利率模型理论和实践 第2版 英文版PDF|Epub|txt|kindle电子书版本网盘下载

利率模型理论和实践 第2版 英文版PDF格式电子书版下载

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Part Ⅰ．BASIC DEFINITIONS AND NO ARBITRAGE1

1．Definitions and Notation1

1.1 The Bank Account and the Short Rate2

1.2 Zero-Coupon Bonds and Spot Interest Rates4

1.3 Fundamental Interest-Rate Curves9

1.4 Forward Rates11

1.5 Interest-Rate Swaps and Forward Swap Rates13

1.6 Interest-Rate Caps/Floors and Swaptions16

2．No-Arbitrage Pricing and Numeraire Change23

2.1 No-Arbitrage in Continuous Time24

2.2 The Change-of-Numeraire Technique26

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 Numeraire37

2.5 The Forward Measure38

2.6 The Fundamental Pricing Formulas39

2.6.1 The Pricing of Caps and Floors40

2.7 Pricing Claims with Deferred Payoffs42

2.8 Pricing Claims with Multiple Payoffs42

2.9 Foreign Markets and Numeraire Change44

Part Ⅱ．FROM SHORT RATE MODELS TO HJM51

3．One-factor short-rate models51

3.1 Introduction and Guided Tour51

3.2 Classical Time-Homogeneous Short-Rate Models57

3.2.1 The Vasicek Model58

3.2.2 The Dothan Model62

3.2.3 The Cox,Ingersoll and Ross(CIR)Model64

3.2.4 Affine Term-Structure Models68

3.2.5 The Exponential-Vasicek(EV)Model70

3.3 The Hull-White Extended Vasicek Model71

3.3.1 The Short-Rate Dynamics72

3.3.2 Bond and Option Pricing75

3.3.3 The Construction of a Trinomial Tree78

3.4 Possible Extensions of the CIR Model80

3.5 The Black-Karasinski Model82

3.5.1 The Short-Rate Dynamics83

3.5.2 The Construction of a Trinomial Tree85

3.6 Volatility Structures in One-Factor Short-Rate Models86

3.7 Humped-Volatility Short-Rate Models92

3.8 A General Deterministic-Shift Extension95

3.8.1 The Basic Assumptions96

3.8.2 Fitting the Initial Term Structure of Interest Rates97

3.8.3 Explicit Formulas for European Options99

3.8.4 The Vasicek Case100

3.9 The CIR++ Model102

3.9.1 The Construction of a Trinomial Tree105

3.9.2 Early Exercise Pricing via Dynamic Programming106

3.9.3 The Positivity of Rates and Fitting Quality106

3.9.4 Monte Carlo Simulation109

3.9.5 Jump Diffusion CIR and CIR++ models(JCIR,JCIR++)109

3.10 Deterministic-Shift Extension of Lognormal Models110

3.11 Some Further Remarks on Derivatives Pricing112

3.11.1 Pricing European Options on a Coupon-Bearing Bond112

3.11.2 The Monte Carlo Simulation114

3.11.3 Pricing Early-Exercise Derivatives with a Tree116

3.11.4 A Fundamental Case of Early Exercise:Bermudan-Style Swaptions121

3.12 Implied Cap Volatility Curves124

3.12.1 The Black and Karasinski Model125

3.12.2 The CIR++ Model126

3.12.3 The Extended Exponential-Vasicek Model128

3.13 Implied Swaption Volatility Surfaces129

3.13.1 The Black and Karasinski Model130

3.13.2 The Extended Exponential-Vasicek Model131

3.14 An Example of Calibration to Real-Market Data132

4．Two-Factor Short-Rate Models137

4.1 Introduction and Motivation137

4.2 The Two-Additive-Factor Gaussian Model G2++142

4.2.1 The Short-Rate Dynamics143

4.2.2 The Pricing of a Zero-Coupon Bond144

4.2.3 Volatility and Correlation Structures in Two-Factor Models148

4.2.4 The Pricing of a European Option on a Zero-Coupon Bond153

4.2.5 The Analogy with the Hull-White Two-Factor Model159

4.2.6 The Construction of an Approximating Binomial Tree162

4.2.7 Examples of Calibration to Real-Market Data166

4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++175

4.3.1 The Basic Two-Factor CIR2 Model176

4.3.2 Relationship with the Longstaff and Schwartz Model(LS)177

4.3.3 Forward-Measure Dynamics and Option Pricing for CIR2178

4.3.4 The CIR2++ Model and Option Pricing179

5．The Heath-Jarrow-Morton(HJM)Framework183

5.1 The HJM Forward-Rate Dynamics185

5.2 Markovianity of the Short-Rate Process186

5.3 The Ritchken and Sankarasubramanian Framework187

5.4 The Mercurio and Moraleda Model191

Part Ⅲ．MARKET MODELS195

6．The LIBOR and Swap Market Models(LFM and LSM)195

6.1 Introduction195

6.2 Market Models:a Guided Tour196

6.3 The Lognormal Forward-LIBOR Model(LFM)207

6.3.1 Some Specifications of the Instantaneous Volatility of Forward Rates210

6.3.2 Forward-Rate Dynamics under Different Numeraires213

6.4 Calibration of the LFM to Caps and Floors Prices220

6.4.1 Piecewise-Constant Instantaneous-Volatility Structures223

6.4.2 Parametric Volatility Structures224

6.4.3 Cap Quotes in the Market225

6.5 The Term Structure of Volatility226

6.5.1 Piecewise-Constant Instantaneous Volatility Structures228

6.5.2 Parametric Volatility Structures231

6.6 Instantaneous Correlation and Terminal Correlation234

6.7 Swaptions and the Lognormal Forward-Swap Model(LSM)237

6.7.1 Swaptions Hedging241

6.7.2 Cash-Settled Swaptions243

6.8 Incompatibility between the LFM and the LSM244

6.9 The Structure of Instantaneous Correlations246

6.9.1 Some convenient full rank parameterizations248

6.9.2 Reduced-rank formulations:Rebonato's angles and eigen-values zeroing250

6.9.3 Reducing the angles259

6.10 Monte Carlo Pricing of Swaptions with the LFM264

6.11 Monte Carlo Standard Error266

6.12 Monte Carlo Variance Reduction:Control Variate Estimator269

6.13 Rank-One Analytical Swaption Prices271

6.14 Rank-r Analytical Swaption Prices277

6.15 A Simpler LFM Formula for Swaptions Volatilities281

6.16 A Formula for Terminal Correlations of Forward Rates284

6.17 Calibration to Swaptions Prices287

6.18 Instantaneous Correlations:Inputs(Historical Estimation) or Outputs(Fitting Parameters)?290

6.19 The exogenous correlation matrix291

6.19.1 Historical Estimation292

6.19.2 Pivot matrices295

6.20 Connecting Caplet and S×1-Swaption Volatilities300

6.21 Forward and Spot Rates over Non-Standard Periods307

6.21.1 Drift Interpolation308

6.21.2 The Bridging Technique310

7．Cases of Calibration of the LIBOR Market Model313

7.1 Inputs for the First Cases315

7.2 Joint Calibration with Piecewise-Constant Volatilities as in TABLE 5315

7.3 Joint Calibration with Parameterized Volatilities as in For-mulation 7319

7.4 Exact Swaptions"Cascade"Calibration with Volatilities as in TABLE 1322

7.4.1 Some Numerical Results330

7.5 A Pause for Thought337

7.5.1 First summary337

7.5.2 An automatic fast analytical calibration of LFM to swaptions.Motivations and plan338

7.6 Further Numerical Studies on the Cascade Calibration Algo-rithm340

7.6.1 Cascade Calibration under Various Correlations and Ranks342

7.6.2 Cascade Calibration Diagnostics:Terminal Correla-tion and Evolution of Volatilities346

7.6.3 The interpolation for the swaption matrix and its im-pact on the CCA349

7.7 Empirically efficient Cascade Calibration351

7.7.1 CCA with Endogenous Interpolation and Based Only on Pure Market Data352

7.7.2 Financial Diagnostics of the RCCAEI test results359

7.7.3 Endogenous Cascade Interpolation for missing swap-tions volatilities quotes364

7.7.4 A first partial check on the calibrated σ parameters stability364

7.8 Reliability:Monte Carlo tests366

7.9 Cascade Calibration and the cap market369

7.10 Cascade Calibration:Conclusions372

8．Monte Carlo Tests for LFM Analytical Approximations377

8.1 First Part.Tests Based on the Kullback Leibler Information(KLI)378

8.1.1 Distance between distributions:The Kullback Leibler information378

8.1.2 Distance of the LFM swap rate from the lognormal family of distributions381

8.1.3 Monte Carlo tests for measuring KLI384

8.1.4 Conclusions on the KLI-based approach391

8.2 Second Part:Classical Tests392

8.3 The"Testing Plan"for Volatilities392

8.4 Test Results for Volatilities396

8.4.1 Case(1):Constant Instantaneous Volatilities396

8.4.2 Case(2):Volatilities as Functions of Time to Maturity401

8.4.3 Case(3):Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Maturity410

8.5 The"Testing Plan"for Terminal Correlations421

8.6 Test Results for Terminal Correlations427

8.6.1 Case(ⅰ):Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Matu-rity,Typical Rank-Two Correlations427

8.6.2 Case(ⅱ):Constant Instantaneous Volatilities,Typical Rank-Two Correlations430

8.6.3 Case(ⅲ):Humped and Maturity-Adjusted Instanta-neous Volatilities Depending only on Time to Matu-rity,Some Negative Rank-Two Correlations432

8.6.4 Case(ⅳ):Constant Instantaneous Volatilities,Some Negative Rank-Two Correlations438

8.6.5 Case(ⅴ):Constant Instantaneous Volatilities,Perfect Correlations,Upwardly Shifted Φ's439

8.7 Test Results:Stylized Conclusions442

Part Ⅳ．THE VOLATILITY SMILE447

9．Including the Smile in the LFM447

9.1 A Mini-tour on the Smile Problem447

9.2 Modeling the Smile450

10．Local-Volatility Models453

10.1 The Shifted-Lognormal Model454

10.2 The Constant Elasticity of Variance Model456

10.3 A Class of Analytically-Tractable Models459

10.4 A Lognormal-Mixture(LM)Model463

10.5 Forward Rates Dynamics under Different Measures467

10.5.1 Decorrelation Between Underlying and Volatility469

10.6 Shifting the LM Dynamics469

10.7 A Lognormal-Mixture with Different Means(LMDM)471

10.8 The Case of Hyperbolic-Sine Processes473

10.9 Testing the Above Mixture-Models on Market Data475

10.10 A Second General Class478

10.11 A Particular Case:a Mixture of GBM's483

10.12 An Extension of the GBM Mixture Model Allowing for Im-plied Volatility Skews486

10.13 A General Dynamics à la Dupire(1994)489

11．Stochastic-Volatility Models495

11.1 The Andersen and Brotherton-Ratclifie(2001)Model497

11.2 The Wu and Zhang(2002)Model501

11.3 The Piterbarg(2003)Model504

11.4 The Hagan,Kumar,Lesniewski and Woodward(2002)Model508

11.5 The Joshi and Rebonato(2003) Model513

12．Uncertain-Parameter Models517

12.1 The Shifted-Lognormal Model with Uncertain Parameters(SLMUP)519

12.1.1 Relationship with the Lognormal-Mixture LVM520

12.2 Calibration to Caplets520

12.3 Swaption Pricing522

12.4 Monte-Carlo Swaption Pricing524

12.5 Calibration to Swaptions526

12.6 Calibration to Market Data528

12.7 Testing the Approximation for Swaptions Prices530

12.8 Further Model Implications535

12.9 Joint Calibration to Caps and Swaptions539

Part Ⅴ．EXAMPLES OF MARKET PAYOFFS547

13．Pricing Derivatives on a Single Interest-Rate Curve547

13.1 In-Arrears Swaps548

13.2 In-Arrears Caps550

13.2.1 A First Analytical Formula(LFM)550

13.2.2 A Second Analytical Formula(G2++)551

13.3 Autocaps551

13.4 Caps with Deferred Caplets552

13.4.1 A First Analytical Formula(LFM)553

13.4.2 A Second Analytical Formula(G2++)553

13.5 Ratchet Caps and Floors554

13.5.1 Analytical Approximation for Ratchet Caps with the LFM555

13.6 Ratchets(One-Way Floaters)556

13.7 Constant-Maturity Swaps(CMS)557

13.7.1 CMS with the LFM557

13.7.2 CMS with the G2++ Model559

13.8 The Convexity Adjustment and Applications to CMS559

13.8.1 Natural and Unnatural Time Lags559

13.8.2 The Convexity-Adjustment Technique561

13.8.3 Deducing a Simple Lognormal Dynamics from the Ad-justment565

13.8.4 Application to CMS565

13.8.5 Forward Rate Resetting Unnaturally and Average-Rate Swaps566

13.9 Average Rate Caps568

13.10 Captions and Floortions570

13.11 Zero-Coupon Swaptions571

13.12 Eurodollar Futures575

13.12.1 The Shifted Two-Factor Vasicek G2++ Model576

13.12.2 Eurodollar Futures with the LFM577

13.13 LFM Pricing with"In-Between"Spot Rates578

13.13.1 Accrual Swaps579

13.13.2 Trigger Swaps582

13.14 LFM Pricing with Early Exercise and Possible Path Dependence584

13.15 LFM:Pricing Bermudan Swaptions588

13.15.1 Least Squared Monte Carlo Approach589

13.15.2 Carr and Yang's Approach591

13.15.3 Andersen's Approach592

13.15.4 Numerical Example595

13.16 New Generation of Contracts601

13.16.1 Target Redemption Notes602

13.16.2 CMS Spread Options603

14．Pricing Derivatives on Two Interest-Rate Curves607

14.1 The Attractive Features of G2++ for Multi-Curve Payoffs608

14.1.1 The Model608

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 Measure611

14.2 Quanto Constant-Maturity Swaps613

14.2.1 Quanto CMS:The Contract613

14.2.2 Quanto CMS:The G2++ Model615

14.2.3 Quanto CMS:Quanto Adjustment621

14.3 Differential Swaps623

14.3.1 The Contract623

14.3.2 Differential Swaps with the G2++ Model624

14.3.3 A Market-Like Formula626

14.4 Market Formulas for Basic Quanto Derivatives626

14.4.1 The Pricing of Quanto Caplets/Floorlets627

14.4.2 The Pricing of Quanto Caps/Floors628

14.4.3 The Pricing of Differential Swaps629

14.4.4 The Pricing of Quanto Swaptions630

14.5 Pricing of Options on two Currency LIBOR Rates633

14.5.1 Spread Options635

14.5.2 Options on the Product637

14.5.3 Trigger Swaps638

14.5.4 Dealing with Multiple Dates639

Part Ⅵ．INFLATION643

15．Pricing of Inflation-Indexed Derivatives643

15.1 The Foreign-Currency Analogy644

15.2 Definitions and Notation645

15.3 The JY Model646

16．Inflation-Indexed Swaps649

16.1 Pricing of a ZCIIS649

16.2 Pricing of a YYIIS651

16.3 Pricing of a YYIIS with the JY Model652

16.4 Pricing of a YYIIS with a First Market Model654

16.5 Pricing of a YYIIS with a Second Market Model657

17．Inflation-Indexed Caplets/Floorlets661

17.1 Pricing with the JY Model661

17.2 Pricing with the Second Market Model663

17.3 Inflation-Indexed Caps665

18．Calibration to market data669

19．Introducing Stochastic Volatility673

19.1 Modeling Forward CPI's with Stochastic Volatility674

19.2 Pricing Formulae676

19.2.1 Exact Solution for the Uncorrelated Case677

19.2.2 Approximated Dynamics for Non-zero Correlations680

19.3 Example of Calibration681

20．Pricing Hybrids with an Inflation Component689

20.1 A Simple Hybrid Payoff689

Part Ⅶ．CREDIT695

21．Introduction and Pricing under Counterparty Risk695

21.1 Introduction and Guided Tour696

21.1.1 Reduced form(Intensity)models697

21.1.2 CDS Options Market Models699

21.1.3 Firm Value(or Structural)Models702

21.1.4 Further Models704

21.1.5 The Multi-name picture:FtD,CDO and Copula Func-tions705

21.1.6 First to Default(FtD)Basket705

21.1.7 Collateralized Debt Obligation(CDO)Tranches707

21.1.8 Where can we introduce dependence?708

21.1.9 Copula Functions710

21.1.10 Dynamic Loss models718

21.1.11 What data are available in the market?719

21.2 Defaultable(corporate)zero coupon bonds723

21.2.1 Defaultable(corporate)coupon bonds724

21.3 Credit Default Swaps and Defaultable Floaters724

21.3.1 CDS payoffs:Different Formulations725

21.3.2 CDS pricing formulas727

21.3.3 Changing filtration:Ft without default VS complete Gt728

21.3.4 CDS forward rates:The first definition730

21.3.5 Market quotes,model independent implied survival probabilities and implied hazard functions731

21.3.6 A simpler formula for calibrating intensity to a single CDS735

21.3.7 Different Definitions of CDS Forward Rates and Anal-gies with the LIBOR and SWAP rates737

21.3.8 Defaultable Floater and CDS739

21.4 CDS Options and Callable Defaultable Floaters743

21.5 Constant Maturity CDS744

21.5.1 Some interesting Financial features of CMCDS745

21.6 Interest-Rate Payoffs with Counterparty Risk747

21.6.1 General Valuation of Counterparty Risk748

21.6.2 Counterparty Risk in single Interest Rate Swaps(IRS)750

22．Intensity Models757

22.1 Introduction and Chapter Description757

22.2 Poisson processes759

22.2.1 Time homogeneous Poisson processes760

22.2.2 Time inhomogeneous Poisson Processes761

22.2.3 Cox Processes763

22.3 CDS Calibration and Implied Hazard Rates/Intensities764

22.4 Inducing dependence between Interest-rates and the default event776

22.5 The Filtration Switching Formula:Pricing under partial in-formation777

22.6 Default Simulation in reduced form models778

22.6.1 Standard error781

22.6.2 Variance Reduction with Control Variate783

22.7 Stochastic Intensity:The SSRD model785

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-rability789

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 SSRD812

22.7.6 Numerical Tests:Gaussian Mapping and Correlation Impact815

22.7.7 The impact of correlation on a few"test payoffs"817

22.7.8 A pricing example:A Cancellable Structure818

22.7.9 CDS Options and Jamshidian's Decomposition820

22.7.10 Bermudan CDS Options830

22.8 Stochastic diffusion intensity is not enough:Adding jumps.The JCIR(++)Model830

22.8.1 The jump-diffusion CIR model(JCIR)831

22.8.2 Bond(or Survival Probability)Formula832

22.8.3 Exact calibration of CDS:The JCIR++ model833

22.8.4 Simulation833

22.8.5 Jamshidian's Decomposition834

22.8.6 Attaining high levels of CDS implied volatility836

22.8.7 JCIR(++)models as a multi-name possibility837

22.9 Conclusions and further research838

23．CDS Options Market Models841

23.1 CDS Options and Callable Defaultable Floaters844

23.1.1 Once-callable defaultable floaters846

23.2 A market formula for CDS options and callable defaultable floaters847

23.2.1 Market formulas for CDS Options847

23.2.2 Market Formula for callable DFRN849

23.2.3 Examples of Implied Vol atilities from the Market852

23.3 Towards a Completely Specified Market Model854

23.3.1 First Choice.One-period and two-period rates855

23.3.2 Second Choice:Co-terminal and one-period CDS rates market model860

23.3.3 Third choice.Approximation:One-period CDS rates dynamics861

23.4 Hints at Smile Modeling863

23.5 Constant Maturity Credit Default Swaps(CMCDS)with the market model864

23.5.1 CDS and Constant Maturity CDS864

23.5.2 Proof of the main result867

23.5.3 A few numerical examples869

Part Ⅷ．APPENDICES877

A．Other Interest-Rate Models877

A.1 Brennan and Schwartz's Model877

A.2 Balduzzi,Das,Foresi and Sundaram's Model878

A.3 Flesaker and Hughston's Model879

A.4 Rogers's Potential Approach881

A.5 Markov Functional Models881

B．Pricing Equity Derivatives under Stochastic Rates883

B.1 The Short Rate and Asset-Price Dynamics883

B.1.1 The Dynamics under the Forward Measure886

B.2 The Pricing of a European Option on the Given Asset888

B.3 A More General Model889

B.3.1 The Construction of an Approximating Tree for r890

B.3.2 The Approximating Tree for S892

B.3.3 The Two-Dimensional Tree893

C．A Crash Intro to Stochastic Differential Equations and Pois-son Processes897

C.1 From Deterministic to Stochastic Differential Equations897

C.2 Ito's Formula904

C.3 Discretizing SDEs for Monte Carlo:Euler and Milstein Schemes906

C.4 Examples908

C.5 Two Important Theorems910

C.6 A Crash Intro to Poisson Processes913

C.6.1 Time inhomogeneous Poisson Processes915

C.6.2 Doubly Stochastic Poisson Processes(or Cox Processes)916

C.6.3 Compound Poisson processes917

C.6.4 Jump-diffusion Processes918

D．A Useful Calculation919

E．A Second Useful Calculation921

F．Approximating Diffusions with Trees925

G．Trivia and Frequently Asked Questions931

H．Talking to the Traders935

References951

Index967