Motivation Aims, Readership and Book Structure Final Word and Acknowledgments Description of Contents by Chapter Abbreviations and Notation Part l.

Transcription

1 Preface vii Motivation vii Aims, Readership and Book Structure xii Final Word and Acknowledgments xiv Description of Contents by Chapter xix Abbreviations and Notation xxxv Part l. BASIC DEFINITIONS AND NO ARBITRAGE 1. Definitions and Notation The Bank Account and the Short Rate Zero-Coupon Bonds and Spot Interest Rates Fundamental Interest-Rate Curves Forward Rates Interest-Rate Swaps and Forward Swap Rates Interest-Rate CapsjFloors and Swaptions No-Arbitrage Pricing and Numeraire Change No-Arbitrage in Continuous Time The Change-of-Numeraire Technique A Change of Numeraire TooIkit (Brigo & Mercurio, 2001c) A helpful notatfon: "DC" The Choice of a Convenient Numeraire The Forward Measure The Fundamental Pricing Formulas The Pricing of Caps and Floors Pricing Claims with Deferred Payoffs Pricing Claims with Multiple Payoffs Foreign Markets and Numeraire Change 44 Part II. FROM SHORT RATE MODELS TO HJM 3. One-factor short-rate models Introduction and Guided Tour Classical Time-Homogeneous Short-Rate Models The Vasicek Model The Dothan Model The Cox, lngersoll and Ross (CIR) Model Affine Term-Structure Models The Exponential- Vasicek (EV) Model The Hull-White Extended Vasicek Model The Short-Rate Dynamics Bond and Option Pricing The Construction of a Trinomial Tree Possible Extensions of the CIR Model The Black-Karasinski Model The Short-Rate Dynamics The Construction of a Trinomial Tree Volatility Structures in One-Factor Short-Rate Models Humped-Volatility Short-Rate Models A General Deterministic-Shift Extension The Basic Assumptions Fitting the Initial Term Structure of Interest Rates Explicit Formulas for European Options The Vasicek Case The CIR++ Model The Construction of a Trinomial Tree 105

2 3.9.2 EarIy Exercise Pricing via Dynamic Programming The Positivity of Rates and Fitting Quality Monte CarIo Simulation Jump Diffusion CIR and CIR++ models (JCIR., JCIR++) Deterministic-Shift Extension of Lognormal Models Some Further Remarks on Derivatives Pricing Pricing European Options on a Coupon-Bearing Bond The Monte CarIo Simulation Pricing Early-Exercise Derivatives with a Tree A Fundamental Case of EarIy Exercise: Bermudan-Style Swaptions Implied Cap Volatility Curves The' Black and Karasinski Model The CIR++ Model The Extended Exponential-Vasicek Model Implied Swaption Volatility Surfaces The Black and Karasinski Model The Extended Exponential-Vasicek Model An Example ol' Calibration to Real-Market Data Two-Factor Short-Rate Models Introduction and Motivation The Two-Additive-Factor Gaussian Model G The Short-Rate Dynamics The Pricing ol' a Zero-Coupon Bond Volatility and 'Correlation Structures in Two-Factor Models The Pricing ol' a European Option On a Zero-Coupon Bond The Analogy with the Hull-White Two-Factor Model The Construction ol' an Approximating Binomial Tree Examples ol' Calibration to Real-Market Data The Two-Additive-Factor Extended CIR/LS Model CIR The Basic Two-Factor CIR2 Model Relationship with the Longstafi' and Schwartz Model (LS) Forward-Measure Dynamics and Opt.ion Pricing l'or CIR The CIR2++ l\iodel and Option Pricing The Heath-Jarrow-Morto(HnJM) Framework The H.JM Forward-Rat.e Dynamics Markovianity ol' the Short-Rate Process The Ritchken and Sankarasllbramanian Framcwork The Mercurio ami Moraleda Modcl 191 Part III. MARKET MODELS 6. The LIBOR and Swap Market Models (LFM and LSM) Introduction l\1arket Models: a Guidcd Tour The Lognormal Forward-LIBOR. JvIodcl (LFM) Some Specifications of' the Instantanceus Volatility of Forward Rates Forward-Rate Dynamics undcr Different Numeraires Calibration of' the LFM to Caps ami Floors Prices Piecewise-Constant Instantaneous- Volatility Structures Parametric Volatility Structures Cap Quotes in the Market The Term Structure ol' Volatility Piecewise-Constant Instantaneolls Volatility Structures Parametric Volatility Structures 231

3 6.6 Instantaneous Correlation and Terminal Correlation Swaptions and the Lognormal Forward-Swap Model (LSM) Swaptions Hedging Cash-Settled Swaptions Incompatibility between the LFM and the LSM The Structure of Instantaneous Correlations Some convenient full rank parameterizations Reduced-rank formulations: Rebonato's angles and eigenvalues zeroing Reducing the angles Monte CarIo Pricing of Swaptions with the LFM Monte Carlo Standard Error Monte CarIo Variance Reduction: Control Variate Estimator Rank-One Analytical Swaption Prices Rank-r Analytical Swaption Prices A Simpler LFM Formula for Swaptions Volatilities A Formula for Terminal Correlations of Forward Rates Calibration to Swaptions Prices Instantaneous Correlations: Inputs or Outputs? The exogenous correlation matrix Historical Estimation Pivot matrices Connecting Caplet and S x l-swaption Volatilities Forward and Spot Rates over Non-Standard Periods Drift Interpolation The Bridging Technique Cases of Calibration of the LIBOR Market Model Inputs for the First Cases Joint Calibration with Piecewise-Constant Volatilities as in TABLE Joint Calibration with Parameterized Volatilities as in Formulation Exact Swaptions "Cascade" Calibration with Volatilities as in TABLE Some Numerical Results A Pause for Thought First summary An automatic fast analytical calibration of LFM to swaptions. Motivations and plan Further Numerical Studies on the Cascade Calibration Algorithm Cascade Calibration under Various Correlations and Ranks Cascade Calibration Diagnostics: Terminal Correlation and Evolution of Volatilities The interpolation for the swaption matrix and its impact on the CCA Empirically efficient Cascade Calibration CCA with Endogenous Interpolation and Based Only on Pure Market Data Financial Diagnostics of the RCCAEI test results Endogenous Cascade Interpolation for missing swaptions volatilities quotes A first partial check on the calibrated a parameters stability Reliability: Monte Carlo tests Cascade Calibration and the cap market Cascade Calibration: Conclusions Monte Carlo Tests for LFM Analytical Approximations First Parto Tests Based on the Kullback Leibler Information (KLI) Distance between distributions: The Kullback Leibler information Distance of the LFM swap rate from the lognormal family of distributions Monte Carlo tests for measuring KLI Conclusions on the KLI-based approach 391

4 8.2 Second Part: Classical Tests The "Testing Plan" for Volatilities Test Results for Volatilities Case (1): Constant Instantaneous Volatilities Case (2): Volatilities as Functions of Time to Maturity Case (3): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity The "Testing Plan" for Terminal Correlations Test Results for TermInal Correlations Case (i): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity, Typical Rank-Two Correlations Case (ii): Constant Instantaneous Volatilities, Typical Rank-Two Correlations Case (iii): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity Some Negative Rank-Two Correlations Case (iv): Constant Instantaneous Volatilities, Some Negative Rank-Two Correlations Case (v): Constant Instantaneous Volatilities, Perfect Correlations, Upwardly Shifted Test Results: Stylized Conclusions 442 Part IV. THE VOLATILITY SMILE 9. Including the Smile in the LFM A Mini-tour on trhe Smile Problem Modeling the Smile Local- VoIatility Models The Shifted-Lognormal Model The Constant Elasticity of Variance Model A Class of Analytically- Tractable Models A Lognormal-Mixture (LM) Model Forward Rates Dynamics under Different Measures Decorrelation Between Underlying and Volatility Shifting the LM Dynamics A Lognormal-Mixture with Different Means (LMDM) The Case of Hyperbolic-Sine Processes Testing the Above Mixture-Models on Market Data A Second General Class A Particular Case: a Mixture of GBM's An Extension of the GBM Mixture Model Leading to Skews A General Dynamics'a la Dupire (1994) Stochastic- Volatility Models The Andersen and Brotherton-Ratcliffe (2001) Model The Wu and Zhang (2002) Model The Piterbarg (2003) Model The Hagan, Kumar, Lesniewski ami Woodward (2002) Model The Joshi and Rebonato (2003) Model Uncertain-Parameter Models The Shifted-Lognormal Model with Uncertain Parameters (SLMUP) Relationship with the Lognormal-Mixture LVM Calibration to Caplets Swaption Pricing Monte-Carlo Swaption Pricing Calibration to Swaptions Calibration to Market Data Testing the Approximation for Swaptions Prices 530

5 12.8 Further Model Implications Joint Calibration to Caps and Swaptions 539 Part V. EXAMPLES OF MARKET PAYOFFS 13. Pricing Derivatives on a Single Interest-Rate Curve In-Arrears Swaps In-Arrears Caps A First Analytical Formula (LFM) A Second Analytical Formula (G2++) Autocaps Caps with Deferred Caplets A First Analytical Formula (LFM) A Second Analytical Formula (G2++) Ratchet Caps and Floors Analytical Approximation for Ratchet Caps with the LFM Ratchets (One-Way Floaters) Constant-Maturity Swaps (GMS) Cr-vISwith the LFM CMS with the G2++ Model The Convexity Adjustment and Applications to Cl\IS Natural ami Unnatural Time Lags The Convexity-Adjust.ment Teclmique Deducing a Simple Lognormal Dynarnics from thc Adjustment Application to Cr-vIS Smile-Consistent Valuat.ion of Cl\IS Adjustrncnts Forward Rate Resetting Unnaturally ami Average-Rate Swaps Average Rate Caps Captions ami Floortions Zere Coupon Swaptions Eurodollar Futuros The Shifted Two-Fact.or Vasicek G2++ Modcl Eurodollar Fut.ures with the LFM LFM Pricing with" In-Betwcen" Sport Rates Accrual Swaps Trigger Swaps LFM Pricing with Early Exercisc ami Possible Path Dependence LFM: Pricing Bermudan Swaptions Least Squarcd Monte Carlo Approach Carr and Yang s Approach Andersen's Approach Numerical Example New Generation of Contracts Target Redemption Notes CMS Spread Options Pricing Derivatives on Two Interest-Rate Curves The Attractive Features of G2++ for Multi-Curve Payoffs The Model Interaction Between Models of the Two Curves "1" and "2" The Two-Models Dynamics under a Unique Convenient Forward Measure Quanto Constant-Maturity Swaps Quanto CMS: The Contract Quanto CMS: The G2++ Model Quanto CMS: Quanto Adjustment 621

6 14.3 Differential Swaps The Contract Differential Swaps with the G2++ Model A Market-Like Formula Market Formulas for Basic Quanto Derivatives The Pricing of Quanto Caplets/Floorlets The Pricing of Quanto Caps/Floors The Pricing of Differential Swaps The Pricing of Quanto Swaptions Pricing of Options on two Currency LIBOR Rates Spread Options Options on the Product Trigger Swaps Dealing with Multiple Dates 639 Part VI. INFLATION 15. Pricing of Inflation-Indexed Derivatives The Foreign-Currency Analogy Definitions and Notation The JY Model Inflation-Indexed Swaps Pricing of a ZCIIS Pricing of a YYIlS Pricing of a YYIlS with the JY Model Pricing of a YYIIS with a First Market Model Pricing of a YYIlS with a Second Market Model Inflation-Indexed Caplets/Floorlets Pricing with the JY Model Pricing with the Second Market Model Inflation-Indexed Caps Calibration to market data Introducing Stochastic Volatility Modeling Forward CPL's with Stochastic Volatility Pricing Formulae Exact Solution for the Uncorrelated Case Approximated Dynamics for Non-zero Correlations Example of Calibration Pricing Hybrids with an Inflation Component A Simple Hybrid Payoff 689 Part VII. CREDIT 21. Introduction and Pricing under Counterparty Risk Introduction and Guided Tour Reduced form (Intensity) models CDS Options l\iarket Models Firm Value (01'Structural) Models Further Models The Multi-name picture: FtD, CDO and Copula Functions First to Default (FtD) Basket Collateralized Debt Obligation (CDO) Tranches Where can we introduce dependence? Copula Functions Dynamic Loss models What data are available in the market'? 719

7 21.2 Defaultable (corporate) zero coupon bonds Defaultable (corporate) coupon bonds Credit Default Swaps and Defaultable Floaters CDS payoffs: Different Formulations CDS pricing formulas Changing filtrat.ion: Ft wit.hout. default. VS complete Qt CDS forward rat.es: The fin;t. definition Market. quot.es, model independent implied survival probabilities and implied hazard functions A simpler formula for calibrating intensity to a single CDS Different Definitions of CDS Forward Rates and Analogies with the LIBOR and SWAP rates Defaultable Floater and CDS CDS Options and Callable Defaultable Floaters Constant Maturity CDS Some interesting Financial features of CMCDS Interest-Rate Payoffs with Counterparty Risk General Valuation of Counterparty Risk COllnterparty Risk in single Interest Rate Swaps (IRS) Hints at contingent CDS Intensity Models Introduction and Chapter Description Poisson processes Time homogeneous Poisson processes Time inhomogeneous Poisson Processes Cox Processes CDS Calibration ami Implied Hazard Ratcs/ Intensitics Inducing dependence between Interest-rates ami the default event The Filtrat.ion Switching Formula: Pricing under partial information Default Simulation in reduced form models Standard error Variance Rcduction with Control Variate Stochll.-;ticInt.ensity: The SSRD model A two-factor shifted square-root diffllsion model for intensity and interest rates (Brigo and Alfonsi (2003) Calibrating the joint stocha.-;tic model to CDS: Separability Discretization schemcsfor simulating Study of the convergencc of the discrctization schemes for simulating CIR processes (Alfonsi (2005)) Gallssian dependcnce mapping: A tractable approximated SSRD Numerical Test.s: Gallssian l\iapping and Correlation Impact The impact of correlation on a few "test payoft.s" A pricing example: A Cancellable Strllcture CDS Options ami.tamshidian's Decomposition Bermudan CDS Options Stocha.-;tic dift'usion intensity is not enough: Adding jumps The jump-diffusion CIR model (JCIR) Bond (or Survival Probability) Formula Exact calibration of CDS: The JCIR++ model Simulation Jamshidian's Decomposition Attaining high levels of CDS implied volatility 836

8 JCIR( ++) models as a multi-name possibility Conclusions and further research CDS Options Market Módels CDS Options ami Callable Defaultable Floaters Once-callable defaultable floaters A market formula for CDS options and callable defaultable floaters Market formulas for CDS Options Market Formula for callable DFRN Examples of Implied Volatilities from the Market Towards a Completely Specified Market Model First Choice. One-period ami two-period rates Second Choice: Co-terminal and one-period CDS rates market model Third choice. Approximation: One-period CDS rates dynamics Hints at Smile Modeling Constant Maturity Credit Default Swaps (Cl\ICDS) with the market model CDS and Constant Maturity CDS Proof of the main result A few numerical examples 869 Part VIII. APPENDICES A. Other Interest-Rate Models 877 A.1 Brennan ami Schwartz 's Model 877 A.2 Balduzzi, Das, Foresi and Sundaralll's Model 878 A.3 Flesaker ami Hllghston'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 Bate ami Asset-Price Dynamics 883 B.1.1 The Dynamics under the Forwanl 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 CrashIntro to Stochastic Differential Equations and Poisson 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 Pl'ocesses (01'Cox Pl'ocesses) 916 C.6.3 Compound Poisson processes 917 C.6.4 Jump-diffusion Processes 918 D. A Useful Calculation 919 E. A Second Use fui 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