B-Spline-Based Monotone Multigrid Methods

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 to 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 pricing 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 price 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.

[1]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[2]  W. Hackbusch,et al.  On multi-grid methods for variational inequalities , 1983 .

[3]  M. Griebel.  Sparse Grids and Related Approximation Schemes for Higher Dimensional Problems , 2006 .

[4]  Rolf Krause,et al.  Monotone Multigrid Methods for Signorini's Problem with Friction , 2001 .

[5]  A. Pinkus.  On L1-Approximation , 1989 .

[6]  J. Mandel.  A multilevel iterative method for symmetric, positive definite linear complementarity problems , 1984 .

[7]  M. Holtz.  The Computation of American Option Price Sensitivities using a Monotone Multigrid Method for Higher Order B – Spline Discretizations , 2004 .

[8]  Ulrich Reif,et al.  Multigrid methods with web-splines , 2002, Numerische Mathematik.

[9]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[10]  A. Pinkus.  On L[1]-approximation , 1991 .

[11]  A. Kunoth,et al.  B–SPLINE–BASED MONOTONE MULTIGRID METHODS — EXTENDED VERSION , 2022 .

[12]  Gabriel Wittum,et al.  On multigrid for anisotropic equations and variational inequalities “Pricing multi-dimensional European and American options” , 2004 .

[13]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities I , 1994 .

[14]  P. Oswald,et al.  Remarks on the Abstract Theory of Additive and Multiplicative Schwarz Algorithms , 1995 .

[15]  William W. Hager,et al.  Error estimates for the finite element solution of variational inequalities , 1978 .

[16]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[17]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[18]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[19]  Christoph Reisinger,et al.  Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben , 2004 .

[20]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[21]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[22]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[23]  K. Höllig.  Finite element methods with B-splines , 1987 .

[24]  Ronald A. DeVore,et al.  One-Sided Approximation of Functions , 1968 .

[25]  A. Brandt,et al.  Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems , 1983 .

[26]  D. Braess,et al.  A cascadic multigrid algorithm for variational inequalities , 2004 .

[27]  R. Kornhuber.  Adaptive monotone multigrid methods for nonlinear variational problems , 1997 .

[28]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[29]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[30]  P. Wilmott,et al.  The Mathematics of Financial Derivatives: Contents , 1995 .

[31]  R. Hoppe.  Multigrid Algorithms for Variational Inequalities , 1987 .