A robust finite difference scheme for pricing American put options with singularity-separating method. (English) Zbl 1192.91190

The determination of the value of an American option is more complicated than for a European option. The former is governed by a linear complementarity problem involving the Black-Scholes differential operator and a constraint on the value of the option. Since analytical solutions can rarely be found for practical problems, numerical methods needed to be developed. There are several methods in the literature for the valuation of European and American options: the lattice technique, classical difference methods, the upwind numerical approach, linearized methods based on differential quadrature methods etc. This paper presents a stable numerical method which is based on a hybrid finite difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. Error estimates are derived for the direct application of the finite difference method to the American option pricing model. The scheme is shown to be second-order convergent with respect to the spatial variable. To overcome the lack of smoothness of the American put option pricing, the singularity-separating method is used.


91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Angermann, L., Wang, S.: Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American option pricing. Numer. Math. 106, 1–40 (2007) · Zbl 1131.65301
[2] Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) · Zbl 1092.91524
[3] Cheng, X., Xue, L.: On the error estimate of finite difference method for the obstacle problem. Appl. Math. Comput. 183, 416–422 (2006) · Zbl 1133.65039
[4] Courtadon, G.: A more accurate finite difference approximation for the valuation of options. J. Fin. Quant. Anal. 17, 697–703 (1982)
[5] Cox, J.C., Ross, S., Rubinstein, M.: Option pricing: a simplified approach. J. Fin. Econ. 7, 229–264 (1979) · Zbl 1131.91333
[6] Forsyth, P.A., Vetzal, K.R.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23(6), 2095–2122 (2002) · Zbl 1020.91017
[7] Glowinski, R., Lions, J.L., Trémolières, T.: Numerical analysis of variational inequality. North-Holland, Amsterdam (1984)
[8] Goeleven, D.: A uniqueness theorem for the generalized-order linear complementary problem associated with M-matrices. Linear Algebra Appl. 235, 221–227 (1996) · Zbl 0845.90119
[9] Hull, J.C., White, A.: The use of the control variate technique in option pricing. J. Fin. Quant. Anal. 23, 237–251 (1988)
[10] Ikonen, S., Toivanen, J.: Operator splitting methods for American option pricing. Appl. Math. Lett. 17, 809–814 (2004) · Zbl 1063.65081
[11] Kangro, R., Nicolaides, R.: Far field boundary conditions for Black-Scholes equations. SIAM J. Numer. Anal. 38, 1357–1368 (2000) · Zbl 0990.35013
[12] Rogers, L.C.G., Talay, D.: Numercial methods in finance. Cambridge University Press, Cambridge (1997) · Zbl 0867.00036
[13] Schwartz, E.: The valuation of warrants: implementing a new approach. J. Fin. Econ. 4, 79–93 (1977)
[14] Stynes, M., Roos, H.-G.: The midpoint upwind scheme. Appl. Numer. Math. 32, 291–308 (1997) · Zbl 0877.65055
[15] Stynes, M., Tobiska, L.: A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms 18, 337–360 (1998) · Zbl 0916.65108
[16] Tangman, D.Y., Gopaul, A., Bhuruth, M.: A fast high-order finite difference algorithm for pricing American options. J. Comput. Appl. Math. 222, 17–29 (2008) · Zbl 1147.91032
[17] Vazquez, C.: An upwind numerical approach for an American and European option pricing model. Appl. Math. Comput. 97, 273–286 (1998) · Zbl 0937.91053
[18] Wang, S., Yang X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129(2), 227–254 (2006) · Zbl 1139.91020
[19] Wilmott, P., Dewynne, J., Howison, S.: Option pricing: mathematical models and computation. Oxford Financial, Oxford (1993) · Zbl 0844.90011
[20] Wu, X., Kong, W.: A highly accurate linearized method for free boundary problems. Comput. Math. Appl. 50, 1241–1250 (2005) · Zbl 1083.65062
[21] Zhu, Y., Chen, B., Ren, H., Xu, H.: Application of the singularity-separating method to American exotic option pricing. Adv. Comput. Math. 19, 147–158 (2003) · Zbl 1042.91029
[22] Zhu, Y., Sun, Y.S.: The singularity-separating method for two factor convertible bonds. J. Comput. Fin. 3, 91–100 (1999)
[23] Zhu, Y., Wu, X., Chen, I.: Derivative securities and difference methods. Springer, New York (2004) · Zbl 1061.91036
[24] Zhu, Y., Zhong, X., Chen, B., Zhang, Z.: Difference methods for initial-boundary-value problems and flow around bodies. Springer and Science Press, Heidelberg (1988) · Zbl 0653.76003
[25] Zvan, R., Forsyth, P.A., Vetzal, K.R.: A general finite element approach for PDE option pricing models. University of Waterloo, Waterloo (1998) · Zbl 0945.65005
[26] Zvan, R., Forsyth, P.A., Vetzal, K.R.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91, 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.