Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in L\'evy Models

TL;DR

This paper introduces a novel numerical method combining optimal damping and hierarchical adaptive quadrature to efficiently price multi-asset options in Lévy models, outperforming traditional methods in speed and complexity.

Contribution

It proposes a new approach that optimizes damping parameters and employs adaptivity techniques to improve Fourier-based option pricing in high dimensions.

Findings

Outperforms COS method in computational time for two-asset options.

Achieves significant speed-up over Monte Carlo methods up to six dimensions.

Demonstrates effectiveness of adaptivity and damping in reducing numerical complexity.

Abstract

Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the frequency space often has a higher regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of high dimensionality. We propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas to address these challenges. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed optimization rule. Second, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some L\'evy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the COS method in terms of computational time. Finally, we show significant speed-up compared to the Monte Carlo method for up to six dimensions.