Part Ⅰ．BASIC DEFINITIONS AND NO ARBITRAGE 1
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
Part Ⅱ．FROM SHORT RATE MODELS TO HJM 51
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
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 195
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
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 eigen-values 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 For-mulation 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 Algo-rithm 340
7.6.1 Cascade Calibration under Various Correlations and Ranks 342
7.6.2 Cascade Calibration Diagnostics:Terminal Correla-tion and Evolution of Volatilities 346
7.6.3 The interpolation for the swaption matrix and its im-pact 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 swap-tions 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 Instanta-neous 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 Instanta-neous Volatilities Depending only on Time to Matu-rity,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 Instanta-neous Volatilities Depending only on Time to Matu-rity,Some Negative Rank-Two Correlations 432
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 447
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 Im-plied Volatility Skews 486
10.13 A General Dynamics à la Dupire(1994) 489
11．Stochastic-Volatility Models 495
11.1 The Andersen and Brotherton-Ratclifie(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
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 547
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 Ad-justment 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
13.14 LFM Pricing with Early Exercise and Possible Path Dependence 584
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
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 Conve-nient 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 643
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
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 695
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 Func-tions 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
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 Anal-gies 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 in-formation 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:Sepa-rability 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 Ganssian dependence mapping:A tractable approxi-mated 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
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 Vol atilities 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 877
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
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 Pois-son 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 Schemes 906
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