Numerical Investigation of the Three-Dimensional HCIR Partial Differential Equation Utilizing a New Localized RBF-FD Method

Abstract

This work is concerned with the computational solution of the time-dependent 3D parabolic Heston–Cox–Ingersoll–Ross (HCIR) PDE, which is of practical importance in mathematical finance. The HCIR dynamic states that the model follows randomness for the underlying asset, the volatility and the rate of interest. Since the PDE formulation has degeneracy and non-smoothness at some area of its domain, we design a new numerical solver via semi-discretization and the radial basis function–finite difference (RBF-FD) scheme. Our scheme is built on graded meshes so as to employ the lowest possible number of discretized nodes. The stability of our solver is proven analytically. Computational testing is conducted to uphold the analytical findings in practice.

Keywords

• HCIR PDE
• hot area
• stochastic rate of interest
• fractional Black–Scholes model
• non-smoothness