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Large deviations for the extended Heston model: the large-time case

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  • Antoine Jacquier
  • Aleksandar Mijatovic

Abstract

We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.

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File URL: http://arxiv.org/pdf/1203.5020
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1203.5020.

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Date of creation: Mar 2012
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Handle: RePEc:arx:papers:1203.5020

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Web page: http://arxiv.org/

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References

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  1. Agnieszka Janek & Tino Kluge & RafaƂ Weron & Uwe Wystup, 2010. "FX Smile in the Heston Model," SFB 649 Discussion Papers SFB649DP2010-047, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  2. Jim Gatheral & Antoine Jacquier, 2011. "Convergence of Heston to SVI," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 11(8), pages 1129-1132.
  3. Viatcheslav Gorovoi & Vadim Linetsky, 2004. "Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 14(1), pages 49-78.
  4. Martin Forde & Antoine Jacquier & Aleksandar Mijatovic, 2009. "Asymptotic formulae for implied volatility in the Heston model," Papers 0911.2992, arXiv.org, revised May 2010.
  5. Martin Forde & Antoine Jacquier, 2011. "The large-maturity smile for the Heston model," Finance and Stochastics, Springer, Springer, vol. 15(4), pages 755-780, December.
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Cited by:
  1. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
  2. Archil Gulisashvili & Josef Teichmann, 2014. "The G\"{a}rtner-Ellis theorem, homogenization, and affine processes," Papers 1406.3716, arXiv.org.

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