introduce an adaptive space-time multigrid method for the pricing of barrier
options. In particular, we consider the numerical valuation of
up-and-out options by the method of lines. We treat both the implicit Euler and
Crank-Nicolson methods. We implement a space-time multigrid
method in which the domain in space and time are treated simultaneously.
We consider an adaptive coarsening technique in which the choice
of restriction operator is dependent on the gridís degree of anisotropy
at each level. We perform local Fourier analysis to find a suitable
choice of our anisotropy constant. We detail the advantages and
disadvantages of our technique. In particular, we stress that our algorithm
is extremely well suited for parallel computing and, with a suitable
smoother, has parallel complexity O(logM + logN), allowing for fast
computation of extremely large scale problems.