## A second-order difference scheme for the penalized Black-Scholes equation governing American put option pricing.(English)Zbl 1254.91744

Summary: In this paper we present a stable finite difference scheme on a piecewise uniform mesh along with a power penalty method for solving the American put option problem. By adding a power penalty term the linear complementarity problem arising from pricing American put options is transformed into a nonlinear parabolic partial differential equation. Then a finite difference scheme is proposed to solve the penalized nonlinear PDE, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit time stepping technique. It is proved that the scheme is stable for arbitrary volatility and arbitrary interest rate without any extra conditions and is second-order convergent with respect to the spatial variable. Furthermore, our method can efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 91G20 Derivative securities (option pricing, hedging, etc.) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text:

### References:

 [1] Angermann L., Wang S. (2007) Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American option pricing. Numerische Mathematik 106: 1–40 · Zbl 1131.65301 [2] Black F., Scholes M. S. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654 · Zbl 1092.91524 [3] Courtadon G. (1982) A more accurate finite difference approximation for the valuation of options. Journal of Financial and Quantitative Analysis 17: 697–703 [4] Cox J. C., Ross S., Rubinstein M. (1979) Option pricing: A simplified approach. Journal of Financial Economics 7: 229–264 · Zbl 1131.91333 [5] Forsyth P. A., Vetzal K. R. (2002) Quadratic convergence for valuing American options using a penalty method. SIAM Journal on Scientific Computing 23(6): 2095–2122 · Zbl 1020.91017 [6] Hull J. C., White A. (1988) The use of the control variate technique in option pricing. Journal of Financial and Quantitative Analysis 23: 237–251 [7] Ikonen S., Toivanen J. (2004) Operator splitting methods for American option pricing. Applied Mathematics Letters 17: 809–814 · Zbl 1063.65081 [8] Jaillet P., Lamberton D., Lapeyre B. (1990) Variational inequalities and the pricing of American options. Acta Applicandae Mathematicae 21: 263–289 · Zbl 0714.90004 [9] Kangro R., Nicolaides R. (2000) Far field boundary conditions for Black-Scholes equations. SIAM Journal on Numerical Analysis 38: 1357–1368 · Zbl 0990.35013 [10] Nielsen B. F., Skavhaug O., Tveito A. (2002) Penalty and front-fixing methods for the numerical solution of American option problems. Journal of Computational Finance 5: 69–97 [11] Rogers L. C. G., Talay D. (1997) Numercial methods in finance. Cambridge University Press, Cambridge [12] Schwartz E. (1977) The valuation of warrants: Implementing a new approach. Journal of Financial Economics 4: 79–93 [13] Vazquez C. (1998) An upwind numerical approach for an American and European option pricing model. Applied Mathematics and Computation 97: 273–286 · Zbl 0937.91053 [14] Wilmott P., Dewynne J., Howison S. (1993) Option pricing: Mathematical models and computation. Oxford Financial Press, Oxford, UK · Zbl 0844.90011 [15] Wu X., Kong W. (2005) A highly accurate linearized method for free boundary problems. Computers and Mathematics with Applications 50: 1241–1250 · Zbl 1083.65062 [16] Zvan R., Forsyth P. A., Vetzal K. R. (1998a) A general finite element approach for PDE option pricing models. University of Waterloo, Canada · Zbl 0945.65005 [17] Zvan R., Forsyth P. A., Vetzal K. R. (1998b) Penalty methods for American options with stochastic volatility. Journal of Computational and Applied Mathematics 91: 199–218 · 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.