DERIVATIVES, RISK MANAGEMENT & VALUE

Transcription

1 DERIVATIVES, RISK MANAGEMENT & VALUE

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3 DERIVATIVES, RISK MANAGEMENT & VALUE Mondher Bellalah Université de Cergy-Pontoise, France World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. DERIVATIVES, RISK MANAGEMENT & VALUE Copyright 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN Typeset by Stallion Press enquiries@stallionpress.com Printed in Singapore.

5 DEDICATION I dedicate this book to the President of the Tunisian Republic, his Excellency Mr. Zine El Abidine Ben Ali, in recognition of his continuous and pivotal support for Science and its men in Tunisia. Over the last twenty-plus years, the scientific achievements of Tunisian researchers in various fields were of high importance. I do believe that the distinction of the scientific research in Tunisia, in relation to other countries in the Euro-Mediterranean region, is due to the particular efforts of Mr. Ben Ali. In the field of Finance, the scientific community and I in Tunisia were lucky to benefit from his particular attention. Indeed, by placing the International Finance Conferences that I organized and headed under his high patronage, they gained a remarkable international reputation. With this opportunity, Ph.D. students, college professors and professionals were able to communicate with professors and experts from the best U.S. universities and institutions, alongside Nobel laureates such as H. Markowitz and J. Heckman. Another important insight of Mr. Ben Ali s role is related to the setting of national awards to strengthen the mechanics of the scientific research at undergraduate and graduate levels. Such awards boost one s willingness to improve the Tunisian economy and its ability to meet the new challenges posed by the international context. Furthermore, the creation of Ben Ali s Chair for the dialogue of civilizations and religions in 2001 had a key role in the enrichment of knowledge and human values in a multi-religions context. In addition, it is considered as the Mecca of researchers from all over the world who are involved in bringing new approaches to make people closer. Mondher Bellalah v

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7 FOREWORD by Edward C. Prescott (Arizona State University; Federal Reserve Bank of Minneapolis) This book covers the main aspects regarding derivatives, risk, and the role of information and financial innovation in capital markets and in the banking system. An analysis is provided regarding financial markets and financial instruments and their role in the financial crisis. This analysis hopefully will be useful in avoiding or at least mitigating future financial crises. The book presents the principal concepts, the basics, the theory, and the practice of virtually all types of financial derivatives and their use in risk management. It covers simple vanilla options as well as structured products and more exotic derivative transactions. Special attention is devoted to risk management, value at risk, credit valuation, credit derivatives, and recent pricing methodologies. This book is not only useful for specific courses in risk management and derivatives, but also is a valuable reference for users and potential users of derivatives and more generally for those with risk management responsibilities. vii

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9 FOREWORD by Harry M. Markowitz (University of California, San Diego) Herein follows a remarkable volume, suitable as both a textbook and a reference book. Mondher Bellalah starts with an introduction to options and basic hedges built from specific options. He then presents an accessible account of the formulae used in valuing options. This account includes historically important formulae as well as the currently most used results of Black Scholes, Merton and others. Bellalah then proceeds to the main task of the volume, to show how to value an endless assortment of exotic options. Mondher Bellalah is to be congratulated for this tour de force of the field. ix

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11 FOREWORD by James J. Heckman (University of Chicago and University College Dublin) Mondher Bellalah offers a lucid and comprehensive introduction to the important field of modern asset pricing. This field has witnessed a remarkable growth over the past 50 years. It is an example of economic science at its best where theory meets data, and shapes and improves on reality. Economic theory has suggested a variety of new and exotic financial instruments to spread risk. Created from the minds of theorists and traders guided by theory, these instruments are traded in large volume and now define modern capital markets. Bellalah offers a step-by-step introduction to this evolving theory starting from its classical foundations. He takes the reader to the frontier by systematically building up the theory. His examples and intuition are splendid and the formal proofs are clearly stated and build on each other. I strongly recommend this book to anyone seeking to gain a deep understanding of the intricacies of asset pricing. xi

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13 FOREWORD by George M. Constantinides (University of Chicago) Both the trading of options and the theory of option pricing have long histories. The first use of option contracts took place during the Dutch tulip mania in the 17th century. Organized trading in calls and puts began in London during the 18th century, but such trading was banned on several occasions. The creation of the Chicago Board Options Exchange (CBOE) in 1973 greatly encouraged the trading of options. Initially, trading took place at the CBOE only in calls of 16 common stocks, but soon expanded to many more stocks, and in 1977, put options were also listed. The great success of option trading at the CBOE contributed to their trading in other exchanges, such as the American, Philadelphia and Pacific Stock Exchanges. Currently, daily option trading is a multibillion-dollar global industry. The theory of option pricing has had a similar history that dates to Bachelier (1900). Sixty-five years after Bachelier s remarkable study, Samuelson (1965) revisited the question of pricing a call. Samuelson recognized that Bachelier s assumption that the price of the underlying asset follows a continuous random walk leads to negative asset prices, and thus makes a correction by assuming a geometric continuous random walk. Samuelson obtained a formula very similar to the Black Scholes Merton formula, but discounted the cash flows of the call at the expected rate of return of the underlying asset. The seminal papers of Black and Scholes (1973) and Merton (1973) ushered in the modern era of derivatives. This is a lucid textbook treatment of the principles of derivatives pricing and hedging. At the same time, it is an exhaustively comprehensive encyclopedia of the vast array of exotic options, fixed-income options, corporate claims, credit derivatives and real options. Written by an expert in the field, Mondher Bellalah s comprehensive and rigorous book is an indispensable reference on any professional s desk. xiii

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15 ABOUT THE AUTHOR After obtaining his Ph.D. in Finance in 1990 at France s leading University Paris-Dauphine, Mondher Bellalah began his career both as a Professor of Finance (HEC, INSEAD, University of Maine, and University of Cergy- Pontoise) and as an international consultant and portfolio manager. He started out as a market maker on the Paris Bourse, before being put in charge of BNP s financial engineering research team as Head of Derivatives and Structured Products. Dr. Mondher has acted as an advisor to various leading financial institutions, including BNP, Rothschild Bank, Euronext, Houlihan Lokey Howard & Zukin, Associés en Finance, the NatWest, Central Bank of Tunisia,DubaiHolding,etc.,andhasbeenChiefRiskOfficer,ManagingDirector inalternativeinvestments,headofcapitalmarketsandheadoftrading. Dr. Mondher has also enjoyed a distinguished academic career as a tenured Professor of Finance at the University of Cergy-Pontoise in Paris for about 20 years. During this time, he has authored more than 14 books and 150 articles in leading academic and professional journals, and was awarded the Turgot Prize for the best French-language book on risk management in English-language books co-written/co-edited by Dr. Mondher include Options, Futures and Exotic Derivatives: Theory, Application and Practice published by John Wiley in 1998, and Risk Management and Value: Valuation and Asset Pricing published by World Scientific in His French-language books include Quantitative Portfolio Management and New Financial Markets; Options, Futures and Risk Management; andrisk Management and Classical and Exotic Derivatives. Dr. Mondher is an associate editor of the International Journal of Finance, Journal of Finance and Banking and International Journal of Business, and has been published in leading academic journals including Financial Review, Journal of Futures Markets, International Journal of Finance, International Journal of Theoretical and Applied Finance as well as in the Harvard Business Review. xv

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17 CONTENTS Dedication Foreword by Edward C. Prescott Foreword by Harry M. Markowitz Foreword by James J. Heckman Foreword by George M. Constantinides About the Author v vii ix xi xiii xv PART I. FINANCIAL MARKETS AND FINANCIAL INSTRUMENTS: BASIC CONCEPTS AND STRATEGIES 1 CHAPTER 1. FINANCIAL MARKETS, FINANCIAL INSTRUMENTS, AND FINANCIAL CRISIS 3 Chapter Outline Introduction Trading Characteristics of Commodity Contracts: The Case of Oil Fixedprices Floating prices Exchange of futures for Physical (EFP) Description of Markets and Instruments: The Case of the International Petroleum Exchange Characteristics of Crude Oils and Properties of Petroleum Products Specific features of some oil contracts Description of Markets and Trading Instruments: The Brent Market xvii

18 xviii Derivatives, Risk Management and Value 1.4. Description of Markets and Trading Instruments: The Case of Cocoa How do the futures and physicals market work? Arbitrage How is the ICCO price for cocoa beans calculated? Information on how prices are affected by changing economic factors? Cocoa varieties Commodities Market participants: The case of cocoa, coffee, and white sugar Trading Characteristics of Options: The Case of Equity Options Options on equity indices Options on index futures Index options markets around the world Stock Index Markets and the underlying indices in Europe Trading Characteristics of Options: The Case of Options on Currency Forwards and Futures Trading Characteristics of Options: The Case of Bonds and BondOptionsMarkets The specific features of classic interest rate instruments The specific features of mortgage-backed securities The specific features of interest rate futures, options, bond options, and swaps Simple and Complex Financial Instruments The Reasons of Financial Innovations Derivatives Markets in the World: Stock Options, Index Options, Interest Rate and Commodity Options and Futures Markets Global overview The main indexes around the world: a historical perspective Summary Questions Exercises Appendix References

19 Contents xix CHAPTER 2. RISK MANAGEMENT, DERIVATIVES MARKETS AND TRADING STRATEGIES 67 Chapter Outline Introduction Introduction to Commodity Markets: The Case of Oil Oilfuturesmarkets Oil futures exchanges Delivery procedures The long-term oil market Pricing Models The pricing of forward and futures oil contracts Relationship to physical market Term structure of prices Pricing swaps The pricing of forward and futures commodity contracts: General principles Forward prices and futures prices: Some definitions Futures contracts on commodities Futures contracts on a security with no income Futures contracts on a security with a known income Futures contracts on foreign currencies Futures contracts on a security with a discrete income Valuation of interest rate futures contracts The pricing of future bond contracts Trading Motives: Hedging, Speculation, and Arbitrage Hedging using futures markets Hedging: The case of cocoa Hedging: The case of oil Hedging: The case of petroleum products futures contracts The use of futures contracts by petroleum products marketers, jobbers, consumers, andrefiners... 82

20 xx Derivatives, Risk Management and Value Speculation using futures markets Arbitrage and spreads in futures markets The Main Bounds on Option Prices Boundary conditions for call options Boundary conditions for put options Some relationships between call options Some relationships between put options Otherproperties Simple Trading Strategies for Options and their Underlying Assets Trading the underlying assets Buying and selling calls Buying and selling puts Some Option Combinations The straddle The strangle OptionSpreads Bull and bear spreads with call options Bull and bear spreads with put options Box spread Definitions and examples Trading a box spread Butterfly Strategies Butterfly spread with calls Butterfly spread with puts CondorStrategies Condor strategy with calls Condor strategy with puts Ratio Spreads Some Combinations of Options with Bonds and Stocks Covered call: short a call and hold the underlying asset Portfolio insurance Mimicking portfolios and synthetic instruments Mimicking the underlying asset Synthetic underlying asset: Long call plus a short put and bonds The synthetic put: put-call parity relationship

21 Contents xxi Conversions and Reversals Case study: Selling Calls (Without Holding the Stocks/as an Alternative to Short Selling Stocks/the Idea of Selling Calls is Also an Alternative to Buying Puts) Data and assumptions Selling calls (without holding the stock) Comparing the strategy of selling calls (with a short portfolio of stocks): the extreme case Selling calls (holding the stock) Leverage in selling call options (without holding the stocks) Selling Call options (without holding the stocks) Leverage in selling Call options (without holding the stocks): The extreme case Selling calls using leverage (and holding the stock) Short sale of the stocks without options Buying Calls on EMA Buying a call as an alternative to buying the stock: (also as an alternative to short sell put options) Data and assumptions Pattern of risk and return Compare buying calls (as an alternative to portfolio ofstocks) Risk return in options Example by changing volatility to 20% Data and assumptions: Compare buying calls (as an alternative toportfolioofstocks.) Leverage in buying call options (without selling the underlying) Summary Questions Case Study: Comparisons Between put and Call Options Buying Puts and Selling Puts Naked Buying puts Selling puts

22 xxii Derivatives, Risk Management and Value 2. Buying and Selling Calls Buying calls Selling a call Strategy of Buying a Put and Hedge and Selling a Put and Hedge Strategy of selling put and hedge: sell delta units of the underlying Strategy of buy put and hedge: buy delta units of the underlying Strategy of Buy Call, Sell Put, and Buy Call, Sell Put and Hedge Strategy of Buy Call, Sell Put: Equivalent to Holding the Underlying Strategy of Buy Call, Sell Put and Hedge: Reduces Profits and Reduces Losses References CHAPTER 3. TRADING OPTIONS AND THEIR UNDERLYING ASSET: RISK MANAGEMENT IN DISCRETE TIME 141 Chapter Outline Introduction Basic Strategies and Synthetic Positions Options and synthetic positions Long or short the underlying asset Long a call Shortcall Long a put Short a put Combined Strategies Long a straddle Short a straddle Long a strangle Short a strangle Long a tunnel Short a tunnel Long a call bull spread Long a put bull spread

23 Contents xxiii Long a call bear spread Selling a put bear spread Long a butterfly Short a butterfly Long a condor Short a condor How Traders Use Option Pricing Models: Parameter Estimation Estimation of model parameters Historical volatility Implied volatilities and option pricing models Trading and Greek letters Summary Case Studies Exercises Questions References PART II. PRICING DERIVATIVES AND THEIR UNDERLYING ASSETS IN A DISCRETE-TIME SETTING 219 CHAPTER 4. OPTION PRICING: THE DISCRETE- TIME APPROACH FOR STOCK OPTIONS 221 Chapter Outline Introduction The CRR Model for Equity Options The mono-periodic model The multiperiodic model Applications and examples Applications of the CRR model within two periods Other applications of the binomial model of CRR for two periods Applications of the binomial model of CRRforthreeperiods Examples with five periods

24 xxiv Derivatives, Risk Management and Value 4.2. The Binomial Model and the Distributions to the Underlying Assets The Put-Call parity in the presence of several cash-distributions Early exercise of American stock options The model Simulations for a small number of periods Simulations in the presence of two dividend dates Simulations for different periods and several dividends: The general case Summary Questions Appendix:TheLatticeApproach References CHAPTER 5. CREDIT RISKS, PRICING BONDS, INTEREST RATE INSTRUMENTS, AND THE TERM STRUCTURE OF INTEREST RATES 259 Chapter Outline Introduction Time Value of Money and the Mathematics of Bonds Single payment formulas Uniform-series present worth factor (USPWF) and the capital recovery factor (CRF) Uniform-series compound-amount factor (USCAF ) and the sinking fund factor (SFF ) Nominal interest rates and continuous compounding Pricing Bonds A coupon-paying bond Zero-coupon bonds Computation of the Yield or the Internal Rate of Return How to measure the yield TheCY The YTM TheYTC The potential yield from holding bonds Price Volatility Measures: Duration and Convexity Duration

25 Contents xxv Duration of a bond portfolio Modified duration Price volatility measures: Convexity The Yield Curve and the Theories of Interest Rates The shapes of the yield curve Theories of the term structure of interest rates The pure expectations theory The YTM and the Theories of the Term Structure of Interest Rates Computing the YTM Market segmentation theory of the term structure Spot Rates and Forward Interest Rates The theoretical spot rate Forward rates IssuingandRedeemingBonds Mortgage-Backed Securities: The Monthly Mortgage Payments for a Level-Payment Fixed-Rate Mortgage Interest Rate Swaps The pricing of interest rate swaps The swap value as the difference between the prices of two bonds The valuation of currency swaps Computing the swap Summary Questions References CHAPTER 6. EXTENSIONS OF SIMPLE BINOMIAL OPTION PRICING MODELS TO INTEREST RATES AND CREDIT RISK 293 Chapter Outline Introduction The Rendleman and Bartter Model (for details, refer to Bellalah et al., 1998) for Interest-Rate Sensitive Instruments Using the model for coupon-paying bonds Ho and Lee Model for Interest Rates and Bond Options The binomial dynamics of the term structure The binomial dynamics of bond prices

26 xxvi Derivatives, Risk Management and Value Computation of bond prices in the Ho and Lee model Option pricing in the Ho and Lee model Deficiency in the Ho and Lee model Binomial Interest-Rate Trees and the Log-Normal Random Walk The Black-Derman-Toy Model (BDT) Examples and applications Trinomial Interest-Rate Trees and the Pricing of Bonds The model Applications of the binomial and trinomial models 316 Summary Questions Appendix A: Ho and Lee model and binomial dynamics of bond prices References CHAPTER 7. DERIVATIVES AND PATH-DEPENDENT DERIVATIVES: EXTENSIONS AND GENERALIZATIONS OF THE LATTICE APPROACH BY ACCOUNTING FOR INFORMATION COSTS AND ILLIQUIDITY 327 Chapter Outline Introduction The Standard Lattice Approach for Equity Options: The StandardAnalysis The model for options on a spot asset with any pay outs The model for futures options The model with dividends A known dividend yield A known proportional dividend yield A known discrete dividend Examples The European put price with dividends The American put price with dividends A Simple Extension to Account for Information Uncertainty in the Valuation of Futures and Options

27 Contents xxvii On the valuation of derivatives and information costs The valuation of forward and futures contracts in the presence of information costs Forward, futures, and arbitrage The valuation of forward contracts in the absence of distributions to the underlying asset The valuation of forward contracts in the presence of a known cash income to the underlying asset The valuation of forward contracts in the presence of a known dividend yield to the underlying asset The valuation of stock index futures The valuation of Forward and futures contracts on currencies The valuation of futures contracts on silver and gold The valuation of Futures on other commodities Arbitrage and information costs in the lattice approach The binomial model for options in the presence of a continuous dividend stream and information costs The binomial model for options in the presence of a known dividend yield and information costs The binomial model for options in the presence of a discrete dividend stream and information costs The binomial model for futures options in the presence of information costs The lattice approach for American options with information costs and several cash distributions The model The Binomial Model and the Risk Neutrality: Some Important Details The binomial parameters and risk neutrality The convergence argument

28 xxviii Derivatives, Risk Management and Value 7.4. The Hull and White Trinomial Model for Interest Rate Options Pricing Path-Dependent Interest Rate Contingent Claims Using a Lattice The framework Valuation of the path-dependent security Fixed-coupon rate security Floating-coupon security Options on path-dependent securities Short-dated options Long-dated options Summary Questions References PART III. OPTION PRICING IN A CONTINUOUS-TIME SETTING: BASIC MODELS, EXTENSIONS AND APPLICATIONS 365 CHAPTER 8. EUROPEAN OPTION PRICING MODELS: THE PRECURSORS OF THE BLACK SCHOLES MERTON THEORY AND HOLES DURING MARKET TURBULENCE 367 Chapter Outline Introduction Precursors to the Black Scholes Model Bachelier formula Sprenkle formula Boness formula Samuelson formula How the Black Scholes Option Formula is Obtained The short story The differential equation The derivation of the formula Publication of the formula Testing the formula Financial Theory and the Black Scholes Merton Theory The Black Scholes Merton theory Analytical formulas

29 Contents xxix 8.4. The Black Scholes Model The Black Scholes model and CAPM An alternative derivation of the Black Scholes model The put-call parity relationship Examples The Black Model for Commodity Contracts The model for forward, futures, and option contracts The put-call relationship Application of the CAPM Model to Forward and Futures Contracts An application of the model to forward and futures contracts An application to the derivation of the commodity option valuation An application to commodity options and commodity futures options The Holes in the Black Scholes Merton Theory and the FinancialCrisis Volatility changes Interest rate changes Borrowing penalties Short-selling penalties Transaction costs Taxes Dividends Takeovers Summary Questions Appendix A. The Cumulative Normal Distribution Function Appendix B. The Bivariate Normal Density Function References CHAPTER 9. SIMPLE EXTENSIONS AND APPLICATIONS OF THE BLACK SCHOLES TYPE MODELS IN VALUATION AND RISK MANAGEMENT 403 Chapter Outline Introduction

30 xxx Derivatives, Risk Management and Value 9.1. Applications of the Black Scholes Model Valuation and the role of equity options Valuation and the role of index options Analysis and valuation Arbitrage between index options and futures Valuation of options on zero-coupon bonds Valuation and the role of short-term options on long-term bonds Valuation of interest rate options Valuation and the role of bond options: the case of coupon-paying bonds The valuation of a swaption Applications of the Black s Model Options on equity index futures Options on currency forwards and options on currency futures Options on currency forwards Options on currency futures The Black s model and valuation of interest rate caps The Extension to Foreign Currencies: The Garman and Kohlhagen Model and its Applications The currency call formula The currency put formula The interest-rate theorem and the pricing of forward currency options The Extension to Other Commodities: The Merton, Barone-Adesi and Whaley Model, and Its Applications The model An application to portfolio insurance The Real World and the Black Scholes Type Models Volatility The hedging strategy The log-normal assumption A world of finite trading Total variance Black Scholes as the limiting case Using the model to optimize hedging

31 Contents xxxi Summary Questions Appendix References CHAPTER 10. APPLICATIONS OF OPTION PRICING MODELS TO THE MONITORING AND THE MANAGEMENT OF PORTFOLIOS OF DERIVATIVES IN THE REAL WORLD 439 Chapter Outline Introduction Option-Price Sensitivities: Some Specific Examples Delta Gamma Theta Vega Rho Elasticity Monitoring and Managing an Option Position in Real Time Simulations and analysis of option price sensitivities using Barone-Adesi and Whaley model Monitoring and adjusting the option position in real time Monitoring and managing the delta Monitoring and managing the gamma Monitoring and managing the theta Monitoring and managing the vega The Characteristics of Volatility Spreads Summary Appendix A: Greek-Letter Risk Measures in Analytical Models A.1. B S model A.2. Black s Model A.3. Garman and Kohlhagen s model A.4. Merton s and Barone-Adesi and Whaley s model. 463 Appendix B: The Relationship Between Hedging Parameters Appendix C: The Generalized Relationship Between the Hedging Parameters Appendix D: A Detailed Derivation of the Greek Letters Questions References

32 xxxii Derivatives, Risk Management and Value PART IV. MATHEMATICAL FOUNDATIONS OF OPTION PRICING MODELS IN A CONTINUOUS-TIME SETTING: BASIC CONCEPTS AND EXTENSIONS 491 CHAPTER 11. THE DYNAMICS OF ASSET PRICES AND THE ROLE OF INFORMATION: ANALYSIS AND APPLICATIONS IN ASSET AND RISK MANAGEMENT 493 Chapter Outline Introduction Continuous Time Processes for Asset Price Dynamics Asset price dynamics and Wiener process Asset price dynamics and the generalized Wiener process Asset price dynamics and the Ito process The log-normal property Distribution of the rate of return Ito s Lemma and Its Applications Intuitive form Applications to stock prices Mathematical form The generalized Ito s formula Other applications of Ito s formula Taylor Series, Ito s Theorem and the Replication Argument The relationship between Taylor series and Ito s differential Ito s differential and the replication portfolio Ito s differential and the arbitrage portfolio Why are error terms neglected? Forward and Backward Equations The Main Concepts in Bond Markets and the General Arbitrage Principle The main concepts in bond pricing Time-dependent interest rates and information uncertainty The general arbitrage principle Discrete Hedging and Option Pricing Discrete hedging

33 Contents xxxiii Pricing the option The real distribution of returns and the hedgingerror Summary Questions Appendix A: Introduction to Diffusion Processes Appendix B: The Conditional Expectation Appendix C: Taylor Series Exercises References CHAPTER 12. RISK MANAGEMENT: APPLICATIONS TO THE PRICING OF ASSETS AND DERIVATIVES IN COMPLETE MARKETS 535 Chapter Outline Introduction Characterization of Complete Markets Pricing Derivative Assets: The Case of Stock Options The problem The PDE method The martingale method Pricing Derivative Assets: The Case of Bond Options and Interest Rate Options Arbitrage-free family of bond prices Time-homogeneous models Time-inhomogeneous models Asset Pricing in Complete Markets: Changing Numeraire and Time Assumptions and the valuation context Valuation of derivatives in a standard Black Scholes Merton economy Changing numeraire and time: The martingale approach and the PDE approach Valuation in an Extended Black and Scholes Economy in the Presence of Information Costs Summary Questions Appendix A: The Change in Probability and the Girsanov Theorem 564

34 xxxiv Derivatives, Risk Management and Value Appendix B: Resolution of the Partial Differential Equation for a European Call Option on a Non-Dividend Paying Stock in the Standard Context Appendix C: Approximation of the Cumulative Normal Distribution 571 Appendix D: Leibniz s Rule for Integral Differentiation Appendix E: Pricing Bonds: Mathematical Foundations Exercises References CHAPTER 13. SIMPLE EXTENSIONS AND GENERALIZATIONS OF THE BLACK SCHOLES TYPE MODELS IN THE PRESENCE OF INFORMATION COSTS 583 Chapter Outline Introduction Differential Equation for a Derivative Security on a Spot Asset in the Presence of a Continuous Dividend Yield and InformationCosts The Valuation of Securities Dependent on Several Variables in the Presence of Incomplete Information: A General Method The General Differential Equation for the Pricing of Derivatives Extension of the Risk-Neutral Argument in the Presence of InformationCosts Extension to Commodity Futures Prices within Incomplete Information Differential equation for a derivative security dependent on a futures price in the presence of information costs Commodity futures prices Convenience yields Summary Questions Appendix A: A General Equation for Derivative Securities Appendix B: Extension to the Risk-Neutral Valuation Argument. 596 Exercises References

35 Contents xxxv PART V. EXTENSIONS OF OPTION PRICING THEORY TO AMERICAN OPTIONS AND INTEREST RATE INSTRUMENTS IN A CONTINUOUS-TIME SETTING: DIVIDENDS, COUPONS AND STOCHASTIC INTEREST RATES 613 CHAPTER 14. EXTENSION OF ASSET AND RISK MANAGEMENT IN THE PRESENCE OF AMERICAN OPTIONS: DIVIDENDS, EARLY EXERCISE, AND INFORMATION UNCERTAINTY 615 Chapter Outline Introduction The Valuation of American Options: The General Problem Early exercise of American calls Early exercise of American puts The American put option and its critical stock price Valuation of American Commodity Options and Futures Options with Continuous Distributions Valuation of American commodity options Examples and applications Valuation of American futures options Examples and applications Valuation of American Commodity and Futures Options with Continuous Distributions within Information Uncertainty Commodity option valuation with information costs Simulation results Valuation of American Options with Discrete Cash-Distributions Early exercise of American options Valuation of American options with dividends Valuation of American Options with Discrete Cash Distributions within Information Uncertainty The model Simulation results The Valuation Equations for Standard and Compound Options with Information Costs The pricing of assets under incomplete information The valuation of equity as a compound option.. 650

36 xxxvi Derivatives, Risk Management and Value Summary Questions Appendix A: An Alternative Derivation of the Compound Option s Formula Using the Martingale Approach Exercises References CHAPTER 15. RISK MANAGEMENT OF BONDS AND INTEREST RATE SENSITIVE INSTRUMENTS IN THE PRESENCE OF STOCHASTIC INTEREST RATES AND INFORMATION UNCERTAINTY: THEORY AND TESTS 667 Chapter Outline Introduction The Valuation of Bond Options and Interest Rate Options The problems in using the B S model for interest-rate options Sensitivity of the theoretical option prices to changes in factors A Simple Non-Parametric Approach to Bond Futures Option Pricing Canonical modeling and option pricing theory Assessing the distribution of the underlying futures price Transforming actual probabilities into risk-neutral probabilities Qualitative comparison of Black and canonical model values One-Factor Interest Rate Modeling and the Pricing of Bonds: The General Case Bond pricing in the general case: The arbitrage argument and information costs Pricing callable bonds within information uncertainty Fixed Income Instruments as a Weighted Portfolio of Power Options Merton s Model for Equity Options in the Presence of Stochastic Interest Rates: Two-Factor Models

37 Contents xxxvii The model in the presence of stochastic interest rates Applications of Merton s model Some Models for the Pricing of Bond Options An extension of the Ho-Lee model for bond options The Schaefer and Schwartz model The Vasicek (1977) model The Ho and Lee model The Hull and White model Summary Questions Appendix A: Government Bond Futures and Implicit Embedded Options A.1. Criteria for the CTD A.2. Yield changes A.3. The value for a futures position A.4. Parallel yield shift A.5. Relative yield shift Appendix B: One-Factor Fallacies for Interest Rate Models B.1. Themodelsinpractice B.2. Spreads between rates Appendix C: Merton s Model in the Presence of Stochastic Interest Rates References CHAPTER 16. MODELS OF INTEREST RATES, INTEREST-RATE SENSITIVE INSTRUMENTS, AND THE PRICING OF BONDS: THEORY AND TESTS 703 Chapter Outline Introduction Interest Rates and Interest-Rate Sensitive Instruments Zero-coupon bonds Term structure of interest rates Forward interest rates Short-term interest rate Coupon-bearing bonds Yield-to-Maturity (YTM) Market conventions

38 xxxviii Derivatives, Risk Management and Value Interest Rates and the Pricing of Bonds The instantaneous interest rates under certainty The instantaneous interest rate under uncertainty Interest Rate Processes and the Pricing of Bonds and Options The Vasicek model The Brennan and Schwartz model The CIR model The Ho and Lee model The HJM model The BDT model The Hull and White model Fong and Vasicek model Longstaff and Schwartz model The Relative Merits of the Competing Models A Comparative Analysis of Term Structure Estimation Models The construction of the term structure and coupon bonds Fitting functions and estimation procedure Term Premium Estimates From Zero-Coupon Bonds: New Evidence on the Expectations Hypothesis Distributional Properties of Spot and Forward Interest Rates: USD, DEM, GBP, and JPY Interest rate levels Interest rate differences and log differences Summary Appendix A: An Application of Interest Rate Models to Account for Information Costs: An Exercise A.1. An application of the HJM model in the presence of informationcosts A.1.1. The forward rate equation A.1.2. The spot rate process A.1.3. The market price of risk A.1.4. Relationship between risk-neutral forward rate drift and volatility A.1.5. Pricing derivatives A.2. An application of the Ho and Lee model in the presence of information cost

39 Contents xxxix Appendix B: Implementation of the BDT Model with Different Volatility Estimators B.1. The BDT model B.2. Estimation results Questions References PART VI. GENERALIZATION OF OPTION PRICING MODELS AND STOCHASTIC VOLATILITY 743 CHAPTER 17. EXTREME MARKET MOVEMENTS, RISK AND ASSET MANAGEMENT: GENERALIZATION TO JUMP PROCESSES, STOCHASTIC VOLATILITIES, AND INFORMATION COSTS 745 Chapter Outline Introduction The Jump-Diffusion and the Constant Elasticity of Variance Models The jump-diffusion model The constant elasticity of variance diffusion (CEV) process On Jumps, Hedging and Information Costs Hedging in the presence of jumps Hedging the jumps Jump volatility On the Smile Effect and Market Imperfections in the Presence of Jumps and Incomplete Information On smiles and jumps On smiles, jumps, and incomplete information Empirical results in the presence of jumps and incomplete information Implied Volatility and Option Pricing Models: The Model and Simulation Results The valuation model Simulation results Model calibration and the smile effect Summary Questions References