Stochastic Volatility Surface Estimation

We propose a method for calibrating a volatility surface that matches option prices using an entropy-inspired framework. Starting with a stochastic volatility model for asset prices, we cast the estimation problem as a variational one and we derive a Hamilton-Jacobi-Bellman (HJB) equation for the volatility surface. We study the asymptotics of the HJB equation assuming that the stochastic volatility model exhibits fast mean-reversion. From the asymptotic solution of the HJB equation we get an estimate of the stochastic volatility surface. We also incorporate uncertainty in quoted derivative prices through a penalty term, i.e. by softening the constraints in the HJB equation. We present numerical solutions of our estimation scheme. We find that, depending on the softness of the constraints, certain parameters of the volatility surface related to the implied volatility smile can be calibrated so that they are stable over time. These parameters are essentially the ones found in previous fast mean-reversion asymptotics papers by Fouque, Papanicolaou and Sircar. We find that our procedure provides a natural way of interpolating between the prior parameters and the parameters of Fouque, Papanicolaou and Sircar.