An Artificial Boundary Method for American Option Pricing under the CEV Model

Abstract

The Black–Scholes asset price dynamics is well known to be inadequate for capturing the volatility smile in the financial market. Therefore, the constant elasticity of variance (CEV) model has become a popular alternative to valuing options fitting to the smile. American option pricing, however, is computationally intensive, as there are no analytical formulas available. This paper proposes an artificial boundary method for partial differential equations (PDEs) to compute American option prices and Greeks under the CEV model. The idea is to reduce the infinite computational domain to a finite one by introducing an artificial boundary on which an exact boundary condition is derived. We then develop a Crank–Nicolson scheme to solve the PDE with the artificial boundary condition implemented with a numerical Laplace inversion. With a finite computational domain, the optimal exercise boundary can be determined efficiently. Our numerical examples show that the proposed scheme is accurate, robust with respect to the truncation size, and more efficient than alternative methods for accurate option prices.