Stochastic Volatility : Option Pricing Using a Multinomial Recombining Tree

We treat the problem of option pricing under the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be meanreverting. Assuming that only discrete past stock information is available, we adapt an interacting particle stochastic filtering algorithm due to Del Moral, Jacod and Protter (Del Moral et al., 2001) to estimate the SV, and construct a quadrinomial tree which samples volatilities from the SV filter's empirical measure approximation at time 0. Proofs of convergence of the tree to continuous-time SV models are provided. Classical arbitrage-free option pricing is performed on the tree, and provides answers that are close to market prices of options on the SP500 or on blue-chip stocks. We compare our results to non-random volatility models, and to models which continue to estimate volatility after time 0. We show precisely how to calibrate our incomplete market, choosing a specific martingale measure, by using a benchmark option