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