  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.
