Numerical solving for generalized Black-Scholes-Merton model with neural finite element method.

In this paper, we propose the neural finite element method (NFEM) with the two-part structure to obtain a highly accurate result for generalized Black-Scholes-Merton equation arising in the financial market. The two-layer rectified linear unit in the first part can be regarded as the adaptive subdivision of the domain into several elements, and the nonlinear activation function in the second is similar to the numerical solution of each element. The shallow and deep neural networks are connected to fit high-frequency and low-frequency components, and the deep residual module is proposed to improve the approximate ability of the NFEM. We apply the NFEM to solve three generalized BSM equations with different boundary conditions, including the European call option and the American put option. Through comparison with other methods by numerical experiments, our work shows that NFEM provides an exciting new tool for this model. [ABSTRACT FROM AUTHOR]