B-Spline-Based Monotone Multigrid Methods

Abstract. For the efficient numerical solution of elliptic variational inequalities on closed convex sets, multigrid methods based on piecewise linear finite elements have been investigated over the past decades. Essential for their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements. On the other hand, there are a number of problems which profit from higher order approximations. Among these are the problem of prizing American options, formulated as a parabolic boundary value problem involving Black–Scholes’ equation with a free boundary. In addition to computing the free boundary, the optimal exercise prize of the option, of particular importance are accurate pointwise derivatives of the value of the stock option up to order two, the so–called Greek letters. In this paper, we propose a monotone multigrid method for discretizations in terms of B–splines of arbitrary order to solve elliptic variational inequalities on a closed convex set. In order to maintain monotonicity (upper bound) and quasi–optimality (lower bound) of the coarse grid corrections, we propose an optimized coarse grid correction (OCGC) algorithm which is based on B–spline expansion coefficients. We prove that the OCGC algorithm is of optimal complexity of the degrees of freedom of the coarse grid and, therefore, the resulting monotone multigrid method is of asymptotically optimal multigrid complexity. Finally, the method is applied to a standard model for the valuation of American options. In particular, it is shown that a discretization based on B–splines of order four enables us to compute the second derivative of the value of the stock option to high precision.