Smoothing the payoff for efficient computation of Basket option prices

TL;DR

This paper introduces a smoothing technique for basket option pricing in high-dimensional models, enabling efficient numerical integration by transforming non-smooth payoffs into smooth functions using conditional expectation, significantly improving computational speed.

Contribution

It proposes a novel payoff smoothing method via conditional expectation to facilitate high-order numerical integration for basket options in complex models.

Findings

High-order methods outperform Monte Carlo in speed by orders of magnitude.

The approach is effective in dimensions up to 35.

Numerical results demonstrate significant efficiency gains.

Abstract

We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 35.