Summary
Pricing American options using partial (integro-)differential equation based methods leads to linear complementarity problems (LCPs). The numerical solution of these problems resulting from the Black-Scholes model, Kou’s jump-diffusion model, and Heston’s stochastic volatility model are considered. The finite difference discretization is described. The solutions of the discrete LCPs are approximated using an operator splitting method which separates the linear problem and the early exercise constraint to two fractional steps. The numerical experiments demonstrate that the prices of options can be computed in a few milliseconds on a PC.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
L. Andersen and J. Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res., 4:231–262, 2000.
A. Almendral and C. W. Oosterlee. Numerical valuation of options with jumps in the underlying. Appl. Numer. Math., 53:1–18, 2005.
Y. Achdou and O. Pironneau. Computational methods for option pricing, volume 30 of Frontiers in Applied Mathematics. SIAM, Philadelphia, PA, 2005.
D. S. Bates. Jumps and stochastic volatility: Exchange rate processes implicit Deutsche mark options. Review Financial Stud., 9:69–107, 1996.
A. Brandt and C. W. Cryer. Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems. SIAM J. Sci. Statist. Comput., 4:655–684, 1983.
F. Black and M. Scholes. The pricing of options and corporate liabilities. J. Polit. Econ., 81:637–654, 1973.
M. J. Brennan and E. S. Schwartz. The valuation of American put options. J. Finance, 32:449–462, 1977.
N. Clarke and K. Parrott. Multigrid for American option pricing with stochastic volatility. Appl. Math. Finance, 6:177–195, 1999.
C. W. Cryer. The solution of a quadratic programming problem using systematic overrelaxation. SIAM J. Control, 9:385–392, 1971.
R. Cont and P. Tankov. Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, FL, 2004.
R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 43:1596–1626, 2005.
Y. d’Halluin, P. A. Forsyth, and G. Labahn. A penalty method for American options with jump diffusion processes. Numer. Math., 97:321–352, 2004.
Y. d’Halluin, P. A. Forsyth, and K. R. Vetzal. Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. Anal., 25:87–112, 2005.
D. Duffie, J. Pan, and K. Singleton. Transform analysis and asset pricing for affine jump-diffusions. Econometrica, 68(6):1343–1376, 2000.
B. Dupire. Pricing with a smile. Risk, 7:18–20, 1994.
J.-P. Fouque, G. Papanicolaou, and K. R. Sircar. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge, 2000.
P. A. Forsyth and K. R. Vetzal. Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput., 23:2095–2122, 2002.
M. B. Giles and R. Carter. Convergence analysis of Crank-Nicolson and Rannacher time-marching. J. Comput. Finance, 9:89–112, 2006.
R. Glowinski. Finite element methods for incompressible viscous flow. In P. G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis, Vol. IX, pages 3–1176. North-Holland, Amsterdam, 2003.
S. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Stud., 6:327–343, 1993.
M. Hintermüller, K. Ito, and K. Kunisch. The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., 13:865–888, 2003.
K. Ito and K. Kunisch. Parabolic variational inequalities: The Lagrange multiplier approach. J. Math. Pures Appl., 85:415–449, 2006.
S. Ikonen and J. Toivanen. Operator splitting methods for American option pricing. Appl. Math. Lett., 17:809–814, 2004.
S. Ikonen and J. Toivanen. Operator splitting methods for pricing American options with stochastic volatility. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing B11/2004, University of Jyväskylä, Jyväskylä, 2004.
S. Ikonen and J. Toivanen. Componentwise splitting methods for pricing American options under stochastic volatility. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing B7/2005, University of Jyväskylä, Jyväskylä, 2005.
S. Ikonen and J. Toivanen. Componentwise splitting methods for pricing American options under stochastic volatility. Int. J. Theor. Appl. Finance, 10(2):331–361, 2007.
K. Ito and J. Toivanen. Lagrange multiplier approach with optimized finite difference stencils for pricing American options under stochastic volatility. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing B6/2006, University of Jyväskylä, Jyväskylä, 2006.
R. Kangro and R. Nicolaides. Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal., 38:1357–1368, 2000.
S. G. Kou. A jump-diffusion model for option pricing. Management Sci., 48:1086–1101, 2002.
P. Lötstedt, J. Persson, L. von Sydow, and J. Tysk. Space-time adaptive finite difference method for European multi-asset options. Comput. Math. Appl., 53(8):1159–1180, 2007.
R. C. Merton. Theory of rational option pricing. Bell J. Econom. and Management Sci., 4:141–183, 1973.
R. Merton. Option pricing when underlying stock returns are discontinuous. J. Financial Econ., 3:125–144, 1976.
A.-M. Matache, C. Schwab, and T. P. Wihler. Fast numerical solution of parabolic integrodifferential equations with applications in finance. SIAM J. Sci. Comput., 27:369–393, 2005.
T. A. Manteuffel and A. B. White, Jr. The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comp., 47:511–535, 1986.
C. W. Oosterlee. On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal., 15:165–185, 2003.
R. Rannacher. Finite element solution of diffusion problems with irregular data. Numer. Math., 43:309–327, 1984.
C. Reisinger and G. Wittum. On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options. Comput. Vis. Sci., 7(3–4):189–197, 2004.
J. Toivanen. Numerical valuation of European and American options under Kou’s jump-diffusion model. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing B11/2006, University of Jyväskylä, Jyväskylä, 2006.
D. Tavella and C. Randall. Pricing financial instruments: The finite difference method. John Wiley & Sons, Chichester, 2000.
P. Wilmott. Derivatives. John Wiley & Sons, Chichester, 1998.
R. Zvan, P. A. Forsyth, and K. R. Vetzal. Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math., 91:199–218, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science + Business Media B.V.
About this chapter
Cite this chapter
Ikonen, S., Toivanen, J. (2008). An Operator Splitting Method for Pricing American Options. In: Glowinski, R., Neittaanmäki, P. (eds) Partial Differential Equations. Computational Methods in Applied Sciences, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8758-5_16
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8758-5_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8757-8
Online ISBN: 978-1-4020-8758-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)