Option pricing under non-Markovian stochastic volatility models: A deep signature approach

TL;DR

This paper introduces a deep signature method to efficiently price options under non-Markovian stochastic volatility models by transforming complex path-dependent problems into manageable stochastic differential equations.

Contribution

It presents a novel deep signature approach that reformulates non-Markovian models, enabling standard analytical techniques and providing convergence guarantees.

Findings

Effective numerical performance on both Markovian and non-Markovian models

Theoretical proof of convergence for the proposed algorithm

Framework offers computational efficiency and accuracy

Abstract

This paper studies the pricing problem in which the underlying asset follows a non-Markovian stochastic volatility model. Classical partial differential equation methods face significant challenges in this context, as the option prices depend not only on the current state, but also on the entire historical path of the process. To overcome these difficulties, we reformulate the asset dynamics as a rough stochastic differential equation and then represent the rough paths via linear or non-linear combinations of time-extended Brownian motion signatures. This representation transforms a rough stochastic differential equation to a classical stochastic differential equation, allowing the application of standard analytical tools. We propose a deep signature approach for both linear and nonlinear representations and rigorously prove the convergence of the algorithm. Numerical examples demonstrate the effectiveness of our approach for both Markovian and non-Markovian volatility models, offering a theoretically grounded and computationally efficient framework for option pricing.