The paper introduces the Compound BSDE method, a deep-learning-based fully forward numerical approach for solving a broad class of option pricing and optimal stopping problems in finance. The method is built upon a novel compound BSDE formulation, which consists of a forward SDE coupled with multiple backward equations defined on successive time intervals. These backward equations interact only at prescribed compounding times through compounding conditions, making the framework particularly suitable for pricing Bermudan options and other early-exercise derivatives. The authors establish well-posedness of the compound BSDE under standard Lipschitz assumptions and derive an a posteriori error estimate that guarantees convergence of the numerical scheme. The method parameterizes the initial values and Z processes of each BSDE using neural networks and minimizes a joint loss function that enforces all compounding and terminal conditions simultaneously. Numerical experiments demonstrate that the approach achieves accurate prices and hedges for high-dimensional Bermudan options, confirming its effectiveness and computational efficiency. The fully forward nature of the algorithm distinguishes it from traditional backward methods, offering potential advantages in parallelization and handling of path-dependent features.