On the Approximation of Optimal Stopping Problems with Application to Financial Mathematics

The determination of the value function associated with a given reward and stochastic process represents an important class of stochastic control problem. In particular, the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom, unbounded domains, and lack of "natural" boundary conditions. In this paper, we introduce two methodologies for computing the value function of optimal stopping associated with general stochastic processes. Our results are implemented utilizing finite elements and are validated using problems taken from financial mathematics.

[1]  René M. Stulz,et al.  Options on the minimum or the maximum of two risky assets : Analysis and applications , 1982 .

[2]  I. Karatzas.  On the pricing of American options , 1988 .

[3]  Sanjiv Ranjan Das.  Discrete-time bond and option pricing for jump-diffusion processes , 1996 .

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

[5]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

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

[7]  M. Schervish.  Multivariate normal probabilities with error bound , 1984 .

[8]  Kuldeep Shastri,et al.  Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques , 1985, Journal of Financial and Quantitative Analysis.

[9]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[10]  Andrew Rennie,et al.  Financial Calculus: An Introduction to Derivative Pricing , 1996 .

[11]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[12]  Xiao Lan Zhang,et al.  Numerical Analysis of American Option Pricing in a Jump-Diffusion Model , 1997, Math. Oper. Res..

[13]  A. Bensoussan.  On the theory of option pricing , 1984, Acta Applicandae Mathematicae.

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

[15]  G. Barles,et al.  CONVERGENCE OF NUMERICAL SCHEMES FOR PARABOLIC EQUATIONS ARISING IN FINANCE THEORY , 1995 .

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

[17]  Robert A. Jarrow,et al.  Pricing foreign currency options under stochastic interest rates , 1991 .

[18]  J. Crank.  Free and moving boundary problems , 1984 .

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

[20]  D. Lamberton,et al.  Variational inequalities and the pricing of American options , 1990 .

[21]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

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

[23]  P. Boyle.  Options: A Monte Carlo approach , 1977 .

[24]  Ivo Babuska,et al.  The p and h-p Versions of the Finite Element Method, Basic Principles and Properties , 1994, SIAM Rev..

[25]  H. Johnson.  Options on the Maximum or the Minimum of Several Assets , 1987, Journal of Financial and Quantitative Analysis.

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

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

[28]  A. Friedman.  Variational principles and free-boundary problems , 1982 .

[29]  R. Hoppe,et al.  Adaptive multilevel methods for obstacle problems , 1994 .

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

[31]  James N. Bodurtha,et al.  Discrete-Time Valuation of American Options with Stochastic Interest Rates , 1995 .

[32]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

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

[34]  P. Moerbeke.  On optimal stopping and free boundary problems , 1973, Advances in Applied Probability.

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

[36]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[37]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[38]  Paul Wilmott,et al.  Pricing Parisian Options , 1999 .

[39]  R. Jarrow,et al.  Pricing Options On Risky Assets In A Stochastic Interest Rate Economy , 1992 .

[40]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .