Interest Rate Models_Theory & Practice_2006 Edition

987 pages

author: Damiano Brigo · Fabio Mercurio

year of publishing: 2006

.pdf (11M)

Table of Contents

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
XLIV Table of Contents
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
Table of Contents XLV
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
8.3 The “Testing Plan” for Volatilities . . . . . . . . . . . . . . . . . . . . . . . . 392
8.4 Test Results for Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
8.4.1 Case (1): Constant Instantaneous Volatilities . . . . . . . . . 396
8.4.2 Case (2): Volatilities as Functions of Time to Maturity 401
8.4.3 Case (3): Humped and Maturity-Adjusted Instantaneous
Volatilities Depending only on Time to Maturity 410
8.5 The “Testing Plan” for Terminal Correlations . . . . . . . . . . . . . . 421
8.6 Test Results for Terminal Correlations . . . . . . . . . . . . . . . . . . . . 427
8.6.1 Case (i): Humped and Maturity-Adjusted Instantaneous
Volatilities Depending only on Time to Maturity,
Typical Rank-Two Correlations . . . . . . . . . . . . . . . . 427
8.6.2 Case (ii): Constant Instantaneous Volatilities, Typical
Rank-Two Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
8.6.3 Case (iii): Humped and Maturity-Adjusted Instantaneous
Volatilities Depending only on Time to Maturity,
Some Negative Rank-Two Correlations. . . . . . . . . . 432
XLVIII Table of Contents
8.6.4 Case (iv): Constant Instantaneous Volatilities, Some
Negative Rank-Two Correlations. . . . . . . . . . . . . . . . . . . . 438
8.6.5 Case (v): Constant Instantaneous Volatilities, Perfect
Correlations, Upwardly Shifted Φ’s . . . . . . . . . . . . . . . . . 439
8.7 Test Results: Stylized Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 442
Part IV. THE VOLATILITY SMILE
9. Including the Smile in the LFM . . . . . . . . . . . . . . . . . . . . . . . . . . 447
9.1 A Mini-tour on the Smile Problem. . . . . . . . . . . . . . . . . . . . . . . . 447
9.2 Modeling the Smile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
10. Local-Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
10.1 The Shifted-Lognormal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
10.2 The Constant Elasticity of Variance Model . . . . . . . . . . . . . . . . 456
10.3 A Class of Analytically-Tractable Models . . . . . . . . . . . . . . . . . . 459
10.4 A Lognormal-Mixture (LM) Model . . . . . . . . . . . . . . . . . . . . . . . 463
10.5 Forward Rates Dynamics under Different Measures . . . . . . . . . 467
10.5.1 Decorrelation Between Underlying and Volatility . . . . . 469
10.6 Shifting the LM Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
10.7 A Lognormal-Mixture with Different Means (LMDM) . . . . . . . 471
10.8 The Case of Hyperbolic-Sine Processes . . . . . . . . . . . . . . . . . . . . 473
10.9 Testing the Above Mixture-Models on Market Data . . . . . . . . . 475
10.10 A Second General Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
10.11 A Particular Case: a Mixture of GBM’s . . . . . . . . . . . . . . . . . . 483
10.12 An Extension of the GBM Mixture Model Allowing for Implied
Volatility Skews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
10.13 A General Dynamics `a la Dupire (1994) . . . . . . . . . . . . . . . . . . 489
11. Stochastic-Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
11.1 The Andersen and Brotherton-Ratcliffe (2001) Model . . . . . . . 497
11.2 The Wu and Zhang (2002) Model . . . . . . . . . . . . . . . . . . . . . . . . . 501
11.3 The Piterbarg (2003) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
11.4 The Hagan, Kumar, Lesniewski and Woodward (2002) Model 508
11.5 The Joshi and Rebonato (2003) Model . . . . . . . . . . . . . . . . . . . . 513
12. Uncertain-Parameter Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
12.1 The Shifted-Lognormal Model with Uncertain Parameters
(SLMUP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
12.1.1 Relationship with the Lognormal-Mixture LVM . . . . . . 520
12.2 Calibration to Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
12.3 Swaption Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
12.4 Monte-Carlo Swaption Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
12.5 Calibration to Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
Table of Contents XLIX
12.6 Calibration to Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
12.7 Testing the Approximation for Swaptions Prices . . . . . . . . . . . . 530
12.8 Further Model Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
12.9 Joint Calibration to Caps and Swaptions . . . . . . . . . . . . . . . . . . 539
Part V. EXAMPLES OF MARKET PAYOFFS
13. Pricing Derivatives on a Single Interest-Rate Curve . . . . . . 547
13.1 In-Arrears Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
13.2 In-Arrears Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
13.2.1 A First Analytical Formula (LFM) . . . . . . . . . . . . . . . . . 550
13.2.2 A Second Analytical Formula (G2++) . . . . . . . . . . . . . . 551
13.3 Autocaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
13.4 Caps with Deferred Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
13.4.1 A First Analytical Formula (LFM) . . . . . . . . . . . . . . . . . 553
13.4.2 A Second Analytical Formula (G2++) . . . . . . . . . . . . . . 553
13.5 Ratchet Caps and Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
13.5.1 Analytical Approximation for Ratchet Caps with the
LFM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
13.6 Ratchets (One-Way Floaters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
13.7 Constant-Maturity Swaps (CMS) . . . . . . . . . . . . . . . . . . . . . . . . . 557
13.7.1 CMS with the LFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
13.7.2 CMS with the G2++ Model . . . . . . . . . . . . . . . . . . . . . . . 559
13.8 The Convexity Adjustment and Applications to CMS . . . . . . . 559
13.8.1 Natural and Unnatural Time Lags . . . . . . . . . . . . . . . . . . 559
13.8.2 The Convexity-Adjustment Technique . . . . . . . . . . . . . . . 561
13.8.3 Deducing a Simple Lognormal Dynamics from the Adjustment
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
13.8.4 Application to CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
13.8.5 Forward Rate Resetting Unnaturally and Average-
Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
13.9 Average Rate Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
13.10 Captions and Floortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
13.11 Zero-Coupon Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
13.12 Eurodollar Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
13.12.1 The Shifted Two-Factor Vasicek G2++ Model . . . . . . 576
13.12.2 Eurodollar Futures with the LFM . . . . . . . . . . . . . . . . . 577
13.13 LFM Pricing with “In-Between” Spot Rates . . . . . . . . . . . . . . 578
13.13.1 Accrual Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
13.13.2 Trigger Swaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
13.14LFM Pricing with Early Exercise and Possible Path Dependence584
13.15 LFM: Pricing Bermudan Swaptions . . . . . . . . . . . . . . . . . . . . . . 588
13.15.1 Least Squared Monte Carlo Approach . . . . . . . . . . . . . . 589
13.15.2 Carr and Yang’s Approach . . . . . . . . . . . . . . . . . . . . . . . 591
L Table of Contents
13.15.3 Andersen’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
13.15.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
13.16 New Generation of Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
13.16.1 Target Redemption Notes . . . . . . . . . . . . . . . . . . . . . . . . 602
13.16.2 CMS Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
14. Pricing Derivatives on Two Interest-Rate Curves . . . . . . . . . 607
14.1 The Attractive Features of G2++ for Multi-Curve Payoffs . . . 608
14.1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
14.1.2 Interaction Between Models of the Two Curves “1”
and “2” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
14.1.3 The Two-Models Dynamics under a Unique Convenient
Forward Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
14.2 Quanto Constant-Maturity Swaps . . . . . . . . . . . . . . . . . . . . . . . . 613
14.2.1 Quanto CMS: The Contract . . . . . . . . . . . . . . . . . . . . . . . 613
14.2.2 Quanto CMS: The G2++ Model . . . . . . . . . . . . . . . . . . . 615
14.2.3 Quanto CMS: Quanto Adjustment . . . . . . . . . . . . . . . . . . 621
14.3 Differential Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
14.3.1 The Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
14.3.2 Differential Swaps with the G2++ Model . . . . . . . . . . . . 624
14.3.3 A Market-Like Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
14.4 Market Formulas for Basic Quanto Derivatives . . . . . . . . . . . . . 626
14.4.1 The Pricing of Quanto Caplets/Floorlets . . . . . . . . . . . . 627
14.4.2 The Pricing of Quanto Caps/Floors . . . . . . . . . . . . . . . . . 628
14.4.3 The Pricing of Differential Swaps . . . . . . . . . . . . . . . . . . . 629
14.4.4 The Pricing of Quanto Swaptions. . . . . . . . . . . . . . . . . . . 630
14.5 Pricing of Options on two Currency LIBOR Rates . . . . . . . . . . 633
14.5.1 Spread Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
14.5.2 Options on the Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
14.5.3 Trigger Swaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
14.5.4 Dealing with Multiple Dates . . . . . . . . . . . . . . . . . . . . . . . 639
Part VI. INFLATION
15. Pricing of Inflation-Indexed Derivatives . . . . . . . . . . . . . . . . . . . 643
15.1 The Foreign-Currency Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
15.2 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
15.3 The JY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
16. Inflation-Indexed Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
16.1 Pricing of a ZCIIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
16.2 Pricing of a YYIIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
16.3 Pricing of a YYIIS with the JY Model . . . . . . . . . . . . . . . . . . . . 652
16.4 Pricing of a YYIIS with a First Market Model . . . . . . . . . . . . . 654
Table of Contents LI
16.5 Pricing of a YYIIS with a Second Market Model . . . . . . . . . . . 657
17. Inflation-Indexed Caplets/Floorlets . . . . . . . . . . . . . . . . . . . . . . . 661
17.1 Pricing with the JY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
17.2 Pricing with the Second Market Model . . . . . . . . . . . . . . . . . . . . 663
17.3 Inflation-Indexed Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
18. Calibration to market data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
19. Introducing Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 673
19.1 Modeling Forward CPI’s with Stochastic Volatility . . . . . . . . 674
19.2 Pricing Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
19.2.1 Exact Solution for the Uncorrelated Case . . . . . . . . . . . . 677
19.2.2 Approximated Dynamics for Non-zero Correlations . . . 680
19.3 Example of Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
20. Pricing Hybrids with an Inflation Component . . . . . . . . . . . . 689
20.1 A Simple Hybrid Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
Part VII. CREDIT
21. Introduction and Pricing under Counterparty Risk . . . . . . . 695
21.1 Introduction and Guided Tour . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
21.1.1 Reduced form (Intensity) models . . . . . . . . . . . . . . . . . . . 697
21.1.2 CDS Options Market Models . . . . . . . . . . . . . . . . . . . . . . 699
21.1.3 Firm Value (or Structural) Models. . . . . . . . . . . . . . . . . . 702
21.1.4 Further Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
21.1.5 The Multi-name picture: FtD, CDO and Copula Functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
21.1.6 First to Default (FtD) Basket. . . . . . . . . . . . . . . . . . . . . . 705
21.1.7 Collateralized Debt Obligation (CDO) Tranches. . . . . . 707
21.1.8 Where can we introduce dependence? . . . . . . . . . . . . . . . 708
21.1.9 Copula Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710
21.1.10 Dynamic Loss models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 718
21.1.11 What data are available in the market? . . . . . . . . . . . . 719
21.2 Defaultable (corporate) zero coupon bonds . . . . . . . . . . . . . . . . 723
21.2.1 Defaultable (corporate) coupon bonds. . . . . . . . . . . . . . . 724
21.3 Credit Default Swaps and Defaultable Floaters . . . . . . . . . . . . . 724
21.3.1 CDS payoffs: Different Formulations . . . . . . . . . . . . . . . . 725
21.3.2 CDS pricing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727
21.3.3 Changing filtration: Ft without default VS complete
Gt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728
21.3.4 CDS forward rates: The first definition . . . . . . . . . . . . . . 730
LII Table of Contents
21.3.5 Market quotes, model independent implied survival
probabilities and implied hazard functions . . . . . . . . . . . 731
21.3.6 A simpler formula for calibrating intensity to a single
CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735
21.3.7 Different Definitions of CDS Forward Rates and Analogies
with the LIBOR and SWAP rates . . . . . . . . . . . . . . 737
21.3.8 Defaultable Floater and CDS . . . . . . . . . . . . . . . . . . . . . . 739
21.4 CDS Options and Callable Defaultable Floaters . . . . . . . . . . . . 743
21.5 Constant Maturity CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
21.5.1 Some interesting Financial features of CMCDS . . . . . . . 745
21.6 Interest-Rate Payoffs with Counterparty Risk . . . . . . . . . . . . . . 747
21.6.1 General Valuation of Counterparty Risk . . . . . . . . . . . . . 748
21.6.2 Counterparty Risk in single Interest Rate Swaps (IRS) 750
22. Intensity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757
22.1 Introduction and Chapter Description . . . . . . . . . . . . . . . . . . . . . 757
22.2 Poisson processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
22.2.1 Time homogeneous Poisson processes . . . . . . . . . . . . . . . 760
22.2.2 Time inhomogeneous Poisson Processes . . . . . . . . . . . . . 761
22.2.3 Cox Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
22.3 CDS Calibration and Implied Hazard Rates/ Intensities . . . . . 764
22.4 Inducing dependence between Interest-rates and the default
event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776
22.5 The Filtration Switching Formula: Pricing under partial information
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
22.6 Default Simulation in reduced form models . . . . . . . . . . . . . . . . 778
22.6.1 Standard error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781
22.6.2 Variance Reduction with Control Variate . . . . . . . . . . . . 783
22.7 Stochastic Intensity: The SSRD model . . . . . . . . . . . . . . . . . . . . 785
22.7.1 A two-factor shifted square-root diffusion model for
intensity and interest rates (Brigo and Alfonsi (2003)) . 786
22.7.2 Calibrating the joint stochastic model to CDS: Separability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
22.7.3 Discretization schemes for simulating (λ, r) . . . . . . . . . . 797
22.7.4 Study of the convergence of the discretization schemes
for simulating CIR processes (Alfonsi (2005)) . . . . . . . . 801
22.7.5 Gaussian dependence mapping: A tractable approximated
SSRD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812
22.7.6 Numerical Tests: Gaussian Mapping and Correlation
Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815
22.7.7 The impact of correlation on a few “test payoffs” . . . . . 817
22.7.8 A pricing example: A Cancellable Structure. . . . . . . . . . 818
22.7.9 CDS Options and Jamshidian’s Decomposition . . . . . . . 820
22.7.10 Bermudan CDS Options. . . . . . . . . . . . . . . . . . . . . . . . . . 830
Table of Contents LIII
22.8 Stochastic diffusion intensity is not enough: Adding jumps.
The JCIR(++) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830
22.8.1 The jump-diffusion CIR model (JCIR) . . . . . . . . . . . . . . 831
22.8.2 Bond (or Survival Probability) Formula. . . . . . . . . . . . . . 832
22.8.3 Exact calibration of CDS: The JCIR++ model . . . . . . . 833
22.8.4 Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833
22.8.5 Jamshidian’s Decomposition. . . . . . . . . . . . . . . . . . . . . . . 834
22.8.6 Attaining high levels of CDS implied volatility . . . . . . . 836
22.8.7 JCIR(++) models as a multi-name possibility . . . . . . . . 837
22.9 Conclusions and further research . . . . . . . . . . . . . . . . . . . . . . . . . 838
23. CDS Options Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841
23.1 CDS Options and Callable Defaultable Floaters . . . . . . . . . . . . 844
23.1.1 Once-callable defaultable floaters . . . . . . . . . . . . . . . . . . . 846
23.2 A market formula for CDS options and callable defaultable
floaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
23.2.1 Market formulas for CDS Options . . . . . . . . . . . . . . . . . . 847
23.2.2 Market Formula for callable DFRN . . . . . . . . . . . . . . . . . 849
23.2.3 Examples of Implied Volatilities from the Market . . . . . 852
23.3 Towards a Completely Specified Market Model . . . . . . . . . . . . . 854
23.3.1 First Choice. One-period and two-period rates . . . . . . . 855
23.3.2 Second Choice: Co-terminal and one-period CDS rates
market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860
23.3.3 Third choice. Approximation: One-period CDS rates
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861
23.4 Hints at Smile Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
23.5 Constant Maturity Credit Default Swaps (CMCDS) with the
market model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864
23.5.1 CDS and Constant Maturity CDS . . . . . . . . . . . . . . . . . . 864
23.5.2 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . 867
23.5.3 A few numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 869
Part VIII. APPENDICES
A. Other Interest-Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
A.1 Brennan and Schwartz’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
A.2 Balduzzi, Das, Foresi and Sundaram’s Model . . . . . . . . . . . . . . . 878
A.3 Flesaker and Hughston’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . 879
A.4 Rogers’s Potential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881
A.5 Markov Functional Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881
LIV Table of Contents
B. Pricing Equity Derivatives under Stochastic Rates . . . . . . . . 883
B.1 The Short Rate and Asset-Price Dynamics . . . . . . . . . . . . . . . . . 883
B.1.1 The Dynamics under the Forward Measure . . . . . . . . . . 886
B.2 The Pricing of a European Option on the Given Asset . . . . . . 888
B.3 A More General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889
B.3.1 The Construction of an Approximating Tree for r . . . . 890
B.3.2 The Approximating Tree for S . . . . . . . . . . . . . . . . . . . . . 892
B.3.3 The Two-Dimensional Tree . . . . . . . . . . . . . . . . . . . . . . . . 893
C. A Crash Intro to Stochastic Differential Equations and Poisson
Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897
C.1 From Deterministic to Stochastic Differential Equations . . . . . 897
C.2 Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904
C.3 Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes906
C.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908
C.5 Two Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910
C.6 A Crash Intro to Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . 913
C.6.1 Time inhomogeneous Poisson Processes . . . . . . . . . . . . . 915
C.6.2 Doubly Stochastic Poisson Processes (or Cox Processes)916
C.6.3 Compound Poisson processes . . . . . . . . . . . . . . . . . . . . . . 917
C.6.4 Jump-diffusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 918
D. A Useful Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919
E. A Second Useful Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921
F. Approximating Diffusions with Trees . . . . . . . . . . . . . . . . . . . . . 925
G. Trivia and Frequently Asked Questions . . . . . . . . . . . . . . . . . . . 931
H. Talking to the Traders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967