IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v16y2013i01ns0219024913500015.html
   My bibliography  Save this article

Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results

Author

Listed:
  • LEIF ANDERSEN

    (Bank of America Merrill Lynch, One Bryant Park, New York, NY 10036, USA)

  • ALEXANDER LIPTON

    (Bank of America Merrill Lynch, 2 King Edward Street, London, EC1A 1HQ, UK)

Abstract

Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. 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, and 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 Lévy processes and derive novel numerical methods based on judicious 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 Lévy 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 is not always practically useful because the domain of validity of many asymptotic expressions is small.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijtafx:v:16:y:2013:i:01:n:s0219024913500015
    DOI: 10.1142/S0219024913500015
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024913500015
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024913500015?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook0501, December.
    2. Antoine Jacquier & Aleksandar Mijatović, 2014. "Large Deviations for the Extended Heston Model: The Large-Time Case," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(3), pages 263-280, September.
    3. Amel Bentata & Rama Cont, 2012. "Short-time asymptotics for marginal distributions of semimartingales," Working Papers hal-00667112, HAL.
    4. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
    5. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    6. Sergei Levendorskii, 2004. "The American put and European options near expiry, under Levy processes," Papers cond-mat/0404103, arXiv.org.
    7. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    8. Amel Bentata & Rama Cont, 2012. "Short-time asymptotics for marginal distributions of semimartingales," Papers 1202.1302, arXiv.org.
    9. Alexey Medvedev & Olivier Scaillet, 2007. "Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 427-459.
    10. Eric Benhamou, 2002. "Option pricing with Levy Process," Finance 0212006, University Library of Munich, Germany.
    11. Michael Roper, 2008. "Implied volatility explosions: European calls and implied volatilities close to expiry in exponential L\'evy models," Papers 0809.3305, arXiv.org, revised Sep 2008.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jorge González Cázares & Aleksandar Mijatović, 2022. "Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation," Finance and Stochastics, Springer, vol. 26(4), pages 671-732, October.
    2. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    3. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    4. 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.
    5. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility," Finance and Stochastics, Springer, vol. 20(1), pages 219-265, January.
    6. José Figueroa-López & Sveinn Ólafsson, 2016. "Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility," Finance and Stochastics, Springer, vol. 20(1), pages 219-265, January.
    7. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps," Finance and Stochastics, Springer, vol. 20(4), pages 973-1020, October.
    8. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    9. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
    10. Caravenna, Francesco & Corbetta, Jacopo, 2018. "The asymptotic smile of a multiscaling stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1034-1071.
    11. Alexander Lipton, 2020. "Old Problems, Classical Methods, New Solutions," Papers 2003.06903, arXiv.org.
    12. 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.
    13. George Bouzianis & Lane P. Hughston, 2019. "Determination Of The Lévy Exponent In Asset Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-18, February.
    14. 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.
    15. Archil Gulisashvili & Josep Vives, 2014. "Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models," Papers 1403.5302, arXiv.org.
    16. Michele Azzone & Roberto Baviera, 2021. "Short-time implied volatility of additive normal tempered stable processes," Papers 2108.02447, arXiv.org.
    17. Svetlana Boyarchenko & Sergei Levendorskii, 2023. "Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models," Papers 2312.03915, arXiv.org.
    18. Philipp Mayer & Natalie Packham & Wolfgang Schmidt, 2015. "Static hedging under maturity mismatch," Finance and Stochastics, Springer, vol. 19(3), pages 509-539, July.
    19. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    20. 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.
    21. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    22. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    23. Ma, Chunsheng, 2023. "Multivariate exponential power Lévy processes and random fields," Statistics & Probability Letters, Elsevier, vol. 197(C).
    24. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.
    25. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Peter Friz & Stefan Gerhold & Arpad Pinter, 2016. "Option Pricing in the Moderate Deviations Regime," Papers 1604.01281, arXiv.org.
    3. Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Papers 1208.4282, arXiv.org.
    4. 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.
    5. Alexander Lipton & Andrey Gal & Andris Lasis, 2013. "Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results," Papers 1312.5693, arXiv.org.
    6. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    7. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    8. Peng Cheng & Olivier Scaillet, 2002. "Linear-Quadratic Jump-Diffusion Modeling with Application to Stochastic Volatility," FAME Research Paper Series rp67, International Center for Financial Asset Management and Engineering.
    9. Alexander Lipton, 2023. "Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results," Papers 2309.04547, arXiv.org.
    10. Likuan Qin & Vadim Linetsky, 2016. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing," Operations Research, INFORMS, vol. 64(1), pages 99-117, February.
    11. Lorenzo Torricelli, 2016. "Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes," Review of Derivatives Research, Springer, vol. 19(1), pages 1-39, April.
    12. Martino Grasselli, 2017. "The 4/2 Stochastic Volatility Model: A Unified Approach For The Heston And The 3/2 Model," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1013-1034, October.
    13. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    14. Alessandro Gnoatto & Martino Grasselli, 2013. "An analytic multi-currency model with stochastic volatility and stochastic interest rates," Papers 1302.7246, arXiv.org, revised Mar 2013.
    15. Alessandro Gnoatto, 2017. "Coherent Foreign Exchange Market Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-29, February.
    16. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    17. Jan Baldeaux & Alexander Badran, 2014. "Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(4), pages 299-312, September.
    18. Masaaki Fukasawa, 2015. "Short-time at-the-money skew and rough fractional volatility," Papers 1501.06980, arXiv.org.
    19. Gerald H. L. Cheang & Len Patrick Dominic M. Garces, 2020. "Representation of Exchange Option Prices under Stochastic Volatility Jump-Diffusion Dynamics," Papers 2002.10202, arXiv.org.
    20. Minqiang Li & Fabio Mercurio, 2015. "Analytic Approximation of Finite‐Maturity Timer Option Prices," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 35(3), pages 245-273, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:ijtafx:v:16:y:2013:i:01:n:s0219024913500015. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.