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| Preface | |
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| Motivation | |
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| Aims, Readership and Book Structure | |
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| Final Word and Acknowledgments | |
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| Description of Contents by Chapter | |
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| Abbreviations and Notation | |
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| Basic Definitions and No Arbitrage | |
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| Definitions and Notation | |
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| The Bank Account and the Short Rate | |
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| Zero-Coupon Bonds and Spot Interest Rates | |
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| Fundamental Interest-Rate Curves | |
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| Forward Rates | |
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| Interest-Rate Swaps and Forward Swap Rates | |
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| Interest-Rate Caps/Floors and Swaptions | |
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| No-Arbitrage Pricing and Numeraire Change | |
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| No-Arbitrage in Continuous Time | |
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| The Change-of-Numeraire Technique | |
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| A Change of Numeraire Toolkit (Brigo & Mercurio 2001c) | |
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| A helpful notation: "DC" | |
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| The Choice of a Convenient Numeraire | |
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| The Forward Measure | |
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| The Fundamental Pricing Formulas | |
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| The Pricing of Caps and Floors | |
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| Pricing Claims with Deferred Payoffs | |
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| Pricing Claims with Multiple Payoffs | |
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| Foreign Markets and Numeraire Change | |
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| From Short Rate Models to HJM | |
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| One-factor short-rate models | |
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| Introduction and Guided Tour | |
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| Classical Time-Homogeneous Short-Rate Models | |
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| The Vasicek Model | |
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| The Dothan Model | |
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| The Cox, Ingersoll and Ross (CIR) Model | |
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| Affine Term-Structure Models | |
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| The Exponential-Vasicek (EV) Model | |
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| The Hull-White Extended Vasicek Model | |
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| The Short-Rate Dynamics | |
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| Bond and Option Pricing | |
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| The Construction of a Trinomial Tree | |
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| Possible Extensions of the CIR Model | |
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| The Black-Karasinski Model | |
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| The Short-Rate Dynamics | |
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| The Construction of a Trinomial Tree | |
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| Volatility Structures in One-Factor Short-Rate Models | |
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| Humped-Volatility Short-Rate Models | |
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| A General Deterministic-Shift Extension | |
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| The Basic Assumptions | |
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| Fitting the Initial Term Structure of Interest Rates | |
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| Explicit Formulas for European Options | |
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| The Vasicek Case | |
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| The CIR++ Model | |
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| The Construction of a Trinomial Tree | |
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| Early Exercise Pricing via Dynamic Programming | |
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| The Positivity of Rates and Fitting Quality | |
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| Monte Carlo Simulation | |
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| Jump Diffusion CIR and CIR++ models (JCIR, JCIR++) | |
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| Deterministic-Shift Extension of Lognormal Models | |
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| Some Further Remarks on Derivatives Pricing | |
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| Pricing European Options on a Coupon-Bearing Bond | |
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| The Monte Carlo Simulation | |
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| Pricing Early-Exercise Derivatives with a Tree | |
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| A Fundamental Case of Early Exercise: Bermudan-Style Swaptions | |
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| Implied Cap Volatility Curves | |
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| The Black and Karasinski Model | |
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| The CIR++ Model | |
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| The Extended Exponential-Vasicek Model | |
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| Implied Swaption Volatility Surfaces | |
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| The Black and Karasinski Model | |
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| The Extended Exponential-Vasicek Model | |
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| An Example of Calibration to Real-Market Data | |
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| Two-Factor Short-Rate Models | |
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| Introduction and Motivation | |
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| The Two-Additive-Factor Gaussian Model G2++ | |
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| The Short-Rate Dynamics | |
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| The Pricing of a Zero-Coupon Bond | |
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| Volatility and Correlation Structures in Two-Factor Models | |
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| The Pricing of a European Option on a Zero-Coupon Bond | |
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| The Analogy with the Hull-White Two-Factor Model | |
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| The Construction of an Approximating Binomial Tree | |
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| Examples of Calibration to Real-Market Data | |
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| The Two-Additive-Factor Extended CIR/LS Model CIR2++ | |
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| The Basic Two-Factor CIR2 Model | |
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| Relationship with the Longstaff and Schwartz Model (LS) | |
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| Forward-Measure Dynamics and Option Pricing for CIR2 | |
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| The CIR2++ Model and Option Pricing | |
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| The Heath-Jarrow-Morton (HJM) Framework | |
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| The HJM Forward-Rate Dynamics | |
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| Markovianity of the Short-Rate Process | |
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| The Ritchken and Sankarasubramanian Framework | |
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| The Mercurio and Moraleda Model | |
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| MArket Models | |
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| The LIBOR and Swap Market Models (LFM and LSM) | |
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| Introduction | |
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| Market Models: a Guided Tour | |
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| The Lognormal Forward-LIBOR Model (LFM) | |
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| Some Specifications of the Instantaneous Volatility of Forward Rates | |
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| Forward-Rate Dynamics under Different Numeraires | |
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| Calibration of the LFM to Caps and Floors Prices | |
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| Piecewise-Constant Instantaneous-Volatility Structures | |
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| Parametric Volatility Structures | |
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| Cap Quotes in the Market | |
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| The Term Structure of Volatility | |
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| Piecewise-Constant Instantaneous Volatility Structures | |
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| Parametric Volatility Structures | |
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| Instantaneous Correlation and Terminal Correlation | |
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| Swaptions and the Lognormal Forward-Swap Model (LSM) | |
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| Swaptions Hedging | |
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| Cash-Settled Swaptions | |
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| Incompatibility between the LFM and the LSM | |
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| The Structure of Instantaneous Correlations | |
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| Some convenient full rank parameterizations | |
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| Reduced-rank formulations: Rebonato's angles and eigen-values zeroing | |
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| Reducing the angles | |
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| Monte Carlo Pricing of Swaptions with the LFM | |
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| Monte Carlo Standard Error | |
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| Monte Carlo Variance Reduction: Control Variate Estimator | |
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| Rank-One Analytical Swaption Prices | |
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| Rank-r Analytical Swaption Prices | |
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| A Simpler LFM Formula for Swaptions Volatilities | |
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| A Formula for Terminal Correlations of Forward Rates | |
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| Calibration to Swaptions Prices | |
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| Instantaneous Correlations: Inputs (Historical Estimation) or Outputs (Fitting Parameters)? | |
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| The exogenous correlation matrix | |
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| Historical Estimation | |
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| Pivot matrices | |
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| Connecting Caplet and S x 1-Swaption Volatilities | |
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| Forward and Spot Rates over Non-Standard Periods | |
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| Drift Interpolation | |
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| The Bridging Technique | |
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| Cases of Calibration of the LIBOR Market Model | |
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| Inputs for the First Cases | |
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| Joint Calibration with Piecewise-Constant Volatilities as in Table 5 | |
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| Joint Calibration with Parameterized Volatilities as in Formulation 7 | |
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| Exact Swaptions "Cascade" Calibration with Volatilities as in Table 1 | |
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| Some Numerical Results | |
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| A Pause for Thought | |
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| First summary | |
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| An automatic fast analytical calibration of LFM to swaptions. Motivations and plan | |
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| Further Numerical Studies on the Cascade Calibration Algorithm | |
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| Cascade Calibration under Various Correlations and Ranks | |
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| Cascade Calibration Diagnostics: Terminal Correlation and Evolution of Volatilities | |
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| The interpolation for the swaption matrix and its impact on the CCA | |
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| Empirically efficient Cascade Calibration | |
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| CCA with Endogenous Interpolation and Based Only on Pure Market Data | |
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| Financial Diagnostics of the RCCAEI test results | |
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| Endogenous Cascade Interpolation for missing swaptions volatilities quotes | |
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| A first partial check on the calibrated [sigma] parameters stability | |
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| Reliability: Monte Carlo tests | |
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| Cascade Calibration and the cap market | |
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| Cascade Calibration: Conclusions | |
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| Monte Carlo Tests for LFM Analytical Approximations | |
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| First Part. Tests Based on the Kullback Leibler Information (KLI) | |
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| Distance between distributions: The Kullback Leibler information | |
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| Distance of the LFM swap rate from the lognormal family of distributions | |
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| Monte Carlo tests for measuring KLI | |
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| Conclusions on the KLI-based approach | |
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| Second Part: Classical Tests | |
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| The "Testing Plan" for Volatilities | |
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| Test Results for Volatilities | |
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| Case (1): Constant Instantaneous Volatilities | |
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| Case (2): Volatilities as Functions of Time to Maturity | |
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| Case (3): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity | |
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| The "Testing Plan" for Terminal Correlations | |
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| Test Results for Terminal Correlations | |
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| Case (i): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity, Typical Rank-Two Correlations | |
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| Case (ii): Constant Instantaneous Volatilities, Typical Rank-Two Correlations | |
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| Case (iii): Humped and Maturity-Adjusted Instantaneous Volatilities Depending only on Time to Maturity, Some Negative Rank-Two Correlations | |
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| Case (iv): Constant Instantaneous Volatilities, Some Negative Rank-Two Correlations | |
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| Case (v): Constant Instantaneous Volatilities, Perfect Correlations, Upwardly Shifted [Phi]'s | |
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| Test Results: Stylized Conclusions | |
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| The Volatility Smile | |
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| Including the Smile in the LFM | |
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| A Mini-tour on the Smile Problem | |
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| Modeling the Smile | |
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| Local-Volatility Models | |
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| The Shifted-Lognormal Model | |
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| The Constant Elasticity of Variance Model | |
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| A Class of Analytically-Tractable Models | |
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| A Lognormal-Mixture (LM) Model | |
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| Forward Rates Dynamics under Different Measures | |
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| Decorrelation Between Underlying and Volatility | |
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| Shifting the LM Dynamics | |
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| A Lognormal-Mixture with Different Means (LMDM) | |
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| The Case of Hyperbolic-Sine Processes | |
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| Testing the Above Mixture-Models on Market Data | |
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| A Second General Class | |
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| A Particular Case: a Mixture of GBM's | |
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| An Extension of the GBM Mixture Model Allowing for Implied Volatility Skews | |
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| A General Dynamics a la Dupire (1994) | |
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| Stochastic-Volatility Models | |
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| The Andersen and Brotherton-Ratcliffe (2001) Model | |
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| The Wu and Zhang (2002) Model | |
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| The Piterbarg (2003) Model | |
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| The Hagan, Kumar, Lesniewski and Woodward (2002) Model | |
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| The Joshi and Rebonato (2003) Model | |
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| Uncertain-Parameter Models | |
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| The Shifted-Lognormal Model with Uncertain Parameters (SLMUP) | |
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| Relationship with the Lognormal-Mixture LVM | |
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| Calibration to Caplets | |
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| Swaption Pricing | |
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| Monte-Carlo Swaption Pricing | |
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| Calibration to Swaptions | |
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| Calibration to Market Data | |
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| Testing the Approximation for Swaptions Prices | |
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| Further Model Implications | |
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| Joint Calibration to Caps and Swaptions | |
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| Examples of Market Payoffs | |
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| Pricing Derivatives on a Single Interest-Rate Curve | |
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| In-Arrears Swaps | |
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| In-Arrears Caps | |
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| A First Analytical Formula (LFM) | |
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| A Second Analytical Formula (G2++) | |
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| Autocaps | |
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| Caps with Deferred Caplets | |
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| A First Analytical Formula (LFM) | |
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| A Second Analytical Formula (G2++) | |
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| Ratchet Caps and Floors | |
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| Analytical Approximation for Ratchet Caps with the LFM | |
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| Ratchets (One-Way Floaters) | |
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| Constant-Maturity Swaps (CMS) | |
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| CMS with the LFM | |
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| CMS with the G2++ Model | |
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| The Convexity Adjustment and Applications to CMS | |
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| Natural and Unnatural Time Lags | |
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| The Convexity-Adjustment Technique | |
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| Deducing a Simple Lognormal Dynamics from the Adjustment | |
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| Application to CMS | |
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| Forward Rate Resetting Unnaturally and Average-Rate Swaps | |
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| Average Rate Caps | |
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| Captions and Floortions | |
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| Zero-Coupon Swaptions | |
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| Eurodollar Futures | |
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| The Shifted Two-Factor Vasicek G2++ Model | |
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| Eurodollar Futures with the LFM | |
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| LFM Pricing with "In-Between" Spot Rates | |
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| Accrual Swaps | |
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| Trigger Swaps | |
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| LFM Pricing with Early Exercise and Possible Path Dependence | |
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| LFM: Pricing Bermudan Swaptions | |
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| Least Squared Monte Carlo Approach | |
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| Carr and Yang's Approach | |
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| Andersen's Approach | |
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| Numerical Example | |
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| New Generation of Contracts | |
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| Target Redemption Notes | |
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| CMS Spread Options | |
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| Pricing Derivatives on Two Interest-Rate Curves | |
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| The Attractive Features of G2++ for Multi-Curve Payoffs | |
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| The Model | |
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| Interaction Between Models of the Two Curves "1" and "2" | |
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| The Two-Models Dynamics under a Unique Convenient Forward Measure | |
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| Quanto Constant-Maturity Swaps | |
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| Quanto CMS: The Contract | |
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| Quanto CMS: The G2++ Model | |
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| Quanto CMS: Quanto Adjustment | |
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| Differential Swaps | |
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| The Contract | |
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| Differential Swaps with the G2++ Model | |
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| A Market-Like Formula | |
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| Market Formulas for Basic Quanto Derivatives | |
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| The Pricing of Quanto Caplets/Floorlets | |
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| The Pricing of Quanto Caps/Floors | |
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| The Pricing of Differential Swaps | |
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| The Pricing of Quanto Swaptions | |
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| Pricing of Options on two Currency LIBOR Rates | |
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| Spread Options | |
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| Options on the Product | |
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| Trigger Swaps | |
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| Dealing with Multiple Dates | |
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| Inflation | |
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| Pricing of Inflation-Indexed Derivatives | |
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| The Foreign-Currency Analogy | |
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| Definitions and Notation | |
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| The JY Model | |
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| Inflation-Indexed Swaps | |
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| Pricing of a ZCIIS | |
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| Pricing of a YYIIS | |
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| Pricing of a YYIIS with the JY Model | |
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| Pricing of a YYIIS with a First Market Model | |
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| Pricing of a YYIIS with a Second Market Model | |
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| Inflation-Indexed Caplets/Floorlets | |
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| Pricing with the JY Model | |
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| Pricing with the Second Market Model | |
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| Inflation-Indexed Caps | |
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| Calibration to market data | |
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| Introducing Stochastic Volatility | |
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| Modeling Forward CPI's with Stochastic Volatility | |
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| Pricing Formulae | |
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| Exact Solution for the Uncorrelated Case | |
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| Approximated Dynamics for Non-zero Correlations | |
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| Example of Calibration | |
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| Pricing Hybrids with an Inflation Component | |
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| A Simple Hybrid Payoff | |
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| Credit | |
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| Introduction and Pricing under Counterparty Risk | |
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| Introduction and Guided Tour | |
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| Reduced form (Intensity) models | |
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| CDS Options Market Models | |
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| Firm Value (or Structural) Models | |
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| Further Models | |
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| The Multi-name picture: FtD, CDO and Copula Functions | |
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| First to Default (FtD) Basket | |
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| Collateralized Debt Obligation (CDO) Tranches | |
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| Where can we introduce dependence? | |
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| Copula Functions | |
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| Dynamic Loss models | |
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| What data are available in the market? | |
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| Defaultable (corporate) zero coupon bonds | |
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| Defaultable (corporate) coupon bonds | |
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| Credit Default Swaps and Defaultable Floaters | |
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| CDS payoffs: Different Formulations | |
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| CDS pricing formulas | |
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| Changing filtration: F[subscript t] without default VS complete G[subscript t] | |
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| CDS forward rates: The first definition | |
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| Market quotes, model independent implied survival probabilities and implied hazard functions | |
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| A simpler formula for calibrating intensity to a single CDS | |
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| Different Definitions of CDS Forward Rates and Analogies with the LIBOR and SWAP rates | |
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| Defaultable Floater and CDS | |
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| CDS Options and Callable Defaultable Floaters | |
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| Constant Maturity CDS | |
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| Some interesting Financial features of CMCDS | |
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| Interest-Rate Payoffs with Counterparty Risk | |
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| General Valuation of Counterparty Risk | |
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| Counterparty Risk in single Interest Rate Swaps (IRS) | |
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| Intensity Models | |
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| Introduction and Chapter Description | |
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| Poisson processes | |
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| Time homogeneous Poisson processes | |
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| Time inhomogeneous Poisson Processes | |
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| Cox Processes | |
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| CDS Calibration and Implied Hazard Rates/Intensities | |
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| Inducing dependence between Interest-rates and the default event | |
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| The Filtration Switching Formula: Pricing under partial information | |
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| Default Simulation in reduced form models | |
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| Standard error | |
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| Variance Reduction with Control Variate | |
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| Stochastic Intensity: The SSRD model | |
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| A two-factor shifted square-root diffusion model for intensity and interest rates (Brigo and Alfonsi (2003)) | |
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| Calibrating the joint stochastic model to CDS: Separability | |
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| Discretization schemes for simulating ([lambda], r) | |
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| Study of the convergence of the discretization schemes for simulating CIR processes (Alfonsi (2005)) | |
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| Gaussian dependence mapping: A tractable approximated SSRD | |
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| Numerical Tests: Gaussian Mapping and Correlation Impact | |
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| The impact of correlation on a few "test payoffs" | |
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| A pricing example: A Cancellable Structure | |
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| CDS Options and Jamshidian's Decomposition | |
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| Bermudan CDS Options | |
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| Stochastic diffusion intensity is not enough: Adding jumps. The JCIR(++) Model | |
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| The jump-diffusion CIR model (JCIR) | |
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| Bond (or Survival Probability) Formula | |
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| Exact calibration of CDS: The JCIR++ model | |
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| Simulation | |
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| Jamshidian's Decomposition | |
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| Attaining high levels of CDS implied volatility | |
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| JCIR(++) models as a multi-name possibility | |
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| Conclusions and further research | |
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| CDS Options Market Models | |
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| CDS Options and Callable Defaultable Floaters | |
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| Once-callable defaultable floaters | |
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| A market formula for CDS options and callable defaultable floaters | |
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| Market formulas for CDS Options | |
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| Market Formula for callable DFRN | |
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| Examples of Implied Volatilities from the Market | |
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| Towards a Completely Specified Market Model | |
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| First Choice. One-period and two-period rates | |
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| Second Choice: Co-terminal and one-period CDS rates market model | |
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| Third choice. Approximation: One-period CDS rates dynamics | |
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| Hints at Smile Modeling | |
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| Constant Maturity Credit Default Swaps (CMCDS) with the market model | |
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| CDS and Constant Maturity CDS | |
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| Proof of the main result | |
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| A few numerical examples | |
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| Appendices | |
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| Other Interest-Rate Models | |
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| Brennan and Schwartz's Model | |
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| Balduzzi, Das, Foresi and Sundaram's Model | |
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| Flesaker and Hughston's Model | |
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| Rogers's Potential Approach | |
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| Markov Functional Models | |
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| Pricing Equity Derivatives under Stochastic Rates | |
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| The Short Rate and Asset-Price Dynamics | |
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| The Dynamics under the Forward Measure | |
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| The Pricing of a European Option on the Given Asset | |
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| A More General Model | |
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| The Construction of an Approximating Tree for r | |
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| The Approximating Tree for S | |
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| The Two-Dimensional Tree | |
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| A Crash Intro to Stochastic Differential Equations and Poisson Processes | |
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| From Deterministic to Stochastic Differential Equations | |
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| Ito's Formula | |
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| Discretizing SDEs for Monte Carlo: Euler and Milstein Schemes | |
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| Examples | |
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| Two Important Theorems | |
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| A Crash Intro to Poisson Processes | |
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| Time inhomogeneous Poisson Processes | |
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| Doubly Stochastic Poisson Processes (or Cox Processes) | |
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| Compound Poisson processes | |
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| Jump-diffusion Processes | |
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| A Useful Calculation | |
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| A Second Useful Calculation | |
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| Approximating Diffusions with Trees | |
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| Trivia and Frequently Asked Questions | |
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| Talking to the Traders | |
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| References | |
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| Index | |