Stochastic Volatility and Epsilon-Martingale Decomposition

We address the problems of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. We show that when volatility is stochastic but fast mean reverting Black-Scholes pricing theory can be corrected. The correction accounts for the effect of stochastic volatility and the associated market price of risk. For European derivatives it is given by explicit formulas which involve parsimonous parameters directly calibrated from the implied volatility surface. The method presented here is based on a martingale decomposition result which enables us to treat nonMarkovian models as well. 1. Stochastic Volatility Models We consider stochastic volatility models where the asset price (Xt)t≥0 satisfies the stochastic differential equation dXt = μXtdt + σtXtdWt, and (σt)t≥0 is called the volatility process. It must satisfy some regularity conditions for the model to be well-defined, but it does not have to be an Itô process: it can be a jump process, a Markov chain, etc. In order for it to be a volatility, it should be positive. Unlike the implied deterministic volatility models for which the volatility is a deterministic function σ(t, Xt) of time and price, the volatility process is not perfectly correlated with the Brownian motion (Wt). Therefore, volatility is modeled to have an independent random component and since σt is not the price of a traded asset, the market is incomplete and there is no longer a unique equivalent martingale measure. We refer to [6], [5] or [3](Ch.2) for reviews of stochastic volatility models. 1.1. Mean-Reverting Stochastic Volatility Models We consider first volatility processes which are Itô processes satisfying stochastic differential equations driven by a second Brownian motion. This is a convenient way to incorporate correlation with stock price changes. One feature that most volatility models seem to like is mean-reversion. The term mean-reverting refers to the characteristic (typical) time it takes for a process 2 J.-P. Fouque, G. Papanicolaou and R. Sircar to get back to the mean-level of its invariant distribution (the long-run distribution of the process). In other words we assume that σt is ergodic with additional mixing properties. From a financial modeling perspective, mean-reverting refers to a linear pull-back term in the drift of the volatility process itself, or in the drift of some (underlying) process of which volatility is a function. Let us denote σt = f(Yt) where f is some positive function. Then mean-reverting stochastic volatility means that the stochastic differential equation for (Yt) looks like dYt = α(m − Yt)dt + βdẐt, where (Ẑt)t≥0 is a Brownian motion correlated with (Wt). Here α is called the rate of mean-reversion and m is the long-run mean-level of Y . The drift term pulls Y towards m and consequently we would expect that σt is pulled towards the mean value of f(Y ), with respect to the long-run distribution of Y . Choosing β > 0 constant corresponds to the Ornstein-Uhlenbeck process which is a Gaussian process with the normal invariant distribution N (m, β/2α). This choice, though not necessary, is particularly convenient to explain the concept of fast mean-reversion and to show through relatively explicit computations how to exploit this property in pricing and hedging problems. It is still very flexible since the function f is unspecified. The second Brownian motion (Ẑt) is correlated with the Brownian motion (Wt) driving the asset price equation. We denote by ρ ∈ [−1, 1] the instantaneous correlation coefficient defined by