Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree

Abstract

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.

Estimation and pricing under long-memory stochastic volatility

“…In preparation for the task of option pricing, we adjust a genetic-type particle filtering algorithm by Del Moral et al (2001, [10]) adapted by Florescu and Viens (2008, [13]) in a stochastic volatility setting with no memory, in order to estimate the empirical distribution of the volatility. In the sequel, following the approach in [13], we use a multinomial recombining tree algorithm for option pricing. Using simulated data we show that this algorithm performs well and captures the underlying memory of the system.…”

Construction of Volatility Indices Using a Multinomial Tree Approximation Method

“…It is observed that when p is close to 1 6 the option values obtained are stable even with few replications (Figure 4 in Florescu and Viens (2008)). In this paper we set p = 0.135 throughout the algorithm.…”

“…ϕ t models the stochastic volatility process. It has been proved that for any proxy of the current stochastic volatility distribution at t the option prices calculated at time t converge to the true option prices (Theorem 4.6 in Florescu and Viens (2008)). When using Stochastic volatility models we are faced with a real problem when trying to come up with ONE number describing the volatility.…”

“…The quadrinomial tree approximation developed by Florescu and Viens (2008) has demonstrated that it can be used to estimate option values and that it produces option chain values which are within bid ask spread. The quadrinomial tree method outperforms other approximating techniques (with the exception of analytical solutions) in terms of both error and time.…”

“…This is used in conjunction with a highly recombining quadrinomial tree method to compute the price of options. The quadrinomial tree method is described in details in Florescu and Viens (2008). In the Appendix B we describe a one-step quadrinomial tree construction.…”