On Sparse Grid Interpolation for American Option Pricing with Multiple Underlying Assets

TL;DR

This paper introduces a new sparse grid polynomial interpolation method for efficiently pricing American options with multiple assets, leveraging domain transformations and boundary singularity removal to improve accuracy and computational performance.

Contribution

The paper presents a novel sparse grid interpolation technique with domain transformation and boundary singularity handling for high-dimensional American option pricing.

Findings

Effective for up to 16 assets

Reduces computational complexity

Accurate pricing demonstrated in numerical experiments

Abstract

In this work, we develop a novel efficient quadrature and sparse grid based polynomial interpolation method to price American options with multiple underlying assets. The approach is based on first formulating the pricing of American options using dynamic programming, and then employing static sparse grids to interpolate the continuation value function at each time step. To achieve high efficiency, we first transform the domain from $\mathbb{R}^d$ to $(-1,1)^d$ via a scaled tanh map, and then remove the boundary singularity of the resulting multivariate function over $(-1,1)^d$ by a bubble function and simultaneously, to significantly reduce the number of interpolation points. We rigorously establish that with a proper choice of the bubble function, the resulting function has bounded mixed derivatives up to a certain order, which provides theoretical underpinnings for the use of sparse grids. Numerical experiments for American arithmetic and geometric basket put options with the number of underlying assets up to 16 are presented to validate the effectiveness of the approach.