Function-Approximation-Based Importance Sampling for Pricing American Options

Abstract

Monte Carlo simulation techniques that use function approximations have been successfully applied to approximately price multi-dimensional American options. However, for many pricing problems the time required to get accurate estimates can still be prohibitive, and this motivates the development of variance reduction techniques. In this paper, we describe a zero-variance importance sampling measure for American options. We then discuss how function approximation may be used to approximately learn this measure; we test this idea in simple examples. We also note that the zero-variance measure is fundamentally connected to a duality result for American options. While our methodology is geared towards developing an estimate of an accurate lower bound for the option price, we observe that importance sampling also reduces variance in estimating the upper bound that follows from the duality.

Approximate zero-variance simulation

“…Example 5 Bolia, Juneja, and Glasserman (2004) use least-squares regression with a given set of basis functions to approximate the value function in the context of pricing an American option, as in the least-square Monte Carlo method (Glasserman 2004), and also approximate the zero-variance IS by plugging in (9) a least-squares approximation of the continuation value function (the value of the option conditional on not exercising it at the current step). In a first-stage, they learn good vectors of weights θ for each of these two value functions, and use them to approximate both the zero-variance IS and the optimal stopping rule.…”

On improving the least squares Monte Carlo option valuation method

“…Please refer toMoreni (2004),Bolia et al (2004), andLemieux and La (2005) for an analysis of this technique in the context of LSMC.…”

Variance reduction techniques for pricing American options using function approximations

“…The additive duality in American options is well known (see Rogers 2002, Haugh and Kogan 2004, Andersen and Broadie 2004. Jamshidian (2003) and Bolia, Glasserman and Juneja (2004) propose multiplicative duality for American options (also see Chen and Glasserman 2007). In this paper, we make an interesting observation that the perfect control variate solves the additive duality problem and the perfect importance sampling estimator solves the multiplicative duality problem.…”

“…Finally we conclude in Section 7 where we also discuss how our methodology specializes to some popular models. Significant portions of the analysis in this paper appeared in Bolia, Glasserman and Juneja (2004) and Bolia and Juneja (2005).…”