Space-time adaptive finite difference method for European multi-asset options

The multi-dimensional Black-Scholes equation is solved numerically for a European call basket option using a priori–a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error at the expiry date. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. The performance of the method is illustrated by examples in one, two, three, and four dimensions.