Preface ...................................................... VII
Motivation ................................................. VII
Aims, Readership and Book Structure ......................... XII
Final Word and Acknowledgments ............................ XIII
Deseription of Contents by Chapter ........................... XVI
Abbreviations and Notation ................................. XXV
Table of Contents ................................ XXIX
Part I. MODELS: THEORY AND IMPLEMENTATION
1. Definitions and Notation ................................. 1
1.1 The Bank Account and the Short Rate .................... 1
1.2 Zero-Coupon Bonds and Spot Interest Rates ............... 3
1.3 Fundamental Interest-Rate Curves ........................ 8
1.4 Forward Rates ......................................... 10
1.5 Interest-Rate Swaps and Forward Swap Rates .............. 13
1.6 Interest-Rate Caps/Floors and Swaptions .................. 15
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 .......................... 28
2.4 The Choice of a Convenient Numeraire .................... 32
2.5 The Forward Measure ................................... 33
2.6 The Fundamental Pricing Formulas ....................... 35
2.6.1 The Pricing of Caps and Floors .................... 36
2.7 Pricing Claims with Deferred Payoffs ..................... 37
2.8 Pricing Claims with Multiple Payoffs ...................... 38
2.9 Foreign Markets and Numeraire Change ................... 40
3. One-factor short-rate models ............................. 43
3.1 Introduction and Guided Tour ........................... 43
3.2 Classical Time-Homogeneous Short-Rate Models ........... 48
3.2.1 The Vasicek Model ............................... 50
3.2.2 The Dothan Model ............................... 54
3.2.3 The Cox, Ingersoll and Ross (CIR) Model ........... 56
3.2.4 Affine Term-Structure Models ...................... 60
3.2.5 The Exponential-Vasicek (EV) Model ............... 61
3.3 The Hull-White Extended Vasicek Model .................. 63
3.3.1 The Short-Rate Dynamics ......................... 64
3.3.2 Bond and Option Pricing .......................... 66
3.3.3 The Construction of a Binomial Tree ............... 69
3.4 Possible Extensions of the CIR Model ..................... 72
3.5 The Black-Karasinski Model ............................. 73
3.5.1 The Short-Rate Dynamics ......................... 74
3.5.2 The Construction of a Binomial Tree ............... 76
3.6 Volatility Structures in One-Factor Short-Rate Models ...... 77
3.7 Numped-Volatility Short-Rate Models ..................... 83
3.8 A General Deterministic-Shift Extension .................. 86
3.8.1 The Basic Assumptions ........................... 87
3.8.2 Fitting the Initial Term Structure of Interest Rates ... 88
3.8.3 Explicit Formulas for European Options ............. 90
3.8.4 The Vasicek Case ................................ 91
3.9 The CIR++ Model ..................................... 93
3.9.1 The Construction of a Trinomial Tree ............... 96
3.9.2 The Positivity of Rates and Fitting Quality .......... 97
3.10 Deterministic-Shift Extension of Lognormal Models ......... 100
3.11 Some Further Remarks on Derivatives Pricing .............. 102
3.11.1 Pricing European Options on a Coupon-Bearing Bond 102
3.11.2 The Monte Carlo Simulation ....................... 103
3.11.3 Pricing Early-Exereise Derivatives with a Tree ....... 106
3.11.4 A Fundamental Case of Early Exercise: Bermudan-
Style Swaptions . ................................. 111
3.12 Implied Cap Volatility Curves ............................ 114
3.12.1 The Black and Karasinski Model .................... 115
3.12.2 The CIR++ Model ............................... 116
3.12.3 The Extended Exponential-Vasicek Model ........... 117
3.13 Implied Swaption Volatility Surfaces ...................... 119
3.13.1 The Black and Karasinski Model ................... 120
3.13.2 The Extended Exponential-Vasicek Model ........... 120
3.14 An Example of Calibration to Real-Market Data ........... 121
4. Two-Factor Short-Rate Models ........................... 127
4.1 Introduction and Motivation ............................. 127
4.2 The Two-Additive-Factor Gaussian Model G2 ++ ............ 132
4.2.1 The Short-Rate Dynamics ......................... 133
4.2.2 The Pricing of a Zero-Coupon Bond ................ 134
4.2.3 Volatility and Correlation Structures in Two-Factor
Models .......................................... 137
4.2.4 The Pricing of a European Option on a Zero-Coupon
Bond ........................................... 143
4.2.5 The Analogy with the Hull-White Two-Factor Model . 149
4.2.6 The Construction of an Approximating Binomial Tree. 152
4.2.7 Exarnples of Calibration to Real-Market Data ........ 156
4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++ 165
4.3.1 The Basic Two-Factor CIR2 Model ................. 166
4.3.2 Relationship with the Longstaff and Schwartz Model
(LS) ............................................ 167
4.3.3 Forward-Measure Dynamics and Option Pricing for
CIR2 ........................................... 168
4.3.4 The C1R2++ Model and Option Pricing ............ 168
5. The Heath-Jarrow-Morton (HJM) Framework ............ 173
5.1 The HJM Forward-Rate Dynamics ........................ 175
5.2 Markovianity of the Short-Rate Process ................... 176
5.3 The Ritchken and Sankarasubramanian Framework ......... 177
5.4 The Mercurio and Moraleda Model ....................... 181
6. The LIBOR and Swap Market Models (LFM and LSM) .. 183
6.1 Introduction ........................................... 183
6.2 Market Models: a Guided Tour ........................... 184
6.3 The Lognormal Forward-LIBOR Model (LFM) ............. 192
6.3.1 Some Specifieations of the Instantaneous Volatility of
Forward Rates ................................... 195
6.3.2 Forward-Rate Dynamics under Different Nurneraires .. 198
6.4 Calibration of the LFM to Caps and Floors Prices .......... 203
6.4.1 Piecewise-Constant Instantaneous-Volatility Structures 206
6.4.2 Parametric Volatility Structures .................... 207
6.4.3 Cap Quotes in the Market ......................... 208
6.5 The Term Structure of Volatility ......................... 210
6.5.1 Piecewise-Constant Instantaneous Volatility Structures 210
6.5.2 Parametrie Volatility Structures .................... 215
6.6 Instantaneous Correlation and Terminal Correlation ........ 217
6.7 Swaptions and the Lognormal Forward-Swap Model (LSM) .. 220
6.7.1 Swaptions Hedging ............................... 224
6.7.2 Cash-Settled Swaptions ........................... 226
6.8 Incompatibility between the LFM and the LSM ............ 227
6.9 The Structure of Instantaneous Correlations ............... 230
6.10 Monte Carlo Pricing of Swaptions with the LFM ........... 233
6.11 Rank-One Analytical Swaption Prices ..................... 236
6.12 Rank-r Analytical Swaption Prices ....................... 242
6.13 A Sirnpler LFM Formula for Swaptions Volatilities .......... 246
6.14 A Formula for Terminal Correlations of Forward Rates ...... 249
6.15 Calibration to Swaptions Prices .......................... 252
6.16 Connecting Caplet and S x 1-Swaption Volatilities .......... 254
6.17 Forward and Spot Rates over Non-Standard Periods ........ 261
6.17.1 Drift Interpolation ................................ 262
6.17.2 The Bridging Technique ........................... 264
6.18 Including the Caplet Smile in the LFM .................... 266
6.18.1 A Mini-tour on the Smile Problem .................. 266
6.18.2 Modeling the Smile ............................... 270
6.18.3 The Shifted-Lognormal Case ....................... 271
6.18.4 The Constant Elastieity of Vmiance (CEV) Model .... 273
6.18.5 A Mixture-of-Lognormals Model .................... 276
6.18.6 Shifting the Lognormal-Mixture Dynamics ........... 280
7. Cases of Calibration of the LIBOR Market Model ........ 283
7.1 The Inputs ............................................ 284
7.2 Joint Calibration with Piecewise-Constant Volatilities as in
TABLE 5 .............................................. 284
7.2.1 Instantaneous Correlations- Narrowing the Angles .... 288
7.2.2 Instantaneous Correlations: Fixing the Angles to Typ-
ical Values ....................................... 290
7.2.3 Instantaneous Correlations: Fixing the Angles to Atyp-
ical Values ............. ......................... 292
7.2.4 Instantaneous Correlations: Collapsing to One Factor . 293
7.3 Joint Calibration with Parameterized Volatilities as in For-
mulation 7 ............................................. 295
7.3.1 Formulation 7. Narrowing the Angles ............... 297
7.3.2 Formulation 7.- Calibrating only to Swaptions ........ 300
7.4 Exact Swaptions Calibration with Volatilities as TABLE 1 ... 303
7.4.1 Some Nurnerical Results ........................... 309
7.5 Conclusions: Where Now? ............................... 314
8. Monte Carlo Tests for LFM Analytical Approximations ... 317
8.1 The Specification of Rates ............................... 317
8.2 The "Testing Plan" for Volatilities ........................ 318
8.3 Test Results for Volatilities .............................. 321
8.3.1 Case (1). Constant Instantaneous Volatilities ......... 322
8.3.2 Case (2): Volatilities as Funetions of Time to Maturity 329
8.3.3 Case (3): Humped and Maturity-Adjusted Instanta-
neous Volatilities Depending only on Time to Maturity 334
8.4 The "Testing Plan" for Terminal Correlations .............. 345
8.5 Test Results for Terminal Correlations .................... 353
8.5.1 Case (i): Humped and Maturity-Adjusted Instanta-
neous Volatilities Depending only on Time to Matu-
rity, Typical Rank-Two Correlations ................ 353
8.5.2 Case (ii): Constant Instantaneous Volatilities, Typical
Rank-Two Correlations ............................ 355
8.5.3 Case (iii): Humped and Maturity-Adjusted Instanta-
neons Volatilities Depending only on Time to Matu-
rity, Some Negative Rank-Two Correlations. 359
8.5.4 Case (iv): Constant Instantaneous Volatilities, Some
Negative Rank-Two Correlations .................... 363
8.5.5 Case (v): Constant Instantaneous Volatilities, Perfect
Correlations, Upwardly Shifted O's ................. 365
8.6 Test Reaults: Stylized Conclusions ........................ 367
9. Other Interest-Rate Models .............................. 369
9.1 Brennan and Schwartz's Model ........................... 369
9.2 Balduzzi, Das, Foresi and Sundaram's Model ............... 370
9.3 Flesaker and Hughston's Model .......................... 371
9.4 Rogers's Potential Approach ............................. 373
9.5 Markov Functional Models ............................... 373
Part II. PRICING DERIVATIVES IN PRACTICE
10. Pricing Derivatives on a Single Interest-Rate Curve ...... 377
10.1 In-Advance Swaps ...................................... 378
10.2 In-Advance Caps ....................................... 379
10.2.1 A First Analytical Formula (LFM) ................. 380
10.2.2 A Second Analytical Formula (G2++) .............. 380
10.3 Autocaps .............................................. 381
10.4 Caps with Deferred Caplets .............................. 382
10.4.1 A First Analytical Formula (LFM) ................. 382
10.4.2 A Second Analytical Formula (G2++) .............. 383
10.5 Ratchets (One-Way Floaters) ............................ 384
10.6 Constant-Maturity Swaps (CMS) ......................... 385
10.6.1 CMS with the LFM .............................. 385
10.6.2 CMS with the G2++ Model ....................... 386
10.7 The Convexity Adjustment and Applications to CMS ....... 386
10.7.1 Natural and Unnatural Time Lags .................. 386
10.7.2 The Convexity-Adjustment Technique ............... 387
10.7.3 Deducing a Simple Lognormal Dynamics from the Ad-
justrnent ........................................ 391
10.7.4 Application to CMS .............................. 392
10.7.5 Forward Rate Resetting Unnaturally and Average-
Rate Swaps ...................................... 393
10.8 Captions and Floortions ................................. 395
10.9 Zero-Coupon Swaptions ................................. 395
10.10 Eurodollar Futures ..................................... 399
10.10.1 The Shifted Two-Factor Vasicek G2++ Model ....... 400
10.10.2 Eurodollar Futures with the LFM .................. 402
10.11 LFM Prieing with "In-Between" Spot Rates ............... 402
10.11.1 Accrual Swaps .................................. 403
10.11.2 Trigger Swaps ................................... 406
10.12 LFM Pricing with Early Exercise and Possible Path Depen-
dence ................................................. 408
10.13 LFM: Pricing Bermudan Swaptions ...................... 412
10.13.1 Longstaff and Schwartz's Approach ................ 413
10.13.2 Carr and Yang's Approach ........................ 415
10.13.3 Andersen's Approach ............................. 416
11. Pricing Derivatives on Two Interest-Rate Curves ......... 421
11.1 The Attractive Features of G2++ for Multi-Curve Payoffs ... 421
11.1.1 The Model ...................................... 421
11.1.2 Interaction Between Models of the Two Curves "l"
and "2" ......................................... 424
11.1.3 The Two-Models Dynamics under a Unique Conve-
nient Forward Measure ............................ 425
11.2 Quanto Constant-Maturity Swaps ........................ 427
11.2.1 Quanto CMS: The Contraet ....................... 427
11.2.2 Quanto CMS: The G2++ Model ................... 429
11.2.3 Quanto CMS: Quanto Adjustment .................. 435
11.3 DifferentiaJ Swaps ...................................... 437
11.3.1 The Contract .................................... 437
11.3.2 Differential Swaps with the G2++ Model ............ 438
11.3.3 A Market-Like Formula ........................... 440
11.4 Market Formulas for Basic Quanto Derivatives ............. 440
11.4.1 The Pricing of Quanto Caplets/Floorlets ............ 440
11.4.2 The Pricing of Quanto Caps/Floors ................. 443
11.4.3 The Pricing of Differential Swaps ................... 444
11.4.4 The Pricieg of Quanto Swaptions ................... 444
12. Pricing Equity Derivatives under Stochastic Rates ........ 453
12.1 The Short Rate and Asset-Price Dynamics ................. 453
12.1.1 The Dynamics under the Forward Measure .......... 456
12.2 The Pricing of a European Option on the Given Asset ...... 458
12.3 A More General Model .................................. 459
12.3.1 The Construction of an Approximating Tree for r .... 460
12.3.2 The Approximating Tree for S ..................... 462
12.3.3 The Two-Dimensional Tree ........................ 463
Part III. APPENDICES
A. A Crash Introduction to Stochastic Differential Equations 469
A.1 From Deterministic to Stochastic Differential Equations ..... 469
A.2 Ito's Formula .......................................... 476
A.3 Discretizing SDEs for Monte Carlo. Euler and Milstein Schemes478
A.4 Examples .............................................. 480
A.5 Two Important Theorems ............................... 482
B. A Useful Calculation ..................................... 485
C. Approximating Diffusions with Trees ..................... 487
D. Talking to the Traders .................................... 493
References .................................................... 501
Index ......................................................... 509