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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? | |