A differential neural network learns stochastic differential equations and the Black-Scholes equation for pricing multi-asset options

Abstract

Neural networks with sufficiently smooth activation functions can approximate values and derivatives of any smooth function, and they are differentiable themselves. We improve the approximation capability of neural networks by utilizing the differentiability of neural networks; the gradient and Hessian of neural networks are used to train the neural networks to satisfy the differential equations of the problems of interest. Several activation functions are also compared in term of effective differentiation of neural networks. We apply the differential neural networks to the pricing of financial options, where stochastic differential equations and the Black-Scholes partial differential equation represent the relation of price of option and underlying assets, and the first and second derivatives, Greeks, of option play important roles in financial engineering. The proposed neural network learns -- (a) the sample paths of option prices generated by stochastic differential equations and (b) the Black-Scholes equation at each time and asset price. Option pricing experiments were performed on multi-asset options such as exchange and basket options. Experimental results show that the proposed method gives accurate option values and Greeks; sufficiently smooth activation functions and the constraint of Black-Scholes equation contribute significantly for accurate option pricing.