Adaptive multi-asset Black–Scholes option pricing model and COMSOL implementation

Accurate valuation of options, among the most significant financial derivatives, is critical for the stability and efficiency of the derivatives market. This paper presents a practical and reproducible implementation workflow for solving multi-asset Black–Scholes (BS) pricing partial differential equations (PDEs) using adaptive finite elements in COMSOL Multiphysics. The backward differentiation formula (BDF) is adopted for the time discretization of the BS equation. The COMSOL built-in adaptive module is employed to improve the computational accuracy of the numerical solutions. Two refinement indicators and four boundary conditions are investigated through benchmark problems. The feature in COMSOL, i.e., Livelink for MATLAB, is employed to call the well-established functions of MATLAB and conveniently evaluate cumulative normal distribution required by certain boundary conditions. Through simulating the cash-or-nothing call option and European put option, the accuracy and robustness of the proposed method are validated. Detailed implementation in COMSOL is given, so it is convenient for readers or researchers without programming experience to employ the proposed method to solve the multi-asset BS equation.