A binomial tree approach to stochastic volatility driven model of the stock price

In this article we attempt to deal with the problem of finding option prices when the volatility component of the price is stochastic. The model we use is: dS(t) = mu S(t)dt + sigma(Y-t)S(t)dW(t), where Y-t is a mean-reverting type process. First, we show how to estimate the distribution of the volatility component, using an algorithm due to Del Moral, Jacod and Protter [6]. Second, using this distributon we are able to construct a binomial tree model which converges to the solution of the given equation. In order to price options on the stock, we use the Monte Carlo method to sample from this tree, and obtain a smaller, recombing tree easier to work with. Finally, we use this method to compute the price of European Call Options on the SP500 index price in April. We use daily data and our method gives good results that are proximate to the reported bid-ask spread.