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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 | |