A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of “easy to implement” numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of “predictor‐corrector type” for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor‐corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor‐corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.

[1]  G. Barles,et al.  Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations , 1990 .

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

[3]  Frank Spitzer.  Optimal Stopping Rules (A. N. Shiryayev) , 1981 .

[4]  H. Ishii.  Hamilton-Jacobi Equations with Discontinuous Hamiltonians on Arbitrary Open Sets , 1985 .

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

[6]  Valery A. Kholodnyi,et al.  A nonlinear partial differential equation for American options in the entire domain of the state variable , 1997 .

[7]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

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

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

[10]  Kenneth H. Karlsen,et al.  A semilinear Black and Scholes partial differential equation for valuing American options , 2003, Finance Stochastics.

[11]  G. Barles,et al.  Numerical Methods in Finance: Convergence of Numerical Schemes for Degenerate Parabolic Equations Arising in Finance Theory , 1997 .

[12]  Dawn Hunter,et al.  Penalty and front-fixing methods for the numerical solution of American option problems , 2002 .

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

[14]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[15]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

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

[17]  Hitoshi Ishii,et al.  A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations , 1989 .

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

[19]  Patrick Jaillet,et al.  Inéquations variationnelles et théorie des options , 1988 .

[20]  Kristin Reikvam.  Viscosity solutions of optimal stopping problems , 1997 .

[21]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1990, 29th IEEE Conference on Decision and Control.

[22]  Huy En Pham.  Optimal Stopping of Controlled Jump Diiusion Processes: a Viscosity Solution Approach , 1998 .

[23]  Eduardo S. Schwartz,et al.  The Valuation of American Put Options , 1977 .

[24]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[25]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .