IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1306.5447.html
   My bibliography  Save this paper

Explicit implied volatilities for multifactor local-stochastic volatility models

Author

Listed:
  • Matthew Lorig
  • Stefano Pagliarani
  • Andrea Pascucci

Abstract

We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, $3/2$ stochastic volatility, and SABR local-stochastic volatility.

Suggested Citation

  • Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Explicit implied volatilities for multifactor local-stochastic volatility models," Papers 1306.5447, arXiv.org, revised Nov 2014.
    • Handle: RePEc:arx:papers:1306.5447
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1306.5447
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584.
    2. Jan Obloj, 2007. "Fine-tune your smile: Correction to Hagan et al," Papers 0708.0998, arXiv.org, revised Mar 2008.
    3. Antoine Jacquier & Matthew Lorig, 2012. "The Smile of certain L\'evy-type Models," Papers 1207.1630, arXiv.org, revised Apr 2013.
    4. 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.
    5. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    6. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    7. Matthew Lorig, 2013. "The exact smile of certain local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 897-905, May.
    8. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    9. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A family of density expansions for L\'evy-type processes," Papers 1312.7328, arXiv.org.
    10. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    11. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    12. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    13. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Pricing approximations and error estimates for local L\'evy-type models with default," Papers 1304.1849, arXiv.org, revised Nov 2014.
    14. Matthew Lorig, 2011. "Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach," Papers 1109.0738, arXiv.org, revised Apr 2012.
    15. Jan Baldeaux & Alexander Badran, 2012. "Consistent Modeling of VIX and Equity Derivatives Using a 3/2 Plus Jumps Model," Research Paper Series 306, Quantitative Finance Research Centre, University of Technology, Sydney.
    16. Martin Forde & Antoine Jacquier, 2009. "Small-Time Asymptotics For Implied Volatility Under The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 861-876.
    Full references (including those not matched with items on IDEAS)

    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. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2014. "Asymptotics for $d$-dimensional L\'evy-type processes," Papers 1404.3153, arXiv.org, revised Nov 2014.
    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. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    4. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
    5. Archil Gulisashvili & Peter Laurence, 2013. "The Heston Riemannian distance function," Papers 1302.2337, arXiv.org.
    6. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
    7. Andrea Barletta & Elisa Nicolato & Stefano Pagliarani, 2019. "The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 928-966, July.
    8. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
    9. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    10. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    11. Recchioni, Maria Cristina & Iori, Giulia & Tedeschi, Gabriele & Ouellette, Michelle S., 2021. "The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications," European Journal of Operational Research, Elsevier, vol. 293(1), pages 336-360.
    12. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A family of density expansions for L\'evy-type processes," Papers 1312.7328, arXiv.org.
    13. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    14. 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.
    15. J.-P. Fouque & Y. F. Saporito, 2018. "Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 1003-1016, June.
    16. Andrew Papanicolaou, 2018. "Consistent Time-Homogeneous Modeling of SPX and VIX Derivatives," Papers 1812.05859, arXiv.org, revised Mar 2022.
    17. Alexander Badran & Beniamin Goldys, 2015. "A Market Model for VIX Futures," Papers 1504.00428, arXiv.org.
    18. Wendong Zheng & Pingping Zeng, 2015. "Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model," Papers 1504.08136, arXiv.org.
    19. Antoine Jacquier & Fangwei Shi, 2016. "The randomised Heston model," Papers 1608.07158, arXiv.org, revised Dec 2018.
    20. Weston Barger & Matthew Lorig, 2017. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-31, June.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:1306.5447. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.