A Stochastic Volatility Forward Libor Model with a Term Structure of Volatility Smiles

Abstract:     
Volatility smiles of European swaptions of various expiries and maturities typically have different slopes. This important feature of interest rate markets has not been incorporated in any of the practical interest rate models available to date. In this paper, we build a model that treats the swaption skew matrix as a market input and is calibrated to it. The model is constructed as an extension of a Stochastic Volatility Forward Libor model, with local volatility functions imposed upon forward Libor rates being time-dependent and Libor-rate specific. The focus of the paper is on deriving efficient European swaption approximation formulas that allow calibration of the model to all European swaptions across all expiries, maturities and strikes. The main conceptual contribution of the paper is its focus on recovering all available market volatility skew information across a full swaption grid within a consistent model. The model we develop has a potential to change the way skew calibration is approached, in the same way the introduction of the log-normal forward Libor model had changed the way volatility calibration is approached. The main technical contribution of the paper is a formula for the "effective" skew in a stochastic volatility model, a formula that relates a total amount of skew generated by the model over a given time period to the time-dependent slope of the instantaneous local volatility function. A new "effective" volatility approximation for stochastic volatility models with time-dependent volatility functions is also derived. The formulas we obtain are simple and intuitive; their applicability goes beyond interest rate modeling.

Keywords: Market model, forward Libor model, Libor market model, LMM, BGM, stochastic volatility, volatility smile, volatility calibration, skew calibration, interest rate models, time-dependent local volatility, effective volatility, effective skew, average skew, homogenization, averaging

Working Paper Series