Adaptive option pricing based on a posteriori error estimates for fully discrete finite difference methods

In this paper we study the a posteriori error estimates of numerical methods for parabolic partial integro-differential equations (PIDEs) which describe jump-diffusion option pricing models in finance. Due to the non-smoothness of the payoff function, the initial singularity of the solution may arise and therefore a posteriori error control and adaptivity are of the utmost importance in numerically solving such type of equations. The implicit-explicit (IMEX) Euler and IMEX Crank-Nicolson Adams-Bashforth methods are used for the time discretization and a second order finite difference method is used for the space discretization. By using the continuous and piecewise reconstructions, the upper and lower bounds of the a posteriori error estimates for the fully discrete finite difference methods are derived, and these error bounds depend on the discretization parameters and the data of the model problems only. Based on these a posteriori error estimates, we further develop a spatial-temporal adaptive algorithm. Numerical experiments are performed with uniform partitions and time-space adaptivity. The new adaptive algorithm significantly reduces the computational cost and provides effective error control for the numerical solutions.