Pricing barrier options with deep backward stochastic differential equation methods

This paper presents a novel and direct approach to solving boundary- and final-value problems, corresponding to barrier options, using forward pathwise deep learning and forward–backward stochastic differential equations (FBSDEs). Barrier instruments are instruments that expire or transform into another instrument if a barrier condition is satisfied before maturity; otherwise they perform like the instrument without the barrier condition. In a partial differential equation, this corresponds to adding boundary conditions to the final-value problem. The deep backward stochastic differential equation (deep BSDE) methods developed so far have not addressed barrier/boundary conditions directly. We extend the pathwise deep BSDE methods to the barrier condition case by adding nodes to the computational graph, in order to explicitly monitor the barrier conditions for each realization of the dynamics, as well as adding nodes that preserve the time, state variables and trading strategy value at the barrier breach, or at maturity otherwise. Given these additional nodes in the computational graph, the forward loss function quantifies the replication of the barrier or final payoff according to a chosen risk measure such as the squared sum of differences. The proposed method can handle any barrier condition in the FBSDE setup and any Dirichlet boundary conditions in the partial differential equation setup, in both low and high dimensions.