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The Black-Scholes equation revisited: asymptotic expansions and singular perturbations. (English) Zbl 1124.91342
Summary: In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.

91G20 Derivative securities (option pricing, hedging, etc.)
35K55 Nonlinear parabolic equations
35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI
[1] DOI: 10.1086/260062 · Zbl 1092.91524 · doi:10.1086/260062
[2] X.Chen, J.Chadam, and R.Stamicar(2000 ): The Optimal Exercise Boundary for American Put Options: Analytic and Numerical Approximations . Preprint, downloaded fromhttp://www.math.pitt.edu/ xfc/Option/CCSFinal.ps.
[3] Crank J., Free and Moving Boundary Problems (1984)
[4] DOI: 10.1111/1467-9965.02008 · Zbl 1031.91047 · doi:10.1111/1467-9965.02008
[5] Hoggard T., Adv. Futures Opt. Res. 7 pp 21– (1994)
[6] Kevorkian J., Multiple Scale and Singular Perturbation Methods (1996) · Zbl 0846.34001 · doi:10.1007/978-1-4612-3968-0
[7] DOI: 10.1111/1467-9590.00183 · Zbl 1152.91522 · doi:10.1111/1467-9590.00183
[8] DOI: 10.1080/135048698334673 · Zbl 1009.91025 · doi:10.1080/135048698334673
[9] Sevcovic D., European J. Appl. Math. 12 pp 2– (2001) · Zbl 1113.91315 · doi:10.1017/S0956792501004338
[10] Stamicar R., Canadian Applied Mathematics Quarterly 7 pp 427– (1999)
[11] Sychev V. V., Asymptotic Theory of Separated Flows (1998) · Zbl 0944.76003 · doi:10.1017/CBO9780511983764
[12] van Dyke M., Perturbation Methods in Fluid Mechanics (1975) · Zbl 0329.76002
[13] Whalley A. E., Risk Magazine 7 pp 82– (1994)
[14] Whalley A. E., Proceedings of the 7th European Conference on Mathematics in Industry pp 427– (1994)
[15] DOI: 10.1111/1467-9965.00034 · Zbl 0885.90019 · doi:10.1111/1467-9965.00034
[16] M.Widdicks(2002 ): PhD Dissertation , University of Manchester.
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