Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Aims, Readership and Book Structure . . . . . . . . . . . . . . . . . . . . . . . . . XII
Final Word and Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV
Description of Contents by Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX
Abbreviations and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .X.X. XV
Part I. BASIC DEFINITIONS AND NO ARBITRAGE
1. Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Bank Account and the Short Rate . . . . . . . . . . . . . . . . . . . . 2
1.2 Zero-Coupon Bonds and Spot Interest Rates . . . . . . . . . . . . . . . 4
1.3 Fundamental Interest-Rate Curves . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Interest-Rate Swaps and Forward Swap Rates . . . . . . . . . . . . . . 13
1.6 Interest-Rate Caps/Floors and Swaptions . . . . . . . . . . . . . . . . . . 16
2. No-Arbitrage Pricing and Numeraire Change . . . . . . . . . . . . . 23
2.1 No-Arbitrage in Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 The Change-of-Numeraire Technique . . . . . . . . . . . . . . . . . . . . . . 26
2.3 A Change of Numeraire Toolkit
(Brigo & Mercurio 2001c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 A helpful notation: “DC”. . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 The Choice of a Convenient Numeraire . . . . . . . . . . . . . . . . . . . . 37
2.5 The Forward Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 The Fundamental Pricing Formulas . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 The Pricing of Caps and Floors . . . . . . . . . . . . . . . . . . . . 40
2.7 Pricing Claims with Deferred Payoffs . . . . . . . . . . . . . . . . . . . . . 42
2.8 Pricing Claims with Multiple Payoffs. . . . . . . . . . . . . . . . . . . . . . 42
2.9 Foreign Markets and Numeraire Change . . . . . . . . . . . . . . . . . . . 44
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Part II. FROM SHORT RATE MODELS TO HJM
3. One-factor short-rate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Introduction and Guided Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Classical Time-Homogeneous Short-Rate Models . . . . . . . . . . . 57
3.2.1 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 The Dothan Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.3 The Cox, Ingersoll and Ross (CIR) Model . . . . . . . . . . . 64
3.2.4 Affine Term-Structure Models . . . . . . . . . . . . . . . . . . . . . . 68
3.2.5 The Exponential-Vasicek (EV) Model . . . . . . . . . . . . . . . 70
3.3 The Hull-White Extended Vasicek Model . . . . . . . . . . . . . . . . . . 71
3.3.1 The Short-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.2 Bond and Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.3 The Construction of a Trinomial Tree . . . . . . . . . . . . . . . 78
3.4 Possible Extensions of the CIR Model . . . . . . . . . . . . . . . . . . . . . 80
3.5 The Black-Karasinski Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.1 The Short-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.2 The Construction of a Trinomial Tree . . . . . . . . . . . . . . . 85
3.6 Volatility Structures in One-Factor Short-Rate Models . . . . . . 86
3.7 Humped-Volatility Short-Rate Models . . . . . . . . . . . . . . . . . . . . . 92
3.8 A General Deterministic-Shift Extension . . . . . . . . . . . . . . . . . . 95
3.8.1 The Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.8.2 Fitting the Initial Term Structure of Interest Rates . . . 97
3.8.3 Explicit Formulas for European Options . . . . . . . . . . . . . 99
3.8.4 The Vasicek Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.9 The CIR++ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.9.1 The Construction of a Trinomial Tree . . . . . . . . . . . . . . . 105
3.9.2 Early Exercise Pricing via Dynamic Programming . . . . 106
3.9.3 The Positivity of Rates and Fitting Quality . . . . . . . . . . 106
3.9.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.9.5 Jump Diffusion CIR and CIR++ models (JCIR, JCIR++)109
3.10 Deterministic-Shift Extension of Lognormal Models . . . . . . . . . 110
3.11 Some Further Remarks on Derivatives Pricing . . . . . . . . . . . . . . 112
3.11.1 Pricing European Options on a Coupon-Bearing Bond 112
3.11.2 The Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 114
3.11.3 Pricing Early-Exercise Derivatives with a Tree . . . . . . . 116
3.11.4 A Fundamental Case of Early Exercise: Bermudan-
Style Swaptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.12 Implied Cap Volatility Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.12.1 The Black and Karasinski Model . . . . . . . . . . . . . . . . . . . 125
3.12.2 The CIR++ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.12.3 The Extended Exponential-Vasicek Model . . . . . . . . . . . 128
3.13 Implied Swaption Volatility Surfaces . . . . . . . . . . . . . . . . . . . . . . 129
3.13.1 The Black and Karasinski Model . . . . . . . . . . . . . . . . . . . 130
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3.13.2 The Extended Exponential-Vasicek Model . . . . . . . . . . . 131
3.14 An Example of Calibration to Real-Market Data . . . . . . . . . . . 132
4. Two-Factor Short-Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2 The Two-Additive-Factor Gaussian Model G2++. . . . . . . . . . . 142
4.2.1 The Short-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2.2 The Pricing of a Zero-Coupon Bond . . . . . . . . . . . . . . . . 144
4.2.3 Volatility and Correlation Structures in Two-Factor
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.4 The Pricing of a European Option on a Zero-Coupon
Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.2.5 The Analogy with the Hull-White Two-Factor Model . 159
4.2.6 The Construction of an Approximating Binomial Tree . 162
4.2.7 Examples of Calibration to Real-Market Data . . . . . . . . 166
4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++ 175
4.3.1 The Basic Two-Factor CIR2 Model . . . . . . . . . . . . . . . . . 176
4.3.2 Relationship with the Longstaff and Schwartz Model
(LS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.3.3 Forward-Measure Dynamics and Option Pricing for
CIR2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.3.4 The CIR2++ Model and Option Pricing . . . . . . . . . . . . 179
5. The Heath-Jarrow-Morton (HJM) Framework . . . . . . . . . . . . 183
5.1 The HJM Forward-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 185
5.2 Markovianity of the Short-Rate Process . . . . . . . . . . . . . . . . . . . 186
5.3 The Ritchken and Sankarasubramanian Framework . . . . . . . . . 187
5.4 The Mercurio and Moraleda Model . . . . . . . . . . . . . . . . . . . . . . . 191
Part III. MARKET MODELS
6. The LIBOR and Swap Market Models (LFM and LSM) . . 195
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.2 Market Models: a Guided Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.3 The Lognormal Forward-LIBOR Model (LFM) . . . . . . . . . . . . . 207
6.3.1 Some Specifications of the Instantaneous Volatility of
Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.3.2 Forward-Rate Dynamics under Different Numeraires . . 213
6.4 Calibration of the LFM to Caps and Floors Prices . . . . . . . . . . 220
6.4.1 Piecewise-Constant Instantaneous-Volatility Structures 223
6.4.2 Parametric Volatility Structures . . . . . . . . . . . . . . . . . . . . 224
6.4.3 Cap Quotes in the Market . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.5 The Term Structure of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.5.1 Piecewise-Constant Instantaneous Volatility Structures 228
XLVI Table of Contents
6.5.2 Parametric Volatility Structures . . . . . . . . . . . . . . . . . . . . 231
6.6 Instantaneous Correlation and Terminal Correlation . . . . . . . . 234
6.7 Swaptions and the Lognormal Forward-Swap Model (LSM) . . 237
6.7.1 Swaptions Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.7.2 Cash-Settled Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.8 Incompatibility between the LFM and the LSM . . . . . . . . . . . . 244
6.9 The Structure of Instantaneous Correlations . . . . . . . . . . . . . . . 246
6.9.1 Some convenient full rank parameterizations . . . . . . . . . 248
6.9.2 Reduced-rank formulations: Rebonato’s angles and eigenvalues
zeroing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
6.9.3 Reducing the angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.10 Monte Carlo Pricing of Swaptions with the LFM . . . . . . . . . . . 264
6.11 Monte Carlo Standard Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.12 Monte Carlo Variance Reduction: Control Variate Estimator . 269
6.13 Rank-One Analytical Swaption Prices . . . . . . . . . . . . . . . . . . . . . 271
6.14 Rank-r Analytical Swaption Prices . . . . . . . . . . . . . . . . . . . . . . . 277
6.15 A Simpler LFM Formula for Swaptions Volatilities . . . . . . . . . . 281
6.16 A Formula for Terminal Correlations of Forward Rates . . . . . . 284
6.17 Calibration to Swaptions Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 287
6.18 Instantaneous Correlations: Inputs (Historical Estimation) or
Outputs (Fitting Parameters)? . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.19 The exogenous correlation matrix. . . . . . . . . . . . . . . . . . . . . . . . . 291
6.19.1 Historical Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
6.19.2 Pivot matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.20 Connecting Caplet and S × 1-Swaption Volatilities . . . . . . . . . . 300
6.21 Forward and Spot Rates over Non-Standard Periods . . . . . . . . 307
6.21.1 Drift Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
6.21.2 The Bridging Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
7. Cases of Calibration of the LIBOR Market Model . . . . . . . . 313
7.1 Inputs for the First Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
7.2 Joint Calibration with Piecewise-Constant Volatilities as in
TABLE 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
7.3 Joint Calibration with Parameterized Volatilities as in Formulation
7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.4 Exact Swaptions “Cascade” Calibration with Volatilities as
in TABLE 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.4.1 Some Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 330
7.5 A Pause for Thought. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.5.1 First summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
7.5.2 An automatic fast analytical calibration of LFM to
swaptions. Motivations and plan . . . . . . . . . . . . . . . . . . . 338
7.6 Further Numerical Studies on the Cascade Calibration Algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Table of Contents XLVII
7.6.1 Cascade Calibration under Various Correlations and
Ranks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
7.6.2 Cascade Calibration Diagnostics: Terminal Correlation
and Evolution of Volatilities . . . . . . . . . . . . . . . . . . . 346
7.6.3 The interpolation for the swaption matrix and its impact
on the CCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
7.7 Empirically efficient Cascade Calibration . . . . . . . . . . . . . . . . . . 351
7.7.1 CCA with Endogenous Interpolation and Based Only
on Pure Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
7.7.2 Financial Diagnostics of the RCCAEI test results . . . . . 359
7.7.3 Endogenous Cascade Interpolation for missing swaptions
volatilities quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.7.4 A first partial check on the calibrated σ parameters
stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.8 Reliability: Monte Carlo tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
7.9 Cascade Calibration and the cap market. . . . . . . . . . . . . . . . . . . 369
7.10 Cascade Calibration: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 372
8. Monte Carlo Tests for LFM Analytical Approximations. . . 377
8.1 First Part. Tests Based on the Kullback Leibler Information
(KLI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
8.1.1 Distance between distributions: The Kullback Leibler
information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
8.1.2 Distance of the LFM swap rate from the lognormal
family of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8.1.3 Monte Carlo tests for measuring KLI . . . . . . . . . . . . . . . 384
8.1.4 Conclusions on the KLI-based approach . . . . . . . . . . . . . 391
8.2 Second Part: Classical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.................................