An Introduction to the Mathematics of Financial Derivatives, Third Edition

456 Pages·2013·9.81 MB·English

by  Ali Hirsa

AN
INTRODUCTION
TO THE
MATHEMATICS OF
FINANCIAL
DERIVATIVES
This page is intentionally left blank
AN
INTRODUCTION
TO THE
MATHEMATICS OF
FINANCIAL
DERIVATIVES
Edited by
Ali Hirsa
salih N. Neftci†
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14 15 16 17 10 9 8 7 6 5 4 3 2 1
Contents
List of Symbols and Acronyms ix
1. Financial Derivatives—A Brief 4. Pricing Derivatives: Models and
Introduction Notation
1.1 Introduction 1 4.1 Introduction 55
1.2 Definitions 2 4.2 Pricing Functions 56
1.3 Types of Derivatives 2 4.3 Application: Another Pricing Model 59
1.4 Forwards and Futures 4 4.4 The Problem 61
1.5 Options 6 4.5 Conclusions 62
1.6 Swaps 8 4.6 References 63
1.7 Conclusion 10 4.7 Exercises 63
1.8 References 10
1.9 Exercises 10 5. Tools in Probability Theory
5.1 Introduction 65
2. A Primer on the Arbitrage 5.2 Probability 66
Theorem 5.3 Moments 67
2.1 Introduction 13 5.4 Conditional Expectations 70
2.2 Notation 14 5.5 Some Important Models 72
2.3 A Numerical Example 22 5.6 Exponential Distribution 77
2.4 An Application: Lattice Models 23 5.7 Gamma distribution 77
2.5 Payouts and Foreign Currencies 25 5.8 Markov Processes and Their Relevance 78
2.6 Some Generalizations 28 5.9 Convergence of Random Variables 81
2.7 Conclusions: A Methodology for Pricing 5.10 Conclusions 84
Assets 28 5.11 References 84
2.8 References 29 5.12 Exercises 84
2.9 Appendix: Generalization of the Arbitrage
Theorem 29 6. Martingales and Martingale
2.10 Exercises 30 Representations
6.1 Introduction 88
3. Review of Deterministic 6.2 Definitions 88
Calculus 6.3 The Use of Martingales in Asset Pricing 89
3.1 Introduction 33 6.4 Relevance of Martingales in Stochastic
3.2 Some Tools of Standard Calculus 34 Modeling 90
3.3 Functions 35 6.5 Properties of Martingale Trajectories 93
3.4 Convergence and Limit 37 6.6 Examples of Martingales 95
3.5 Partial Derivatives 46 6.7 The Simplest Martingale 98
3.6 Conclusions 51 6.8 Martingale Representations 99
3.7 References 51 6.9 The First Stochastic Integral 103
3.8 Exercises 51 6.10 Martingale Methods and Pricing 104
v
vi Contents
6.11 A Pricing Methodology 105 10.6 Integral Form of ITO’s Lemma 172
6.12 Conclusions 109 10.7 ITO’s Formula in More Complex
6.13 References 109 Settings 173
6.14 Exercises 109 10.8 Conclusion 176
10.9 References 177
7. Differentiation in Stochastic 10.10 Exercises 177
Environments
7.1 Introduction 111 11. The Dynamics of Derivative
7.2 Motivation 112 Prices
7.3 A Framework for Discussing 11.1 Introduction 179
Differentiation 114 11.2 A Geometric Description of Paths Implied by
7.4 The “Size” of Incremental Errors 116 SDEs 181
7.5 One Implication 118
11.3 Solution of SDEs 181
7.6 Putting the Results Together 119
11.4 Major Models of SDEs 188
7.7 Conclusion 120
11.5 Stochastic Volatility 191
7.8 References 121
11.6 Conclusions 195
7.9 Exercises 121
11.7 References 195
11.8 Exercises 195
8. The Wiener Process, Lévy Processes,
and Rare Events in Financial Markets 12. Pricing Derivative Products: Partial
8.1 Introduction 123
Differential Equations
8.2 Two Generic Models 125
12.1 Introduction 197
8.3 SDE in Discrete Intervals, Again 130
12.2 Forming Risk-Free Portfolios 198
8.4 Characterizing Rare and Normal Events 131
12.3 Accuracy of the Method 200
8.5 A Model for Rare Events 135
12.4 Partial Differential Equations 202
8.6 Moments That Matter 136
12.5 Classification of PDEs 203
8.7 Conclusions 138
12.6 A Reminder: Bivariate, Second-Degree
8.8 Rare and Normal Events in Practice 139
Equations 207
8.9 References 142
12.7 Types of PDEs 208
8.10 Exercises 143
12.8 Pricing Under Variance Gamma Model 209
9. Integration in Stochastic Environments 12.9 Conclusions 212
12.10 References 212
9.1 Introduction 145
12.11 Exercises 212
9.2 The ITO Integral 148
9.3 Properties of the ITO Integral 155
13. PDEs and PIDEs—An Application
9.4 Other Properties of the ITÔ Integral 160
9.5 Integrals with Respect to Jump Processes 160 13.1 Introduction 215
9.6 Conclusion 161 13.2 The Black–Scholes PDE 216
9.7 References 161 13.3 Local Volatility Model 217
9.8 Exercises 161 13.4 Partial Integro-Differential Equations
(PIDEs) 218
10. ITÔ’s Lemma 13.5 PDEs/PIDEs in Asset Pricing 220
10.1 Introduction 163 13.6 Exotic Options 221
10.2 Types of Derivatives 164 13.7 Solving PDEs/PIDEs in Practice 222
10.3 ITO’s Lemma 165 13.8 Conclusions 227
10.4 The ITO Formula 170 13.9 References 227
10.5 Uses of ITO’s Lemma 170 13.10 Exercises 227
Contents vii
14. Pricing Derivative Products: Equivalent 18.3 A Bond Pricing Equation 305
Martingale Measures 18.4 Forward Rates and Bond Prices 309
18.5 Conclusions: Relevance of the
14.1 Translations of Probabilities 231
Relationships 311
14.2 Changing Means 233
18.6 References 312
14.3 The Girsanov Theorem 238
18.7 Exercises 312
14.4 Statement of the Girsanov Theorem 243
14.5 A Discussion of the Girsanov Theorem 244
19. Classical and HJM Approach to
14.6 Which Probabilities? 246
Fixed Income
14.7 A Method for Generating Equivalent
Probabilities 247 19.1 Introduction 315
14.8 Conclusion 250 19.2 The Classical Approach 316
14.9 References 251 19.3 The HJM Approach to Term Structure 321
14.10 Exercises 251 19.4 How to Fit rt to Initial Term Structure 327
19.5 Conclusion 328
15. Equivalent Martingale Measures 19.6 References 329
15.1 Introduction 253 19.7 Exercises 329
15.2 A Martingale Measure 254
20. Classical PDE Analysis for Interest
15.3 Converting Asset Prices into Martingales 256
15.4 Application: The Black–Scholes Formula 259 Rate Derivatives
15.5 Comparing Martingale and PDE 20.1 Introduction 333
Approaches 261 20.2 The Framework 335
15.6 Conclusions 266 20.3 Market Price of Interest Rate Risk 336
15.7 References 266 20.4 Derivation of the PDE 337
15.8 Exercises 266 20.5 Closed-Form Solutions of the PDE 339
20.6 Conclusion 342
16. New Results and Tools for Interest- 20.7 References 343
Sensitive Securities 20.8 Exercises 343
16.1 Introduction 269
21. Relating Conditional Expectations
16.2 A Summary 270
16.3 Interest Rate Derivatives 271 to PDEs
16.4 Complications 273 21.1 Introduction 345
16.5 Conclusions 275 21.2 From Conditional Expectations to PDEs 347
16.6 References 275 21.3 From PDEs to Conditional Expectations 353
16.7 Exercises 275 21.4 Generators, Feynman–KAC Formula,
and Other Tools 355
17. Arbitrage Theorem in a New Setting 21.5 Feynman–KAC Formula 358
17.1 Introduction 277 21.6 Conclusions 358
17.2 A Model for New Instruments 278 21.7 References 358
17.3 Other Equivalent Martingale Measures 290 21.8 Exercises 358
17.4 Conclusion 297
17.5 References 298 22. Pricing Derivatives via Fourier
17.6 Exercises 298 Transform Technique
22.1 Derivatives Pricing via the Fourier
18. Modeling Term Structure and
Transform 365
Related Concepts 22.2 Findings and Observations 370
18.1 Introduction 301 22.3 Conclusions 370
18.2 Main Concepts 302 22.4 Problems 371
viii Contents
23. Credit Spread and Credit Derivatives 24.6 A Simple Example 408
23.1 Standard Contracts 373 24.7 Stopping Times and Martingales 411
23.2 Pricing of Credit Default Swaps 379 24.8 Conclusions 412
23.3 Pricing Multi-Name Credit Products 388 24.9 References 412
23.4 Credit spread Obtained from Options 24.10 Exercises 412
Market 394
25. Overview of Calibration and
23.5 Problems 399
Estimation Techniques
25.1 Calibration Formulation 416
24. Stopping Times and American-Type
25.2 Underlying Models 417
Securities
25.2 Overview of Filtering and Estimation 427
24.1 Introduction 401
25.2 Exercises 435
24.2 Why Study Stopping Times? 402
24.3 Stopping Times 403 References 437
24.4 Uses of Stopping Times 404
24.5 A Simplified Setting 405 Index 439