Interest Rate Models - Theory and Practice (eBook)

Preface 7
Abbreviations and Notation 34
Contents 41
BASIC DEFINITIONS AND NO ARBITRAGE 53
1. Definitions and Notation 54
1.1 The Bank Account and the Short Rate 55
1.2 Zero-Coupon Bonds and Spot Interest Rates 57
1.3 Fundamental Interest-Rate Curves 62
1.4 Forward Rates 64
1.5 Interest-Rate Swaps and Forward Swap Rates 66
1.6 Interest-Rate Caps/Floors and Swaptions 69
2. No-Arbitrage Pricing and Numeraire Change 76
2.1 No-Arbitrage in Continuous Time 77
2.2 The Change-of-Numeraire Technique 79
2.3 A Change of Numeraire Toolkit ( Brigo & Mercurio 2001c)
2.4 The Choice of a Convenient Numeraire 90
2.5 The Forward Measure 91
2.6 The Fundamental Pricing Formulas 92
2.7 Pricing Claims with Deferred Payoffs 95
2.8 Pricing Claims with Multiple Payoffs 95
2.9 Foreign Markets and Numeraire Change 97
FROM SHORT RATE MODELS TO HJM 101
3. One-factor short-rate models 102
3.1 Introduction and Guided Tour 102
3.2 Classical Time-Homogeneous Short-Rate Models 108
3.3 The Hull-White Extended Vasicek Model 122
3.4 Possible Extensions of the CIR Model 131
3.5 The Black-Karasinski Model 133
3.6 Volatility Structures in One-Factor Short-Rate Models 137
3.7 Humped-Volatility Short-Rate Models 143
3.8 A General Deterministic-Shift Extension 146
3.9 The CIR++ Model 153
3.10 Deterministic-Shift Extension of Lognormal Models 161
3.11 Some Further Remarks on Derivatives Pricing 163
3.12 Implied Cap Volatility Curves 175
3.13 Implied Swaption Volatility Surfaces 180
3.14 An Example of Calibration to Real-Market Data 183
4. Two-Factor Short-Rate Models 188
4.1 Introduction and Motivation 188
4.2 The Two-Additive-Factor Gaussian Model G2++ 193
4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++ 226
5. The Heath-Jarrow-Morton (HJM) Framework 233
5.1 The HJM Forward-Rate Dynamics 235
5.2 Markovianity of the Short-Rate Process 236
5.3 The Ritchken and Sankarasubramanian Framework 237
5.4 The Mercurio and Moraleda Model 241
MARKET MODELS 243
6. The LIBOR and Swap Market Models ( LFM and LSM) 244
6.1 Introduction 244
6.2 Market Models: a Guided Tour 245
6.3 The Lognormal Forward-LIBOR Model (LFM) 256
6.4 Calibration of the LFM to Caps and Floors Prices 269
6.5 The Term Structure of Volatility 275
6.6 Instantaneous Correlation and Terminal Correlation 283
6.7 Swaptions and the Lognormal Forward-Swap Model ( LSM) 286
6.8 Incompatibility between the LFM and the LSM 293
6.9 The Structure of Instantaneous Correlations 295
6.10 Monte Carlo Pricing of Swaptions with the LFM 313
6.11 Monte Carlo Standard Error 315
6.12 Monte Carlo Variance Reduction: Control Variate Estimator 318
6.13 Rank-One Analytical Swaption Prices 320
6.14 Rank-r Analytical Swaption Prices 326
6.15 A Simpler LFM Formula for Swaptions Volatilities 330
6.16 A Formula for Terminal Correlations of Forward Rates 333
6.17 Calibration to Swaptions Prices 336
6.18 Instantaneous Correlations: Inputs (Historical Estimation) or Outputs ( Fitting Parameters)? 339
6.19 The exogenous correlation matrix 340
6.20 Connecting Caplet and S × 1-Swaption Volatilities 349
6.21 Forward and Spot Rates over Non-Standard Periods 356
7. Cases of Calibration of the LIBOR Market Model 362
7.1 Inputs for the First Cases 364
7.2 Joint Calibration with Piecewise-Constant Volatilities as in TABLE 5 364
7.3 Joint Calibration with Parameterized Volatilities as in Formulation 7 368
7.4 Exact Swaptions Cascade Calibration with Volatilities as in TABLE 1 371
7.5 A Pause for Thought 386
7.6 Further Numerical Studies on the Cascade Calibration Algorithm 389
7.7 Empirically efficient Cascade Calibration 400
7.8 Reliability: Monte Carlo tests 415
7.9 Cascade Calibration and the cap market 418
7.10 Cascade Calibration: Conclusions 421
8. Monte Carlo Tests for LFM Analytical Approximations 425
8.1 First Part. Tests Based on the Kullback Leibler Information ( KLI) 426
8.2 Second Part: Classical Tests 440
8.3 The Testing Plan for Volatilities 440
8.4 Test Results for Volatilities 444
8.5 The Testing Plan for Terminal Correlations 469
8.6 Test Results for Terminal Correlations 475
8.7 Test Results: Stylized Conclusions 490
THE VOLATILITY SMILE 492
9. Including the Smile in the LFM 493
9.1 A Mini-tour on the Smile Problem 493
9.2 Modeling the Smile 496
10. Local-Volatility Models 499
10.1 The Shifted-Lognormal Model 500
10.2 The Constant Elasticity of Variance Model 502
10.3 A Class of Analytically-Tractable Models 505
10.4 A Lognormal-Mixture (LM) Model 509
10.5 Forward Rates Dynamics under Different Measures 513
10.6 Shifting the LM Dynamics 515
10.7 A Lognormal-Mixture with Different Means ( LMDM) 517
10.8 The Case of Hyperbolic-Sine Processes 519
10.9 Testing the Above Mixture-Models on Market Data 521
10.10 A Second General Class 524
10.11 A Particular Case: a Mixture of GBM’s 529
10.12 An Extension of the GBM Mixture Model Allowing for Implied Volatility Skews 532
10.13 A General Dynamics a la Dupire (1994) 535
11. Stochastic-Volatility Models 541
11.1 The Andersen and Brotherton-Ratcliffe (2001) Model 543
11.2 The Wu and Zhang (2002) Model 547
11.3 The Piterbarg (2003) Model 550
11.4 The Hagan, Kumar, Lesniewski and Woodward ( 2002) Model 554
11.5 The Joshi and Rebonato (2003) Model 559
12. Uncertain-Parameter Models 563
12.1 The Shifted-Lognormal Model with Uncertain Parameters ( SLMUP) 565
12.2 Calibration to Caplets 566
12.3 Swaption Pricing 568
12.4 Monte-Carlo Swaption Pricing 570
12.5 Calibration to Swaptions 572
12.6 Calibration to Market Data 574
12.7 Testing the Approximation for Swaptions Prices 576
12.8 Further Model Implications 581
12.9 Joint Calibration to Caps and Swaptions 585
EXAMPLES OF MARKET PAYOFFS 591
13. Pricing Derivatives on a Single Interest- Rate Curve 592
13.1 In-Arrears Swaps 593
13.2 In-Arrears Caps 595
13.3 Autocaps 596
13.4 Caps with Deferred Caplets 597
13.5 Ratchet Caps and Floors 599
13.6 Ratchets (One-Way Floaters) 601
13.7 Constant-Maturity Swaps (CMS) 602
13.8 The Convexity Adjustment and Applications to CMS 604
13.9 Average Rate Caps 613
13.10 Captions and Floortions 615
13.11 Zero-Coupon Swaptions 616
13.12 Eurodollar Futures 620
13.13 LFM Pricing with In-Between Spot Rates 623
13.14 LFM Pricing with Early Exercise and Possible Path Dependence 629
13.15 LFM: Pricing Bermudan Swaptions 633
13.16 New Generation of Contracts 646
14. Pricing Derivatives on Two Interest-Rate Curves 651
14.1 The Attractive Features of G2++ for Multi-Curve Payoffs 652
14.2 Quanto Constant-Maturity Swaps 657
14.3 Differential Swaps 667
14.4 Market Formulas for Basic Quanto Derivatives 670
14.5 Pricing of Options on two Currency LIBOR Rates 677
INFLATION 685
15. Pricing of Inflation-Indexed Derivatives 686
15.1 The Foreign-Currency Analogy 687
15.2 Definitions and Notation 688
15.3 The JY Model 689
16. Inflation-Indexed Swaps 691
16.1 Pricing of a ZCIIS 691
16.2 Pricing of a YYIIS 693
16.3 Pricing of a YYIIS with the JY Model 694
16.4 Pricing of a YYIIS with a First Market Model 696
16.5 Pricing of a YYIIS with a Second Market Model 699
17. Inflation-Indexed Caplets/Floorlets 702
17.1 Pricing with the JY Model 702
17.2 Pricing with the Second Market Model 704
17.3 Inflation-Indexed Caps 706
Appendix: IICapFloor Pricing with the LFM 706
18. Calibration to market data 709
19. Introducing Stochastic Volatility 713
19.1 Modeling Forward CPI’s with Stochastic Volatility 714
19.2 Pricing Formulae 716
19.3 Example of Calibration 721
Appendix A: Heston PDE 724
Appendix B: Floorlet Pricing 726
20. Pricing Hybrids with an Inflation Component 728
20.1 A Simple Hybrid Payoff 728
CREDIT 732
21. Introduction and Pricing under Counterparty Risk 733
21.1 Introduction and Guided Tour 734
21.2 Defaultable (corporate) zero coupon bonds 761
21.3 Credit Default Swaps and Defaultable Floaters 762
21.4 CDS Options and Callable Defaultable Floaters 781
21.5 Constant Maturity CDS 782
21.6 Interest-Rate Payoffs with Counterparty Risk 785
22. Intensity Models 794
22.1 Introduction and Chapter Description 794
22.2 Poisson processes 796
22.3 CDS Calibration and Implied Hazard Rates/ Intensities 801
22.4 Inducing dependence between Interest-rates and the default event 813
22.5 The Filtration Switching Formula: Pricing under partial information 814
22.6 Default Simulation in reduced form models 815
22.7 Stochastic Intensity: The SSRD model 822
22.8 Stochastic diffusion intensity is not enough: Adding jumps. The JCIR(++) Model 867
22.9 Conclusions and further research 875
23. CDS Options Market Models 877
23.1 CDS Options and Callable Defaultable Floaters 880
23.2 A market formula for CDS options and callable defaultable floaters 883
23.3 Towards a Completely Specified Market Model 890
23.4 Hints at Smile Modeling 899
23.5 Constant Maturity Credit Default Swaps ( CMCDS) with the market model 900
APPENDICES 911
A. Other Interest-Rate Models 912
A.1 Brennan and Schwartz’s Model 912
A.2 Balduzzi, Das, Foresi and Sundaram’s Model 913
A.3 Flesaker and Hughston’s Model 914
A.4 Rogers’s Potential Approach 916
A.5 Markov Functional Models 916
B. Pricing Equity Derivatives under Stochastic Rates 918
B.1 The Short Rate and Asset-Price Dynamics 918
B.2 The Pricing of a European Option on the Given Asset 923
B.3 A More General Model 924
C. A Crash Intro to Stochastic Differential Equations and Poisson Processes 931
C.1 From Deterministic to Stochastic Differential Equations 931
C.2 Ito’s Formula 938
C.3 Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes 940
C.4 Examples 942
C.5 Two Important Theorems 944
C.6 A Crash Intro to Poisson Processes 947
D. A Useful Calculation 953
E. A Second Useful Calculation 955
F. Approximating Diffusions with Trees 959
G. Trivia and Frequently Asked Questions 965
H. Talking to the Traders 969
References 985
Index 1001