Analysis of Markov Chain Approximation for Diffusion Models with Non-Smooth Coefficients

Abstract

Diffusion models with non-smooth coefficients often appear in financial applications, with examples including but not limited to threshold models for financial variables, the pricing of occupation time derivatives and shadow rate models for interest rate dynamics. To calculate the expected value of a discounted payoff under general state-dependent discounting and monitoring of barrier crossing, continuous time Markov chain (CTMC) approximation can be applied. In a recent work, Zhang and Li (2018, Operations Research, forthcoming) established sharp convergence rates of CTMC approximation for diffusion models with smooth coefficients but non-smooth payoff functions, and proposed grid design principles to ensure nice convergence behaviors. However, their theoretical analysis fails to obtain sharp convergence rates when model coefficients lack smoothness. Moreover, it is unclear how to design the grid of CTMC to remedy the inferior convergence behaviors resulting from non-smooth model coefficients. In this paper, we introduce new ways for the theoretical analysis of CTMC approximation for general diffusion models with non-smooth coefficients. We prove that convergence of option price is only first order in general. However, strikingly, if all the discontinuous points of the model coefficients and the payoff function are in the midway between two grid points, second order convergence in the maximum norm is restored and in this case, delta and gamma have second order convergence at almost all grid points except those next to the discontinuous points. Numerical experiments are conducted that confirm the validity of our theoretical results. We also compare the CTMC approximation approach with properly designed grids to a classical numerical PDE scheme for diffusion models with non-smooth coefficients, where the finite difference method is applied separately in each region with smooth coefficients and continuous pasting of the value function is enforced at the discontinuities. We show that our approach is superior to the latter in terms of both the convergence rate and the simplicity of implementation.