Journal of Computational Finance

Risk.net

Pricing barrier options with deep backward stochastic differential equation methods

Narayan Ganesan, Yajie Yu and Bernhard Hientzsch

  • The method uses deep learning and pathwise forward–backward stochastic differential equations to solve boundary value problems (eg, for barrier options) by adding nodes to the computational graph to monitor for barrier breaches and boundary conditions.
  • This method, based on deep backward stochastic differential equations, can price instruments in higher dimensions with several underliers where traditional methods based on partial differential equations are limited in applicability.
  • The approach presented is capable of handling boundary value problems with nonuniform, time-varying and multiple boundaries.
  • The hedging profit and loss obtained by this method is naturally robust due to the definition and optimization of the problem compared with traditional discrete-time hedging approaches.

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.

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