Augmented Lagrangian method applied to American option pricing

The American option pricing problem is originally formulated as a stochastic optimal stopping time problem. It is also equivalent to a variational inequality problem or a complementarity problem involving the Black-Scholes partial differential operator. In this paper, the corresponding variational inequality problem is discretized by using a fitted finite volume method. Based on the discretized form, an algorithm is developed by applying augmented Lagrangian method (ALM) to the valuation of the American option. Convergence properties of ALM are considered. By empirical numerical experiments, we conclude that ALM is more effective than penalty method and Lagrangian method, and comparable with the projected successive overrelaxation method (PSOR). Furthermore, numerical results show that ALM is more robust in terms of computation time under changes in market parameters: interest rate and volatility.

[1]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[2]  Curt Randall,et al.  Pricing Financial Instruments: The Finite Difference Method , 2000 .

[3]  Peter A. Forsyth,et al.  Quadratic Convergence for Valuing American Options Using a Penalty Method , 2001, SIAM J. Sci. Comput..

[4]  Peter A. Forsyth,et al.  Penalty methods for American options with stochastic volatility , 1998 .

[5]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[6]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[7]  R. Myneni.  The Pricing of the American Option , 1992 .

[8]  Kazufumi Ito,et al.  Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces , 2000 .

[9]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[10]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[11]  Karl Kunisch,et al.  Augmented Lagrangian Active Set Methods for Obstacle Problems , 2003 .

[12]  Song Wang,et al.  A novel fitted finite volume method for the Black-Scholes equation governing option pricing , 2004 .

[13]  J. Pang,et al.  Option Pricing and Linear Complementarity , 1998 .

[14]  Rüdiger U. Seydel.  Tools for Computational Finance , 2002 .

[15]  Carlos Vázquez,et al.  An upwind numerical approach for an American and European option pricing model , 1998, Appl. Math. Comput..

[16]  Alain Bensoussan,et al.  Applications of Variational Inequalities in Stochastic Control , 1982 .

[17]  Suresh Sethi,et al.  Optimal Financing of a Corporation Subject To Random Returns , 2002 .

[18]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[19]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[20]  Kok Lay Teo,et al.  Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation , 2006 .

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

[22]  M. Dempster,et al.  Pricing American Stock Options by Linear Programming , 1996 .

[23]  Kazufumi Ito,et al.  Semi–Smooth Newton Methods for Variational Inequalities of the First Kind , 2003 .

[24]  D. N. De G. Allen,et al.  RELAXATION METHODS APPLIED TO DETERMINE THE MOTION, IN TWO DIMENSIONS, OF A VISCOUS FLUID PAST A FIXED CYLINDER , 1955 .

[25]  H. P. Jr. Mackean,et al.  Appendix : A free boundary problem for the heat equation arising from a problem in mathematical economics , 1965 .

[26]  P. G. Ciarlet,et al.  Introduction to Numerical Linear Algebra and Optimisation , 1989 .