Malliavin calculus for signatures with applications to finance

TL;DR

This paper develops explicit algebraic formulas for Malliavin calculus operators applied to signature-based random variables, enabling practical computation of Greeks in path-dependent options.

Contribution

It introduces explicit formulas for Malliavin derivatives and related operators for signature-based variables, enhancing tractability and applications in finance.

Findings

Derived closed-form expressions for Malliavin derivatives of signatures.

Provided algebraic formulations for Clark--Ocone and Ornstein--Uhlenbeck operators.

Numerically compared Malliavin weights for computing Greeks in signature models.

Abstract

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous It\^o processes. As a consequence, we obtain closed-form expressions for the Clark--Ocone representation, the Ornstein--Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.