Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree

Abstract

The problem of option pricing is treated using the Stochastic Volatility (SV) model: the volatility of the underlying asset is a function of an exogenous stochastic process, typically assumed to be mean‐reverting. Assuming that only discrete past stock information is available, an interacting particle stochastic filtering algorithm due to Del Moral et al. (Del Moral et al., 2001 Del Moral, P., Jacod, J. and Protter, P. 2001. The Monte‐Carlo method for filtering with discrete time observations.. Probability Theory and Related Fields, 120: 346–368. [Crossref], [Web of Science ®] , [Google Scholar]) is adapted to estimate the SV, and a quadrinomial tree is constructed 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. Results obtained here are compared with those from non‐random volatility models, and from models which continue to estimate volatility after time 0. It is shown precisely how to calibrate the incomplete market, choosing a specific martingale measure, by using a benchmark option.

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