Deep Signature Algorithm for Multidimensional Path-Dependent Options

SIAM Journal on Financial Mathematics, Volume 15, Issue 1, Page 194-214, March 2024.
Abstract. In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [C. Huré, H. Pham, and X. Warin, Math. Comp., 89 (2020), pp. 1547–1579] 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 Shiryaev’s optimal stopping problem.