A Semi-Analytical Approach to Solving the Black-Scholes Equation via Reproducing Kernel Hilbert Spaces (RKHS)

This paper presents a semi–analytical method for solving the Black–Scholes equation by embedding its deterministic and stochastic components into a Reproducing Kernel Hilbert Space (RKHS). The deterministic term is approximated via regularized kernel regression, while the stochastic term is modeled using an autoregressive representation in RKHS. The method is validated on both synthetic geometric Brownian motion trajectories and real adjusted closing prices of Apple Inc. (AAPL), comparing the RKHS approach against the Euler–Maruyama scheme. Results show that the proposed method achieves lower RMSE with fewer anchor points, demonstrating its efficiency and robustness for data–driven financial modeling under uncertainty.