Optimal Convergence Rate of the Binomial Tree Scheme for American Options with Jump Diffusion and Their Free Boundaries

Abstract

An American put option with jump diffusion can be modeled as an integro-variational inequality. With a penalization approximation and under the stability condition $\frac{\sigma^2 \Delta t}{\Delta x^2}\le 1$, where $\Delta x ={\rm ln}\,\frac {S_{n+1}}{S_n}$ ($S_t$-underlying asset price), we obtain the optimal convergence rate $O((\Delta x)+(\Delta t)^{1/2})$ of the binomial tree scheme for this variational inequality. Moreover, we define an approximate optimal exercise boundary within the framework of the binomial tree scheme and derive the convergence rate estimate $O((\Delta t)^{1/4})$ to the actual free boundary.