The Mixing Approach to Stochastic Volatility

This article introduces mixing theorems, which offer both a theoretical and computational approach to certain advanced option models. Before explaining them, we first review a little background about option pricing theory. The Black-Scholes-Merton family of models is a wellknown and sensible starting framework for understanding option prices. The framework relies on the assumption that the underlying stock price (or security price) follows a process known as geometric Brownian motion (GBM). This model has some very strong points in its favor: (i) it’s consistent with stocks as limited liability securities (and so the prices never fall below zero), (ii) it has uncorrelated returns, which are a compelling consequence of highly efficient markets with strong statistical support over many time scales, and (iii) it’s very tractable computationally.