Stochastic Volatility: Option Pricing Using a Multinomial Recombining Tree
38 Pages Posted: 26 May 2012 Last revised: 23 Mar 2018
Date Written: May 25, 2005
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.
Keywords: incomplete markets, Monte-Carlo method, options market, option pricing, particle method, random tree, stochastic ﬁltering, stochastic volatility
JEL Classification: C2
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