On the convergence order of a binary tree approximation of symmetrized diffusion processes

Abstract

The price of a barrier option is often computed numerically. Due to the path dependency, the convergence rate of such numerical approximation is generally of order 1/2. In this paper, we show that the convergence order can be achieved at 1 under certain condition. This confirms a numerical analysis done previously by the third author with others.We consider the case where the underlying process is a Brownian motion with drift. The price of a barrier option coincides with the price of a vanilla option of the “symmetrized” diffusion, which has a discontinuous drift. The symmetrized diffusion is then approximated by a Markov chain and the corresponding option price is calculated. This approximation to the barrier option is shown to have a convergence order of 1 under some mild condition on the initial value of the process and the payoff function.