##
**A new predictor-corrector scheme for valuing American puts.**
*(English)*
Zbl 1237.91236

Summary: We present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [H. G. Landau, Q. Appl. Math. 8, 81–94 (1950; Zbl 0036.13902)] to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with S.-P. Zhu’s analytical solution [Quant. Finance 6, No. 3, 229–242 (2006; Zbl 1136.91468)], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.

### MSC:

91G60 | Numerical methods (including Monte Carlo methods) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

PDF
BibTeX
XML
Cite

\textit{S.-P. Zhu} and \textit{J. Zhang}, Appl. Math. Comput. 217, No. 9, 4439--4452 (2011; Zbl 1237.91236)

Full Text:
DOI

### References:

[1] | Albanese, C.; Campolieti, G., Advanced derivatives pricing and risk management, (2006), Elsevier · Zbl 1140.91034 |

[2] | Allegretto, W.; Lin, Yanping; Yang, H., A fast and highly accurate numerical method for the valuation of American options discrete and continuous dynamical systems, Applications and algorithm, 8, 27-136, (2001) |

[3] | Barles, Guy; Burdeau, Julien; Romano, Marc; Samscen, Nicolas, Critical stock price near expiration, Mathematical finance, 5, 2, 77-95, (1995) · Zbl 0866.90029 |

[4] | Barone-Adesi, G.; Elliott, R., Approximations for the values of American options, Stochastic analysis and applications, 9, 115-131, (1991) · Zbl 0729.60056 |

[5] | Benth, F.E.; Karlsen, K.H.; Reikvam, K., A semilinear black and Scholes partial differential equation for valuing American options: approximate solutions and convergence, Interface and free boundaries, 6, 379-404, (2004) · Zbl 1068.35190 |

[6] | Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524 |

[7] | Bunch, D.S.; Johnson, H., The American put option and its critical stock price, The journal of finance, 5, 2333-2356, (2000) |

[8] | Carr, P.; Jarrow, R.; Myneni, R., Alternative characterizations of American put options, Mathematical finance, 2, 87-106, (1992) · Zbl 0900.90004 |

[9] | Cho, Chung-Ki; Kang, Sunbu; Kim, Taekkeun; Kwon, YongHoon, A new approach for numerical identification of optimal exercise curve, Computational science and its applications, 1, 926-934, (2004) · Zbl 1116.91325 |

[10] | Cox, J.; Ross, S.; Rubinstein, M., Option pricing – a simplified approach, Journal of financial economics, 7, 229-236, (1979) · Zbl 1131.91333 |

[11] | d’Halluin, Y.; Forsyth, P.A.; Labahn, G., A penalty method for American options with jump diffusion processes, (2003) · Zbl 1126.91036 |

[12] | Geske, R.; Johnson, H., The American put option valued analytically, The journal of finance, 39, 1511-1524, (1984) |

[13] | Golub, Gene H.; Ortega, James M., Scientific computing and differential equations, (1992), Academic Press, Inc. · Zbl 0749.65041 |

[14] | Huang, J.Z.; Subrahmanyam, M.G.; Yu, G.G., Pricing and hedging American options: a recrusive integration method, The review of financial studies, 9, 277-300, (1996) |

[15] | Hull, John; White, Alan, Valuing derivative securities using the explicit finite difference method, The journal of financial and quantitative analysis, 25, 1, 87-100, (1990) |

[16] | Ikonen, S.; Toivanen, J., Operator splitting methods for American option pricing, Applied mathematics letters, 17, 809-814, (2004) · Zbl 1063.65081 |

[17] | Jacka, S.D., Optimal stopping and the American put, Journal of mathematical finance, 1, 1-14, (1991) · Zbl 0900.90109 |

[18] | Kim, I.J., The analytic valuation of American puts, The review of financial studies, 3, 547-572, (1990) |

[19] | Landau, H.G., Heat conduction in a melting solid, Quarterly applied mathematics, 8, 81, (1950) · Zbl 0036.13902 |

[20] | Longstaff, F.A.; Schwartz, E.X., Valuing American options by simulation: a simple least-squares approach, Review of financial studies, 14, 1, 113-147, (2001) |

[21] | MacMillan, L.W., Analytic approximation for the American put option, Advances futures option, 1, 119-139, (1986) |

[22] | Merton, R.C., The theory of rational option pricing, Bell journal of economics and management science, 1, 141-183, (1973) · Zbl 1257.91043 |

[23] | Oosterlee, Cornelis W.; Coenraad, C.; Leentvaar, W.; Huang, Xinzheng, Accurate American option pricing by grid stretching and high order finite differences, (2005), Delft Institute of Applied Mathematics, Delft University of Technology The Netherlands |

[24] | Pantazopoulos, K.N.; Houstis, E.N.; Kortesis, S., Frong-tracking finite difference methods for the valuation of American options, Computational economics, 12, 255-273, (1998) · Zbl 0913.90022 |

[25] | Samuelson, P.A., Rational theory of warrant pricing, Industrial management review, 6, 13, (1979) |

[26] | Sullivan, A.M., Valuing American put options using Gaussian quadrature, Review of financial studies, 1, 75-94, (2000) |

[27] | Tavella, D.; Randall, C., Pricing financial instruments: the finite difference method, (2000), John Wiley and Sons New York |

[28] | Tavella, Domingo, Pricing financial instruments, the finite difference method, (2000), John Wiley and Sons, Inc. |

[29] | Tran, T., The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates, Mathematics of computation, 64, 210, 501-513, (1995) · Zbl 0831.65121 |

[30] | Tran, T., Local error estimates for the Galerkin method applied to strongly elliptic integral equations on open curves, SIAM J. numerical analysis, 33, 4, 1484-1493, (1996) · Zbl 0855.65118 |

[31] | Wilmott, Paul; Dewynne, Jeff; Howison, Sam, Option pricing, (1993), Oxford Financial Press · Zbl 0844.90011 |

[32] | Wu, Lixin; Kwok, Y.K., A front-fixing finite difference method for the valuation of American options, Journal of financial engineering, 6, 2, 83-97, (1997) |

[33] | Zakamouline, Valeri L., American option pricing and exercising with transaction costs, The journal of computational finance, 8, 3, 81-113, (2005) |

[34] | Zhao, Jichao; Corless, Robert M.; Davison, Matt, Financial applications of symbolically generated compact finite difference formulae, International workshop on symbolic numerical computation Proceedings, 1, 220-234, (2005) · Zbl 1284.91578 |

[35] | Zhu, S.-P., An exact and explicit solution for the valuation of American put options, Quantitative finance, 6, 3, 229-242, (2006) · Zbl 1136.91468 |

[36] | Zhu, S.-P., A new analytical-approximation formula for the optimal exercise boundary of American put options, International journal of theoretical and applied finance, 9, 7, 1141-1177, (2006) · Zbl 1140.91415 |

[37] | Zhu, S.-P.; He, Z.-W., Calculating the early exercise boundary of American put options with an approximation formula, International journal of theoretical and applied finance, 10, 7, 1203-1227, (2007) · Zbl 1153.91581 |

[38] | -P Zhu, S.; Zhang, Jin; Hong, Y.C., Numerical valuation of American puts on a dividend paying asset, Financial systems engineering IV, 9, 211-220, (2006) |

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.