Interest-rate option models : understanding, analysing and using models for exotic interest-rate options

목차 (Table of Contents)

• 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
• 2.3 American (Bermudan) swaptions = 36
• 2.4 Trigger swaps = 41
• 2.5 One-way floaters = 44
• 2.6 Captions = 47
• 3 A statistical approach to yield curve models = 51
• 3.1 Statistical analysis of the evolution of rates = 51
• 3.2 The effects of model dimensionality on option pricing = 63
• 3.3 A framework for option pricing = 72
• 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
• 5 A motivation for yield curve models = 105
• 5.1 Introduction = 105
• 5.2 Hedging a bond option with the underlying forward contract = 106
• 5.3 Hedging a path-dependent bond option with forward contracts = 109
• PART TWO THE THEORETICAL TOOLS = 117
• 6 Establishing a pricing framework = 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 = 136
• 6.9 Self-financing tradine strategies = 139
• 6.10 Fair prices as expectations = 141
• 6.11 Switching numeraires and relating expectations under different measures = 144
• 6.12 Justifying the two-state branching procedure = 150
• 6.13 The nature of the transformation between measures - Girsanov's theorem = 153
• 7 The conditions of no-arbitrage = 157
• 7.1 First no-arbitrage condition : the Vasicek approach = 157
• 7.2 Second no-arbitrage condition : the martingale approach = 160
• 7.3 The case of a deterministic-interest-rates economy = 162
• 7.4 First choice of numeraire : the money market account = 165
• 7.5 Second choice of numeraire : discount bonds = 170
• 7.6 An intuitive discussion = 174
• 7.7 A worked-out example : valuing a LIBOR-in-arrears swap = 176
• 7.8 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 = 201
• 9.2 Finite-differences(FD) approximations to parabolic PDEs = 205
• 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
• 11 The CIR and Vasicek models = 233
• 11.1 General features of desirable interest-rate processes = 233
• 11.2 Derivation of the CIR and Vasicek models = 239
• 11.3 Analytic Properties of the CIR discount function = 243
• 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 = 288
• 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
• Appendix 13.1 = 312
• 14 The Longstaff and Schwartz model = 131
• 14.1 Motivation = 313
• 14.2 The LS economy = 314
• 14.3 The PDE obeyed by contingent claims = 315
• 14.4 The dynamics of the transformed variables r and V = 316
• 14.5 The equilibrium term structure = 320
• 14.6 Term structure of volatilities = 321
• 14.7 Correlation between rates = 323
• 14.8 Option pricing = 325
• 14.9 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
• 15 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
• 16 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
• 17 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
• 18 The Brace-Gatarek-Musiela／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
• 19 Affine models = 413
• 19.1 Definition of affine models = 413
• 19.2 Time-homogeneous affine models = 414
• 19.3 Time-inhomogeneous affine models = 418
• 19.4 General considerations = 420
• 20 Markovian and non-Markovian interest-rate models = 423
• 20.1 Definition of Markovian rate processes = 423
• 20.2 Conditions for the rate process to be Markovian = 425
• 20.3 Non-Markovian models on recombining trees = 428
• 20.4 Implications for the choice of interest-rate models = 429
• 21 Calibration to cap prices of mean-reverting log-normal short-rate models = 435
• 21.1 Introduction = 435
• 21.2 Statement of the problem = 436
• 21.3 The unconditional variance of the short rate in BDT - the discrete case = 437
• 21.4 The unconditional variance of the short rate in BDT - the continuous-time equivalent = 440
• 21.5 Extension to two-factor approaches = 441
• 21.6 Conclusions = 443
• Appendix 21.1 = 444
• Appendix A Elements of probability and stochastic calculus = 447
• A.1 Fundamental results and definitions about set theory = 447
• A.2 Fundamental probabilistic definitions = 448
• A.3 Representing the flow of information = 452
• A.4 Brownian motions and random walks = 466
• A.5 Martingales and Ito integrals = 474
• A.6 Ito's lemma and the rules of stochastic differentiation = 481
• A.7 The link between stochastic differential equations(SDEs) and parabolic partial differential equations(PDEs) = 485
• A.8 Switching between measures - the Radon-Nikodym derivative and Girsanov's theorem = 486
• Appendix B The securities market = 491
• B.1 Prices and strategies = 491
• B.2 Definition of arbitrage in a discrete complete market = 496
• B.3 Replication of contingent claims = 502
• Bibliography = 509
• Index = 515