In this paper we study the option pricing problem under dynamic elasticity of variance (DEV) model with counterparty risk. The counterparty risk induces a drop in the asset price and the asset can still be traded after this default time. There are no explicit solutions for the value function of the options. The value functions are governed by two joint partial differential equations (PDEs) which are connected at the default time. The PDEs are discretized by the finite difference methods (FDMs) and the second-order convergence rates both in time and space are derived. Numerical examples are carried out to verify the convergence results.