Deep Signature Algorithm for Multi-dimensional Path-Dependent Options

TL;DR

This paper introduces a deep signature algorithm for pricing multi-dimensional path-dependent options, extending previous methods to handle more complex path-dependent features with proven convergence and demonstrated effectiveness through numerical examples.

Contribution

The work extends the backward scheme for FBSDEs to include path-dependent options using signature layers, providing a novel approach for complex option pricing.

Findings

Algorithm successfully prices various path-dependent options.

Convergence of the algorithm is proven with explicit error bounds.

Numerical experiments validate the method's accuracy and applicability.

Abstract

In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Hur\'e-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided including: Amerasian option under the Black-Scholes model, American option with a path-dependent geometric mean payoff function, and the Shiryaev's optimal stopping problem.