Numerical solving of the generalized Black-Scholes differential equation using Laguerre neural network

Abstract

Reasonable pricing of options in the financial derivatives market is crucial. For American options, or when volatility and interest rate are not constant, it is often difficult to obtain analytical solutions to the Black-Scholes (BS) equation. In this paper, the Laguerre neural network was proposed as a novel numerical algorithm with three layers of neurons for solving BS equations. The validity period and stock price are the input of the network, and the option price is the only output layer. Laguerre functions are used as the activation function of the neuron in the hidden layer. The BS equation and boundary conditions are set as penalty function, the training points are uniformly selected in the domain, and the improved extreme learning machine algorithm is used to optimize the network connection weights. Three experiments calculated the numerical solutions of BS equations for European options and generalized option pricing models. Compared with existing algorithms such as the finite element method and radial basis function neural network, the numerical solutions obtained by Laguerre neural network have higher accuracy and smaller errors, which illustrates the feasibility and superiority of the proposed method for solving BS equations.