Numerical Solution of Time Fractional Black–Scholes Model Based on Legendre Wavelet Neural Network with Extreme Learning Machine

Xiaoning Zhang, Jianhui Yang, Yuxin Zhao

In this paper, the Legendre wavelet neural network with extreme learning machine is proposed for the numerical solution of the time fractional Black–Scholes model. In this way, the operational matrix of the fractional derivative based on the two-dimensional Legendre wavelet is derived and employed to solve the European options pricing problem. This scheme converts this problem into the calculation of a set of algebraic equations. The Legendre wavelet neural network is constructed; meanwhile, the extreme learning machine algorithm is adopted to speed up the learning rate and avoid the over-fitting problem. In order to evaluate the performance of this scheme, a comparative study with the implicit differential method is constructed to validate its feasibility and effectiveness. Experimental results illustrate that this scheme offers a satisfactory numerical solution compared to the benchmark method.