Solving barrier options under stochastic volatility using deep learning

Abstract: We develop an unsupervised deep learning method to solve the barrier options under the Bergomi model. The neural networks serve as the approximate option surfaces and are trained to satisfy the PDE as well as the boundary conditions. Two singular terms are added to the neural networks to deal with the non-smooth and discontinuous payoff at the strike and barrier levels so that the neural networks can replicate the asymptotic behaviors of barrier options at short maturities. After that, vanilla options and barrier options are priced in a single framework. Also, neural networks are employed to deal with the high dimensionality of the function input in the Bergomi model. Once trained, the neural network solution yields fast and accurate option values.