Financial derivatives in theory and practice

424 Pages·2004·2.578 MB·English

by  P J Hunt, J E Kennedy

Financial Derivatives
in Theory and Practice
Revised Edition
P. J. HUNT
WestLB AG, London, UK
J. E. KENNEDY
University of Warwick, UK
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Contents
Preface to revised edition xv
Preface xvii
Acknowledgements xxi
Part I: Theory 1
1 Single-Period Option Pricing 3
1.1 Option pricing in a nutshell 3
1.2 The simplest setting 4
1.3 General one-period economy 5
1.3.1 Pricing 6
1.3.2 Conditions for no arbitrage: existence of Z 7
1.3.3 Completeness: uniqueness of Z 9
1.3.4 Probabilistic formulation 12
1.3.5 Units and numeraires 15
1.4 A two-period example 15
2 Brownian Motion 19
2.1 Introduction 19
2.2 Definition and existence 20
2.3 Basic properties of Brownian motion 21
2.3.1 Limit of a random walk 21
2.3.2 Deterministic transformations of Brownian motion 23
2.3.3 Some basic sample path properties 24
2.4 Strong Markov property 26
2.4.1 Reflection principle 28
3 Martingales 31
3.1 Definition and basic properties 32
3.2 Classes of martingales 35
3.2.1 Martingales bounded in L1 35
3.2.2 Uniformly integrable martingales 36
3.2.3 Square-integrable martingales 39
3.3 Stopping times and the optional sampling theorem 41
3.3.1 Stopping times 41
3.3.2 Optional sampling theorem 45
3.4 Variation, quadratic variation and integration 49
3.4.1 Total variation and Stieltjes integration 49
3.4.2 Quadratic variation 51
3.4.3 Quadratic covariation 55
viii Contents
3.5 Local martingales and semimartingales 56
3.5.1 The space c 56
loc
M
3.5.2 Semimartingales 59
3.6 Supermartingales and the Doob—Meyer decomposition 61
4 Stochastic Integration 63
4.1 Outline 63
4.2 Predictable processes 65
4.3 Stochastic integrals: the L2 theory 67
4.3.1 The simplest integral 68
4.3.2 The Hilbert space L2(M) 69
4.3.3 The L2 integral 70
4.3.4 Modes of convergenceto H M 72
•
4.4 Properties of the stochastic integral 74
4.5 Extensions via localization 77
4.5.1 Continuous local martingales as integrators 77
4.5.2 Semimartingales as integrators 78
4.5.3 The end of the road! 80
4.6 Stochastic calculus: Itoˆ’s formula 81
4.6.1 Integration by parts and Itoˆ’s formula 81
4.6.2 Differential notation 83
4.6.3 Multidimensional version of Itoˆ’s formula 85
4.6.4 L´evy’s theorem 88
5 Girsanov and Martingale Representation 91
5.1 Equivalent probability measures and the Radon—Nikody´m derivative 91
5.1.1 Basic results and properties 91
5.1.2 Equivalent and locally equivalent measures on a filtered space 95
5.1.3 Novikov’scondition 97
5.2 Girsanov’stheorem 99
5.2.1 Girsanov’s theorem for continuous semimartingales 99
5.2.2 Girsanov’s theorem for Brownian motion 101
5.3 Martingale representation theorem 105
5.3.1 The space 2(M) and its orthogonal complement 106
I
5.3.2 Martingale measures and the martingale representation
theorem 110
5.3.3 Extensions and the Brownian case 111
6 Stochastic Differential Equations 115
6.1 Introduction 115
6.2 Formal definition of an SDE 116
6.3 An aside on the canonical set-up 117
6.4 Weak and strong solutions 119
6.4.1 Weak solutions 119
Contents ix
6.4.2 Strong solutions 121
6.4.3 Tying together strong and weak 124
6.5 Establishing existence and uniqueness: Itoˆ theory 125
6.5.1 Picard—Lindelo¨f iteration and ODEs 126
6.5.2 A technical lemma 127
6.5.3 Existence and uniqueness for Lipschitz coefficients 130
6.6 Strong Markovproperty 134
6.7 Martingale representation revisited 139
7 Option Pricing in Continuous Time 141
7.1 Asset price processes and trading strategies 142
7.1.1 A model for asset prices 142
7.1.2 Self-financing trading strategies 144
7.2 Pricing European options 146
7.2.1 Option value as a solution to a PDE 147
7.2.2 Option pricing via an equivalent martingale measure 149
7.3 Continuous time theory 151
7.3.1 Information within the economy 152
7.3.2 Units, numeraires and martingale measures 153
7.3.3 Arbitrage and admissible strategies 158
7.3.4 Derivative pricing in an arbitrage-freeeconomy 163
7.3.5 Completeness 164
7.3.6 Pricing kernels 173
7.4 Extensions 176
7.4.1 General payout schedules 176
7.4.2 Controlled derivative payouts 178
7.4.3 More general asset price processes 179
7.4.4 Infinite trading horizon 180
8 Dynamic Term Structure Models 183
8.1 Introduction 183
8.2 An economy of pure discount bonds 183
8.3 Modelling the term structure 187
8.3.1 Pure discount bond models 191
8.3.2 Pricing kernel approach 191
8.3.3 Numeraire models 192
8.3.4 Finite variation kernel models 194
8.3.5 Absolutely continuous (FVK) models 197
8.3.6 Short-rate models 197
8.3.7 Heath—Jarrow—Mortonmodels 200
8.3.8 Flesaker—Hughstonmodels 206
x Contents
Part II: Practice 213
9 Modelling in Practice 215
9.1 Introduction 215
9.2 The real world is not a martingale measure 215
9.2.1 Modelling via infinitesimals 216
9.2.2 Modelling via macro information 217
9.3 Product-based modelling 218
9.3.1 A warning on dimension reduction 219
9.3.2 Limit cap valuation 221
9.4 Local versus global calibration 223
10 Basic Instruments and Terminology 227
10.1 Introduction 227
10.2 Deposits 227
10.2.1 Accrual factors and LIBOR 228
10.3 Forward rate agreements 229
10.4 Interest rate swaps 230
10.5 Zero coupon bonds 232
10.6 Discount factors and valuation 233
10.6.1 Discount factors 233
10.6.2 Deposit valuation 233
10.6.3 FRA valuation 234
10.6.4 Swap valuation 234
11 Pricing Standard Market Derivatives 237
11.1 Introduction 237
11.2 Forward rate agreements and swaps 237
11.3 Caps and floors 238
11.3.1 Valuation 240
11.3.2 Put—call parity 241
11.4 Vanilla swaptions 242
11.5 Digital options 244
11.5.1 Digital caps and floors 244
11.5.2 Digital swaptions 245
12 Futures Contracts 247
12.1 Introduction 247
12.2 Futures contract definition 247
12.2.1 Contract specification 248
12.2.2 Market risk without credit risk 249
12.2.3 Mathematical formulation 251
12.3 Characterizing the futures price process 252
12.3.1 Discrete resettlement 252
Contents xi
12.3.2 Continuous resettlement 253
12.4 Recovering the futures price process 255
12.5 Relationship between forwards and futures 256
Orientation: Pricing Exotic European Derivatives 259
13 Terminal Swap-Rate Models 263
13.1 Introduction 263
13.2 Terminal time modelling 263
13.2.1 Model requirements 263
13.2.2 Terminal swap-ratemodels 265
13.3 Example terminal swap-rate models 266
13.3.1 The exponential swap-rate model 266
13.3.2 The geometric swap-ratemodel 267
13.3.3 The linear swap-rate model 268
13.4 Arbitrage-free property of terminal swap-rate models 269
13.4.1 Existence of calibrating parameters 270
13.4.2 Extension of model to [0, ) 271
∞
13.4.3 Arbitrage and the linear swap-rate model 273
13.5 Zero coupon swaptions 273
14 Convexity Corrections 277
14.1 Introduction 277
14.2 Valuation of ‘convexity-related’products 278
14.2.1 Affine decomposition of convexity products 278
14.2.2 Convexity correctionsusing the linear swap-rate model 280
14.3 Examples and extensions 282
14.3.1 Constant maturity swaps 283
14.3.2 Options on constant maturity swaps 284
14.3.3 LIBOR-in-arrearsswaps 285
15 Implied Interest Rate Pricing Models 287
15.1 Introduction 287
15.2 Implying the functional form D 288
TS
15.3 Numerical implementation 292
15.4 Irregular swaptions 293
15.5 Numerical comparison of exponential and implied swap-ratemodels 299
16 Multi-Currency Terminal Swap-Rate Models 303
16.1 Introduction 303
16.2 Model construction 304
16.2.1 Log-normalcase 305
16.2.2 General case: volatility smiles 307
16.3 Examples 308
xii Contents
16.3.1 Spread options 308
16.3.2 Cross-currencyswaptions 311
Orientation: Pricing Exotic American and Path-Dependent
Derivatives 315
17 Short-Rate Models 319
17.1 Introduction 319
17.2 Well-known short-rate models 320
17.2.1 Vasicek—Hull—White model 320
17.2.2 Log-normalshort-rate models 322
17.2.3 Cox—Ingersoll—Rossmodel 323
17.2.4 Multidimensional short-rate models 324
17.3 Parameterfitting within the Vasicek—Hull—White model 325
17.3.1 Derivation of φ, ψ and B 326
T
·
17.3.2 Derivation of ξ, ζ and η 327
17.3.3 Derivation of µ, λ and A 328
T
·
17.4 Bermudan swaptions via Vasicek—Hull—White 329
17.4.1 Model calibration 330
17.4.2 Specifying the ‘tree’ 330
17.4.3 Valuation through the tree 332
17.4.4 Evaluation of expected future value 332
17.4.5 Erroranalysis 334
18 Market Models 337
18.1 Introduction 337
18.2 LIBOR market models 338
18.2.1 Determining the drift 339
18.2.2 Existence of a consistent arbitrage-freeterm structure model 341
18.2.3 Example application 343
18.3 Regular swap-marketmodels 343
18.3.1 Determining the drift 344
18.3.2 Existence of a consistent arbitrage-freeterm structure model 346
18.3.3 Example application 346
18.4 Reverse swap-marketmodels 347
18.4.1 Determining the drift 348
18.4.2 Existence of a consistent arbitrage-freeterm structure model 349
18.4.3 Example application 350
19 Markov-Functional Modelling 351
19.1 Introduction 351
19.2 Markov-functionalmodels 351
19.3 Fitting a one-dimensional Markov-functionalmodel to swaption
prices 354
Contents xiii
19.3.1 Deriving the numeraire on a grid 355
19.3.2Existenceofaconsistentarbitrage-freetermstructuremodel 358
19.4 Example models 359
19.4.1 LIBOR model 359
19.4.2 Swap model 361
19.5 Multidimensional Markov-functional models 363
19.5.1 Log-normally driven Markov-functional models 364
19.6 Relationship to market models 365
19.7 Mean reversion, forward volatilities and correlation 367
19.7.1 Mean reversion and correlation 367
19.7.2 Mean reversion and forward volatilities 368
19.7.3 Mean reversion within the Markov-functional LIBOR model 369
19.8 Some numerical results 370
20 Exercises and Solutions 373
Appendix 1 The Usual Conditions 417
Appendix 2 L2 Spaces 419
Appendix 3 Gaussian Calculations 421
References 423
Index 427
Preface to revised edition
Since this book first appeared in 2000, it has been adopted by a number of
universitiesasastandardtextforagraduatecourseinfinance.Asaresultwe
have produced this revised edition. The only differences in content between
this text and its predecessor are the inclusion of an additional chapter of
exercises with solutions, and the corrections of a number of errors.
Many of the exercise are variants of ones given to students of the M.Sc.
in Mathematical Finance at the University of Warwick, so they have been
triedandtested.Manyprovidedrillinroutinecalculationsintheinterest-rate
setting.
Since the book first appeared there has been further development in the
modelling of interest rate derivatives. The modelling and approximation of
market models has progressed further, as has that of Markov-functional
modelsthatarenowused,intheirmulti-factorform,inanumberofbanks.A
notable exclusion from this revised edition is any coverage of these advances.
Those interested in this area can find some of this in Rebonato (2002) and
Bennett and Kennedy (2004).
A few extra acknowledgements are due in this revised edition: to Noel
Vaillant and Jørgen Aase Nielsen for pointing out a number of errors, and
to Stuart Price for typing the exercise chapter and solutions.