Tensor train representations of Greeks for Fourier-based pricing of multi-asset options

TL;DR

This paper introduces a tensor train-based framework for efficiently computing Greeks of multi-asset options, significantly speeding up calculations compared to Monte Carlo methods while maintaining accuracy.

Contribution

It presents a novel tensor train approach to compute Greeks from Fourier-based pricing functions with a single evaluation, reducing computational complexity.

Findings

Speed-ups of up to 10^5 times over Monte Carlo methods.

The numerical differentiation approach is more efficient and as accurate as the analytical approach.

The method is demonstrated on a five-asset min-call option in the Black-Scholes model.

Abstract

Efficient computation of Greeks for multi-asset options remains a key challenge in quantitative finance. While Monte Carlo (MC) simulation is widely used, it suffers from the large sample complexity for high accuracy. We propose a framework to compute Greeks in a single evaluation of a tensor train (TT), which is obtained by compressing the Fourier transform (FT)-based pricing function via TT learning using tensor cross interpolation. Based on this TT representation, we introduce two approaches to compute Greeks: a numerical differentiation (ND) approach that applies a numerical differential operator to one tensor core and an analytical (AN) approach that constructs the TT of closed-form differentiation expressions of FT-based pricing. Numerical experiments on a five-asset min-call option in the Black-Sholes model show significant speed-ups of up to about $10^{5} \times$ over MC while maintaining comparable accuracy. The ND approach matches or exceeds the accuracy of the AN approach and requires lower computational complexity for constructing the TT representation, making it the preferred choice.