Brigo / Mercurio Interest Rate Models - Theory and Practice

With Smile, Inflation and Credit
2. Auflage 2006
ISBN: 978-3-540-34604-3
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 982 Seiten

Reihe: Springer Finance

The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced.

The old sections devoted to the smile issue in the LIBOR market model have been enlarged into several new chapters. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered. 

The fast-growing interest for hybrid products has led to new chapters. A special focus here is devoted to the pricing of inflation-linked derivatives. 

The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.

Autoren/Hrsg.

Brigo, Damiano

Mercurio, Fabio

Inhaltsverzeichnis

1;Preface;7
2;Abbreviations and Notation;34
3;Contents;41
4;BASIC DEFINITIONS AND NO ARBITRAGE;53
4.1;1. Definitions and Notation;54
4.1.1;1.1 The Bank Account and the Short Rate;55
4.1.2;1.2 Zero-Coupon Bonds and Spot Interest Rates;57
4.1.3;1.3 Fundamental Interest-Rate Curves;62
4.1.4;1.4 Forward Rates;64
4.1.5;1.5 Interest-Rate Swaps and Forward Swap Rates;66
4.1.6;1.6 Interest-Rate Caps/Floors and Swaptions;69
4.2;2. No-Arbitrage Pricing and Numeraire Change;76
4.2.1;2.1 No-Arbitrage in Continuous Time;77
4.2.2;2.2 The Change-of-Numeraire Technique;79
4.2.3;2.3 A Change of Numeraire Toolkit ( Brigo & Mercurio 2001c);81
4.2.4;2.4 The Choice of a Convenient Numeraire;90
4.2.5;2.5 The Forward Measure;91
4.2.6;2.6 The Fundamental Pricing Formulas;92
4.2.7;2.7 Pricing Claims with Deferred Payoffs;95
4.2.8;2.8 Pricing Claims with Multiple Payoffs;95
4.2.9;2.9 Foreign Markets and Numeraire Change;97
5;FROM SHORT RATE MODELS TO HJM;101
5.1;3. One-factor short-rate models;102
5.1.1;3.1 Introduction and Guided Tour;102
5.1.2;3.2 Classical Time-Homogeneous Short-Rate Models;108
5.1.3;3.3 The Hull-White Extended Vasicek Model;122
5.1.4;3.4 Possible Extensions of the CIR Model;131
5.1.5;3.5 The Black-Karasinski Model;133
5.1.6;3.6 Volatility Structures in One-Factor Short-Rate Models;137
5.1.7;3.7 Humped-Volatility Short-Rate Models;143
5.1.8;3.8 A General Deterministic-Shift Extension;146
5.1.9;3.9 The CIR++ Model;153
5.1.10;3.10 Deterministic-Shift Extension of Lognormal Models;161
5.1.11;3.11 Some Further Remarks on Derivatives Pricing;163
5.1.12;3.12 Implied Cap Volatility Curves;175
5.1.13;3.13 Implied Swaption Volatility Surfaces;180
5.1.14;3.14 An Example of Calibration to Real-Market Data;183
5.2;4. Two-Factor Short-Rate Models;188
5.2.1;4.1 Introduction and Motivation;188
5.2.2;4.2 The Two-Additive-Factor Gaussian Model G2++;193
5.2.3;4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++;226
5.3;5. The Heath-Jarrow-Morton (HJM) Framework;233
5.3.1;5.1 The HJM Forward-Rate Dynamics;235
5.3.2;5.2 Markovianity of the Short-Rate Process;236
5.3.3;5.3 The Ritchken and Sankarasubramanian Framework;237
5.3.4;5.4 The Mercurio and Moraleda Model;241
6;MARKET MODELS;243
6.1;6. The LIBOR and Swap Market Models ( LFM and LSM);244
6.1.1;6.1 Introduction;244
6.1.2;6.2 Market Models: a Guided Tour;245
6.1.3;6.3 The Lognormal Forward-LIBOR Model (LFM);256
6.1.4;6.4 Calibration of the LFM to Caps and Floors Prices;269
6.1.5;6.5 The Term Structure of Volatility;275
6.1.6;6.6 Instantaneous Correlation and Terminal Correlation;283
6.1.7;6.7 Swaptions and the Lognormal Forward-Swap Model ( LSM);286
6.1.8;6.8 Incompatibility between the LFM and the LSM;293
6.1.9;6.9 The Structure of Instantaneous Correlations;295
6.1.10;6.10 Monte Carlo Pricing of Swaptions with the LFM;313
6.1.11;6.11 Monte Carlo Standard Error;315
6.1.12;6.12 Monte Carlo Variance Reduction: Control Variate Estimator;318
6.1.13;6.13 Rank-One Analytical Swaption Prices;320
6.1.14;6.14 Rank-r Analytical Swaption Prices;326
6.1.15;6.15 A Simpler LFM Formula for Swaptions Volatilities;330
6.1.16;6.16 A Formula for Terminal Correlations of Forward Rates;333
6.1.17;6.17 Calibration to Swaptions Prices;336
6.1.18;6.18 Instantaneous Correlations: Inputs (Historical Estimation) or Outputs ( Fitting Parameters)?;339
6.1.19;6.19 The exogenous correlation matrix;340
6.1.20;6.20 Connecting Caplet and S × 1-Swaption Volatilities;349
6.1.21;6.21 Forward and Spot Rates over Non-Standard Periods;356
6.2;7. Cases of Calibration of the LIBOR Market Model;362
6.2.1;7.1 Inputs for the First Cases;364
6.2.2;7.2 Joint Calibration with Piecewise-Constant Volatilities as in TABLE 5;364
6.2.3;7.3 Joint Calibration with Parameterized Volatilities as in Formulation 7;368
6.2.4;7.4 Exact Swaptions Cascade Calibration with Volatilities as in TABLE 1;371
6.2.5;7.5 A Pause for Thought;386
6.2.6;7.6 Further Numerical Studies on the Cascade Calibration Algorithm;389
6.2.7;7.7 Empirically efficient Cascade Calibration;400
6.2.8;7.8 Reliability: Monte Carlo tests;415
6.2.9;7.9 Cascade Calibration and the cap market;418
6.2.10;7.10 Cascade Calibration: Conclusions;421
6.3;8. Monte Carlo Tests for LFM Analytical Approximations;425
6.3.1;8.1 First Part. Tests Based on the Kullback Leibler Information ( KLI);426
6.3.2;8.2 Second Part: Classical Tests;440
6.3.3;8.3 The Testing Plan for Volatilities;440
6.3.4;8.4 Test Results for Volatilities;444
6.3.5;8.5 The Testing Plan for Terminal Correlations;469
6.3.6;8.6 Test Results for Terminal Correlations;475
6.3.7;8.7 Test Results: Stylized Conclusions;490
7;THE VOLATILITY SMILE;492
7.1;9. Including the Smile in the LFM;493
7.1.1;9.1 A Mini-tour on the Smile Problem;493
7.1.2;9.2 Modeling the Smile;496
7.2;10. Local-Volatility Models;499
7.2.1;10.1 The Shifted-Lognormal Model;500
7.2.2;10.2 The Constant Elasticity of Variance Model;502
7.2.3;10.3 A Class of Analytically-Tractable Models;505
7.2.4;10.4 A Lognormal-Mixture (LM) Model;509
7.2.5;10.5 Forward Rates Dynamics under Different Measures;513
7.2.6;10.6 Shifting the LM Dynamics;515
7.2.7;10.7 A Lognormal-Mixture with Different Means ( LMDM);517
7.2.8;10.8 The Case of Hyperbolic-Sine Processes;519
7.2.9;10.9 Testing the Above Mixture-Models on Market Data;521
7.2.10;10.10 A Second General Class;524
7.2.11;10.11 A Particular Case: a Mixture of GBM’s;529
7.2.12;10.12 An Extension of the GBM Mixture Model Allowing for Implied Volatility Skews;532
7.2.13;10.13 A General Dynamics a la Dupire (1994);535
7.3;11. Stochastic-Volatility Models;541
7.3.1;11.1 The Andersen and Brotherton-Ratcliffe (2001) Model;543
7.3.2;11.2 The Wu and Zhang (2002) Model;547
7.3.3;11.3 The Piterbarg (2003) Model;550
7.3.4;11.4 The Hagan, Kumar, Lesniewski and Woodward ( 2002) Model;554
7.3.5;11.5 The Joshi and Rebonato (2003) Model;559
7.4;12. Uncertain-Parameter Models;563
7.4.1;12.1 The Shifted-Lognormal Model with Uncertain Parameters ( SLMUP);565
7.4.2;12.2 Calibration to Caplets;566
7.4.3;12.3 Swaption Pricing;568
7.4.4;12.4 Monte-Carlo Swaption Pricing;570
7.4.5;12.5 Calibration to Swaptions;572
7.4.6;12.6 Calibration to Market Data;574
7.4.7;12.7 Testing the Approximation for Swaptions Prices;576
7.4.8;12.8 Further Model Implications;581
7.4.9;12.9 Joint Calibration to Caps and Swaptions;585
8;EXAMPLES OF MARKET PAYOFFS;591
8.1;13. Pricing Derivatives on a Single Interest- Rate Curve;592
8.1.1;13.1 In-Arrears Swaps;593
8.1.2;13.2 In-Arrears Caps;595
8.1.3;13.3 Autocaps;596
8.1.4;13.4 Caps with Deferred Caplets;597
8.1.5;13.5 Ratchet Caps and Floors;599
8.1.6;13.6 Ratchets (One-Way Floaters);601
8.1.7;13.7 Constant-Maturity Swaps (CMS);602
8.1.8;13.8 The Convexity Adjustment and Applications to CMS;604
8.1.9;13.9 Average Rate Caps;613
8.1.10;13.10 Captions and Floortions;615
8.1.11;13.11 Zero-Coupon Swaptions;616
8.1.12;13.12 Eurodollar Futures;620
8.1.13;13.13 LFM Pricing with In-Between Spot Rates;623
8.1.14;13.14 LFM Pricing with Early Exercise and Possible Path Dependence;629
8.1.15;13.15 LFM: Pricing Bermudan Swaptions;633
8.1.16;13.16 New Generation of Contracts;646
8.2;14. Pricing Derivatives on Two Interest-Rate Curves;651
8.2.1;14.1 The Attractive Features of G2++ for Multi-Curve Payoffs;652
8.2.2;14.2 Quanto Constant-Maturity Swaps;657
8.2.3;14.3 Differential Swaps;667
8.2.4;14.4 Market Formulas for Basic Quanto Derivatives;670
8.2.5;14.5 Pricing of Options on two Currency LIBOR Rates;677
9;INFLATION;685
9.1;15. Pricing of Inflation-Indexed Derivatives;686
9.1.1;15.1 The Foreign-Currency Analogy;687
9.1.2;15.2 Definitions and Notation;688
9.1.3;15.3 The JY Model;689
9.2;16. Inflation-Indexed Swaps;691
9.2.1;16.1 Pricing of a ZCIIS;691
9.2.2;16.2 Pricing of a YYIIS;693
9.2.3;16.3 Pricing of a YYIIS with the JY Model;694
9.2.4;16.4 Pricing of a YYIIS with a First Market Model;696
9.2.5;16.5 Pricing of a YYIIS with a Second Market Model;699
9.3;17. Inflation-Indexed Caplets/Floorlets;702
9.3.1;17.1 Pricing with the JY Model;702
9.3.2;17.2 Pricing with the Second Market Model;704
9.3.3;17.3 Inflation-Indexed Caps;706
9.3.4;Appendix: IICapFloor Pricing with the LFM;706
9.4;18. Calibration to market data;709
9.5;19. Introducing Stochastic Volatility;713
9.5.1;19.1 Modeling Forward CPI’s with Stochastic Volatility;714
9.5.2;19.2 Pricing Formulae;716
9.5.3;19.3 Example of Calibration;721
9.5.4;Appendix A: Heston PDE;724
9.5.5;Appendix B: Floorlet Pricing;726
9.6;20. Pricing Hybrids with an Inflation Component;728
9.6.1;20.1 A Simple Hybrid Payoff;728
10;CREDIT;732
10.1;21. Introduction and Pricing under Counterparty Risk;733
10.1.1;21.1 Introduction and Guided Tour;734
10.1.2;21.2 Defaultable (corporate) zero coupon bonds;761
10.1.3;21.3 Credit Default Swaps and Defaultable Floaters;762
10.1.4;21.4 CDS Options and Callable Defaultable Floaters;781
10.1.5;21.5 Constant Maturity CDS;782
10.1.6;21.6 Interest-Rate Payoffs with Counterparty Risk;785
10.2;22. Intensity Models;794
10.2.1;22.1 Introduction and Chapter Description;794
10.2.2;22.2 Poisson processes;796
10.2.3;22.3 CDS Calibration and Implied Hazard Rates/ Intensities;801
10.2.4;22.4 Inducing dependence between Interest-rates and the default event;813
10.2.5;22.5 The Filtration Switching Formula: Pricing under partial information;814
10.2.6;22.6 Default Simulation in reduced form models;815
10.2.7;22.7 Stochastic Intensity: The SSRD model;822
10.2.8;22.8 Stochastic diffusion intensity is not enough: Adding jumps. The JCIR(++) Model;867
10.2.9;22.9 Conclusions and further research;875
10.3;23. CDS Options Market Models;877
10.3.1;23.1 CDS Options and Callable Defaultable Floaters;880
10.3.2;23.2 A market formula for CDS options and callable defaultable floaters;883
10.3.3;23.3 Towards a Completely Specified Market Model;890
10.3.4;23.4 Hints at Smile Modeling;899
10.3.5;23.5 Constant Maturity Credit Default Swaps ( CMCDS) with the market model;900
11;APPENDICES;911
11.1;A. Other Interest-Rate Models;912
11.1.1;A.1 Brennan and Schwartz’s Model;912
11.1.2;A.2 Balduzzi, Das, Foresi and Sundaram’s Model;913
11.1.3;A.3 Flesaker and Hughston’s Model;914
11.1.4;A.4 Rogers’s Potential Approach;916
11.1.5;A.5 Markov Functional Models;916
11.2;B. Pricing Equity Derivatives under Stochastic Rates;918
11.2.1;B.1 The Short Rate and Asset-Price Dynamics;918
11.2.2;B.2 The Pricing of a European Option on the Given Asset;923
11.2.3;B.3 A More General Model;924
11.3;C. A Crash Intro to Stochastic Differential Equations and Poisson Processes;931
11.3.1;C.1 From Deterministic to Stochastic Differential Equations;931
11.3.2;C.2 Ito’s Formula;938
11.3.3;C.3 Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes;940
11.3.4;C.4 Examples;942
11.3.5;C.5 Two Important Theorems;944
11.3.6;C.6 A Crash Intro to Poisson Processes;947
11.4;D. A Useful Calculation;953
11.5;E. A Second Useful Calculation;955
11.6;F. Approximating Diffusions with Trees;959
11.7;G. Trivia and Frequently Asked Questions;965
11.8;H. Talking to the Traders;969
12;References;985
13;Index;1001