Stochastic Volatility for Interest Rate Derivatives

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introduce appropriate adjustments.The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different $\beta$s) and Heston stochastic volatility models informs about what different specifications of the driving SDEs has to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model ($\beta=0$) satisfactorily passes all tests and seems to provide a good match to the market. In contrast we find the Heston model does not.The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two factors that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified and used to construct a simple terminal Markov-functional type model under a single measure. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model