Two Methods for Solving the Two‐Dimensional Black–Scholes Equation

ABSTRACT The Crank–Nicolson radial basis function (CN‐RBF) method, as a time‐stepping mesh‐less approach, is widely adopted for multi‐asset option pricing. However, its accuracy is highly dependent on shape parameters, which limits its practical applicability. To address this, we propose two polynomial‐based space‐time methods for pricing European exchange options: one is the space‐time polynomial particular solutions method, and the other is the space‐time interpolations method which directly uses polynomial functions as basis functions. These two methods treat the time variable as a common spatial variable, thereby eliminating the need for the time‐stepping process. Their implementation requires only interpolation points, simplifying the solution procedure by removing shape parameter selection and time‐step scheme dependencies. Numerical results demonstrate that both space‐time methods achieve good performance in terms of accuracy and stability.