A unified model of SABR and mean-reverting stochastic volatility for derivative pricing

Abstract

The SABR model is popularly used by practitioners in the financial industry due to a fairly simple implied volatility formula but it wouldn't capture the mean reverting nature of the volatility as a drawback. This paper proposes a stochastic-local volatility model that unifies SABR volatility and mean reverting stochastic volatility for pricing derivatives. We obtain an explicit pricing formula in convolution form through the combination of asymptotics and the Mellin transform method. The formula allows us to compute the derivative price in terms of a single integral calculation (Mellin convolution) instead of a double integral. Further, we obtain a closed-form pricing formula that can be calculated by using the three Greeks (Delta, Gamma, and Speed) of the Black-Scholes derivative price in a reasonably practical situation. The accuracy of the derived formula is tested through Monte Carlo simulation. The validity of the formula is demonstrated through an empirical analysis of a foreign exchange option, as incorporating a mean-reverting volatility feature into the SABR model aids in calibrating the model to real market instruments by reproducing the U-shaped structure of the implied volatility.