The Compounded BSDE method: A fully-forward method for option pricing and optimal stopping problems in finance
Zhipeng Huang, Cornelis W. Oosterlee
TL;DR
The paper introduces the Compound BSDE method, a fully forward deep-learning framework for solving multi-stage option pricing and optimal stopping problems. By coupling a forward SDE with $M$ interlinked backward equations through compounding times $T_j$ and payoff mappings $g_j$, it unifies pricing of compound options and Bermudan-style stopping within a single learning objective. The authors establish well-posedness, $L^2$-regularity, and an a posteriori error bound that relates discretization and training loss to numerical accuracy. Numerical experiments demonstrate accurate pricing and hedging in high dimensions for plain and $M$-fold compound options, as well as Bermudan-type problems, validating the method’s convergence properties and scalability. The framework offers a flexible, forward-driven alternative to backward schemes, with broad applicability to complex payoff structures and stopping problems in finance.
Abstract
We propose the Compound BSDE method, a fully forward, deep-learning-based approach for solving a broad class of problems in financial mathematics, including optimal stopping. The method is based on a reformulation of option pricing problems in terms of a system of backward stochastic differential equations (BSDEs), which offers a new perspective on the numerical treatment of compound options and optimal stopping problems such as Bermudan option pricing. Building on the classical deep BSDE method for a single BSDE, we develop an algorithm for compound BSDEs and establish its convergence properties. In particular, we derive an \emph{a posteriori} error estimate for the proposed method. Numerical experiments demonstrate the accuracy and computational efficiency of the approach, and illustrate its effectiveness for high-dimensional option pricing and optimal stopping problems.