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The Random-Time Binomial Model

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  • Leisen, Dietmar
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    Abstract

    In this paper we study Binomial Models with random time steps. We explain, how calculating values for European and American Call and Put options is straightforward for the Random-Time Binomial Model. We present the conditions to ensure weak-convergence to the Black-Scholes setup and convergence of the values for European and American put options. Differently to the CRR-model the convergence behaviour is extremely smooth in our model. By using extrapolation we therefore achieve order of convergence two. This way it is an efficient tool for pricing purposes in the Black-Scholes setup, since the CRR model and its extrapolations typically achieve order one. Moreover our model allows in a straightforward manner to construct approximations to jump-diffusions. The simple valuation approaches and the convergence properties carry immediately over from the Black-Scholes case.

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    File URL: http://www.wiwi.uni-bonn.de/bgsepapers/bonsfb/bonsfb399.pdf
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    Bibliographic Info

    Paper provided by University of Bonn, Germany in its series Discussion Paper Serie B with number 399.

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    Length: pages
    Date of creation: Feb 1997
    Date of revision:
    Handle: RePEc:bon:bonsfb:399

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    Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
    Fax: +49 228 73 6884
    Web page: http://www.bgse.uni-bonn.de

    Related research

    Keywords: binomial model; order of convergence; smoothing; extrapolation; jump-diffusion;

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    References

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    1. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    2. Sondermann, Dieter, 1987. "Currency options: Hedging and social value," European Economic Review, Elsevier, vol. 31(1-2), pages 246-256.
    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    4. Ball, Clifford A & Torous, Walter N, 1985. " On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-73, March.
    5. Leisen, Dietmar, 1996. "Pricing the American Put Option: A Detailed Convergence Analysis for Binomial Models," Discussion Paper Serie B 366, University of Bonn, Germany, revised Jul 1996.
    6. L.C.G. Rogers & E.J. Stapleton, 1997. "Fast accurate binomial pricing," Finance and Stochastics, Springer, vol. 2(1), pages 3-17.
    7. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-51, July.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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