Martingale Methods in Financial Modelling (eBook)

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Preface to the First Edition 6
Preface to the Second Edition 7
Table of Contents 9
Part I Spot and Futures Markets 18
1. An Introduction to Financial Derivatives 20
1.1 Options 20
1.2 Futures Contracts and Options 23
1.4 Call and Put Spot Options 25
1.4.1 One-period Spot Market 27
1.4.2 Replicating Portfolios 28
1.4.3 Martingale Measure for a Spot Market 29
1.4.4 Absence of Arbitrage 31
1.4.5 Optimality of Replication 32
1.4.6 Put Option 35
1.5 Futures Call and Put Options 36
1.5.1 Futures Contracts and Futures Prices 37
1.5.2 One-period Futures Market 37
1.5.3 Martingale Measure for a Futures Market 39
1.5.4 Absence of Arbitrage 39
1.5.5 One-period Spot/Futures Market 41
1.6 Forward Contracts 42
1.6.1 Forward Price 42
1.7 Options of American Style 44
1.8 Universal No-arbit rage Inequalities 49
2. Discrete-time Security Markets 52
2.1 The Cox-Ross-Rubinstein Model 53
2.1.1 Binomial Lattice for the Stock Price 53
2.1.2 Recursive Pricing Procedure 55
2.1.3 CRR Option Pricing Formula 60
2.2 Martingale Properties of the CRR Model 63
2.2.1 Martingale Measures 64
2.2.2 Risk-neutral Valuation Formula 67
2.3 The Black-Scholes Option Pricing Formula 68
2.4 Valuation of American Options 73
2.4.1 American Call Options 73
2.4.2 American Put Options 75
2.4.3 American Claim 77
2.5 Options on a Dividend-paying Stock 78
2.6 Finite Spot Markets 80
2.6.1 Self-financing Trading Strategies 80
2.6.2 Arbitrage Opportunities 82
2.6.3 Arbitrage Price 83
2.6.4 Risk-neutral Valuation Formula 84
2.6.5 Price Systems 87
2.6.6 Completeness of a Finite Market 90
2.6.7 Change of a Numeraire 91
2.7 Finite Futures Markets 92
2.8 Futures Prices Versus Forward Prices 96
2.9 Discrete-time Models with Infinite State Space 99
3. Benchmark Models in Continuous Time 100
3.1 The Black-Scholes Model 101
3.1.1 Risk-free Bond 101
3.1.2 Stock Price 101
3.1.3 Self-financing Trading Strategies 105
3.1.4 Martingale Measure for the Spot Market 106
3.1.5 Black-Scholes Option Pricing Formula 110
3.1.6 Case of Time-dependent Coefficients 117
3.1.7 Merton's Model 118
3.1.8 Put-Call Parity for Spot Options 119
3.1.9 Black-Scholes PDE 120
3.1.10 A Riskless Portfolio Method 122
3.1 .ll Black-Scholes Sensitivities 125
3.1.12 Market Imperfections 130
3.1.13 Numerical Methods 131
3.2 A Dividend-paying Stock 133
3.2.1 Case of a Constant Dividend Yield 133
3.2.2 Case of Known Dividends 135
3.3 Bachelier Model 139
3.3.1 Bachelier Option Pricing Formula 140
3.3.2 Bachelier's PDE 141
3.3.3 Bachelier Sensitivities 142
3.4 Black Model 143
3.4.1 Self-financing Futures Strategies 144
3.4.2 Martingale Measure for the Futures Market 144
3.4.3 Black's Futures Option Formula 145
3.4.4 Options on Forward Contracts 149
3.4.5 Forward and Futures Prices 151
3.5 Robustness of the Black-Scholes Approach 152
3.5.1 Uncertain Volatility 152
3.5.2 European Call and Put Options 153
3.5.3 Convex Path-independent European Claims 156
3.5.4 General Path-independent European Claims 161
4. Foreign Market Derivatives 164
4.1 Cross- currency Market Model 164
4.2 Currency Forward Contracts and Options 169
4.3 Foreign Equity Forward Contracts 174
4.3.1 Forward Price of a Foreign Stock 174
4.3.2 Quanto Forward Contracts 176
4.4 Foreign Market Futures Contracts 177
4.5 Foreign Equity Options 181
4.5.1 Options Struck in a Foreign Currency 181
4.5.2 Options Struck in Domestic Currency 183
4.5.3 Quanto Options 184
4.5.4 Equity-linked Foreign Exchange Options 186
5. American Options 188
5.1 Valuation of American Claims 189
5.2 American Call and Put Options 197
5.3 Early Exercise Representation of an American Put 199
5.4 Analytical Approach 202
5.5 Approximations of the American Put Price 205
5.6 Option on a Dividend-paying Stock 208
6. Exotic Options 210
6.1 Packages 211
6.2 Forward-start Options 212
6.3 Chooser Options 213
6.4 Compound Options 214
6.5 Digital Options 215
6.6 Barrier Options 216
6.7 Lookback Options 219
6.8 Asian Options 223
6.9 Basket Options 226
6.10 Quantile Options 230
6.11 Other Exotic Options 233
7. Volatility Risk 234
7.1 Implied Volatilities of Traded Options 236
7.1.1 Historical Volatility 236
7.1.2 Implied Volatility 237
7.1.3 Implied Volatility Versus Historical Volatility 238
7.1.4 Approximate Formulas 239
7.1.5 Implied Volatility Surface 240
7.1.6 Asymptotic Behavior of t h e Implied Volatility 243
7.1.7 Marked-to-Market Models 246
7.1.8 Vega Hedging 247
7.1.9 Correlated Brownian Motions 249
7.1.10 Forward-start Options 251
7.2 Extensions of the Black-Scholes Model 254
7.2.1 CEV Model 254
7.2.2 Shifted Lognormal Models 258
7.3 Local Volatility Models 259
7.3.1 Implied Risk-Neutral Probability Law 259
7.3.2 Local Volatility 262
7.3.3 Mixture Models 268
7.3.4 Advantages and Drawbacks of LV Models 271
7.4 Stochastic Volatility Models 272
7.4.1 PDE Approach 273
7.4.2 Examples of SV Models 274
7.4.3 Hull and White Model 275
7.4.4 Heston's Model 280
7.4.5 SABR Model 282
7.5 Dynamical Models of Volatility Surfaces 284
7.5.1 Dynamics of the Local Volatility Surface 284
7.5.2 Dynamics of the Implied Volatility Surface 285
7.6 Alternative Approaches 289
7.6.1 Modelling of Asset Returns 289
8. Continuous-time Security Markets 296
8.1 Standard Market Models 297
8.1.1 Standard Spot Market 297
8.1.2 Futures Market 306
8.1.3 Choice of a Numeraire 308
8.1.4 Existence of a Martingale Measure 312
8.1.5 Fundamental Theorem of Asset Pricing 313
8.2 Multidimensional Black-Scholes Model 315
8.2.1 Market Completeness 317
8.2.2 Variance-minimizing Hedging 319
8.2.3 Risk-minimizing Hedging 320
8.2.4 Market Imperfections 327
Part II Fixed-income Markets 330
9. Interest Rates and Related Contracts 332
9.1 Zero-coupon Bonds 332
9.1.1 Term Structure of Interest Rates 333
9.1.2 Forward Interest Rates 334
9.1.3 Short-term Interest Rate 335
9.2 Coupon-bearing Bonds 335
9.2.1 Yield-to-Maturity 336
9.2.2 Market Conventions 338
9.3 Interest Rate Futures 339
9.3.1 Treasury Bond Futures 339
9.3.2 Bond Options 341
9.3.3 Treasury Bill Futures 341
9.3.4 Eurodollar Futures 343
9.4 Interest Rate Swaps 344
9.4.1 Forward Rate Agreements 345
9.5 Stochastic Models of Bond Prices 348
9.5.1 Arbitrage-free Family of Bond Prices 348
9.5.2 Expectations Hypotheses 349
9.5.3 Case of It6 Processes 350
9.5.4 Market Price for Interest Rate Risk 353
9.6 Forward Measure Approach 354
9.6.1 Forward Price 356
9.6.2 Forward Martingale Measure 357
9.6.3 Forward Processes 360
9.6.4 Choice of a Numeraire 361
10. Short-Term Rate Models 364
10.1 Single-factor Models 365
10.1.1 Time-homogeneous Models 365
10.1.2 Time-inhomogeneous Models 376
10.1.3 Model Choice 380
10.1.4 American Bond Options 382
10.1.5 Options on Coupon-bearing Bonds 383
10.2 Multi-factor Models 384
10.2.1 State Variables 384
10.2.2 Affine Models 385
10.2.3 Yield Models 386
10.3 Extended CIR Model 388
10.3.1 Squared Bessel Process 388
10.3.2 Model Construction 389
10.3.4 Zero-coupon Bond 390
10.3.5 Case of Constant Coefficients 392
10.3.6 Case of Piecewise Constant Coefficients 392
10.3.7 Dynamics of Zero-coupon Bond 394
10.3.8 Transition Densities 395
10.3.9 Bond Option 397
11. Models of Instantaneous Forward Rates 398
11.1 Heath- Jarrow-Morton Methodology 399
11.1.1 Ho and Lee Model 400
11.1.2 Heath-Jarrow-Morton Model 401
11.1.3 Absence of Arbitrage 403
11.1.4 Short-term Interest Rate 408
11.2 Gaussian HJM Model 409
11.2.1 Markovian Case 411
11.3 European Spot Options 415
11.3.1 Bond Options 416
11.3.2 Stock Options 419
11.3.3 Option on a Coupon-bearing Bond 422
11.3.4 Pricing of General Contingent Claims 425
11.3.5 Replication of Options 427
11.4 Volatilities and Correlations 430
11.4.1 Volatilities 430
11.4.2 Correlations 432
11.5 Futures Price 433
11.5.1 Futures Options 434
11.6 PDE Approach to Interest Rate Derivatives 438
11.6.1 PDEs for Spot Derivatives 438
11.6.2 PDEs for Futures Derivatives 442
11.7 Recent Developments 446
12. Market LIBOR Models 448
12.1 Forward and Futures LIBORs 450
12.1.1 One-period Swap Settled in Arrears 450
12.1.2 One-period Swap Settled in Advance 452
12.1.3 Eurodollar Futures 453
12.1.4 LIBOR in the Gaussian HJM Model 454
12.2 Interest Rate Caps and Floors 456
12.3 Valuation in the Gaussian HJM Model 458
12.3.1 Plain-vanilla Caps and Floors 458
12.3.2 Exotic Caps 460
12.3.3 Captions 462
12.4 LIBOR Market Models 463
12.4.1 Black's Formula for Caps 463
12.4.2 Miltersen, Sandmann and Sondermann Approach 465
12.4.3 Brace, Gqtarek and Musiela Approach 465
12.4.4 Musiela and Rutkowski Approach 468
12.4.5 Jamshidian’s Approach 472
12.5 Properties of the Lognormal LIBOR Model 475
12.5.1 Transition Density of the LIBOR 476
12.5.2 Transition Density of the Forward Bond Price 478
12.6 Valuation in the Lognormal LIBOR Model 481
12.6.1 Pricing of Caps and Floors 481
12.6.2 Hedging of Caps and Floors 483
12.6.3 Valuation of European Claims 485
12.6.4 Bond Options 488
12.7 Extensions of the LLM Model 490
13. Alternative Market Models 492
13.1 Swaps and Swaptions 493
13.1.1 Forward Swap Rates 493
13.1.2 Swaptions 497
13.1.3 Exotic Swap Derivatives 499
13.2 Valuation in the Gaussian HJM Model 502
13.2.1 Swaptions 502
13.2.2 CMS Spread Options 502
13.2.3 Yield Curve Swaps 504
13.3 Co-terminal Swap Rates 505
13.3.1 Jamshidian's Approach 510
13.3.2 Valuation of Co-terminal Swaptions 511
13.3.3 Hedging of Swaptions 512
13.3.4 Bermudan Swaptions 513
13.4 Co-initial Swap Rates 514
13.4.1 Valuation of Co-initial Swaptions 517
13.4.2 Valuation of Exotic Options 518
13.5 Co-sliding Swap Rates 519
13.5.1 Modelling of Co-sliding Swap Rates 519
13.5.2 Valuation of Co-sliding Swaptions 523
13.6 Swap Rate Model Versus LIBOR Model 525
13.6.1 Swaptions in the LLM Model 526
13.6.2 Caplets in the Co-terminal Swap Market Model 530
13.7 Markov-functional Models 531
13.7.1 Terminal Swap Rate Model 532
13.7.2 Calibration of Markov-functional Models 534
13.8 Flesaker and Hughston Approach 538
13.8.1 Rational Lognormal Model 540
13.8.2 Valuation of Caps and Swaptions 541
14. Cross-currency Derivatives 544
14.1 Arbitrage-free Cross-currency Markets 545
14.1.1 Forward Price of a Foreign Asset 547
14.1.2 Valuation of Foreign Contingent Claims 551
14.1.3 Cross-currency Rates 552
14.2 Gaussian Model 552
14.2.1 Currency Options 553
14.2.2 Foreign Equity Options 554
14.2.3 Cross-currency Swaps 559
14.2.4 Cross-currency Swaptions 570
14.2.5 Basket Caps 573
14.3 Model of Forward LIBOR Rates 574
14.3.1 Quanto Cap 575
14.3.2 Cross-currency Swap 577
14.4 Concluding Remarks 578
Part III Appendices 580
A. Conditional Expectat ions 582
B. Itö Stochastic Calculus 586
B.1 Itö Integral 586
B.2 Girsanov's Theorem 593
B.3 Itö-Tanaka-Meyer Formula 595
B.4 Laws of Certain Functionals of a Brownian Motion 596
References 600
Index 647

5. American Options (p. 171-172)

In contrast to the holder of a European option, the holder of an American op- tion is allowed to exercise his right to buy (or sell) the underlying asset at any time before or at the expiry date. This special feature of American-style op- tions - and more generally of American claims - makes the arbitrage pricing of American options much more involved than the valuation of standard Eu- ropean claims. We know already that arbitrage valuation of American claims is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, un- der risk-neutral probability). Thus, the uniqueness of the arbitrage price of an American claim holds. One of the earliest works to examine the rela- tionship between the early exercise feature of American options and optimal stopping problems was the paper by McKean (1965).

As the arbitrage valu- ation of derivative securities was not yet discovered at this time, the optimal stopping problem associated with the optimal exercise of American put was studied by McKean (1965) under an actual probability IP, rather than under the martingale measure IP*, as is done nowadays. For further properties of the optimal stopping boundary, we refer the reader toVan Moerbeke (1976). Ba- sic features of American options, within the framework of arbitrage valuation theory, were already examined in Merton (1973). However, mathematically rigorous valuation results for American claims were first established by means of arbitrage arguments in Bensoussan (1984) and Karatzas (1988, 1989). An exhaustive survey of results and techniques related to the arbitrage pricing of American options was given by Myneni (1992). For an innovative approach to American options and related issues, see Bank and Follmer (2003).

The purpose of this chapter is to provide the most fundamental results concerning the arbitrage valuation of American claims within the continuous- time framework of the Black-Scholes financial model. Firstly, we discuss the concept of the arbitrage price of American contingent claims and its ba- sic properties. As a consequence, we present the well-known result that an American call option with a constant strike price, written on a non-dividend- paying stock, is equivalent to the corresponding European call option. Sub- sequently, we focus on the features of the optimal exercise policy associ- ated with the American put option. Next, the analytical approach to the pricing of American options is presented. The free boundary problem as- sociated with the optimal exercise of American put options was studied by, among others, McKean (1965) and Van Moerbeke (1976). More recently, Jaillet et al. (1990) applied the general theory of variational inequalities to study the optimal stopping problem associated with American claims.
Finally, the most widely used numerical procedures related to the ap- proximate valuation of American contingent claims are reviewed. An an- alytic approximation of the American put price on a non-dividend-paying stock was examined by Brennan and Schwartz (1977a), Johnson (1983) and MacMillan (1986). We close this chapter with an analysis of an American call written on a dividend-paying stock (this was examined in Roll (1977)).