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High-order short-time expansions for ATM option prices under the CGMY model

  • Jos\'e E. Figueroa-L\'opez
  • Ruoting Gong
  • Christian Houdr\'e
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    The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the present work, a novel second-order approximation for ATM option prices under the CGMY L\'evy model is derived, and then extended to a model with an additional independent Brownian component. Our results shed light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In case of an additional Brownian component, the second-order term, in time-t, is of the form $ d_{2} t^{(3-Y)/2}$, with the coefficient $d_{2}$ depending only on the overall jump intensity parameter C and the tail-heaviness parameter Y. This extends the known result that the leading term is $(\sigma/\sqrt{2\pi})t^{1/2}$, where $\sigma$ is the volatility of the continuous component. In contrast, under a pure-jump CGMY model, the dependence on the two parameters C and Y is already reflected in the leading term, which is of the form $d_{1} t^{1/Y}$. Information on the relative frequency of negative and positive jumps appears only in the second-order term, which is shown to be of the form $d_{2} t$ and whose order of decay turns out to be independent of Y. The third-order asymptotic behavior of the option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed. Our numerical results show that in most cases the second-order term significantly outperform the first-order approximation.

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    File URL: http://arxiv.org/pdf/1112.3111
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    Paper provided by arXiv.org in its series Papers with number 1112.3111.

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    Date of creation: Dec 2011
    Date of revision: Aug 2012
    Handle: RePEc:arx:papers:1112.3111
    Contact details of provider: Web page: http://arxiv.org/

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    1. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    2. Martin Forde & Antoine Jacquier, 2009. "Small-Time Asymptotics For Implied Volatility Under The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 861-876.
    3. Rama Cont & Marc Potters & Jean-Philippe Bouchaud, 1997. "Scaling in stock market data: stable laws and beyond," Science & Finance (CFM) working paper archive 9705087, Science & Finance, Capital Fund Management.
    4. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    5. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    6. 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.
    7. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55.
    8. Peter Carr & Liuren Wu, 2002. "What Type of Process Underlies Options? A Simple Robust Test," Finance 0207019, EconWPA.
    9. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    10. S. James Press, 1967. "A Compound Events Model for Security Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 317.
    11. H. Berestycki & J. Busca & I. Florent, 2002. "Asymptotics and calibration of local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 61-69.
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