The SABR/LIBOR Market Model

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 model,this 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. 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