Stochastic Volatility and Correction to the Heat Equation

Abstract

From a probabilist’s point of view the Twentieth Century has been a century of Brownian motion with major inputs from — and applications to — Economics and Physics starting with Bachelier’s thesis in 1900 and Einstein’s work in 1905. There has been a tremendous and rapidly increasing number of mathematical works on the Brownian motion and its many applications during the century, starting with Wiener’s construction in 1923 and followed by the development of the stochastic calculus. It is only during the Sixties and early Seventies that it became an essential modelling tool in Economics with the works of Samuelson, Merton and the famous Black-Scholes formula for pricing options. The need for more complicated nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Nineties. In particular a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additional Brownian motion. The authors of this paper have shown that in presence of separation of time scales, between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple universal corrections to constant volatility models. From the point of view of partial differential equations this method corresponds to a singular perturbation analysis. The aim of this paper is to recast this approach in the context of the heat equation and to propose a universal correction to its solution. This is an attempt to show that financial mathematics may also contribute to new ideas in Physics.