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) 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.

Keywords:

• Incomplete markets
• Monte Carlo method
• options market
• option pricing
• particle method
• random tree
• stochastic filtering
• stochastic volatility

Notes

1. We mention that for our later construction of the tree the particle filtering step is not necessary. Any good approximation of the distribution of the hidden process Y would serve for the construction of the tree. The reason we choose the specific method described in Section 3 is that it is the best method currently available. In the future if simpler or better algorithms should be developed the adaptation to these new methods is straightforward. We thank one of our anonymous referees for this observation.