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DerivativesAbdulla Al-Othman

About the Author2Name: Abdulla Nouri Abdulla Abdulatif AlOthman

Biography:Education: London School of Economics (B.Sc. First Class Honours, Mathematics & Economics); Masschussets Institute of Technology (M.Sc. Operations Research and Finance -Thesis with Distinction);)

Work Experience: Merrill Lynch (NY / Tokyo ):Proprietary Trader ; Ivy Software (Boston) : CO-CEO; KMBS ( Kuwait) : Adjunct Professor of Economics and Finance.Abdulla Alothman

Table of Contents

Day 1: The Structures

Basic Derivatives: Options, ForwardsStrategies: Yield Enhancement , Trading, Hedging Exotic Derivatives: Digitals, Knockouts, Quantos Complex Derivatives: Ratchets

Day 2: Valuation (Technical)

Methodology : Trading Exercise / Market Price of Risk / Replication Valuation: Model Choice = Market Price of Risk Specification Models: Black and Scholes Model

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Cont

Day 3: Applications

Trading Game (Simulator)The Kuwaiti Dinar as a DerivativeThe Subprime Debt Crisis

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Motivating Example5Abdulla Alothman

Premium6Abdulla Alothman

IQAMASIQAMAS

Payoff Structure8Abdulla Alothman

Subsidized Fuel / Free Public Transportation

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Free* Public Transportation

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Free Electricity

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Free Water

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Free Health

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Day 1The Basics19Abdulla Alothman

BASIC DERIVATIVES Abdulla Alothman20

Call OptionsPut OptionsForward ContractsStrategies

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Call Options22Are contracts, providing the holder, in return for a premium, with the right but not the obligation, to buy the underlying asset, for a fixed price (strike price), at some future date T (expiration date).Abdulla Alothman

={H,T} , P={ 0.5, 0.5}S is a Random VA22

Payoff Diagram c(x)=max (x-K,0)Abdulla Alothman23$KXT

Payoff Table24

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={H,T} , P={ 0.5, 0.5}S is a Random VA24

DerivativesWhom, us?25Abdulla Alothman

JasoomDecides to replace his GT

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And take out a consumer loan / sign an IOU (Bond)27

Yabeela Ferrari

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For 5 years at 6% 286%Abdulla Alothman

One year elapses...29i

Rates fall to 5% Note becomes more valuable. 5%6%

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Jasoom pays off the loan..30

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Jasoom

With a new loan ...31

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For 4 years at 5%... 325%Abdulla Alothman

Jasooms benefits (long a Call Option) c(x)=max (P(5%,4)-P(6%,4),0)Abdulla Alothman33$ Pb(6%)B1 PB(5%)

NBKs losses (short a Call Option) c(x)=-max (P(5%,4)-P(6%,4),0)Abdulla Alothman34$ Pb(6%)B1 PB(5%)

In Summary

Since all banks in Kuwait derive a large part of their revenues from consumer loans; all are, effectively, up to their eye balls in derivatives.

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Put Options36Are contracts providing the holder , in return for a premium, with the right but not the obligation, to sell the underlying asset for a fixed price K (strike price), at some future date T (expiration date).Abdulla Alothman

={H,T} , P={ 0.5, 0.5}S is a Random VA36

Payoff Diagramp(x)=max (K-x,0)Abdulla Alothman37$KXT

Payoff Table38

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Forwards39Are contracts, in which the holder has obligation to buy the underlying asset for a fixed price K (strike price), at some future date T (expiration date).Abdulla Alothman

={H,T} , P={ 0.5, 0.5}S is a Random VA39

Payoff Diagramf(x)=x-KAbdulla Alothman40$KXT

Payoff Table41

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Trading Strategies42We often combine Call and Put options to create assets with interesting payoff structures.

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={H,T} , P={ 0.5, 0.5}S is a Random VA42

Long Call/Short Put43

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={H,T} , P={ 0.5, 0.5}S is a Random VA43

Payoff (Synthetic Forward!!)44

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={H,T} , P={ 0.5, 0.5}S is a Random VA44

Long Put/Short Put45

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={H,T} , P={ 0.5, 0.5}S is a Random VA45

Payoff 46

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={H,T} , P={ 0.5, 0.5}S is a Random VA46

Long/Short Puts47

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Payoff 48

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={H,T} , P={ 0.5, 0.5}S is a Random VA48

Yield Enhancement Strategies49Portfolio managers, often use options to enhance yields on their portfolios

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={H,T} , P={ 0.5, 0.5}S is a Random VA49

Covered Calls50

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={H,T} , P={ 0.5, 0.5}S is a Random VA50

Covered Calls 51

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={H,T} , P={ 0.5, 0.5}S is a Random VA51

Hedging Strategies52Portfolio managers and firms often use options to preserve profits / hedge exposure

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={H,T} , P={ 0.5, 0.5}S is a Random VA52

Long Put53

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={H,T} , P={ 0.5, 0.5}S is a Random VA53

Payoff 54

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={H,T} , P={ 0.5, 0.5}S is a Random VA54

Long Call / Short Put55

K2 K Abdulla Alothman58$K1.00XT

Digital Putsp(x)=1 X < K Abdulla Alothman59$K1XT

Jasoom pays his premiumAbdulla Alothman60

Asset DepreciatesAbdulla Alothman61

Jasoom is hedged! Abdulla Alothman62

Wanassa!

Jasoom(long a digital put option) Abdulla Alothman63$Asset State

Warba (short a digital put option) Abdulla Alothman64$Asset State

Quanto Callsc(x,y)=y*Max(x-K,0)Abdulla Alothman65KY1.25001.351.45

$1.351.25YXT

Quanto Puts c(x,y)=y*Max(K-x,0)Abdulla Alothman66XKY1.25001.351.45

$1.351.25

Knockout CallsThe KnockOUT EFFECTAbdulla Alothman67Knock Out Barrier L$Ktime

Xt

TXT

Strike KBarrier not hit, same as regular callBarrier hit, option knocked out

Knockout Puts The KnockOUT EFFECTAbdulla Alothman68$Ktime (t)Xt

TXT

Barrier hit, option knocked out

Barrier not hit, same as regular putKL

Knockout Digital CallsThe KnockOUT EFFECTAbdulla Alothman69Knock Out Barrier L$Ktime (t)

Xt

TXT

Strike KBarrier not hit, same as regular digital callBarrier hit, option knocked out

Knockout Digital PutsThe KnockOUT EFFECTAbdulla Alothman70$Ktime (t)Xt

TXT

Barrier hit, option knocked out

Barrier not hit, same as regular digital putKL

Ratchets (Gulf Banks Web Site) Abdulla Alothman71

A Knockout Ratchet Building Blocks

KnockOut F.X. Call OptionsKnockOut F.X. Digital Call OptionsCall Option Strike Price $1.5400KnockOut Level $1.2400

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Example Cont....EUR/USDAbdulla Alothman73 $1.24

EURt$1.54sTRUCTURE (Simplified)Payoff Diagram

Time t1t2t3 t1t2t3$1.56$1.60Scenario

$0.01$0.01$0.01$0.00$0.00$0.06Payoff:$0.01 $0.01$0.07Payoff:$0.01 $0.03$0.00$0.01$0.01

$0.02$0.02$0.01$0.00$0.00$0.00Remaining payments Knocked Out12

A Call /Put RatchetBuilding Blocks

(Call Option + Put Option )/2Call Option Strike Price $1.6500Put Option Strike Price $1.3500

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Example 2 Cont....EUR/USDAbdulla Alothman75 $1.35

EURt$1.65sTRUCTURE (Simplified)Payoff Diagram

Time t1t2t3 t1t2t3Scenario

$0.00$0.00$0.00$0.00$0.00$0.05Payoff:$0.00 $0.00$0.05

$1.75 $1.25$0.03$0.03$0.0012$0.04$0.04$0.04Payoff: $0.04$0.07 $0.07

Day 2Valuation76Abdulla Alothman

At a time whenCash was the only asset77Abdulla Alothman

The King decidedTo introduce another78Abdulla Alothman

With Payoffs Determined by the flip of a fair coin79Abdulla Alothman

And pricing Left to Market Forces 80Abdulla Alothman

Asset Valuation (Group Exercise)

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Analysis

Arbitrage Free Market Price Range

Risk Averse Market Price Range

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={H,T} , P={ 0.5, 0.5}S is a Random VA82

Utility of WealthAsset Payoff

Market Price of Risk and Utility Theory ...

Class MarketPrice of Risk = 583

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Market Price of Risk and Risk Adjusted Probabilities

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={H,T} , P={ 0.5, 0.5}S is a Random VA84

Asset Pricing FormulaAbdulla Alothman85

The Delighted KingNow decides to introduce a Call Option 86Abdulla Alothman

With Payoff Determined by the flip of a fair coin87Abdulla Alothman

88Option Valuation (Group Exercise)Abdulla Alothman

Replication

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Option Pricing FormulaAbdulla Alothman90

45o

Modeling in Practice

M1Q1Q3M3Q2M2Q4M4

91Implied Risk Neutral Probability MeasureModel ChoiceAbdulla Alothman

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Cont...

Q2M2Q3M3Millions of Models to Plough and to SowThe Future s UncertainThe Price We Dont Know!!!

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MORAL TO THE STORY

WITH PROBABILITY ONEANY MODEL YOU CHOOSE WILL BE WRONG!!!

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Black and Scholes Model 197394Abdulla Alothman

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Black and Scholes Model 197395Abdulla AlothmanIn the Black and Scholes Economy, the parameters are constant and so we can solve the above equations explicitly to obtain:

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Black and Scholes Model 197396Abdulla AlothmanThe risk adjusted probability measure in this economy is give by:

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Black and Scholes Model 197397Abdulla AlothmanBut then, mimicking the Al-Othman* analysis, mutatis mutandis, we see that:

*In the previous analysis it was implicitly assumed the r = 0 and so B =1

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Black and Scholes Model 197398Abdulla Alothman

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Black and Scholes Model 197399Abdulla AlothmanSo Value of the Option is:

Since, in the Black and Scholes Model:

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Black and Scholes Model 1973100Abdulla Alothman

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The Formula (Yawn, Yawn)101Abdulla Alothman

Substituting 2 and 3 into 1 gives:

The Black and Scholes formula for a European Call Option!!

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Black and Scholes Model The Greeks102Abdulla Alothman

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BINOMIAL MODELS

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The Two Period Binomial Model

NO ARBITRAGE CONDITION:104

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The Risk Adjusted Probabilities in the Binomial Model:

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Asset Valuation in the Two Period Binomial Model

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Step1: Set Up a Replicating Portfolio

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Step2: Calculate the Replicating Portfolio Weights:

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Step3: Use Market Data to Value the Replicating Portfolio:

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Summary:110

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ARBITRAGE111Time 0PricesArbitrageStrategyPayoffState uPayoff State dCase1Case2Profit

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Ok. Lets Test Drive....112

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Market Data + Model Parameters:

NO ARBITRAGE CONDITION:

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Calculating the Risk Adjusted Probabilities:

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Objective: Value the Below

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Step1: Set Up a Replicating Portfolio

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Step2: Calculate Portfolio Weights:117

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Step3: Use Market Data to Price the Replicating Portfolio:

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Step4: Check Results:

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Market Data + Model Parameters:

NO ARBITRAGE CONDITION:

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121Exercise 1 : Using the information in the previous slide , find the replicating portfolio for the Option below and use this to determine it's fair value. Check your result against that obtained by taking the expected payoff with respect to the risk adjusted probabilities .

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122Exercise 2 : Using the information in the previous slide , find the replicating portfolio for the Option below and use this to determine it's fair value. Check your result against that obtained by taking the expected payoff with respect to the risk adjusted probabilities .

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Day 3Applications123Abdulla Alothman

Abdulla Alothman124Option Valuation Game: Arbitrage, Replication and Hedging

Multi Dimensional Derivatives: The Kuwaiti Dinar

Structured Assets: Mortgage Backed Securities and the Global Debt Crisis