We are interested in the numerical solution of the multi-dimensional Black-Scholes equation to determine the arbitrage free price F(t,x) of an option. We will consider European call basket options on several underlying assets (e.g. stocks). This problem can e.g. be solved by finding a numerical solution to a multi-dimensional PDE. This way to determine the arbitrage free price was introduced independently by F. Black and M. Scholes (who gave name to the PDE) and R.C. Merton in 1973.

In a previous paper an adaptive finite difference method was developed with full control of the local discretization errors. The method was shown to be very efficient. In this paper we develop a space-time adaptive FD-method with control of the global error.

The Black-Scholes equation is discretized by second order accurate finite difference stencils on a Cartesian grid. An error equation is derived for the global error E(t, x) in the solution. The driving right hand side in the error equation is the discretization error in the original PDE. This error is estimated a posteriori and the grid is adapted so that the Cartesian structure of the grid is maintained and the time step is adapted in every time step. The step sizes are chosen so that a linear functional of the solution error at maturity of the option satisfies an accuracy constraint. This means that the integral over the error, E(x,t) times a function g(x) is smaller than some positive constant.

The time step is adjusted to comply with the bound on the local error and the space grid is changed at a few pre-specified time points. The weights for the local error bounds in each time interval are solutions of the adjoint equation of the multidimensional Black-Scholes PDE. The growth of the error in the intervals between the grid adaptations is estimated a priori by the maximum principle for parabolic equations. In the same manner estimates of the solution of the adjoint equation is bounded.

The advantage with our procedure is that we obtain estimates of the numerical errors and we have an algorithm to choose the computational grid so that bounds like the functional explained above on the errors are satisfied also for multi-dimensional equations. For more dimensions than five (or so), the solution with finite difference approximations on a grid suffers severly from the 'curse of dimensionality' with an exponential growth in the number of grid points and other alternatives must be considered.