| PrefaceMotivationAims, Readership and Book StructureFinal Word and AcknowledgmentsDescription of Contents by ChapterAbbreviations and NotationPart I. BASIC DEFINITIONS AND NO ARBITRAGE1. Definitions and Notation1.1 The Bank Account and the Short Rate1.2 Zero-Coupon Bonds and Spot Interest Rates1.3 Fundamental Interest-Rate Curves1.4 Forward Rates1.5 Interest-Rate Swaps and Forward Swap Rates1.6 Interest-Rate Caps/Floors and Swaptions2. No-Arbitrage Pricing and Numeraire Change2.1 No-Arbitrage in Continuous Time2.2 The Change-of-Numeraire Technique2.3 A Change of Numeraire Toolkit(Brigo & Mercurio 2001c)2.3.1 A helpful notation: "DC"2.4 The Choice of a Convenient Numeraire2.5 The Forward Measure2.6 The Fundamental Pricing Formulas2.6.1 The Pricing of Caps and Floors2.7 Pricing Claims with Deferred Payoffs2.8 Pricing Claims with Multiple Payoffs2.9 Foreign Markets and Numeraire ChangePart II. FROM SHORT RATE MODELS TO HJM3. One-factor short-rate models3.1 Introduction and Guided Tour3.2 Classical Time-Homogeneous Short-Rate Models3.2.1 The Vasicek Model3.2.2 The Dothan Model3.2.3 The Cox, Ingersoll and Ross (CIR) Model3.2.4 Affine Term-Structure Models3.2.5 The Exponential-Vasicek (EV) Model3.3 The Hull-White Extended Vasicek Model3.3.1 The Short-Rate Dynamics3.3.2 Bond and Option Pricing3.3.3 The Construction of a Trinomial Tree3.4 Possible Extensions of the CIR Model3.5 The Black-Karasinski Model3.5.1 The Short-Rate Dynamics3.5.2 The Construction of a Trinomial Tree3.6 Volatility Structures in One-Factor Short-Rate Models3.7 Humped-Volatility Short-Rate Models3.8 A General Deterministic-Shift Extension3.8.1 The Basic Assumptions3.8.2 Fitting the Initial Term Structure of Interest Rates3.8.3 Explicit Formulas for European Options3.8.4 The Vasicek Case3.9 The CIR++ Model3.9.1 The Construction of a Trinomial Tree3.9.2 Early Exercise Pricing via Dynamic Programming3.9.3 The Positivity of Rates and Fitting Quality3.9.4 Monte Carlo Simulation3.9.5 Jump Diffusion CIR and CIR++ models (JCIR, JCIR++)3.10 Deterministic-Shift Extension of Lognormal Models3.11 Some Further Remarks on Derivatives Pricing3.11.1 Pricing European Options on a Coupon-Bearing Bond3.11.2 The Monte Carlo Simulation3.11.3 Pricing Early-Exercise Derivatives with a Tree3.11.4 A Fundamental Case of Early Exercise: BermudanStyle Swaptions.3.12 Implied Cap Volatility Curves3.12.1 The Black and Karasinski Model3.12.2 The CIR++ Model3.12.3 The Extended Exponential-Vasicek Model3.13 Implied Swaption Volatility Surfaces3.13.1 The Black and Karasinski Model3.13.2 The Extended Exponential-Vasicek Model3.14 An Example of Calibration to Real-Market Data Two-Factor Short-Rate Models4.1 Introduction and Motivation4.2 The Two-Additive-Factor Gaussian Model G2++4.2.1 The Short-Rate Dynamics4.2.2 The Pricing of a Zero-Coupon Bond4.2.3 Volatility and Correlation Structures in Two-Factor Models4.2.4 The Pricing of a European Option on a Zero-Coupon Bond4.2.5 The Analogy with the Hull-White Two-Factor Model4.2.6 The Construction of an Approximating Binomial Tree4.2.7 Examples of Calibration to Real-Market Data4.3 The Two-Additive-Factor Extended CIR/LS Model CIR2++4.3.1 The Basic Two-Factor CIR2 Model4 3 2 Relationship with the Longstaff and Schwartz Model (LS)4.3.3 Forward-Measure Dynamics and Option Pricing for CIR24.3.4 The CIR2++ Model and Option Pricing5. The Heath-Jarrow-Morton (HJM) Framework5.1 The HJM Forward-Rate Dynamics5.2 Markovianity of the Short-Rate Process5.3 The Ritchken and Sankarasubramanian Framework5.4 The Mercurio and Moraleda ModelPart III. MARKET MODELS6. The LIBOR and Swap Market Models (LFM and LSM)6.1 Introduction6.2 Market Models: a Guided Tour.6.3 The Lognormal Forward-LIBOR Model (LFM)6.3.1 Some Specifications of the Instantaneous Volatility of Forward Rates6.3.2 Forward-Rate Dynamics under Different Numeraires6.4 Calibration of the LFM to Caps and Floors Prices6.4.1 Piecewise-Constant Instantaneous-Volatility Structures6.4.2 Parametric Volatility Structures6.4.3 Cap Quotes in the Market6.5 The Term Structure of Volatility6.5.1 Piecewise-Constant Instantaneous Volatility Structures6.5.2 Parametric Volatility Structures6.6 Instantaneous Correlation and Terminal Correlation6.7 Swaptious and the Lognormal Forward-Swap Model (LSM)6.7.1 Swaptions Hedging6.7.2 Cash-Settled Swaptions6.8 Incompatibility between the LFM and the LSM6.9 The Structure of Instantaneous Correlations6.9.1 Some convenient full rank parameterizations6.9.2 Reduced-rank formulations: Rebonato's angles and eigen- values zeroing6.9.3 Reducing the angles6.10 Monte Carlo Pricing of Swaptions with the LFM6.11 Monte Carlo Standard Error6.12 Monte Carlo Variance Reduction: Control Variate Estimator6.13 Rank-One Analytical Swaption Prices6.14 Rank-r Analytical Swaption Prices6.15 A Simpler LFM Formula for Swaptions Volatilities6.16 A Formula for Terminal Correlations of Forward Rates6.17 Calibration to Swaptions Prices6.18 Instantaneous Correlations: Inputs (Historical Estimation) or Outputs (Fitting Parameters)?6.19 The exogenous correlation matrix6.19.1 Historical Estimation6.19.2 Pivot matrices6.20 Connecting Caplet and S x 1-Swaption Volatilities6.21 Forward and Spot Rates over Non-Standard Periods6.21.1 Drift Interpolation6.21.2 The Bridging Technique7. Cases of Calibration of the LIBOR Market Model7.1 Inputs for the First Cases7.2 Joint Calibration with Piecewise-Constant Volatilities as in TABLE 57.3 Joint Calibration with Parameterized Volatilities as in Formulation 77.4 Exact Swaptions "Cascade" Calibration with Volatilities as in TABLE 17.4.1 Some Numerical Results7.5 A Pause for Thought7.5.1 First summary7.5.2 An automatic fast analytical calibration of LFM to swaptions. Motivations and plan7.6 Further Numerical Studies on the Cascade Calibration Algorithm……8.Monte Carlo Tests for LFM Analytical Approximations Part Ⅳ.THE VOLATILITY SMILF 9.Including the Smile in the LFM 10.Local-Volatility Models 11.Stochasti-Volatility Models 12.Uncertain-Parameter Models Part Ⅴ.EXAMPLES OF MARKET PAYOFFS 13.Pricing Derivatives on a Single Interest-Rate Curve 14.Pricing Derivatives on Two Interest-Rate Curves Part Ⅵ.INFLATION 15.Pricing of Inflation-Indexed Derivatives 16.Inflation Indexed Swaps 17.Inflation-Indexed Caplets/Floorlets 18.Calibration to market data 19.Introducing Stochastic Volatility 20.Pricing Hybrids with an Inflation Component Part Ⅶ.CREDIT 21.Introduction and Pricing under Counterparty Risk 22.Intensity Models 23.CDS Options Market Models Part Ⅷ.APPENDICES A.Other Interest-Rate Models B.Pricing Equity Derivatives under Stochastic Rates C.A Crash Intro to Stochastic Differential Equations and Poisson Processes D.A Useful Calculation E.A Second Useful Calculation F.Approximating Diffusions with Trees G.Trivia and Frequently Asked Questions H.Talking to the Traders References Index |
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