Sequential Monte Carlo pricing of American-style options under stochastic volatility models

TL;DR

This paper develops a sequential Monte Carlo method for pricing American options under stochastic volatility models without observing volatility, using conditional distributions and approximations to solve the dynamic programming problem.

Contribution

It introduces a novel Monte Carlo-based approach that approximates optimal decision functions via conditional distributions, extending existing methods for stochastic volatility models.

Findings

Effective in pricing American options under SV models

Can estimate posterior distributions of volatility risk

Demonstrated on real equity data

Abstract

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.