##
**A moving boundary approach to American option pricing.**
*(English)*
Zbl 1181.91300

Summary: This paper describes a method to solve the free-boundary problem that arises in the pricing of American options. Most numerical methods for American option pricing exploit the representation of the option price as the expected pay-off under the risk-neutral measure and calculate the price for a given time to expiration and stock price. They do not solve the related free-boundary problem explicitly. The advantage of solving the free-boundary problem is that it provides the entire price function as well as the optimal exercise boundary explicitly. Our approach, which we term the moving boundary approach, is based on using a boundary guess and the value associated with the guess to construct an improved boundary. It is also shown that on iteration, the sequence of boundaries converge monotonically to the optimal exercise boundary. Examples illustrating the convergence behavior as well as discussions providing insight into the method are also presented. Finally, we compare runtimes and speeds with other methods that solve the free-boundary problem and compute the optimal boundaries explicitly, like the front-fixing method, penalty method, method based on the integral representations and the method by M. Brennan and E. Schwartz [“The valuation of American put options”, J. Finance 32 (2), 449–462 (1977)].

### MSC:

91G10 | Portfolio theory |

91G80 | Financial applications of other theories |

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

### Keywords:

American option pricing; stochastic control; Hamilton-Jacobi-Bellman equation; free-boundary
PDF
BibTeX
XML
Cite

\textit{K. Muthuraman}, J. Econ. Dyn. Control 32, No. 11, 3520--3537 (2008; Zbl 1181.91300)

Full Text:
DOI

### References:

[1] | AitSahlia, F.; Carr, P., American options: a comparison of numerical methods, (), 67-87 · Zbl 0898.90028 |

[2] | Bensoussan, A.; Lions, J., Impulsive control and quasi-variational inequalities, (1982), Dunod Paris |

[3] | Black, F., The pricing of commodity contracts, Journal of financial economics, 3, 167-179, (1976) |

[4] | Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 3, 637-654, (1973) · Zbl 1092.91524 |

[5] | Brennan, M.J.; Schwartz, E.S., The valuation of American put options, Journal of finance, 32, 2, 449-462, (1977) |

[6] | Broadie, M.; Detemple, J., American option valuation: approximations, and a comparison of existing methods, Review of financial studies, 9, 4, 1211-1250, (1996) |

[7] | Broadie, M.; Detemple, J., Option pricing: valuation models and applications, Management science, 50, 9, 1145-1177, (2004) |

[8] | Broadie, M.; Glasserman, P., Pricing American-style securities by simulation, Journal of economic dynamics and control, 21, 1323-1352, (1997) · Zbl 0901.90009 |

[9] | Burnett, D.S., Finite element analysis: from concepts to applications, (1987), Addison-Wesley Reading, MA · Zbl 0664.65104 |

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

[11] | Chiarella, C.; El-Hassan, N.; Kucera, A., Evaluation of American option prices in a path integral framework using fourier – hermite series expansions, Journal of economic dynamics and control, 23, 1387-1424, (1999) · Zbl 1016.91049 |

[12] | Courtadon, G.R., A more accurate finite difference approximation for the valuation of options, Journal of financial and quantitative analysis, 17, 697-703, (1982) |

[13] | Cox, J.C.; Ross, S.A.; Rubinstein, M., Option pricing: a simplified approach, Journal of financial economics, 7, 229-263, (1979) · Zbl 1131.91333 |

[14] | Duffie, D., Dynamic asset pricing theory, (2001), Princeton University Press New York · Zbl 1140.91041 |

[15] | Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903 |

[16] | Friedman, A., Variational principles and free-boundary problems, (1982), Wiley New York · Zbl 0564.49002 |

[17] | Garman, M.; Kohlhagen, S., Foreign currency option values, Journal of international money and finance, 2, 231-237, (1983) |

[18] | Geske, R.; Johnson, H.E., The American put option valued analytically, Journal of finance, 39, 5, 1511-1524, (1984) |

[19] | Grabbe, J.O., The pricing of call and put options on foreign exchange, Journal of international money and finance, 22, 239-254, (1983) |

[20] | Harrison, J.M.; Kreps, D., Martingales and arbitrage in multiperiod security markets, Journal of economic theory, 20, 381-408, (1979) · Zbl 0431.90019 |

[21] | Harrison, J.M.; Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic processes and applications, 11, 215-260, (1981) · Zbl 0482.60097 |

[22] | Huang, J.; Subrahmanyam, M.; Yu, G., Pricing and hedging American options: a recursive integration method, Review of financial studies, 9, 277-300, (1996) |

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

[24] | Jaillet, P.; Lamberton, D.; Lapeyre, B., Variational inequalities and the pricing of American options, Acta applicandae mathematicae, 21, 263-289, (1990) · Zbl 0714.90004 |

[25] | Ju, N., Pricing an American option by approximating its early exercise boundary as a multipiece exponential function, Review of financial studies, 11, 3, 627-646, (1998) |

[26] | Karatzas, I.; Shreve, S.E., Methods of mathematical finance, (1998), Springer Berlin · Zbl 0941.91032 |

[27] | Kim, I., The analytic valuation of American options, Review of financial studies, 3, 547-572, (1990) |

[28] | Kumar, S.; Muthuraman, K., A numerical method for solving stochastic singular control problems, Operations research, 52, 4, 563-582, (2004) · Zbl 1165.49305 |

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

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

[31] | McDonald, R.; Schroder, M., A parity result for American options, Journal of computational finance, 1, 5-13, (1998) |

[32] | McKean, H.P., Appendix: a free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial management review, 6, 32-39, (1965) |

[33] | Moerbeke, P.L.J.V., On optimal stopping and free boundary problems, Archive for rational mechanics and analysis, 60, 101-148, (1976) · Zbl 0336.35047 |

[34] | Muthuraman, K.; Kumar, S., Multi-dimensional portfolio optimization with proportional transaction costs, Mathematical finance, 16, 2, 301-335, (2006) · Zbl 1145.91352 |

[35] | Myneni, R., The pricing of the American option, The annals of applied probability, 2, 1, 1-23, (1992) · Zbl 0753.60040 |

[36] | 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) |

[37] | Oden, J.T.; Reddy, J.N., An introduction to the mathematical theory of finite elements, (1978), Academic Press New York · Zbl 0336.35001 |

[38] | Schwartz, E., The valuation of warrants: implementing a new approach, Journal of financial economics, 4, 79-93, (1977) |

[39] | Tilley, J.A., Valuing American options in a path simulation model, Transactions of the society of actuaries, 45, 83-104, (1993) |

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

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.