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Introduction. | |
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The Theoretical Set-Up. | |
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The LIBOR Market Model. | |
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Definitions | |
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The Volatility Functions | |
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Separating the Correlation from the Volatility Term | |
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The Caplet-Pricing Condition Again | |
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The Forward-Rate/Forward-Rate Correlation | |
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Possible Shapes of the Doust Correlation Function | |
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The Covariance Integral Again | |
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The SABR Model. | |
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The SABR Model (and Why It Is a Good Model | |
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Description of the Model | |
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The Option Prices Given by the SABR Model | |
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Special Cases | |
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Qualitative Behaviour of the SABR Model | |
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The Link Between the Exponent, _, and the Volatility of Volatility, _ | |
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Volatility Clustering in the (LMM)-SABR Model | |
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The Market | |
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How Do We Know that the Market Has Chosen _ = 0:5? | |
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The Problems with the SABR Model | |
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The LMM-SABR Model. | |
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The Equations of Motion | |
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The Nature of the Stochasticity Introduced by Our Model | |
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A Simple Correlation Structure | |
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A More General Correlation Structure | |
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Observations on the Correlation Structure | |
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The Volatility Structure | |
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What We Mean by Time Homogeneity | |
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The Volatility Structure in Periods of Market Stress | |
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A More General Stochastic Volatility Dynamics | |
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Calculating the No-Arbitrage Drifts | |
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IMPLEMENTATION AND CALIBRATION. | |
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Calibrating the LMM-SABR model to Market Caplet Prices. | |
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The Caplet-Calibration Problem | |
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Choosing the Parameters of the Function, g (_), and the Initial Values, kT 0 | |
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Choosing the Parameters of the Function h(_ | |
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Choosing the Exponent, _, and the Correlation, _SABR | |
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Results | |
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Calibration in Practice: Implications for the SABR Model | |
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Implications for Model Choice | |
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Calibrating the LMM-SABR model to Market Swaption Prices. | |
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The Swaption Calibration Problem | |
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Swap Rate and Forward Rate Dynamics | |
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Approximating the Instantaneous Swap Rate Volatility, St | |
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Approximating the Initial Value of the Swap Rate Volatility, _0 (First Route | |
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Approximating _0 | |
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Approximating the Swap-Rate/Swap-Rate-Volatility Correlation, RSABR | |
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Approximating the Swap Rate Exponent, B | |
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Results | |
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Conclusions and Suggestions for Future Work | |
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Appendix: Derivation of Approximate Swap Rate Volatility | |
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Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation, RSABR | |
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Appendix: Approximation of | |
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Calibrating the Correlation Structure. | |
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Statement of the Problem | |
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Creating a Valid Model Matrix | |
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A Case Study: Calibration Using the Hypersphere Method | |
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Which Method Should One Choose? | |
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Appendix1 | |
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EMPIRICAL EVIDENCE. | |
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The Empirical Problem. | |
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Statement of the Empirical Problem | |
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What Do We know from the Literature? | |
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Data Description | |
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Distributional Analysis and Its Limitations | |
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What Is the True Exponent _? | |
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Appendix: Some Analytic Results | |
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Estimating the Volatility of the Forward Rates. | |
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Expiry-Dependence of Volatility of Forward Rates | |
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Direct Estimation | |
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Looking at the Normality of the Residuals | |
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Maximum-Likelihood and Variations on the Theme | |
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Information About the Volatility from the Options Market | |
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Overall Conclusions | |
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Estimating the Correlation Structure. | |
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What We Are Trying To Do | |
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Some Results from Random Matrix Theory | |
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Empirical Estimation | |
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Descriptive Statistics | |
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Signal and Noise in the Empirical Correlation Blocks | |
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What Does Random Matrix Theory Really Tell Us? | |
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Calibrating the Correlation Matrices | |
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How Much Information Do the Proposed Models Retain? | |
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HEDGING. | |
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Various Types of Hedging. | |
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Statement of the Problem | |
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Three Types of Hedging | |
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Definitions | |
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First-Order Derivatives with Respect to the Underlyings | |
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Second-Order Derivatives with Respect to the Underlyings | |
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Generalizing Functional-Dependence Hedging | |
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How Does the Model Know about Volga and Vanna? | |
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Choice of Hedging Instrument | |
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Hedging Against Moves in the Forward Rate and in the Volatility. | |
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Delta Hedging in the SABR-(LMM) Model | |
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Vega Hedging in the SABR-(LMM) Model | |
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(LMM)-SABR Hedging in Practice: Evidence from Market Data. | |
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Purpose of this Chapter | |
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Notation | |
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Hedging Results for the SABR Model | |
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Hedging Results for the LMM-SABR Model | |
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Conclusions | |
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Hedging the Correlation Structure. | |
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The Intuition Behind the Problem | |
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Hedging the Forward-Rate Block | |
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Hedging the Volatility-Rate Block | |
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Hedging the Forward-Rate/Volatility Block | |
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Final Considerations | |
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Hedging in Conditions of Market Stress. | |
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Statement of the Problem | |
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The Volatility Function | |
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The Case Study | |
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Hedging | |
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Results | |
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Are We Getting Something for Nothing? | |