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)	81
	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