In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Hur\'e-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.

翻译：本文研究用于路径依赖期权的深度签名算法。我们将[Huré-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)]中针对带反射的状态依赖正向随机微分方程的后向方案，通过添加签名层扩展至带反射的路径依赖正向随机微分方程。该算法适用于收益函数依赖于标的股票价格过程完整路径的欧式和美式期权定价问题。我们证明了数值算法的收敛性分析，并显式刻画了签名截断阶数与神经网络逼近误差的依赖关系。数值算例包括：Black-Scholes模型下的亚式美式期权、具有路径依赖几何平均收益函数的美式期权，以及Shiryaev最优停时问题。