Series expansions and direct inversion for the Heston model

TL;DR

This paper introduces a novel series expansion and direct inversion algorithms for efficient, accurate sampling of the Heston stochastic volatility model's integrated variance, improving simulation methods in financial modeling.

Contribution

The paper develops a new series expansion for the integrated variance in the Heston model and proposes direct inversion algorithms that are efficient and applicable under any market conditions.

Findings

Series expansion has exponentially decaying truncation errors.

Sampling scheme is efficient and nearly exact.

Algorithms are applicable regardless of market conditions.

Abstract

Efficient sampling for the conditional time integrated variance process in the Heston stochastic volatility model is key to the simulation of the stock price based on its exact distribution. We construct a new series expansion for this integral in terms of double infinite weighted sums of particular independent random variables through a change of measure and the decomposition of squared Bessel bridges. When approximated by series truncations, this representation has exponentially decaying truncation errors. We propose feasible strategies to largely reduce the implementation of the new series to simulations of simple random variables that are independent of any model parameters. We further develop direct inversion algorithms to generate samples for such random variables based on Chebyshev polynomial approximations for their inverse distribution functions. These approximations can be used under any market conditions. Thus, we establish a strong, efficient and almost exact sampling scheme for the Heston model.