This book presents a major innovation in the interest rate space. It explains a financially motivated extension of the LIBOR Market model which accurately reproduces the prices for plain vanilla hedging instruments (swaptions and caplets) of all strikes and maturities produced by the SABR model. The authors show how to accurately recover the whole of the SABR smile surface using their extension of the LIBOR market model. This is not just a new model, this is a new way of option pricing that takes into account the need to calibrate as accurately as possible to the plain vanilla reference hedging instruments and the need to obtain prices and hedges in reasonable time whilst reproducing a realistic future evolution of the smile surface. It removes the hard choice between accuracy and time because the framework that the authors provide reproduces today's market prices of plain vanilla options almost exactly and simultaneously gives a reasonable future evolution for the smile surface. The authors take the SABR model as the starting point for their extension of the LMM because it is a good model for European options. The problem, however with SABR is that it treats each European option in isolation and the processes for the various underlyings (forward and swap rates) do not talk to each other so it isn't obvious how to relate these processes into the dynamics of the whole yield curve. With this new model, the authors bring the dynamics of the various forward rates and stochastic volatilities under a single umbrella. To ensure the absence of arbitrage they derive drift adjustments to be applied to both the forward rates and their volatilities. When this is completed, complex derivatives that depend on the joint realisation of all relevant forward rates can now be priced. Contents THE THEORETICAL SET-UP The Libor Market model The SABR Model The LMM-SABR Model IMPLEMENTATION AND CALIBRATION Calibrating the LMM-SABR model to Market Caplet prices Calibrating the LMM/SABR model to Market Swaption Prices Calibrating the Correlation Structure EMPIRICAL EVIDENCE The Empirical problem Estimating the volatility of the forward rates Estimating the correlation structure Estimating the volatility of the volatility HEDGING Hedging the Volatility Structure Hedging the Correlation Structure Hedging in conditions of market stress

This book presents a major innovation in the interest rate space.It explains a financially motivated extension of the LIBOR Marketmodel which accurately reproduces the prices for plain vanillahedging instruments (swaptions and caplets) of all strikes andmaturities produced by the SABR model. The authors show how toaccurately recover the whole of the SABR smile surface using theirextension of the LIBOR market model. This is not just a new modelthis is a new way of option pricing that takes into account theneed to calibrate as accurately as possible to the plain vanillareference hedging instruments and the need to obtain prices andhedges in reasonable time whilst reproducing a realistic futureevolution of the smile surface. It removes the hard choice betweenaccuracy and time because the framework that the authors providereproduces today's market prices of plain vanilla options almostexactly and simultaneously gives a reasonable future evolution forthe smile surface.The authors take the SABR model as the starting point for theirextension of the LMM because it is a good model for Europeanoptions. The problem, however with SABR is that it treats eachEuropean option in isolation and the processes for the variousunderlyings (forward and swap rates) do not talk to each other soit isn't obvious how to relate these processes into the dynamics ofthe whole yield curve. With this new model, the authors bring thedynamics of the various forward rates and stochastic volatilitiesunder a single umbrella. To ensure the absence of arbitrage theyderive drift adjustments to be applied to both the forward ratesand their volatilities. When this is completed, complex derivativesthat depend on the joint realisation of all relevant forward ratescan now be priced.ContentsTHE THEORETICAL SET-UPThe Libor Market modelThe SABR ModelThe LMM-SABR ModelIMPLEMENTATION AND CALIBRATIONCalibrating the LMM-SABR model to Market Caplet pricesCalibrating the LMM/SABR model to Market Swaption PricesCalibrating the Correlation StructureEMPIRICAL EVIDENCEThe Empirical problemEstimating the volatility of the forward ratesEstimating the correlation structureEstimating the volatility of the volatilityHEDGINGHedging the Volatility StructureHedging the Correlation StructureHedging in conditions of market stress

Acknowledgements xi1 Introduction 1I The Theoretical Set-Up 72 The LIBOR Market Model 93 The SABR Model 254 The LMM-SABR Model 51II Implementation and Calibration 795 Calibrating the LMM-SABR Model to Market Caplet Prices 816 Calibrating the LMM-SABR Model to Market Swaption Prices 1017 Calibrating the Correlation Structure 125III Empirical Evidence 1418 The Empirical Problem 1439 Estimating the Volatility of the Forward Rates 15910 Estimating the Correlation Structure 181IV Hedging 20311 Various Types of Hedging 20512 Hedging against Moves in the Forward Rate and in the Volatility 22113 (LMM)-SABR Hedging in Practice: Evidence from Market Data 23114 Hedging the Correlation Structure 24715 Hedging in Conditions of Market Stress 257References 271Index 275