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An Alternative Formula to Price American Options

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  • Rocío Elizondo
  • Pablo Padilla
  • Mogens Bladt
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    Abstract

    We give a new way to price American options, using Samuelson´s formula. We first obtain the option price corresponding to a European option at time t, weighting it by the probability that the underlying asset takes the value S at time t. This factor is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can introduce systematically the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamic system in the form of a set of ordinary differential equations we only have to solve a partial differential equation, for the transition probability density. In this context, we verify, for the sake of consistency, that this formula is consistent with the Black-Scholes model.

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    File URL: http://www.banxico.org.mx/publicaciones-y-discursos/publicaciones/documentos-de-investigacion/banxico/%7BF98179CF-D9C5-9B47-1BB9-EE032F102577%7D.pdf
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    Bibliographic Info

    Paper provided by Banco de México in its series Working Papers with number 2009-06.

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    Date of creation: Aug 2009
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    Handle: RePEc:bdm:wpaper:2009-06

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    Related research

    Keywords: American options; Fokker-Planck; Black-Scholes; Samuelson; density probability function.;

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    1. Peter Carr & Robert Jarrow & Ravi Myneni, 1992. "Alternative Characterizations Of American Put Options," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 87-106.
    2. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    3. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-24, December.
    4. Robert A. Jarrow, 1988. "Preferences, Continuity, and the Arbitrage Pricing Theory," Review of Financial Studies, Society for Financial Studies, vol. 1(2), pages 159-172.
    5. Francesc Llerena-Garrés, 2000. "Una nota sobre valoración de opciones americanas y arbitraje," Investigaciones Economicas, Fundación SEPI, vol. 24(1), pages 207-218, January.
    6. Bladt, Mogens & Rydberg, Tina Hviid, 1998. "An actuarial approach to option pricing under the physical measure and without market assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 65-73, May.
    7. Sergei Levendorskii, 2004. "The American put and European options near expiry, under Levy processes," Papers cond-mat/0404103, arXiv.org.
    8. Broadie, Mark & Detemple, Jerome, 1996. "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," Review of Financial Studies, Society for Financial Studies, vol. 9(4), pages 1211-50.
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