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Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results

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  • Leif Andersen
  • Alexander Lipton

Abstract

Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Levy processes and derive novel numerical methods based on judicial finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the L\'evy process based models is particularly clear when the Lewis-Lipton formula is used. Our main conclusion is that studying asymptotic properties of the implied volatility, while theoretically exciting, is not always practically useful because the domain of validity of many asymptotic expressions is small.

Suggested Citation

  • Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    • Handle: RePEc:arx:papers:1206.6787
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    References listed on IDEAS

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    Cited by:

    1. Antoine Jacquier & Matthew Lorig, 2012. "From characteristic functions to implied volatility expansions," Papers 1207.0233, arXiv.org, revised Jun 2014.
    2. Francesco Caravenna & Jacopo Corbetta, 2015. "The asymptotic smile of a multiscaling stochastic volatility model," Papers 1501.03387, arXiv.org, revised Jul 2017.
    3. 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.
    4. Sergio Albeverio & Francesco Cordoni & Luca Persio & Gregorio Pellegrini, 2019. "Asymptotic expansion for some local volatility models arising in finance," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 527-573, December.
    5. Antoine Jacquier & Lorenzo Torricelli, 2019. "Anomalous diffusions in option prices: connecting trade duration and the volatility term structure," Papers 1908.03007, arXiv.org, revised Apr 2020.

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