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Short-time asymptotics for marginal distributions of semimartingales

Author

Listed:
  • Amel Bentata

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Rama Cont

    (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

Abstract

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale.

Suggested Citation

  • Amel Bentata & Rama Cont, 2012. "Short-time asymptotics for marginal distributions of semimartingales," Working Papers hal-00667112, HAL.
    • Handle: RePEc:hal:wpaper:hal-00667112
    Note: View the original document on HAL open archive server: https://hal.science/hal-00667112v2
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    References listed on IDEAS

    as
    1. 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.
    2. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
    3. Amel Bentata & Rama Cont, 2009. "Forward equations for option prices in semimartingale models," Working Papers hal-00445641, HAL.
    4. Johannes Muhle-Karbe & Marcel Nutz, 2010. "Small-Time Asymptotics of Option Prices and First Absolute Moments," Papers 1006.2294, arXiv.org, revised Jun 2011.
    5. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
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    Citations

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    Cited by:

    1. Torben G. Andersen & Nicola Fusari & Viktor Todorov, 2017. "Short-Term Market Risks Implied by Weekly Options," Journal of Finance, American Finance Association, vol. 72(3), pages 1335-1386, June.
    2. Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Papers 1208.4282, arXiv.org.
    3. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
    4. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    5. Peter Friz & Stefan Gerhold & Arpad Pinter, 2016. "Option Pricing in the Moderate Deviations Regime," Papers 1604.01281, arXiv.org.

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    More about this item

    Keywords

    semimartingale; short-time asymptotics; marginal distribution; short maturity asymptotics; Levy process; option pricing;
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