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.

金融衍生产品市场中期权的合理定价至关重要。对于美国期权，或者当波动率和利率不是恒定不变时，通常很难获得布莱克-舒尔斯（BS）方程的解析解。本文提出了Laguerre神经网络作为具有三层神经元的新型数值算法，用于求解BS方程。有效期和股票价格是网络的输入，而期权价格是唯一的输出层。Laguerre函数用作隐藏层中神经元的激活函数。将BS方程和边界条件设置为惩罚函数，在域中均匀选择训练点，并使用改进的极限学习机算法来优化网络连接权重。三个实验计算了欧洲期权和广义期权定价模型的BS方程的数值解。与有限元法和径向基函数神经网络等现有算法相比，拉盖尔神经网络获得的数值解具有较高的精度和较小的误差，说明了该方法求解BS方程的可行性和优越性。