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

作者：Zhang, Xiaoning; Yang, Jianhui^*; Zhao, Yuxin

来源：Fractal and Fractional, 2022, 6(7): 401.

DOI：10.3390/fractalfract6070401

• Legendre wavelet neural network (LWNN)
• operational matrix
• extreme learning machine (ELM)
• Black-Scholes model
• Caputo's fractional derivative

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.