Penalty methods for the numerical solution of American multi-asset option problems. (English) Zbl 1152.91542

Summary: We derive and analyze a penalty method for solving American multi-asset option problems. A small, non-linear penalty term is added to the Black-Scholes equation. This approach gives a fixed solution domain, removing the free and moving boundary imposed by the early exercise feature of the contract. Explicit, implicit and semi-implicit finite difference schemes are derived, and in the case of independent assets, we prove that the approximate option prices satisfy some basic properties of the American option problem. Several numerical experiments are carried out in order to investigate the performance of the schemes. We give examples indicating that our results are sharp. Finally, the experiments indicate that in the case of correlated underlying assets, the same properties are valid as in the independent case.


91G60 Numerical methods (including Monte Carlo methods)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
91G20 Derivative securities (option pricing, hedging, etc.)
60G40 Stopping times; optimal stopping problems; gambling theory
Full Text: DOI


[1] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-659, (1973) · Zbl 1092.91524
[2] Boyle, P.; Evnine, J.; Gibbs, S., Numerical evaluation of multivariate contingent claims, Review of financial studies, 2, 2, 241-250, (1989)
[3] Bruaset, A.M., A survey of preconditioned iterative methods, (1995), Longman Scientific & Technical · Zbl 0834.65014
[4] Choi, S.; Marcozzi, M., A numerical approach to American currency option valuation, Journal of derivatives, 9, 2, 19-29, (2001)
[5] Choi, S.; Marcozzi, M., The valuation of foreign currency options under stochastic interest rates, Computers & mathematics with applications, 45, 741-749, (2003) · Zbl 1073.91029
[6] N. Clarke, A.K. Parrot, The multigrid solution of two-factor American put options, Tech. rep., Oxford Computing Laboratory, 1996
[7] d’Halluin, Y.; Forsyth, P.A.; Labah, G., A penalty method for American options with jump-diffusion processes, Numerische Mathematik, 97, 2, 321-352, (2004) · Zbl 1126.91036
[8] Duffie, D., Dynamic asset pricing theory, (1996), Princeton University Press
[9] Fasshauer, G.; Khaliq, A.Q.M.; Voss, D.A.; Chen, C.S., Using mesh free approximation for multi asset American options, Mesh free methods, Journal of Chinese institute of engineers, 27, 563-571, (2004), (special issue)
[10] Forsyth, P.A.; Vetzal, K.R., Quadratic convergence for valuing American options using a penalty method, SIAM journal on scientific computing, 23, 6, 2095-2122, (2002) · Zbl 1020.91017
[11] Forsyth, P.A.; Vetzal, K.R.; Zvan, R., A finite element approach to the pricing of discrete lookbacks with stochastic volatility, Applied mathematical finance, 6, 2, 87-106, (1999) · Zbl 1009.91030
[12] A. Ibanez, F. Zapatero, Monte Carlo valuation of American options through computation of the optimal exercise frontier, Tech. rep., Finance and Business Economics Department, Marshall School of Business, 1999
[13] W. Joubert, Iterative methods for the solution of nonsymmetric systems of linear equations, Tech. Rep. CNA-242, Center for Numerical Analysis, University of Texas at Austin, 1990
[14] Kangro, R.; Nicolaides, R., Far field boundary conditions for Black-Scholes equations, SIAM journal on numerical analysis, 38, 4, 1357-1368, (2000) · Zbl 0990.35013
[15] Khaliq, A.Q.M.; Voss, D.A.; Kazmi, S.H.K., A linearly implicit predictor-corrector scheme for pricing American options using a penalty method approach, Journal of banking and finance, 30, 489-502, (2006)
[16] Kwok, Y.K., Mathematical models of financial derivatives, (1998), Springer-Verlag · Zbl 0931.91018
[17] Marcozzi, M., On the approximation of optimal stopping problems with application to financial mathematics, SIAM journal on scientific computing, 22, 5, 1865-1884, (2001) · Zbl 0980.60047
[18] Marcozzi, M.; Choi, S.; Chen, C.S., On the use of boundary conditions for variational formulations arising in financial mathematics, Applied mathematics and computation, 124, 2, 197-214, (2001) · Zbl 1047.91033
[19] Nielsen, B.F.; Skavhaug, O.; Tveito, A., Penalty and front-fixing methods for the numerical solution of American option problems, The journal of computational finance, 5, 4, 69-97, (2002)
[20] PyCC, Software framework under development, URL: http://www.simula.no/pycc, 2007
[21] Sapariuc, I.; Marcozzi, M.; Flaherty, J.E., A numerical analysis of variational valuation techniques for derivative securities, Applied mathematics and computation, 159, 1, 171-198, (2004) · Zbl 1080.91040
[22] van der Vorst, H., Bi-CGSTAB: A fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems, SIAM journal on scientific and statistical computing, 13, 2, 631-644, (1992) · Zbl 0761.65023
[23] Voss, D.A.; Khaliq, A.Q.M.; Kazmi, S.H.K.; He, H., A fourth order L-stable method for the Black-Scholes model with barrier options, (), 199-207
[24] Wilmott, P., Derivatives, the theory and practice of financial engineering, (1998), John Wiley & Sons
[25] Wilmott, P.; Dewynne, J.; Howison, S., Option pricing, mathematical models and computation, (1993), Oxford Financial Press · Zbl 0797.60051
[26] Zhu, Y.I.; Sun, Y., The singularity-separating method for two-factor convertible bonds, The journal of computational finance, 3, 1, 91-110, (1999)
[27] Zvan, R.; Forsyth, P.; Vetzal, K.R., A finite volume approach for contingent claims valuation, IMA journal of numerical analysis, 21, 3, 703-731, (2001) · Zbl 1004.91032
[28] Zvan, R.; Forsyth, P.A.; Vetza, K.R., Penalty methods for American options with stochastic volatility, Journal of computational and applied mathematics, 91, 2, 199-218, (1998) · Zbl 0945.65005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.