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A new look at short-term implied volatility in asset price models with jumps

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  • Aleksandar Mijatovi'c
  • Peter Tankov

Abstract

We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential L\'evy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a non-constant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal-Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model-independent slope. This result gives a theoretical justification for the preference of the infinite variation L\'evy models over the finite variation ones in the calibration based on short-maturity option prices.

Suggested Citation

  • Aleksandar Mijatovi'c & Peter Tankov, 2012. "A new look at short-term implied volatility in asset price models with jumps," Papers 1207.0843, arXiv.org, revised Jul 2012.
  • Handle: RePEc:arx:papers:1207.0843
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    File URL: http://arxiv.org/pdf/1207.0843
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    References listed on IDEAS

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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, vol. 58(6), pages 2581-2610, December.
    2. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    3. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12.
    4. Jos'e E. Figueroa-L'opez & Ruoting Gong & Christian Houdr'e, 2011. "High-order short-time expansions for ATM option prices under the CGMY model," Papers 1112.3111, arXiv.org, revised Aug 2012.
    5. Antoine Jacquier & Martin Keller-Ressel & Aleksandar Mijatovic, 2011. "Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models," Papers 1108.3998, arXiv.org.
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    Citations

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    Cited by:

    1. Masaaki Fukasawa, 2015. "Short-time at-the-money skew and rough fractional volatility," Papers 1501.06980, arXiv.org.
    2. Jacquier, Antoine & Roome, Patrick, 2016. "Large-maturity regimes of the Heston forward smile," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1087-1123.
    3. Jos'e E. Figueroa-L'opez & Sveinn 'Olafsson, 2014. "Short-time expansions for close-to-the-money options under a L\'evy jump model with stochastic volatility," Papers 1404.0601, arXiv.org, revised Oct 2014.
    4. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    5. Jos'e E. Figueroa-L'opez & Sveinn 'Olafsson, 2015. "Short-time asymptotics for the implied volatility skew under a stochastic volatility model with L\'evy jumps," Papers 1502.02595, arXiv.org, revised Dec 2015.
    6. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    7. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
    8. Stefan Gerhold & I. Cetin Gulum & Arpad Pinter, 2013. "Small-maturity asymptotics for the at-the-money implied volatility slope in L\'evy models," Papers 1310.3061, arXiv.org, revised May 2016.

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