1、AN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVESThis page is intentionally left blankAN INTRODUCTION TO THE MATHEMATICS OF FINANCIAL DERIVATIVES Edited by Ali Hirsa s alih N. Neftci AMSTERDAM BOST ON HEIDELBERG LONDON NEW YORK OXFORD P ARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY T OKYO
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6、guing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-384682-2 For information on all Academic Press publications visi
7、t our Web site at Typeset by sps www.sps.co.in Printed and bound in the United States of America 14 15 16 17 10 9 8 7 6 5 4 3 2 1 v 1. Financial DerivativesA Brief Introduction 1.1 Introduction 1 1.2 Definitions 2 1.3 Types of Derivatives 2 1.4 Forwards and Futures 4 1.5 Options 6 1.6 Swaps 8 1.7 C
8、onclusion 10 1.8 References 10 1.9 Exercises 10 2. A Primer on the Arbitrage Theorem 2.1 Introduction 13 2.2 Notation 14 2.3 A Numerical Example 22 2.4 An Application: Lattice Models 23 2.5 Payouts and Foreign Currencies 25 2.6 Some Generalizations 28 2.7 Conclusions: A Methodology for Pricing Asset
9、s 28 2.8 References 29 2.9 Appendix: Generalization of the Arbitrage Theorem 29 2.10 Exercises 30 3. Review of Deterministic Calculus 3.1 Introduction 33 3.2 Some Tools of Standard Calculus 34 3.3 Functions 35 3.4 Convergence and Limit 37 3.5 Partial Derivatives 46 3.6 Conclusions 51 3.7 References
10、51 3.8 Exercises 51 4. Pricing Derivatives: Models and Notation 4.1 Introduction 55 4.2 Pricing Functions 56 4.3 Application: Another Pricing Model 59 4.4 The Problem 61 4.5 Conclusions 62 4.6 References 63 4.7 Exercises 63 5. Tools in Probability Theory 5.1 Introduction 65 5.2 Probability 66 5.3 Mo
11、ments 67 5.4 Conditional Expectations 70 5.5 Some Important Models 72 5.6 Exponential Distribution 77 5.7 Gamma distribution 77 5.8 Markov Processes and Their Relevance 78 5.9 Convergence of Random Variables 81 5.10 Conclusions 84 5.11 References 84 5.12 Exercises 84 6. Martingales and Martingale Re
12、presentations 6.1 Introduction 88 6.2 Definitions 88 6.3 The Use of Martingales in Asset Pricing 89 6.4 Relevance of Martingales in Stochastic Modeling 90 6.5 Properties of Martingale Trajectories 93 6.6 Examples of Martingales 95 6.7 The Simplest Martingale 98 6.8 Martingale Representations 99 6.9
13、The First Stochastic Integral 103 6.10 Martingale Methods and Pricing 104 Contents List of Symbols and Acronyms ixvi Contents 6.11 A Pricing Methodology 105 6.12 Conclusions 109 6.13 References 109 6.14 Exercises 109 7. Differentiation in Stochastic Environments 7.1 Introduction 111 7.2 Motivation 1
14、12 7.3 A Framework for Discussing Differentiation 114 7.4 The “Size” of Incremental Errors 116 7.5 One Implication 118 7.6 Putting the Results Together 119 7.7 Conclusion 120 7.8 References 121 7.9 Exercises 121 8. The Wiener Process, Lvy Processes, and Rare Events in Financial Markets 8.1 Introduct
15、ion 123 8.2 Two Generic Models 125 8.3 SDE in Discrete Intervals, Again 130 8.4 Characterizing Rare and Normal Events 131 8.5 A Model for Rare Events 135 8.6 Moments That Matter 136 8.7 Conclusions 138 8.8 Rare and Normal Events in Practice 139 8.9 References 142 8.10 Exercises 143 9. Integration in
16、 Stochastic Environments 9.1 Introduction 145 9.2 The ITO Integral 148 9.3 Properties of the ITO Integral 155 9.4 Other Properties of the IT Integral 160 9.5 Integrals with Respect to Jump Processes 160 9.6 Conclusion 161 9.7 References 161 9.8 Exercises 161 10. ITs Lemma 10.1 Introduction 163 10.2
17、Types of Derivatives 164 10.3 ITOs Lemma 165 10.4 The ITO Formula 170 10.5 Uses of ITOs Lemma 170 10.6 Integral Form of ITOs Lemma 172 10.7 ITOs Formula in More Complex Settings 173 10.8 Conclusion 176 10.9 References 177 10.10 Exercises 177 11. The Dynamics of Derivative Prices 11.1 Introduction 17
18、9 11.2 A Geometric Description of Paths Implied by SDEs 181 11.3 Solution of SDEs 181 11.4 Major Models of SDEs 188 11.5 Stochastic Volatility 191 11.6 Conclusions 195 11.7 References 195 11.8 Exercises 195 12. Pricing Derivative Products: Partial Differential Equations 12.1 Introduction 197 12.2 Fo
19、rming Risk-Free Portfolios 198 12.3 Accuracy of the Method 200 12.4 Partial Differential Equations 202 12.5 Classification of PDEs 203 12.6 A Reminder: Bivariate, Second-Degree Equations 207 12.7 Types of PDEs 208 12.8 Pricing Under Variance Gamma Model 209 12.9 Conclusions 212 12.10 References 212
20、12.11 Exercises 212 13. PDEs and PIDEsAn Application 13.1 Introduction 215 13.2 The BlackScholes PDE 216 13.3 Local Volatility Model 217 13.4 Partial Integro-Differential Equations (PIDEs) 218 13.5 PDEs/PIDEs in Asset Pricing 220 13.6 Exotic Options 221 13.7 Solving PDEs/PIDEs in Practice 222 13.8 C
21、onclusions 227 13.9 References 227 13.10 Exercises 227 Contents vii 14. Pricing Derivative Products: Equivalent Martingale Measures 14.1 Translations of Probabilities 231 14.2 Changing Means 233 14.3 The Girsanov Theorem 238 14.4 Statement of the Girsanov Theorem 243 14.5 A Discussion of the Girsano
22、v Theorem 244 14.6 Which Probabilities? 246 14.7 A Method for Generating Equivalent Probabilities 247 14.8 Conclusion 250 14.9 References 251 14.10 Exercises 251 15. Equivalent Martingale Measures 15.1 Introduction 253 15.2 A Martingale Measure 254 15.3 Converting Asset Prices into Martingales 256 1
23、5.4 Application: The BlackScholes Formula 259 15.5 Comparing Martingale and PDE Approaches 261 15.6 Conclusions 266 15.7 References 266 15.8 Exercises 266 16. New Results and Tools for Interest- Sensitive Securities 16.1 Introduction 269 16.2 A Summary 270 16.3 Interest Rate Derivatives 271 16.4 Com
24、plications 273 16.5 Conclusions 275 16.6 References 275 16.7 Exercises 275 17. Arbitrage Theorem in a New Setting 17.1 Introduction 277 17.2 A Model for New Instruments 278 17.3 Other Equivalent Martingale Measures 290 17.4 Conclusion 297 17.5 References 298 17.6 Exercises 298 18. Modeling Term Stru
25、cture and Related Concepts 18.1 Introduction 301 18.2 Main Concepts 302 18.3 A Bond Pricing Equation 305 18.4 Forward Rates and Bond Prices 309 18.5 Conclusions: Relevance of the Relationships 311 18.6 References 312 18.7 Exercises 312 19. Classical and HJM Approach to Fixed Income 19.1 Introduction
26、 315 19.2 The Classical Approach 316 19.3 The HJM Approach to Term Structure 321 19.4 How to Fit r tto Initial Term Structure 327 19.5 Conclusion 328 19.6 References 329 19.7 Exercises 329 20. Classical PDE Analysis for Interest Rate Derivatives 20.1 Introduction 333 20.2 The Framework 335 20.3 Mark
27、et Price of Interest Rate Risk 336 20.4 Derivation of the PDE 337 20.5 Closed-Form Solutions of the PDE 339 20.6 Conclusion 342 20.7 References 343 20.8 Exercises 343 21. Relating Conditional Expectations to PDEs 21.1 Introduction 345 21.2 From Conditional Expectations to PDEs 347 21.3 From PDEs to
28、Conditional Expectations 353 21.4 Generators, FeynmanKAC Formula, and Other Tools 355 21.5 FeynmanKAC Formula 358 21.6 Conclusions 358 21.7 References 358 21.8 Exercises 358 22. Pricing Derivatives via Fourier Transform Technique 22.1 Derivatives Pricing via the Fourier Transform 365 22.2 Findings a
29、nd Observations 370 22.3 Conclusions 370 22.4 Problems 371viii Contents 23. Credit Spread and Credit Derivatives 23.1 Standard Contracts 373 23.2 Pricing of Credit Default Swaps 379 23.3 Pricing Multi-Name Credit Products 388 23.4 Credit spread Obtained from Options Market 394 23.5 Problems 399 24.
30、Stopping Times and American-Type Securities 24.1 Introduction 401 24.2 Why Study Stopping Times? 402 24.3 Stopping Times 403 24.4 Uses of Stopping Times 404 24.5 A Simplified Setting 405 24.6 A Simple Example 408 24.7 Stopping Times and Martingales 411 24.8 Conclusions 412 24.9 References 412 24.10
31、Exercises 412 25. Overview of Calibration and Estimation Techniques 25.1 Calibration Formulation 416 25.2 Underlying Models 417 25.2 Overview of Filtering and Estimation 427 25.2 Exercises 435 References 437 Index 439 ix set of real numbers set of positive real numbers set of integers set of natural
32、 numbers p(t, T) price of a zero-coupon at time t maturing at T r t instantaneous short rate at calendar time t B t money market account at time t starting with $1 at time 0 and rolling at the instantaneous short rate f(t, T) instantaneous forward rate at calendar time t for the forward period T, T
33、+ dt real-world (physical) measure Risk neutral measure expectation of x under some measure expectation of x under some measure conditional on knowing all information up to t List of Symbols and AcronymsThis page is intentionally left blankCHAPTER 1 FinancialDerivativesABrief Introduction OUTLINE 1.
34、1 Introduction 1 1.2 Denitions 2 1.3 Types of Derivatives 2 1.3.1 Cash-and-Carry Markets 3 1.3.2 Price-Discovery Markets 3 1.3.3 Expiration Date 3 1.4 Forwards and Futures 4 1.4.1 Forwards 4 1.4.2 Futures 5 1.4.3 Repos, Reverse Repos, and Flexible Repos 5 1.5 Options 6 1.5.1 Some Notation 7 1.6 Swap
35、s 8 1.6.1 A Simple Interest Rate Swap 9 1.6.2 Cancelable Swaps 9 1.7 Conclusion 10 1.8 References 10 1.9 Exercises 10 1.1 INTRODUCTION This book is an introduction to quantitative toolsusedinpricingnancialderivatives.Hence, itismainlyaboutmathematics.Itisasimpleand heuristic introduction to mathemat
36、ical concepts that have practical use in nancial markets. Such an introduction requires a discussion of the logic behind asset pricing. In addition, at various points we provide examples that also require an understanding of formal asset pric- ingmethods.Allthesenecessitateabriefdiscus- sion of the
37、securities under consideration. This introductory chapter has that aim. Readers can consult other books to obtain more background onderivatives.Hull(2009)isanexcellentsource forderivatives.JarrowandTurnbull(1996)gives anotherapproach.Themoreadvancedbooksby Financial Derivatives, Third Edition. 2014E
38、lsevierInc. http:/dx.doi.org/10.1016/B978-0-12-384682-2.00001-3 12 1. FINANCIALDERIVATIVESABRIEFINTRODUCTION Ingersoll(1987)andDufe(1996)providestrong links to the underlying theory. The manual by Das (1994) provides a summary of the practi- cal issues associated with derivative contracts. A compreh
39、ensive new source is Wilmott (1998). This chapter rst deals with the two basic building blocks of nancial derivatives: options and forwards (futures). Next, we introduce the more complicated class of derivatives known as swaps. The chapter concludes by showing that a complicated swap can be decompos
40、ed into a numberofforwardsandoptions.Thisdecompo- sitionisverypractical.Ifonesucceedsinpricing forwards and options, one can then reconsti- tute any swap and obtain its price. This chapter alsointroducessomeformalnotationthatwillbe used throughout the book. 1.2 DEFINITIONS In the words of practition
41、ers, “Derivative securities are nancial contracts that derive their value from cash market instruments such as stocks, bonds, currencies and commodities.” 1 The academic denition of a “derivative instrument” is more precise. Denition 1 (Ingersoll, 1987). A nancialcon- tract is a derivative security,
42、 or a contingent claim, if its value at expiration date T is deter- mined exactly by the market price of the under- lying cash instrument at time T. Hence, at the time of the expiration of the derivativecontract,denotedbyT,theprice F(T) of a derivative asset is completely determined by S T , the val
43、ue of the “underlying asset.” After that date, the security ceases to exist. This sim- plecharacteristicofderivativeassetsplaysavery important role in their valuation. In the rest of this book, the symbols F(t) and F(S t ,t) will be used alternately to denote the price of a derivative product writte
44、n on 1 See pages 23, Klein and Lederman (1994). the underlying asset S t at time t. The nancial derivative is sometimes assumed to yield a pay- out dt. At other times, the payout is zero. T will always denote the expiration date. 1.3 TYPES OF DERIVATIVES Wecangroupderivativesecuritiesunderthree gene
45、ral headings: 1. Futures and forwards. 2. Options. 3. Swaps. Forwards and options are considered basic building blocks. Swaps and some other compli- catedstructuresareconsideredhybridsecurities, whichcaneventuallybedecomposedintosetsof basic forwards and options. We let S t denote the price of the r
46、elevant cash instrument, which we call the underlying security. We can list ve main groups of underlying assets: 1. Stocks: These are claims to “real” returns gen- erated in the production sector for goods and services. 2. Currencies:Theseareliabilitiesofgovernments or, sometimes, banks. They are no
47、t direct claims on real assets. 3. Interest rates: In fact, interest rates are not assets. Hence, a notional asset needs to be devised so that one can take a position on the direction of future interest rates. Futures on Eurodollars is one example. In this category, we can also include derivatives o
48、n bonds, notes,andT-bills,whicharegovernmentdebt instruments. They are promises by govern- mentstomakecertainpaymentsonsetdates. By dealing with derivatives on bonds, notes, and T-bills, one takes positions on the direc- tion of various interest rates. In most cases, 2 2 There is a signicant amount
49、of trading on “notional” French government bonds in Paris.1.3 TYPESOFDERIVATIVES 3 these derivative instruments are not notionals andcanresultinactualdeliveryoftheunder- lying asset. 4. Indexes: The S&P 500 and the FTSE 100 are twoexamplesofstockindexes.TheCRBcom- modityindexisanindexofcommodityprices. Again, these are not “assets” themselves. But derivativecontractscanbewrittenonnotional amounts and a position taken with respect to the direction
