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The American put and European options near expiry, under Levy processes

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  • Sergei Levendorskii
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    Abstract

    We derive explicit formulas for time decay, for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a non-vanishing margin on the interval [0,T). As the riskless rate vanishes and the drift decreases accordingly so that the stock remains a martingale, the optimal exercise price goes to zero uniformly over the interval [0, T). The implications for parameters' fitting are discussed.

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    File URL: http://arxiv.org/pdf/cond-mat/0404103
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    Paper provided by arXiv.org in its series Papers with number cond-mat/0404103.

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    Date of creation: Apr 2004
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    Handle: RePEc:arx:papers:cond-mat/0404103

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    1. Peter Carr & Liuren Wu, 2003. "What Type of Process Underlies Options? A Simple Robust Test," Journal of Finance, American Finance Association, American Finance Association, vol. 58(6), pages 2581-2610, December.
    2. Andrew Matacz, 1997. "Financial modeling and option theory with the truncated Lévy process," Science & Finance (CFM) working paper archive 500035, Science & Finance, Capital Fund Management.
    3. 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, Anderson Graduate School of Management, UCLA qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    4. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 9(1), pages 31-53.
    5. Yacine Aït-Sahalia, 2002. "Telling from Discrete Data Whether the Underlying Continuous-Time Model Is a Diffusion," Journal of Finance, American Finance Association, American Finance Association, vol. 57(5), pages 2075-2112, October.
    6. Rama Cont & Marc Potters & Jean-Philippe Bouchaud, 1997. "Scaling in stock market data: stable laws and beyond," Papers cond-mat/9705087, arXiv.org.
    7. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
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    Cited by:
    1. Michael Roper, 2008. "Implied volatility explosions: European calls and implied volatilities close to expiry in exponential L\'evy models," Papers 0809.3305, arXiv.org, revised Sep 2008.
    2. Rocío Elizondo & Pablo Padilla & Mogens Bladt, 2009. "An Alternative Formula to Price American Options," Working Papers, Banco de México 2009-06, Banco de México.

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