A Highly Accurate Finite Element Method to Price Discrete Double Barrier Options

We develop a highly accurate numerical method for pricing discrete double barrier options under the Black–Scholes (BS) model. To this aim, the BS partial differential equation is discretized in space by the parabolic finite element method, which is based on a variational formulation and thus is well-suited for dealing with the non-smoothness of the discrete barrier option solutions. In addition, the approximation in time is performed using the implicit Euler scheme, which allows us to remove spurious oscillations that may occur at each monitoring date, and whose convergence rate is enhanced by means of a repeated Richardson extrapolation procedure. Numerical experiments are carried out which reveal that the method proposed achieves fourth-order accuracy in both space and time (even if the solutions being approximated are non-smooth), and performs hundredths of times better than a finite difference scheme in Wade et al. (J Comput Appl Math 204:144–158, 2007 ). To the best of our knowledge, the one developed in the present paper is the first lattice-based approach for discrete barrier options which is empirically shown to be fourth-order accurate in both space and time.