weekly data. However, the results do not always support the model, especially when daily data are used.

97-82 Hedging Option Position Risk: An Empirical Examination Les Clewlow, Stewart Hodges, Rodrigo Martinez, Michael Selby, Chris Strickland and

Xinzhong Xu (Written in May 1993) This paper presents an empirical study of the effectiveness of various methods of hedging the risk of options positions. The data used are eight years of daily data for the CME S&P 500 index options contract. The paper examines the performance of delta-gamma and delta-kappa hedges as well as simple delta hedges. The hedging errors are also decomposed and attributed to the effect of the changes in the underlying, (including components from the portfolio’s delta and gamma), the effect of changes in volatility and the effect of the changes in the interest rate. This decomposition is (loosely) based on the analysis given in Bookstaber (1991). The analysis shows that although the (kappa) risk attributable to changes in volatility is small relative to that (delta) due to changes in the underlying, it becomes very significant after delta hedging has been done. Naive kappa hedging is less effective than gamma hedging because the variation of implied volatilities depends on the option maturity.

97-81

The Potential for Profitable Stock Market Manipulation in the Presence of Positive Feedback Trading

Roger Courtenay The aim of this paper is to investigate the implications of positive feedback trading. In particular, the paper investigates the potential for manipulators to profit from the existence of such behaviour. It is shown that prices in the presence of feedback trading may or may not oscillate, and may be stable or unstable. The potential for manipulation depends on the strength of the feedback trading, the delay with which it responds to price changes, and the strength of value-based investor demand.

97-80 Kalman Filtering of Generalized Vasicek Term Structure Models Simon H Babbs and K Ben Nowman

We present a subclass of Langetieg’s (1980) linear Gaussian models of the term structure. The bond price is derived in terms of a finite set of state variables with correlated innovations. The subclass contains a reformulation of the double decay model of Beaglehole and Tenney (1991), enabling us to clarify interpretation of their parameters. We apply Kalman filtering to a state space formulation of the model, allowing measurement errors in the data. One and two factor models are estimated on US data over 1987 - 1996. Results indicate our subclass of models can explain movements in the US term structure.

97-79 Can a Diffusion Recover a Lognormal Jump Process? Luca Pappalardo

Option prices observed in the market do not agree with the assumptions of the Black and Scholes model. In particular, the volatility, which is assumed constant in the model, takes different values for options with different maturities or strike prices. Dupire showed that, if we relax the hypotheses of the Black and Scholes model and we assume the stock prices are


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driven by a general diffusion, it is always possible to recover such a process that mimics a given price system (provided that the price system is sufficiently regular). The aim of this paper is to investigate the behaviour of the system of option prices whose underlying is driven by a discontinuous process (more precisely, by a lognormal jump process): we want to check whether this prices system can be recovered by a diffusion or not, and we want to analyse its local volatility function.

97-78 Numerical Procedures for Pricing Interest Rate Exotics Using Markovian Short Rate

Models Les Clewlow, Kin Pang and Chris Strickland

In this paper we discuss the pricing of exotic (American featured and path dependent) interest rate derivatives. We work within the framework of a discrete time and state lattice model or trinomial tree for the evolution of the short term interest rate. Examples of models in this class are Hull and White [1990], Black, Derman and Toy [1991], and Black and Karasinski [1992] in the one factor case, and Hull and White [1994] in the two factor case. We describe numerical procedures that can be used to efficiently price and hedge American featured, and in particular path dependent interest rate derivatives. Examples we consider are American swaptions into new and existing swaps, down and out barrier caps, lookback caps, average rate caps, and index amortising rate swaps

97-77 American Featured Options Silio Aparicio and Les Clewlow

In this paper we discuss the pricing and hedging of options in which the holder can make a decision at some point during the life of the option which can alter the payoff of the option. We consider as specific examples: standard American and Bermudan options, compound options, chooser options and shout options. We use these examples to show how more complex American featured options can be analysed to determine appropriate pricing and hedging methods.

97-76 Efficient Pricing of Caps and Swaptions in a Multi-Factor Gaussian Interest Rate

Model Les Clewlow, Kin Pang and Chris Strickland

In this paper we describe a formulation of the multi-factor Gaussian interest rate model of Carverhill (1995) closely related to the Heath, Jarrow and Morton model (1992). Our formulation allows us to efficiently price complex interest rate derivatives. We illustrate this by describing analytical solutions for caps and floors and efficient numerical integration and approximation formulae for European swaptions. Finally we present some numerical results for the pricing of European coupon bond options in a three factor implementation of the model.

Hedging Barrier Options in Incomplete Markets with Transactions Costs Silio Aparicio and Les Clewlow

Analytical forumlae exist for continuously fixed barrier options under the standard Black-Scholes assumptions and this implies a hedging strategy equivalent to the Black-Scholes delta hedging strategy for standard options. However, the key assumptions of geometric Brownian motion, continuous trading and no transactions costs do not hold in reality. In real markets jumps in asset prices occur, volatility changes randomly, the hedging strategy can only be implemented in discrete time, and transaction costs are incurred each time the hedge is rebalanced. For barrier options reasonable risk reduction implies such frequent rebalancing of the hedge, particularly near the barrier, that the transaction costs make this impractical. Recently, the construction of static portfolios of standard options which replicate the value of a barrier option within a binomial or trinomial tree have been proposed. However, this approach has problems; the tree cannot fully reflect the jumps and stochastic volatility, they do not take into account transaction costs and most importantly they do not allow for adjusting the portfolio. The requirement to adjust the replicating portfolio implies a trade-off between the risk reduction and the transactin costs


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incurred. In this paper we propose a Monte Carlo simulation based yopic optimisation model to control transaction costs and we compare the performance of this model against naive Black-Scholes delta and delta-gamma hedging, and static replicating portfolios in the presence of jumps or stochastic volatility.

97-74 Computational Aspects of Term Structure Models and Pricing Interest Rate

Derivatives Les Clewlow, Stewart Hodges, Kin Pang and Chris Strickland

This paper, which appears as a chapter in ‘Option Embedded Bonds’ edited by Israel Nelken (published by Irwin), gives an overview of some of the computational aspects involved in implementing interest rate models for pricing and hedging derivatives. We describe methodologoies for both the time-dependent short rate approach favoured by authors such as Black-Derman-Toy, Black-Karasinski, and Hull-White, as well as the arbitrage free approach favoured by amongst others Heath-Jarrow-Morton, and Carverhill.

For the short rate approach we describe a general procedure for building trees which are consistent with the observed yield curve and yield volatilities. We then describe how a range of standard interest rate derivatives; coupon bond options, caps, floor, collars, and swaptions can be priced using the short rate tree. Finally, for this approach we show how path dependent interest rate exotics can be priced using the example of an Index Amortising Swap (IAS). For the HJM approach we describe efficient simulation techniques for swaption pricing.

97-73 Mathematical Programming and Risk Management of Derivative Securities Les Clewlow, Stewart Hodges and Ana Pascoa

In this paper we review and discuss the application of mathematical programming techniques such as linear programming, dynamic programming, and goal programming to the problem of the risk management of derivative securities, in particular complex or exotic options. Our aim is to present the ideas in a way which is accessable to a wider audience in order to help stimulate interdisciplinary research. However we also hope to provide a useful review for researchers in the derivatives area.

97-72 Simple Resettable Cap and Floor Pricing Formulae Kin Pang

Using results from recent papers on the market Libor model, we show how simple expressions for an upper and lower bound for the value of resettable caps and floors can be obtained. Using an additional approximation, we show how resettable caps and floors can be priced approximately using forward Libor rates and standard caplet and floorlet volatilities.

96-71 Multi-Factor Gaussian HJM Approximation to Kennedy and Calibration to Caps and Swaptions Prices

Kin Pang We calibrate a Gaussian Random Field Term Structure of Kennedy (1994) to market caps and swaptions prices using an exact cap pricing formula and an approximate swaption pricing formula. The implied correlation structure of yield changes of the fitted model is analysed. We find a multi-factor Gaussian HJM approximation to the fitted model and compare the resulting HJM model with previous calibrations.

96-70 Option Pricing and Smile Effect when Underlying Stock Prices are Driven By a

Jump Process Luca Pappalardo

Black and Scholes values of volatility, implied from market prices, show a strong dependence on both the strike price and the maturity of a given European call option. This dependence is called the Smile effect. If we believe that stock prices are driven by a diffusion process in which volatility is allowed to be dependent in either the stock price or time,


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Dupire showed how it is possible to recover the process from option prices. We assume, instead, a model in which stock prices undergo a diffusion plus a jump term (which is driven by a Poisson process). Option pricing is possible in this model (Merton 1973), but since the model is no more complete some hypothesis on the structure of the market is necessary - such as the CAPM. The aim of this work is to provide a method to recover a jump process from the prices observed in the market. It can be done modifying in a suitable way the procedure followed by Dupire. Some care is needed since the spot process is not continuous. Hence, deriving the Kolmogorov Equation for the process, Ito lemma has to be applied in its general form. The equation for the Smile surface is derived in some particular cases of jump distribution and the problem of the parameter estimation - from which difficulties may arise - is discussed.

96-69 Pricing by Arbitrage in Incomplete Markets Simon Babbs and Michael Selby

A substantial literature on pricing by arbitrage has restricted information, so that it arises solely from securities markets, and return distributions are governed by past prices. We reconsider pricing by arbitrage in markets rendered incomplete by arbitrary information, which, moreover, influences required returns. Specific martingale representation results, playing a key role in existing analyses, depend on the restricted information, and so become inapplicable. We show that pricing by arbitrage alone, of claims depending on the market history, depends precisely on the agreement of all possible risk-adjusted probabilities. In particular, we rederive existing results in our generalised economy.

In a wider class of models, arbitrage arguments must be supplemented by specifying market price(s) of risk(s). We characterise viable models in terms of “risk pricing measures”, which generalise Harrison and Kreps’ [1971] (HK’) “equivalent martingale measures”, to handle primitive processes not representing securities prices - a situation not contemplated by HK. Our analysis leads us to conclude explicitly that the pricing of risks provides a more fundamental interpretation of such reassignments of probabilities than making securities ‘fair bets’.

96-68 Optimal Delta-Hedging Under Transactions Costs Les Clewlow and Stewart Hodges

This paper examines the problem of delta-hedging portfolios of options under transactions costs, using the stochastic optimal control approach first described by Hodges and Neuberger (1989). Rather than seeking a strategy for exact replication, which is liable to be expensive and may be dominated by other strategies, this approach obtains the optimal hedging strategy to maximise expected utility (or to minimise a loss function defined on the replication error). In this paper we extend the work of Hodges and Neuberger (1989) to study the optimal strategy under a general cost function with a constant cost per transaction, a cost per unit of the asset transacted and a cost proportional to the value transacted. A computational procedure for solving this problem is described and we develop an efficient computational method for the case of proportional transaction costs. We examine the nature of the solution close to the expiry date which depends critically on whether asset and cash settlement occurs. Using simulation we compare the performance of the optimal strategies against typical discrete rebalancing strategies such as Black-Scholes and Leland for a variety of portfolios of mixed long and short positions.

The simulations show that the optimal control approach is substantially more effective than the discretely rebalanced strategies, and that while the target delta and the band around it are both important, good hedges can be obtained by hedging using a control band of the right width but based around an incorrect central delta.


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96-67 Monte Carlo Valuation of Interest Rate Derivatives Under Stochastic Volatility Les Clewlow and Chris Strickland

This paper describes flexible and efficient Monte Carlo techniques for the valuation of interest rate derivatives in a world with stochastic interest rates and interest rate volatility. The particular model we study is due to Fong and Vasicek (1992) but the techniques are generally applicable. We extend the model of Fong and Vasicek [1992] allowing the valuation of a wide range of interest rate derivatives within their framework. We also extend the control variate technique described by Clewlow and Carverhill [1994] to efficiently value options on a wide range of derivatives, including bond options, caps, floors, collars, and swaptions. We analyse the benefits of different control variates and the effect of different numbers of time steps and simulations.

96-66 A Note on the Efficiency of the Binomial Option Pricing Model Les Clewlow and Andrew Carverhill

We discuss the efficiency of the binomial option pricing model for single and multivariate American style options. We demonstrate how the efficiency of lattice techniques such as the binomial model can be analysed in terms of their computational cost. For the case of a single underlying asset the most efficient implementation is the extrapolated jump-back method. That is to value a series of options with nested discrete sets of early exercise opportunities by jumping across the lattice between the early exercise times and then extrapolating from these values to the limit of a continuous exercise opportunity set. For the multivariate case, the most efficient method depends on the computational cost of the early exercise test. But, for typical problems, the most efficient method is the standard step-back method. That is performing the early exercise test at each time step.

96-65 The Risk Premium in Trading Equilibria Which Support Black-Scholes Option

Pricing Stewart Hodges and Michael J P Selby

This paper provides further analysis of the behaviour of the risk premium on the market portfolio of risky assets. Earlier work by Hodges and Carverhill (1993), and by others, has characterised the evolution of the market risk premium in economies where the variance of the return on the market has constant variance and market index options can be priced using the Black Scholes model. In such economies the risk premium satisfies a non-linear partial differential equation called Burgers' equation. This paper provides some significant new insights into this analysis. First we describe the nature of the existing results and provide a much simpler and more intuitive derivation. Next, we consider the time homogeneous case. Our original objective was to find a time homogeneous economy which allows the risk premium to vary so that some kind of mean reversion could take place. Sadly, this is impossible. We obtain the interesting, but negative, result that the risk premium must be constant for time homogenous equilibria which rule out arbitrage. Finally, this result is shown to tie in to earlier work on asymptotic portfolio selection. The paper illustrates that caution is required in this kind of modelling to avoid writing down models which admit arbitrage. The analysis also shows the limitations of the representative investor paradigm.

96-64 Equilibrium and the Role of Options in an Economy with Stochastic Volatility Anthony Neuberger and Stewart Hodges

The paper develops a model of market equilibrium in an economy where a single risky asset evolves with stochastic volatility. Our work is in the spirit of Bick (1987). In our economy, unlike Bick's which is complete and supports Black-Scholes option values, the market is incomplete unless some kind of volatility sensitive (ie option-like) asset is traded. The model we present enables us to examine the role options play in enabling investors (other than the representative investor) to optimize their investments. In steady state equilibria involving investors with the same horizons but different degrees of risk aversion the gains from options appear to be relatively minor.


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95-63 Martingale Restriction Tests of Option Pricing Models and Their Interpretation Silio Aparicio and Stewart Hodges

Most, if not all, option pricing models imply some parameterization of the risk neutral density function of the price of the underlying asset. Longstaff (1993) has proposed that the specification of options models may be tested by estimating the risk neutral density function from options data, and then checking whether the mean differs from the forward price of the asset. The purpose of our paper is to investigate why this test seems to be a sensitive one, and to clarify correct and incorrect interpretations of the approach. Our conclusions are reinforced by means of both simulations, and the empirical analysis of an extensive and different data set.

95-62 Non-Negative Affine Yield Models of the Term Structure Kin Pang and Stewart Hodges

Affine yield models of the term structure (e.g. as described by Duffie & Kan, 1995) form an important general class which encompasses many popular models. However, it also includes models with the unfortunate property that interest rates can go negative. In practice, it would be difficult to estimate the coefficients of this form of model to preclude negative rates occurring. In this paper we show that those affine yield models which guarantee only positive interest rates are equivalent to generalised versions of the Cox, Ingersoll and Ross (1985) model. Working with the model in this framework provides an easier way to guarantee that all yields remain non-negative.

95-61 Term Structure Modelling Under Alternative Official Regimes Simon Babbs and Nick Webber

Monetary authorities exercise control of domestic short term interest rates. We argue that models of the term structure of interest rates must take into account the consequences of this control if they are to capture important empirical features of interest rate dynamics. In particular, previous one or two factor models of the term structure may not adequately describe the short rate process. Interest rate regimes in the United Kingdom, the United States, France and Germany are described. A common characteristic is the presence of ‘non-effective’ official interest rates used for signalling purposes. We construct a term structure model, consistent with general equilibrium, that captures, in idealised form, some important features of these regimes. The paper concludes with an illustrative example of a regime in which the short rate is constrained to lie within an officially controlled corridor.

95-60 A Comparison of Alternative Covariance Matrices for Models with Over-Lapping

Observations Jeremy Smith and Sanjay Yadav

This paper investigates the relative performance of alternative covariance matrices for models with over-lapping observations commonly used in the finance literature. The alternative covariance matrices used are those of Hansen (1982), Newey and West (1987) (Bartlett and Quadratic Sprectral (QS) weights) and Andrews and Monahan (1992) (QS weights). All matrices produce standard errors which are too small, yielding empirical size probabilities above their corresponding theoretical values, even in large samples. Empirical examples, such as testing efficiency in the foreign exchange market and mean reversion in stock prices, show that the choice of covariance matrix can affect the outcome of a hypothesis test. (JEL C15 and C22)

95-59 Can Dividend Yields Forecast Returns? Mitual Kotecha and Sanjay Yadav

There have been many studies on the use of dividend yields to predict returns, most concentrating on US data. There is a paucity of studies based on data from the UK stock market. We investigate the ability of dividend yields to forecast nominal returns on the value weighted FTA All Share Index, for return horizons ranging from one month to four


30

years. We use non-overlapping as well as overlapping observations in our study and the sample period runs from 1965 to 1992.

Following Goetzmann and Jorion (1993), we allow for the lagged price relation between returns and dividend yields in our regression specification and use the bootstrap methodology to model the distribution of test statistics under the null hypothesis that returns and dividend yields are independent. We extend Goetzmann and Jorion’s methodology by using WLS rather than OLS to estimate our model and thus correct explicitly for heteroscedasticity in the regression residuals. We also study the yield -return relationship using a methodology that does not impose a particular functional form on the data, a priori.

Our overall conclusion is that when we take adequate account of potential nonlinearities in the yield-return relationship, evidence supportive of return predictability largely disappears. This is consistent with the findings of Goetzmann and Jorion for US data, although their results were obtained within the framework of a linear model. The results obtained from applying the joint tests are in close agreement with the results obtained using individual return horizon statistics. Finally, none of the alternative covariance matrix estimators evaluated in this study are satisfactory; all of them yield standard error estimates which are substantially downward biased in the presence of overlapping observations. Therefore, researchers ought to focus primarily on the regression slope coefficients to conduct inference.

95-58 An Econometric Analysis of Long Horizon Mean Reversion in UK Stock Prices Stewart Hodges and Sanjay Yadav

This paper is concerned with examining the mean reversion hypothesis for UK stock prices for the 1925-1991 period. Specifically, this paper incorporates recent innovations in time series techniques which are not subject to the flaws which have plagued other studies. To the best of our knowledge, the phenomenon of mean reversion in UK stock prices has not been studied using a data set which covers the period 1925-1991. Most UK studies use data that extends from the mid 1950s onwards. In this paper, we eschew reliance on asymptotic distributions to conduct inference. Instead, our preferred strategy is to use simulations to generate the empirical distributions of test statistics in order to conduct inference. We use Monte Carlo as well as Randomisation methods to obtain significant levels. Using somewhat related testing methodologies, we do not find any evidence that is consistent with mean reverting behaviour in UK stock prices. This conclusion is robust with respect to heteroscedasticity in stock returns and different choice of sample periods. By and large, the random walk model for stock prices is accepted. There are a few instances when we come close to rejecting the random walk model, but in every case, rejection is in favour of mean aversion or positive serial correlation in stock returns. Unlike for the US, there is no evidence of statistically significant mean reversion in UK stock prices.

95-57 Arbitrage in a Fractional Brownian Motion Market Stewart Hodges

Various commentators have suggested that Fractional Brownian Motion may be an appropriate model for financial markets. However, this seems a surprising choice, since arbitrage opportunities exist within this kind of structure. This paper explores the nature of the arbitrage opportunities in a Fractional Brownian Motion market. We describe why the risk neutral probabilities are degenerate and investigate the numbers of transactions required to obtain essentially riskless profits. We conclude that for a market with a Hurst exponent outside the range 0.4 to 0.6 less than 300 transactions would be required.


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Fractional Brownian Motion is thus a quite inappropriate model for a financial market unless that market really is grossly inefficient.

95-56 On the Simulation of Contingent Claims Les Clewlow and Andrew Carverhill

For many complex option valuation problems analytical solutions are not possible. In these cases Monte Carlo simulation is an important numerical solution tool. Furthermore, simulation can reveal important insights into the hedging of these complex options. In its basic form, however, Monte Carlo simulation is computationally inefficient. In this article we address this problem. We describe a new approach to constructing control variates, based on the “Greek letter” risk exposures of the option, which can improve the efficiency of the simulation dramatically. We illustrate the use of the technique for a standard European call option, and for a realistic example involving valuing and hedging a look-back call with discrete fixing of the minimum price under stochastic volatility.

Journal of Derivatives, 2(2), Winter 1994, pp 66-74 94-55 Efficient and Flexible Bond Option Valuation in the Heath Jarrow and Morton

Framework Andrew Carverhill and Kin Pang

The HJM bond option valuation framework is very flexible; we present an efficient numerical implementation, which uses a Monte Carlo simulation technique, with carefully chosen Martingale Variance Reduction variates. These variates make the simulation technique up to about 16 times faster, to achieve a given standard error. We also show how to ensure that the model avoids negative interest rates in this context.

Journal of Fixed Income, 5, (2), September 1995, pp. 70-77 94-54 Pricing Exotic Options in a Black-Scholes World Les Clewlow, Javier Llanos and Chris Strickland

In this paper we present derivations of pricing formulae for Collars, Break Forwards, Range Forwards, Forward Start Options, Compound Options, Chooser Options, Asian Options, Lookback Options, Barrier Options, Binary Options, Asset Exchange Options, and Quanto Options. The purpose of this paper is to provide a set of tools with which to value exotic options. The formulae are made as general as possible and are derived in the Black-Scholes world (Black and Scholes (1973)). The results unify and generalise many published and unpublished results and provide a general notational structure.

94-53 The Dynamics of Stochastic Volatility Les Clewlow and Xinzhong Xu

This paper presents a theoretical and empirical study of the dynamical behaviour of the volatility of the Chicago Mercantile Exchange Standard and Poor 500 index futures. The time series properties of the underlying instrument, its realised volatility and volatilities implied from options prices are examined. We study a range of estimators of both realised volatility and implied volatility and draw some conclusions concerning their utility. We show how B-splines can be used to fit smooth curves to the biases in the implied volatilites across strike prices. The properties of volatility which are revealed are compared with those assumed in some recent models for pricing options under stochastic volatility. We then examine the ability of a particular, fairly general yet tractable, diffusion model (Hull and White (1988)) to explain the biases observed in the S&P 500 index futures option market prices.

94-52 Gamma Hedging in Incomplete Markets Under Transactions Costs Les Clewlow and Stewart Hodges


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This paper explores the role of gamma hedging strategies in situations where there are either jumps or stochastic volatility, and it is costly to make options transactions in order to hedge. The motivation for this work stems from earlier empirical work we have completed on hedging options portfolios. A previous empirical analysis (Hodges, Clewlow et al (1993)) of hedging CME options on the S&P futures confirmed that gamma hedging provided a significant risk reduction compared to delta hedging, and did so even when rebalancing was done daily. The reason why the improvement is so marked appears to be because real asset processes involve jumps and stochastic volatility. However, the study also showed that this kind of standard gamma hedging involved high turnover of options positions - which with transactions costs would make the hedge prohibitively costly. We therefore have the following problem: What is the best way to manage gamma when options are costly to transact and we have jumps and/or stochastic volatility?

We have some insights into this problem from the literature on optimal delta hedging under transactions costs. The gamma hedging problem, though, is very much harder since, unlike delta, there are many different candidate instruments that could be used to modify gamma. We would have to have many state variables to be able to solve the problem using a dynamic programming formulation (as we did, for example, in Hodges and Neuberger (1989) and Hodges and Clewlow (1993)), though it could of course be written in that framework in a formal sense. Instead, we have therefore used a simulation approach to search for optimal methods of hedging based on heuristics which are consistent with myopic policies.

94-51 Multi-Period Minimax Hedging Strategies Melendres Howe, Berc Rustem, and Michael Selby

Option pricing and hedging under transaction costs are of major importance to marketmakers and investors. In this paper we present the basic minimax strategy which determines the optimum number of shares that minimizes the worst-case potential hedging error under transaction costs for the next period. We present two extensions of this strategy. The first extension is the two-period minimax where the worst case is defined over a two-period setting. The objective function of the basic minimax strategy is augmented to include the hedging error for the second period. The second extension is the variable minimax strategy where early rebalancing is triggered by the minimax hedging error. Simulation results suggest that the basic minimax strategy and its two extensions are superior in performance to delta hedging and that the variable minimax strategy is superior to both the basic and the two-period strategies. This result is due to the opportunity provided by the variable minimax strategy to rebalance early.

European Journal Operational Research, 1995 94-50 Minimax Hedging Strategy Melendres Howe, Berc Rustem, and Michael Selby

We present several variants of a robust risk management strategy based on minimax for the writer of a European call option on a stock and show that it performs at least as well as the standard hedging strategy, delta hedging. When using the minimax strategy, the hedger specifies a worst case scenario in terms of the price of the underlying stock. The minimax strategy recommends the number of shares in the underlying stock the hedger should hold in order to minimize the hedging error against the worst case occurring. The minimax hedging error may correspond to an extreme point of the price range being considered or to a mid-range solution. Simulation and empirical results suggest that the minimax strategy is particularly powerful for hedging the risk of writing an option when the price of the underlying stock is both highly volatile and crosses over the exercise price frequently.

Computational Economics, Vol. 7, No.4, pp.245-275, 1994. 94-49 A Theory of the Term Structure with an Official Short Rate Simon Babbs and Nick Webber


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Many of the larger movements in market interest rates are associated with changes in official interest rate policy. This applies especially in the UK, where the Bank of England effectively controls very short term rates, and moves its key dealing rate in multiples of a discrete unit. We present a theory of the term structure, with the short rate following a pure jump process with stochastic intensities depending on the short rate and a diffusion state variable. An illustrative model mimics empirical regularities of UK rates. We suggest how to extend our approach to other countries, most readily the US.

94-48 A Model of UK Libor as a Jump-Diffusion Process Leticia Veruete and Nick Webber

This paper presents a jump-diffusion model of the UK sterling interbank short rate (‘libor’). We are able to decompose libor into the sum of a mean-reverting continuous part, and a jump process. He latter is identified with the clearing bank base rate. In addition, including a transition function allows libor to ‘anticipate’ base rate movements. If this decomposition is a reasonable description of the process followed by libor then models that do not take this behaviour into account may be mis-specified. In particular the models and the methodology used by Chan, Karolyi, Longstaff and Sanders (1992) in their empirical study of American short rates may not be applicable to UK libor. Implications are drawn for the valuation of derivative securities on libor. The valuation of the base rate process. Long maturity options reflect the conditional distribution of the base rate process. Short term options are influenced by the probability of single base rate jumps within the horizon of their time to maturity.

94-47 A Comparison of Models for Pricing Interest Rate Derivative Securities Chris Strickland

This paper looks at the different approaches and different models that have been developed to value interest rate-dependent securities, providing a survey of pricing procedures which are based on mathematical models of the term structure. It can be viewed as a reference for the different interest rate models with explicit representations, where they exist, for prices of derivative instruments and an analysis of their respective advantages and disadvantages.

94-46 A Comparison of Models of the Term Structure Chris Strickland

This paper looks at a number of different continuous-time approaches that have been developed to model the term structure of interest rates. These techniques span the interest rate literature over the last 20 years or so, and are the most commonly used among both academics and practitioners. We view this paper as a reference for the different term structure models, aiming to bring together the three most commonly used approaches, emphasising their differences, analysing their respective advantages and disadvantages, and with explicit representations, where they exist, for prices of discount bonds.

94-45 Option Prices as Predictors of Stock Prices: Intraday Adjustments to Information Releases

Paula Varson and Michael Selby This study tests for intraday lead/lag relationships between a given stock price and the stock value implied by the prices of call options on that stock. The results indicate that throughout the five trading-days preceding earnings announcements with significant unanticipated information content, implied stock values lead their corresponding observed stock prices by about fifteen minutes. On the announcement day itself, this lead lengthens to the point that call option prices usually adjust at least one hour before the public announcement. Under most circumstances, evidence of this lead disappears immediately after the announcement and prices remain synchronous between the two markets.

Proceedings of Annual INQUIRE Conference, 1994


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94-44 On a Free Boundary Problem That Arises in Portfolio Management Stanley Pliska and Michael Selby

This paper studies a model for the optimal management of a portfolio when there are transaction costs proportional to a fixed fraction of the portfolio value. The risky securities are modelled as correlated geometric Brownian motions, there is a riskless bank account, and the objective is to maximise the long-run growth rate. It is known that the optimal trading strategy is characterised by the solution of a certain PDE free boundary problem. This paper explains how to transform this free boundary problem for the case of three securities into a much simpler one that is feasible to solve with numerical methods. Philosophical Transactions of the Royal Society, Series A (physical sciences and engineering), 347,

May 1994, pp.555-561. Also reprinted in: Mathematical Models in Finance, chapter 13, Howison, S.D., F.P. Kelly & P. Wilmott (Eds), Chapman and Hall, London, 1995

93-43 Dynamic Asset Allocation: Insights From Theory Stewart Hodges

This paper provides a survey of the now considerable academic theory relating to the practice of dynamic asset allocation. This work is scattered through the literature and many of the key ideas are not as accessible or well known as they deserve to be. The paper begins by providing a definition of what is meant by dynamic asset allocation and a description of its most significant features. Next it develops the concept of path independence and its relationship to efficient diversification through time. it is shown that this principle also applies to funds whose performance is appraised relative to an index benchmark. he final sections of the paper describe the implications of recent work on market equilibrium and on performance measurement.

Philosophical Transactions of the Royal Society of London Series A, 1994, 347, pp.587-598. Also reprinted in: Mathematical Models in Finance, chapter 13, Howison, S.D., F.P. Kelly & P. Wilmott

(Eds), Chapman and Hall, London, 1995 93-42 Computing the Fong and Vasicek Pure Discount Bond Price Formula Michael Selby and Chris Strickland

Fong and Vasicek [1992] developed a model of the term structure which results in closed-form expressions for pure discount bond prices, and which reflects both the current level of interest rates as well as the level of interest rate volatility. Although a number of articles have appeared detailing the Fong-Vasicek model (Fong and Vasicek [1991], Fong and Vasicek [1992a], Fong and Vasicek [1992b]), there appears to be very little reported practitioner interest. One of the reasons for this relatively minor practitioner approval, we feel, is the computational difficulty in obtaining prices from their formula. Two of the three functions of time involved in the model’s solution require computation of the confluent hypergeometric function. As described by Fong and Vasicek, this involved complex (as opposed to real) algebra. In this article we propose an alternative solution to the ordinary differential equations which lead to the need to solve the confluent hypergeometric function. Our method involves a series solution which is both computationally efficient and can be easily implemented in a programming language or spreadsheet.

Journal of Fixed Income, 5 (2), September 1995, pp. 78-85 93-41 A Note on Parameter Estimation in the Two Factor Longstaff and Schwartz Interest

Rate Model


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Les Clewlow and Chris Strickland This note concerns the parameter estimation technique outlined by Longstaff and Schwartz [1993], for their two-factor interest rate model (Longstaff and Schwartz [1992a]). We show that the technique of using historical times series data has practical implementation problems given the length of financial time series available to practitioners. We conclude that this problem is generic to models of this type.

Journal of Fixed Income, March 1994, pp. 95-100 93-40 The Magnitude of Implied Volatility Smiles: Theory and Empirical Evidence for

Exchange Rates Stephen Taylor and Gary Xu

Theoretical methods are used to show that implied volatilities are approximately a quadratic function of the forward price divided by the exercise price when volatility is stochastic, asset price and volatility differentials are uncorrelated and volatility risk is not priced. The curvature of the quadratic function depends on the time to maturity of the option and several volatility parameters including the present level, the long-run median level and the variance of future average volatility. The magnitude of the ‘smile effect’ is a decreasing function of time to maturity. Empirical evidence for exchange rate options supports the theoretical predictions, although the empirical smiles are approximately twice as large as those predicted by theory.

Review of Futures Markets

93-39 Convergence to Efficiency of the Nikkei Put Warrant Market of 1989-1990 Julian Shaw, Edward Thorp and William Ziemba

This paper discusses the Nikkei put warrant market in Toronto and New York during 1989-1990. The classes of long term American puts were traded which when evaluated in yen are ordinary, product and exchange asset puts, respectively. Type I do not involve exchange rates for yen investors. Type II fix in advance the exchange rate to be use don expiry in the home currency. Type III evaluate the strike and spot prices of the Nikkei Stock Average in the home currency rather than in yen. For typically observed parameters, Type I are theoretically more valuable than Type II which in turn are more valuable than Type III. In late 1989 and early 1990 there were significant departures from fair values in various markets. This was a market with a set of complex financial instruments that even sophisticated investors needed time to learn about to price properly. Investors in Canada were willing to pay far more than fair value for their puts. In addition, US investors overpriced Type II puts fixed in dollars rather than yen compared to Type I. This led to cross border and US traded (on the same exchange) low risk hedges. The market’s convergence to efficiency took about one month after the introduction of the US puts in early 1990 leading to significant profits for the hedgers. The underreaction to this new information about cheaper nearly equivalent securities is analogous to that of the price delay of stocks to new earnings announcements.

93-38 Contingent Claims Analysis Simon Babbs and Michael Selby

The purpose of this article is to give a concise overview of the modern theory of contingent claims analysis (CCA). CCA is possibly the most significant development in financial economics over the last twenty years. From its origins in option pricing and the valuation of corporate liabilities, it has become a major approach to intertemporal general analysis equilibrium under uncertainty. It has made important contributions to economic theory. Moreover, its focus on the pricing and replication of contingent payoffs offers insight into the role of financial intermediaries.

The New Palgrave Dictionary of Money and Finance, Eds: J Eatwell, M Milgate and P Newman, Macmillan (1992), pp 437-440

93-37 Interest Rate Volatility and the Term Structure of Interest Rates


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Chris Strickland Interest rate volatility is a key element when valuing fixed income securities. This paper looks at how the traditional models of interest rate movements handle the uncertainty of interest rates, and analyses recent attempts to incorporate more realistic interest rate movements into models for pricing interest rate dependent securities. We find that the earlier, single factor models of the term structure place quite severe restrictions on the shape and form of possible yield curves, due to a large extent to their assumptions about interest rate volatility. Recent attempts to model the term structure have followed two main approaches. The first approach has been to incorporate interest rate volatility as a second stochastic factor in equilibrium type models of the term structure with the advantage that prices are analytically tractable, but with the disadvantages that the resulting term structures belong to a limited family which will generally be inconsistent with the observed term structures. The second approach focuses on building models that use information contained in the prices of traded securities about the whole term structure of both spot or forward rates and rate volatilities. We show that this latter structure may be constrained to evolve over time in an unstable way that was not originally intended by the user. The link between the uncertainty of interest rates and the value of differing maturity bonds, suggests a relationship between the level of interest rate volatility and the shape of the term structure of interest rates. We analyse this relationship and develop a method to empirically test it using the prices of traded securities.

93-36 Optimal Delta-Hedging Under Transactions Costs Stewart Hodges and Les Clewlow

The paper examines the problem of delta-hedging under transactions costs, using the stochastic optimal control approach first described by Hodges and Neuberger (1989). Rather than seeking a strategy for exact replication, which is liable to be expensive and may be dominated by other strategies, this approach obtains the optimal hedging strategy to maximise expected utility (or to minimise a loss function defined on the replication error). Under proportional transactions costs this results in policies characterised by control bands within which the hedge delta must be maintained. Only with a fixed cost component would it be appropriate to make large transactions to jump into the interior of the control region. This method has the advantage over Leland’s (1985) approach that it works just as well for hedging miexed portfolios of long and short positions, and also mixed maturity dates. The apper describes the basic approach, and derives a new computational method which substantially reduces the storage required for the calculation. Characteristics of the optimal policies are discussed, and a simulation study is completed to compare the hedging performance of some alternative policies. The strategies we tested were chosen so that we could examine which features of a hedging strategy are most important; hedging to the “correct” delta or hedging only to within a band in order to conserve transactions costs. The simulations show that hte optimal control approach is substantially more effective than Lenald’s method, and that while the target delta and the band around it are both important, surprisingly good hedges can be obtained by hedging using a control region of the right width but based around an incorrect central delta.

92-35 A Review of Option Pricing with Stochastic Volatility Les Clewlow and Xinzhong Xu


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In this paper we consider the problem of option pricing when the volatility is changing randomly. We review some of the major advances in option pricing with stochastic volatility which have appeared in the literature. We discuss their advantages and disadvantages from both a theoretical and practical viewpoint. In particular we focus on the key issues of how the models deal with the risk premium on volatility, hedging the volatility risk and the correlation between the asset and its volatility. Finally, we attempt to reach some conclusions regarding the most promising and useful approaches.

92-34 Recent Developments in Derivative Securities Les Clewlow, Stewart Hodges, Michael Selby, Chris Strickland and Xinzhong Xu

The paper reviews recent developments in modelling derivative securities. It describes new models for valuing interest rate derivatives based on ‘whole term structure models’ which are consistent with an arbitrary initial term structure. it also discusses the tension between this approach and other recent studies of the relationship between the shape of the yield curve and internal rate volatility. The issue of stochastic volatility in pricing and hedging other kinds of options is also examined in its own right. The usual Black-Scholes replication argument requires frictionless markets. New results for valuing options when transactions costs affect trading in the underlying asset are reviewed. The fourth topic examined concerns the role of general equilibrium analysis in the field of contingent claims. Brief mention is made of some other topics in the options literature. The paper concludes with the thought that work on options has focused major new resources at the problems of understanding the nature of the stochastic processes for underlying security market assets.

Financial Markets Institutions and Instruments 1, No 5, 1992 92-33 Conditional Volatility and the Informational Efficiency of the PHLX Currency

Options Market Xinzhong Xu and Stephen Taylor

The relative importance of implied volatility and historical volatility predictors is evaluated by including both predictors in the conditional variance equation of an ARCH model. The implied predictors come from the term structure model developed in our earlier paper. Results for foreign exchange volatility from 1985 to 1989 show that for three of the four exchange rates considered (£/$, DM/$, YEN/$, SF/$) the implied volatility gives an optimal predictor of the next day’s conditional volatility. Thus information from past currency returns has no incremental value. This within-sample conclusion is consistent with the informational efficiency of the Philadelphia currency options market.

Out-of-sample volatility forecasts are also evaluated for 1990 and 1991 with forecasts made for an average volatility measure over a four-week period. Once more the implied predictors are found to be superior to historical predictors.

Journal of Banking and Finance, Vol 19, 1995, pp 803-821 92-32 The Term Structure of Volatility Implied by Foreign Exchange Options Xinzhong Xu and Stephen Taylor

This paper illustrates methods for estimating the time-varying term structure of volatility expectations, as revealed by options prices. Short and long-term expectations can be estimated using the Kalman filter. These expectations are estimated for four currencies from 1985 to 1989 using daily PHLX options prices. Throughout the five-year period there were important differences between short and long-term expectations. The slope of the term structure changed frequently and there were significant variations in long-term volatility expectations. The four currencies had very similar term structures, particularly in 1988 and 1989.

Journal of Financial and Quantitative Analysis 92-31 Quasi-Mean-Reversion in an Efficient Stock Market: The Characterization of


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Economic Equilibria Which Support Black-Scholes Option Pricing Stewart Hodges and Andrew Carverhill

This paper is concerned with the behaviour of the risk premium on the market portfolio of risky assets. The paper provides a characterization of the evolution of the market risk premium in economies where the variance of the return ont he market has constant variance and market index options can be priced using the 1973 Black Scholes model. It is shown that the risk premium satisfies a non linear partial differential equation called Burgers equation. The analysis has potentially importnat implications for empirical work, where for example, it is undecided whether observed mean reversion in stock prices can be explained by time varying risk premia within an efficient market.

Economic Journal, March 1993 92-30 Efficient Monte Carlo Valuation and Hedging of Contingent Claims Les Clewlow and Andrew Carverhill

In this paper we discuss the use of antithetic and control variates for reducing the variance or error in a Monte Carlo valuation. We describe a choice of control variates which have an economic interpretation. By combining these two techniques we obtain an increase in the efficiency of the Monte Carlo method of many orders of magnitude. We also consider how the hedge ratios (DELTA, GAMMA, THETA, KAPPA, RHO) can be computed efficiently within our variance reduced Monte Carlo simulation. Finally we describe how recent ideas of Breen (1991) and Ho et al (1991) can be used to obtain approximate values for American style options by Monte Carlo simulation.

92-29 Current Research on Derivative Products Stewart Hodges

Derivative products have undergone intensive development and have been the subject of a great deal of research in the last few years. This is an area in which current academic research lies particularly close to the needs and interests of participants in the financial markets. This article sketches the main themes of current research in the derivatives area that are already affecting the products available to the corporate treasurer, or which are likely to do so in the near future.

The Treasurer, November 1991, 6-9 92-20 The Term Structure of Spot Rate Volatility and the Behaviour of Interest Rate

Processes Nick Webbe r (Revised 1992)

The results presented in this paper show how, under certain conditions, the parameters of the short interest rate process are related to one another. Short rate volatility, as a function of the short rate, is shown to be critical in determining the deterministic part of short rate movements. Sufficient conditions are established for the short rate to be mean-reverting. The long rate is investigated. Conditions for it to be constant are stated.

92-19 Do Derivative Instruments Increase Market Volatility? Stewart Hodges

The October 1987 crash and other more recent hiccups have raised important questions about the volatility of security markets. These events occurred after a period of rapid growth in derivative instruments, and in the use of ‘program trading’ techniques, so it is hardly surprising that some observers have singled them out as a likely cause of recent high volatility.

A major report, Market Volatility and Investor Confidence, was published by the New York Stock Exchange in June 1990. Much of the report is concerned with attitudes to and the effect of program trading. The NYSE defines program trading as the simultaneous purchase or sale of 15 or more stocks with a total market value of at least $1 million’, but misconceptions abound (for instance, that it means trading triggered by computers).


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Program trades can thus be either index arbitrage transactions, or transactions related to portfolio rebalancing that may or may not involve futures or options.

The subject of whether the introduction of derivatives tends to increase the volatility of the underlying market is therefore one of great topical interest. The purpose of this paper is to review what we know about this issue. The paper describes and synthesises both theoretical and empirical work.

Options: Recent Advances in Theory and Practice Volume 2, (Ed: S D Hodges), Manchester University Press, 1992, 194-214

91-28 A Note on the Breen Binomial Acceleration Technique Les Clewlow and Andrew Carverhill

We discuss the theoretical efficiency of the binomial technique in its standard and Breen accelerated forms for American style options. The choice of the early exercise opportunity set for the Breen method is considered. Firstly we consider the case of a single underlying instrument, demonstrating the theoretical improvement in efficiency it yields. We then consider the application of Breen’s acceleration technique to the multiple asset case and show that here it is only more efficient if the early exercise test is relatively computationally costly.

91-27 Interest Rate Swaps and Default-Free Bonds: A Joint Term Structure Model Simon Babbs

Typical interest rate swaps are contracts contingent on future borrowing rates of banks then of good credit worthiness. Differences between the term structures for default-free bonds and for swaps are more appropriately modelled in terms of “required return differentials” rather than of default risks. We present a theory for the joint evolution of the two term structures, consistent with their initial positions, and for pricing contingent claims on either or both structures. We explore a class of models within our general framework, and exhibit a readily computable valuation formula for European-style contingent claims; we provide a concrete example.

91-26 Behaviour of the FTSE 100 Basis Chris Strickland and Xinzhong Xu

Using data on the London International Financial Futures Exchange FTSE 100 index contract, from the beginning of 1988 until the end of 1989, we look at deviations of transaction prices from calculated ‘fair’ values. Taking into account the fortnightly accounting periods used by the London Stock Exchange we examine the size, direction, and persistency of mispricing. Our results show that the fair value pricing formula is frequently violated, often by large amounts, with the tendency for mispricing to be negative and persistent. We also show that there are significant profitable opportunities for arbitrageurs who unwind the simple buy and hold cash/futures strategy early, or who roll forward the futures position involved in the arbitrage into the next nearest contract during the expiry month. Finally, we provide evidence to suggest that the mispricing time series follows an AR2 process.

Review of Futures Markets, September 1993

91-25 The Delivery Options in Bond Futures Contracts: An Empirical Analysis of the LIFFE Long Gilt Futures Contract

Chris Strickland Using data on the London International Financial Futures Exchange long gilt futures contract, we look at the implicit quality option contained in that contract. We look at three different ‘valuations’ of the quality option; an ex-ante value given by the excess of the forward price of the cheapest to deliver bond over its conversion factor multiplied by the futures price; an ex-post value of the payoffs to a strategy of buying and holding a short-futures long-forward position and delivering the cheapest bond at the expiration of the futures contract; and a Monte Carlo simulated value. We show that the delivery experience


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for the LIFFE contract is consistent with the delivery incentives created by the calculation of the LIFFE price factors.

The Review of Futures Markets, Vol 11, No 1, 1992, pp 83-102 91-24 A Family of Ito Process Models for the Term Structure of Interest Rates Simon Babbs

A family of Ito process models is constructed for the dynamics of the term structure of interest rates on default-free bonds, consistent with whatever term structure is initially observed. The results of Harrison and Kreps [1979] are extended to cover term structure models. It is shown that the family of models constructed in this paper can be supported in general equilibrium; in particular arbitrage opportunities are absent. a general formula is provided for the valuation of contingent claims. Consideration of sub-families sheds fresh light, in a generalised setting, on the term structure dynamics under which conventional duration is the correct risk measure for bond portfolios. a number of other models are shown to be special cases.

91-23 Ex-Post Evaluation of Dynamic Portfolio Strategies (or How to Tell Whether a

Million Dollars has been Thrown Away) Stewart Hodges

It has been demonstrated, in a paper by Dybvig [1988a] with a related title to this, that a number of portfolio (asset mix) strategies followed by practitioners are significantly inefficient. The inefficiencies arise not because of imperfect diversification across stocks, but because of poor diversification over time. Dybvig showed how the costs of following such strategies may be calculated, provided we know the parameters of the market return generating process and full details of the strategies concerned. The costs of following policies such as stop-loss, lock-in, random market timing or repeated portfolio insurance were shown to be substantial. However, in terms of applying this approach to practical problems of measuring the performance of investment portfolios, there remains the difficulty that usually we can only observe a single time series of what the fund manager actually did, and we are unlikely to know what policy would have been pursued along other possible unrealised paths of the probability tree. The purpose of this paper is to develop a powerful and robust technique which will enable Dybvig’s payoff distribution pricing approach to be applied for performance measurement purposes with realistically limited amounts of observational data.

91-22 An Introduction to Parallel Processing for Financial Valuation Problems Les Clewlow (Revised 1992)

The major disadvantage of serial computers is that the set of computations or tasks required to solve a given problem umust be performed sequentially. Some or all of these may be computationally independent and could hterefore be performed simultaneously. Performing the computations sequentially is obviously sub-optimal. The solution is a computer which can perform many computations simultaneously. This is the recent and rapidly developing field of parallel processing.

In this paper we introduce parallel processing terminology and concepts and examine some examples of applying these techniques to financial valuation problems with particular emphasis on contingent claims related problems.

91-21 The Term Structure of Interest Rates and Associated Options; Equilibrium vs

Evolutionary Models Andrew Carverhill

Among models of the term strucuture of interest rates we distinguish two classed, which we will refer to collectively as the “Equilibrium Model” and “Evolutionary Model”. The Equilibrium Model is represented by the papers [CIR 1985] (Cox, Ingersoll and Ross), [Vasicek 1977], [Courtadon 1982]. It seeks to characterise the term strucuture by assuming that it is in economic equilibrium, and is determined (‘driven’) by a given set of parameters. This set of parameters might typically comprise the long and short interest rates, or just the


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short rate, and being in econoimic equilibrium means that there are not arbitrage opportunities amond the interest rate instruments whose prices make up the term structure. The Evolutionary Model is represented by the papers [HL 1986] (Ho and Lee), [HJM 1987] (Heath, Jarrow and Morton), [Carverhill 1989a], [Carverhill 1989b], [Babbs 1990]. It concentrates on the evolution of the term structure, rather than the term structure itself, from an imprically given initial shape. Again, it assumes tha thtese are no arbitrage opportunities amond the interest rat einstruments. These fundamental differences give rise to the contrasting strenghts and weaknesses of the two models: the Equilibrium Model predicts a shape for the term structure at a given time, but which one expects to be a reasonable match for any time; the Evoluitionary Model predicts a term structure which decreases in accuracy as it is pushed further into the future.

Our aim in this paper is first to present the two models and the pricing procedures for contingent claims which they entail, from a unified perspective. This enables us to discover the similarities, differencees, and conflicts between these models. Also we discuss the work of some other authors (notably [Dybvig 1989], [Dybvig Ingersoll Ross 1989], [Hull White 1990a], [Jamshidian 1989]) and its relationship to the ideas of this paper. Finally, our perspective allows us to clarify and extend much of the work to which we make reference. This paper includes the material of ther two preprints [Carverhill 1989a] and [Carverhill 1989b].

91-18 Testing for Overreaction in Short Sterling Options Peter Bates and Les Clewlow

This paper examines the hypothesis that Short Sterling options are rationally prcied, against the alternative that the options market overreacts (or underreacts) to new volatility information in pricing longer-dated options. Two forms of explanatory analysis have been used; firstly, following Stein (1989), the jont behaviour of contemporaneous implied volatilties and secondly, the comparison of implied volatilities with realised volatilities. We are unable, using either of these forms of analysis, to reject the hypotheses that Short Sterling options are rationally priced. The methods of analysis used hwoever are simple, approximate and weak. Suggestions for developing improved and alternative methods of analysis are made. We believe that use of such methods would lead to sharper, and quite possibly different, conclusions.

Options: Recent Advances in Theory and Practice Volume 2 (Ed: S D Hodges), Manchester University Press, 1992, 104-132

90-17 Interest-Rate Derivatives: Evolutionary Valuation and Hedging Andrew Carverhill

Our aim in this article is to show how to apply the evolutionary model of the term structure of interest rates to the valuation and hedging of interest rate caps in the LIBOR-based money market. We will use the formulation of the evolutionary model as given in Carverhill (1991) and briefly reviewed in Section III; this model can be regarded as a reformulation of Heath et al (1987) or of Ho and Lee (1986). Also we will concentrated on the single-factor version of the model, and will make the assumption, associated with the Vasicek model (which is an ‘Equilibrium Model’ in the terminology of Carverhill (1991)), that the evolution of the term structure as prescribed by the model is independent of the level of the interest rates.

The evolutionary model is contrasted in Carverhill (1991) with the equilibrium model which is represented in such papers as Vasicek (1997) and Cox et al (1985). The basic difference is that the equilibrium model seeks to characterise the term structure at any time, assuming that it is in economic equilibrium, whereas the Evolutionary Model concentrates on the evolution of the term structure from an exogenously given initial condition, rather than on the term structure itself. This basic difference leads to the contrasting strengths and weaknesses of the models; the equilibrium model predicts a shape for the term structure which may not accurately match the actual term structure at any given time. but which one


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hopes to be a reasonable match over all time; whereas the prediction of the evolutionary model will initially be perfect, but will decrease in accuracy as it is pushed further into the future.


90-11 Valuing Average Rate ('Asian') Options Andrew Carverhill and Les Clewlow

The convolution method is an efficient and flexible means of valuing average-rate (Asian) option. It allows us to answer the questions of how often the underlying should be read in taking the average and how the price is affected if underlying returns are not normal.

We will discuss two types of Asian option - floating-strike and fixed strike - but concentrate on the latter. As the name implies, the floating-strike option pays the difference - if positive - between the average value of the underlying on which it is written and the spot value of the underlying when it is exercised. The fixed-strike option pays the difference between the average and a previously agreed strike price. The floating-strike is easier to deal with, but less widely used. We will take all our Asian options to be European-style, and the average to be over the entire life of the option.

Risk, Vol 4, No 3, April 1990 90-10 Finite Difference Techniques for One and Two Dimension Option Valuation

Problems Les Clewlow

Finite difference methods represent an important numerical technique in the valuation of options for which analytical solutions can be obtained.

In this paper we review the finite difference methods which have appeared in the finance literature and we compare their advantages and disadvantages in terms of stability, convergence and efficiency.

The Motivation for this paper is to present a logical development from the simplest one-dimensional method through to more complex one- and two-dimensional methods. We deal with the problems of stability and convergence as they arise naturally within the logical development. We also attempt to show the theoretical relationship between the various methods and their relationship to discrete stochastic processes and iterative methods.

90-09 A Survey of Elementary Techniques for Pricing Options on Bonds and Interest Rates Andrew Carverhill

The pricing techniques that we will survey in this article are the most commonly used by practitioners who write and trade options on bonds and interest rates, and they are all based on the techniques of Black and Scholes as applied to the more elementary problem of pricing options on equities. To implement these techniques we only need to make an appropriate assumption about the volatility of the bond or interest rate in question. We will distinguish between, and we will treat separately, options on bonds and options on interest rates. He former relate to actual Government securities or their associated futures contracts and the latter relate to short money market interest rates such as 3-month or 6-month LIBOR, or their associated futures contracts.

The pricing procedures of this article should be contrasted with the more sophisticated procedures which are based on mathematical models of the term structure of interest rates, ie, models of the comovements of the interest rates over the whole spectrum of maturities. In a planned sequel to this article we will survey these more sophisticated pricing procedures and compare them with the procedures of this article. The term structure models for these more sophisticated valuations are separated into two classes, which we


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refer to as ‘equilibrium’ models (represented by the models of [V] and [CIR]), and ‘evolutionary’ models (represented by the models of [HL], [HJM], [C1] and [C2]).

89-08 Two Factor Models in Option Pricing Les Clewlow, Stewart Hodges and Nick Webber

Practitioners and others are developing and using increasingly complex models for option valuation. It is now commonplace to encounter models involving two or more factors in operational use. Though we might prefer to employ simple single-factor models (such as Black-Scholes or numeral methods which take account of early exercise, eg, the binomial method), they may become unsatisfactorily crude when faced with situations where the usual Black and Scholes assumptions are violated. Such situations are all too common, for example, in valuing long-term warrants or convertibles where the interest rate plays a role of similar importance to that of the asset value. Stochastic volatility models provide another important example: to model asset distributions with fat tails we allow the volatility of the asset to follow a stochastic process itself.

These extensions are not without their cost. Care is needed at all stages of the modelling: setting up an appropriate model, estimating its parameters, calculating model values, and finally interpreting the outputs. The purpose of the current paper is to review the issues involved in working with two-factor models. We don’t claim any originality for the results presented here, but nevertheless we hope it may play a useful role in reviewing an important literature and set of problems.

The paper consists of three main sections. In Section II we discuss the choice and formulation of models. We provide a unified framework within which a variety of models are discussed, and we also consider the issue of under what circumstances the added complexity of a two-factor model is worthwhile. In this section we also discuss some issues concerning the estimation of model parameters: an area where further research is needed. Section III describes the main numerical techniques available. In general they are not difficult to program, but consume much more computing power than, say, the simple binomial method. Finally, in Section IV we use the modelling of stochastic volatility as a case study on two-factor models.


89-07 Optimal Replication of Contingent Claims Under Transactions Costs Stewart Hodges and Anthony Neuberger

The construction of hedging strategies that best replicate the outcomes from options (and other contingent claims) in the presence of transactions costs is an important problem. Leland (1985) presents and describes properties of a method for hedging call options when, in addition to the usual Black and Scholes assumptions, there is a proportional transactions cost. However, this method is in no sense an optimal one. Other important papers, by Davis (1988), Davis and Norman (1988), and Taksar, Klass and Assaf (1988), describe optimal portfolio policies to maximize expected utility over an infinite horizon. While these papers are concerned with optimal policies, they are not concerned with the “financial engineering” problem of replicating a given contingent claim. This paper describes the general problem of best replication of a contingent claim under proportional transactions costs, and other cost structures. The context in which the contingent claim is to be hedged is as follows. It is assumed that a financial intermediary or individual already has selected an optimal portfolio of assets and liabilities. An opportunity occurs to issue a contingent claim (presumably at a favourable price) and to hedge the risk involved by means of transactions in either a futures contract or the underlying asset itself, and risk-free bonds. Exact replication at finite cost is generally either impossible or too expensive to be desirable. The replication problem must therefore


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be defined relative to a loss function. The precise formulation of the problem is conditioned by the optimality of the original portfolio. The problem is one of stochastic optimal control and can be characterized by a dynamic programming (Bellam-Hamilton-Jacobi) equation. In general, it is necessary to solve this equation numerically. Numerical results are provided for a realistic situation of replicating a conventional call option. He paper also derives some further insights into optimal replicating strategies by considering an alternative and simpler contingent claim. The optimal strategies are shown to be considerably better than the alternatives given by the strategies described by Leland. Finally, some general properties are described, and some further extensions are suggested.

The Review of Futures Markets, Vol 8, No 2, pp 223-242 89-06 Financial Engineering: New Approaches to Managing Risk Exposure Stewart Hodges

We are in the middle of a revolution. A quiet revolution, but a revolution nevertheless. It concerns how financial risk is managed. How it is managed by companies, by investment funds and particularly by financial institutions. I shall describe a number of features of this revolution. This will include a brief introduction to the markets and analytical methods involved, and a rather more detailed description of some key ideas in managing money through time, in hedging interest rate risks, and in coping with market imperfections such as transactions costs and thin markets. By way of introduction, let us first take a look at some recent advertising by the industry. An advertisement published by Refco states ‘Risk is everywhere. That’s why you need Refco.’ this kind of advertising would have been almost inconceivable in the UK only five or ten years ago. Refco is advertising itself as a world leader in ‘financial risk management through the use of futures and options’. These few words describe the essence of the new approach to managing financial risk. An important component of this revolution concerns the development of markets in new securities called financial options, and in other derivative instruments, and in the use of these securities. This is what this chapter is about. Futures, although important, have been highlighted elsewhere (for example, in Merton Miller’s (1986) presidential address to the Western Finance Association) and they will not be discussed here and neither will there by much about related markets in swaps. Academic research has played an important role in facilitating many of the current developments. This is also a field in which there is an increasing amount of direct collaboration between the financial and academic communities. I shall cover three kinds of material. First, a brief summary of what has been happening to financial markets will be given. Second, key academic insights that are directly related to the solving of valuation problems and managing funds efficiently through time will be discussed. Finally, two of the areas of current research at the University of Warwick - models of interest rate movements, and the problems of hedging in markets where transactions are costly - will be covered.

New Issues in Financial Engineering, Chapter 6 (Ed: R Kinsella), Blackwell, 1992 89-05 Valuing Interest Rate Options via a Primitive Theory of the Term Structure Andrew Carverhill

The Primitive Theory of the term structure of interest rates has been presented in the paper [C 1989]. It is so called because it is based on a primitive, intuitively easy assumption about how the term structure, or yield curve, evolves. In fact in [C 1989] the theory is worked out in 1-factor form, that factor being the short interest rate, and in this form of the theory, the primitive assumption is as follows: the yield on a bond at a short time into the future will be given by the forward yield implied by current prices, plus a random element which is perfectly correlated to the change in the short rate. This assumption will be stated in full mathematical brutality in Section 2.


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The Primitive Theory is closely related to the theories of [HL] and [HJM], which also deals with the evolution of the term structure. Also the Primitive Theory uses the ideas of the factor analysis of the evolution, which are presented in [S 1989b] and [D]. “Factor Analysis” is a technique which naively identifies the principal components of the evolution. By “naively” here we mean that the technique does not rely on any economic theories about the term structure, but merely looks at the magnitudes involved and calculates various covariances and eigenvectors. In fact one might say that the Primitive Theory is a reformulation of the theory of [HJM], using the frame-of-reference of Factor Analysis, which makes it easy to understand in terms of elasticities, covariances, etc.

89-04 A Primitive Theory of the Term Structure of Interest Rates Andrew Carverhill

Among theories of the term structure of interest rates we distinguish two approaches: that of Cox, Ingersoll and Ross ([CIR]) and Vasicek ([V]), which we identify together and refer to as the “CIR/V theory”, and that of Ho and Lee ([HL]) and Heath, Jarrow and Morton ([HJM]), which we refer to as the “HL/HJM theory”. See also ([C]). Both theories assume that there is just one random factor which drives the term structure, but they differ fundamentally in that the CIR/V theory assumes that the whole of the term structure at a given time is determined by the short rate at that time, and the HL/HJM theory studies the evolution of the term structure from an arbitrary initial structure. These fundamental differences give rise to the contrasting strengths and weaknesses of the two theories; HL/HJM seems preferable in that it is empirically clear that the term structure evolves smoothly but that it does not depend only on the short rate, also CIR/V seems preferable in that it is transparent, and gives a sensible prediction for the long-run behaviour of the term structure, whereas the HL/HJM can easily lead to absurd results in the long-run, such as predicting negative or unbounded interest rates (see [C], [HJM]). Our aim in this paper is to present a “primitive” term structure theory, which captures the attractive features of both the CIR/V and the HL/HJM theories, and yet is not too primitive to be empirically meaningful. The basic assumption of the primitive theory is simply that as time evolves from t t+ (with small), then the difference between the forward price at time t, to run from t+ and mature at time q, and the spot price at time t+ to mature at time q, is perfectly correlated with the evolution of the short rate between times t and t+ . this assumption actually subsumes that of the CIR/V theory, and is very much like that of HL/HJM.

89-03 Expected Turnover in a Binomial Tree Stewart Hodges

NOT CURRENTLY AVAILABLE - UNDER REVISION

The amount of turnover involved in hedging options positions by means of a delta hedge on the underlying asset can be very important when transactions costs are involved. Practitioners often use hedges based on either a Black Scholes (1973) model or a Binomial model, (eg Cox, Ross, Rubinstein (1979)), and make revisions to the hedge at discrete time intervals. The purpose of this paper is to examine how the expected level of transactions turnover implied by delta hedging using a simple (additive) binomial tree depends on the number of time intervals used to characterize the tree. We have found that the relationship between the frequency of revisions of a delta hedging strategy and the expected level of turnover is often poorly understood by even quite sophisticated participants in options markets. The paper shows that the volume of turnover depends on the square root of the number of intervals, and is therefore unbounded as we approach the continuous time case. Leland (1985) contains a similar result. The direct approach adopted here provides some additional insights.

89-02 Modelling the Dynamics of the Term Structure of Interest Rates


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James Steeley In order to provide a tractable bond pricing formulae, the arbitrage theories of the term structure make specific assumptions as to the number, identity and process generating the underlying forcing variables. This paper assesses the empirical plausibility of these common assumptions. It is found that there are three underlying factors, one more than is usually permitted. However, by careful examination of the dynamics of suitable instrumental variables to these factors, it is found that the further factor may be represented by the autogressive conditional volatility of one of these factors. Thus, it can be readily integrated into existing two factor models.

The Economic and Social Review, Vol 21, No 4, (Symposium on Finance), July 1990, pp 337-361

89-01 Estimating the Gilt-Edged Term Structure: Basis Splines and Confidence Intervals James Steeley

Studies estimating the term structure using a linear approximation to the discount function frequently use spline functions. Many of these functions can be shown to generate a regressors’ matrix which is nearly perfectly collinear. This paper presents a form of spline function, “B-splines” which avoid this problem. Procedures to aid application to term structure estimation are given, and are used to estimate the term structure for British Government securities. Rarely are estimation errors reported, which casts doubt upon the results of many studies. Simple, yet robust, formulae are derived and used to demonstrate the strengths of the B-spline methodology.

Journal of Business Finance and Accounting, Vol 18, No 4, June 1991, pp 513-529 88-05 The Ho and Lee Term Structure Theory: A Continuous Time Version Andrew Carverhill

In their paper [HL], Ho and Lee present an innovatory theory of the term structure of interest rates. Their theory differs from those of their predecessors (notably [V]), and [CIR] is concentrating on the evolution of the term structure from its initial shape, rather than on an equilibrium characterisation of what the shape of the term structure should be. It works in discrete time, and with the simplifying (though apparently naive) assumption that the term structure evolves binomially, ie, given the structure at a certain time point, then the structure at the next time point can be one of only two alternatives. However, upon this basis, and using some very reasonable arguments, they are able to characterise completely the evolution of the term structure.

Ho and Lee’s assumption of binomial evolution is not as naive as it might seem; it is equivalent to the assumption that the random input to the evolution is a binomial random walk, and since a random walk is a good approximation to a Brownian Motion when the discrete time increments are small, the binomial assumption is close to the assumption that the term structure is driven by a single Brownian Motion as its random input.

Our aim is to present a continuous time analogue of the arguments and conclusions of the Ho and Lee Theory. This article is meant to stand in relation to that of Ho and Lee in the same way as the usual treatment of the Black-Scholes formula for equity options (see [BS]) stands in relation to the binomial approach to option valuation of [CRR]. Our motivation and conclusions are similar to those of [HJM], and our debt to [HJM] is clear. However, our techniques are different, being directly based on those of [HL].

88-04 Financial Options: New Markets and New Valuation Techniques Stewart Hodges

We are in the middle of a quiet revolution in the way financial risk is managed by companies, investment funds, and particularly financial institutions. In this paper I wish to


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concentrate on one of the most important aspects of this revolution, the development of markets in new securities called financial options, and the use of these securities. I shall not concern myself with the other major development, that of futures, for, while they are important, they have been highlighted elsewhere - for example, in a recent paper by Miller (1986). Academic research has played an important role in facilitating many of the current developments; indeed, this is a field in which there is increasing direct collaboration between the financial and academic communities. I shall describe, first, what has been happening to financial markets; second, what has been happening in terms of academic research; and finally, an agenda for further research.

Options: Recent Advances in Theory and Practice (Ed: S D Hodges), Manchester University Press, 1990, 3-9

88-03 Valuation and Hedging: Theoretical Issues Stewart Hodges

This paper reviews the main theoretical issues related to valuing and hedging options. We first consider the assumptions made by valuation models, and the general form of these models. Option valuations can only be made conditional on inputs which describe the probability distribution of future values on the underlying asset. Our ability to value contingent claims is mostly limited by our ignorance of these distributions. The effectiveness of extensions of Black an Scholes (for example, to take account of dividends, jump-diffusion or CEV processes, or of stochastic volatility) depends on how much they improve the modelling of the distributional characteristics of the underlying asset. Issues concerning valuation and model choice are discussed in the light of this perspective.

Valuations which may be accurate in an equilibrium sense do not necessarily enable price discrepancies to be turned into profitable arbitrage if transactions costs must be incurred or markets are incomplete. The paper reviews various sources of risk in hedged positions and discusses their relative importance. Recent work on hedging in imperfect markets (subject to incompleteness or transaction costs) is reviewed. The difficulties of exactly replicating options positions suggest that we should put different bounds around the value of an instrument, depending on the market conditions, the assumptions, and the securities that would be used to arbitrage any mispricing.


88-02 American Options: Theory and Numerical Analysis Andrew Carverhill and Nick Webber

The aim of this paper is to develop a formula for the value of an American option, and to give a simple and efficient procedure for determining this value numerically. Many of the ideas that we use have been presented before (see references); we see the value of this paper largely as presenting these ideas in a coherent and rigorous way. We will indicate the points of contact with these papers, and our work can serve as an introduction to and review of them.

We will deal mostly with the American put: the option to sell a stock for a price, c, at any time before T. The case of the call is similar and in some ways simpler than that of the put, because optimally it will not be exercised between dividend dates. Our standing assumption about the stock is that it pays no dividend and its price follows a geometric Brownian motion with constant drift and variance parameter.


88-01 Numerical Methods Andrew Carverhill

The aim of this paper is to review and compare the standard approaches to the numerical evaluation of options. We will pay particular attention to accuracy and efficiency, and to


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dividends and the early exercise opportunity in the case of American options. This report can be regarded as an introduction to Chapter 6b, and it leans heavily on Geske and Shastri (1985). We will make the usual assumptions about market behaviour, namely that continuous trading is possible with no transaction costs, and that there is no penalty for selling short, and no taxes. Also, we will assume that the risk-free continuously compounded interest rate is a constant. Our standing assumption about the stock price on which the option is written is that it has constant proportional drift (expected rate of return) and constant proportional volatility.


I:RZM/wfri/forc/pre-print abstract - 03/12/04

options inv research abstract pp_abstract-dec_04

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