Deep Signature Algorithm for Multi-dimensional Path-Dependent Options
This work addresses the challenge of pricing complex financial derivatives for quantitative finance, offering a method that handles path-dependence and reflections, though it is incremental as it builds on existing schemes.
The authors tackled the problem of pricing multi-dimensional path-dependent options by extending a backward scheme to handle path-dependent forward-backward stochastic differential equations with reflections, incorporating a signature layer, and proved convergence with explicit error bounds. They demonstrated the algorithm's effectiveness on examples like Amerasian options and Shiryaev's optimal stopping problem, achieving competitive numerical results.
In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Huré-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.