Explicit Heston Solutions and Stochastic Approximation for Path-dependent Option Pricing

TL;DR

This paper introduces new simulation methods for path-dependent options that improve speed and accuracy by using explicit solutions, stochastic approximation, and importance sampling, addressing limitations of traditional approaches.

Contribution

It develops explicit weak solutions for the Heston model, replaces regression with stochastic approximation in the LSM algorithm, and applies importance sampling to enhance simulation efficiency.

Findings

Up to 100x speed improvements over standard Monte Carlo methods

Effective handling of path-dependence in option execution strategies

Potential extension of methods beyond the Heston model

Abstract

New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: Stochastic approximation replaces regression in the LSM algorithm; Explicit weak solutions to stochastic differential equations are developed and applied to Heston model simulation; and Importance sampling expands these explicit solutions. The approach complements Heston (1993) and Broadie and Kaya (2006) by handling the case of path-dependence in the option's execution strategy. Numeric comparison against standard Monte Carlo methods demonstrate up to two orders of magnitude speed improvement. The general ideas will extend beyond the important Heston setting.