A Space-Time Multigrid Method for the Numerical Valuation of Barrier Options

We 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 grid’s 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. Mathematics Subject Classification: 65M55