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本附件包括：- Riccardo Rebonato.Interest-Rate Option Models - Understanding, Analysing and Using Models for Exotic Interest-Rate Options.2nd ed.John Wiley & Sons.1998.djvu
Contents
Preface to the Second Edition xiii
Preface to the First Edition xv
Acknowledgements xix
List of symbols and abbreviations xxi
PART ONE THE NEED FOR YIELD CURVE OPTION PRICING
MODELS 1
1 Definition and valuation of the underlying instruments 3
1.1 Introduction 3
1.2 Definition of spot rates, forward rates, swap rates and par coupon
rates 5
1.3 The valuation of plain-vanilla swaps and FRAs 8
1.4 Obtaining the discount function from a set of spanning forward or
swap rates 14
1.5 The valuation of caps, floors and European swaptions 15
1.6 Determination of the discount function: the case of bonds — linear
models 21
1.7 Determination of the discount function: the case of bonds — non-linear
models 25
1.8 Determination of the discount function: the case of the LIBOR
curve 27
2 Exotic interest-rate instruments: description and valuation issues 29
2.1 Introduction 29
2.2 LIBOR-in-arrears swaps 30
Contents
2.3 American (Bermudan) swaptions
2.4 Trigger swaps 41
2.5 One-way floaters 44
2.6 Captions 47
36
51
A statistical approach to yield curve models
3.1 Statistical analysis of the evolution of rates 51
3.2 The effects of model dimensionality on option pricing
3.3 A framework for option pricing 72
63
4 Correlation, average and instantaneous volatilities, and their impact on
the pricing of LIBOR options 75
4.1 Introduction and motivation 75
4.2 Instantaneous and average volatilities 77
4.3 Pricing European swaptions with instantaneous volatilities 80
4.4 Term decorrelation 83
4.5 Relationships between average, instantaneous and term structure of
volatilities 91
4.6 Conclusions 100
Appendix 4.1 101
Appendix 4.2 102
A motivation for yield curve models 105
5.1 Introduction 105
5.2 Hedging a bond option with the underlying forward contract
5.3 Hedging a path-dependent bond option with forward contracts
106
109
PART TWO THE THEORETICAL TOOLS
6 Establishing a pricing framework
117
119
6.1 Introduction and motivation 119
6.2 First approach — 'Replication Strategy' 121
6.3 Second approach — 'Naive Expectation' 123
6.4 Third approach — 'Market Price of Risk' 125
6.5 Fourth approach — Risk-neutral valuation 129
6.6 Pseudo-probabilities 130
6.7 A pricing framework 134
6.8 Evaluation of a contingent claim in a multi-period setting
6.9 Self-financing trading strategies 139
6.10 Fair prices as expectations 141
Contents
IX
6.11
6.12
6.13
Switching numeraires and relating expectations under different
measures 144
Justifying the two-state branching procedure 150
The nature of the transformation between measures — Girsanov's
theorem 153
136
7 The conditions of no-arbitrage 157
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
160
First no-arbitrage condition: the Vasicek approach 157
Second no-arbitrage condition: the martingale approach
The case of a deterministic-interest-rates economy 162
First choice of numeraire: the money market account 165
Second choice of numeraire: discount bonds 170
An intuitive discussion 174
A worked-out example: valuing a LIBOR-in-arrears swap 176
Switching between measures — the Vaillant brackets 179
PART THREE THE IMPLEMENTATION TOOLS 185
8 Lattice methods 187
8.1 Justification of lattice models 187
8.2 Implementation of lattice models: backward induction 194
8.3 Implementation of lattice models: forward induction 197
9 The partial differential equation (PDE) approach 201
9.1 The underlying parabolic equation and the calibration issues
9.2 Finite-differences (FD) approximations to parabolic PDEs
9.3 The explicit finite-differences scheme 208
9.4 The implicit finite-differences scheme 212
10 Monte Carlo methods 215
10.1 Introduction 215
10.2 The method 216
10.3 Variance-reduction techniques 222
10.4 Handling American options 227
PART FOUR ANALYSIS OF SPECIFIC MODELS 231
201
205
11 The CIR and Vasicek models 233
11.1 General features of desirable interest-rate processes
11.2 Derivation of the CIR and Vasicek models 239
233
Contents
243
11.3 Analytic properties of the CIR discount function
11.4 Bond options in the CIR model 246
11.5 Parametrisation of the CIR model 249
11.6 The CIR model: empirical results 251
12 The Black Derman and Toy model 259
12.1 Introduction 259
12.2 Analytic characterisation 260
12.3 Assessing the realism of the BDT model 262
12.4 Derivatives in one-factor models: the BDT case 268
12.5 Calibrating the BDT model: pricing FRAs, caps and swaptions using
lattice models 270
13 The Hull and White approach 281
13.1 Introduction and motivation 281
13.2 Specification of the one-factor version of the model 283
13.3 Exact fitting of the model to the term structure of volatilities
13.4 Constructing the HW tree for constant reversion speed and
volatility 289
13.5 Best-fit calibration of the one-dimensional HW model to market
data 295
13.6 The two-dimensional formulation of the HW model 301
13.7 Calibrating a two-factor HW model 306
13.8 Numerical implementation 308
13.9 Conclusions 311
Contents
288
Appendix 13.1 312
14
313
316
The Longstaff and Schwartz model
14.1 Motivation 313
14.2 The LS economy 314
The PDE obeyed by contingent claims 315
The dynamics of the transformed variables r and V
The equilibrium term structure 320
Term structure of volatilities 321
Correlation between rates 323
Option pricing 325
Calibrating the LS model 327
14.10 Fitting the yield curve using the implied approach 328
14.11 Tests of the joint dynamics using the implied approach 332
14.12 Calibration to the yield curve using the historical approach 337
14.13 Conclusions 339
14.3
14.4
14.5
14.6
14.7
14.8
14.9
XI
15
16
17
The Brennan and Schwartz model 341
15.1 Introduction 341
15.2 The condition of no-arbitrage and the market price of long yield
risk 342
15.3 The specific model 345
15.4 Conclusions 351
A class of arbitrage-free log-normal short-rate two-factor
models 353
16.1 Introduction and motivation 353
16.2 Description of the model 355
16.3 Implementation and numerical issues 358
16.4 Calibration and parametrisation 361
16.5 Computational results 364
16.6 Conclusions 367
Appendix 16.1 368
The Heath Jarrow and Morton approach 371
17.1 Introduction 371
17.2 The HJM approach 373
17.3 Specifications of the HJM model consistent with log-normal bond
prices or forward rates 378
17.4 General constraints on the volatilities of discount bond prices 380
17.5 The process for the short rate 384
17.6 Conclusions 389
The Brace GatarekMusiela/Jamshidian approach 393
18.1 Observable and unobservable state variables 393
18.2 The discretely-compounded money-market account — forward
rates 395
18.3 The discretely-compounded money-market account — swap
rates 399
18.4 The choice of the most suitable pricing framework 402
18.5 Do models still exist? 406
PART FIVE GENERAL TOPICS 411
18
19 Affine models 413
19.1 Definition of affine models 413
19.2 Time-homogeneous affine models
414
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