Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree

Ionuţ Florescu, Frederi G. Viens
2008 Applied Mathematical Finance  
In this article we treat the problem of option pricing when the volatility component of the underlying asset price is stochastic. The basic model we consider is commonly known as the Stochastic Volatility model: where Y t is an exogenous mean-reverting-type process. We seek a discrete approximation of this model, one that will converge in distribution to the above continuous model, and that will allow us to calculate the option price numerically. First, we show how to estimate the distribution
more » ... f the volatility component, using an interacting particle filtering algorithm due to Del Moral, Jacod and Protter (Del Moral et al., 2001) . Then we use this distribution to construct two different models which converge to the solution of the original model. The first model is our main contribution to the research in Mathematical Finance, a recombining tree for the stock process, featuring four successors at every branch. The second model uses an Euler method to generate future stock prices and calculate option price. We are in the incomplete market situation, and in order to price options on the stock, we use classical arbitrage-free valuation, combined with resampling and Monte-Carlo methods to generate versions of the stock price and compute expectations. Finally, we compare our methods with the classical Black-Scholes prices, using daily
doi:10.1080/13504860701596745 fatcat:2a3sxhfqkjfbfctw4nxobxj7ia