A Meshless Method for Nonlinear High-Dimensional PDEs: Application to Financial and Electricity Options

High-dimensional option pricing partial differential equations are computationally prohibitive for grid-based discretizations due to the curse of dimensionality. We propose a meshless radial basis function (RBF) collocation method in which spatial centers are generated by a quasi-Monte Carlo acceptance-rejection strategy. The resulting non-uniform center cloud places more centers near the initial state, where the underlying process is expected to have high probabilistic relevance over the finite maturity horizon, and fewer centers in peripheral regions of the truncated computational domain. This leads to a more efficient allocation of degrees of freedom by concentrating RBF centers in the pricing-relevant regions of the domain, thereby alleviating some of the computational burden associated with dimensionality while preserving the flexibility and accuracy of meshless radial basis function approximations. The method is applied to two nonlinear pricing problems under the Leland transaction cost model, namely a spark-spread option with stochastic volatility and a European call option on the maximum of multiple assets. Numerical experiments show stable behavior and decreasing errors in the linear benchmark tests as the number of centers increases, and indicate robust performance in nonlinear Leland-type settings at moderate computational cost. Finally, the proposed RBF approach is not specific to option pricing and can be viewed as a flexible framework for extensions to more general financial models, as well as to high-dimensional nonlinear PDEs arising in other scientific applications.