Pricing Options Under Rough Volatility with Backward SPDEs

TL;DR

This paper develops a framework using backward SPDEs to price options under rough volatility models, addressing non-Markovian challenges and proposing deep learning methods for numerical solutions.

Contribution

It introduces a novel approach to option pricing with rough volatility via backward SPDEs and establishes existence and uniqueness results for these equations.

Findings

Successfully models European and American options under rough Bergomi type volatility.

Demonstrates the effectiveness of deep learning methods for solving non-Markovian BSPDEs.

Provides numerical examples validating the proposed approach.

Abstract

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solution is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep leaning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options.