Neural network method for fractional order black scholesdifferential equation

This dissertation presents a novel approach using neural networks (NNM) to tackle the fractional-order ‎Black-Scholes differential equation (FOBSDE). This equation is crucial in financial mathematics for option ‎pricing, as it takes into account memory effects and non-local dynamics. The proposed method cleverly combines ‎spectral approximation with shifted Legendre polynomials and neural network optimization, allowing for effective ‎handling of fractional derivatives and boundary conditions. The solution is expressed as a series of Legendre basis ‎functions, with weights that are fine-tuned iteratively through gradient descent to reduce the Frobenius norm of the ‎residual error matrix. Extensive numerical tests demonstrate the method's accuracy and efficiency, proving it ‎outperforms traditional numerical techniques. For instance, with parameters r = 0.05, σ = 0.2, and fractional order ‎α = 0.5, the method achieves a maximum error of 1.23×〖10〗^(-3) and a relative L^2-error of 2.87×〖10〗^(-4). ‎These results underscore the potential of neural networks in solving complex fractional partial differential ‎equations, laying a strong foundation for financial modeling. This research bridges spectral techniques and ‎machine learning, providing a flexible and scalable tool for modern financial engineering.‎