IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1208.5802.html
   My bibliography  Save this paper

Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration

Author

Listed:
  • Jean-Pierre Fouque
  • Matthew Lorig
  • Ronnie Sircar

Abstract

Multiscale stochastic volatility models have been developed as an efficient way to capture the principle effects on derivative pricing and portfolio optimization of randomly varying volatility. The recent book Fouque, Papanicolaou, Sircar and S{\o}lna (2011, CUP) analyzes models in which the volatility of the underlying is driven by two diffusions -- one fast mean-reverting and one slow-varying, and provides a first order approximation for European option prices and for the implied volatility surface, which is calibrated to market data. Here, we present the full second order asymptotics, which are considerably more complicated due to a terminal layer near the option expiration time. We find that, to second order, the implied volatility approximation depends quadratically on log-moneyness, capturing the convexity of the implied volatility curve seen in data. We introduce a new probabilistic approach to the terminal layer analysis needed for the derivation of the second order singular perturbation term, and calibrate to S&P 500 options data.

Suggested Citation

  • Jean-Pierre Fouque & Matthew Lorig & Ronnie Sircar, 2012. "Second Order Multiscale Stochastic Volatility Asymptotics: Stochastic Terminal Layer Analysis & Calibration," Papers 1208.5802, arXiv.org, revised Sep 2015.
  • Handle: RePEc:arx:papers:1208.5802
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1208.5802
    File Function: Latest version
    Download Restriction: no
    ---><---

    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, October.
    2. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
    3. B. LeBaron, 2001. "Stochastic volatility as a simple generator of apparent financial power laws and long memory," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 621-631.
    4. 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.
    5. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    6. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    7. K. Ronnie Sircar & George Papanicolaou, 1999. "Stochastic volatility, smile & asymptotics," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(2), pages 107-145.
    8. Blake LeBaron, 2001. "Volatility," Computing in Economics and Finance 2001 108, Society for Computational Economics.
    9. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    10. Max O. Souza & Jorge P. Zubelli, 2007. "On The Asymptotics Of Fast Mean-Reversion Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(05), pages 817-835.
    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. Jean-Pierre Fouque & Matthew Lorig & Ronnie Sircar, 2016. "Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration," Finance and Stochastics, Springer, vol. 20(3), pages 543-588, July.
    2. Park, Sang-Hyeon & Kim, Jeong-Hoon, 2013. "A semi-analytic pricing formula for lookback options under a general stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2537-2543.
    3. Martin Tegnér & Rolf Poulsen, 2018. "Volatility Is Log-Normal—But Not for the Reason You Think," Risks, MDPI, vol. 6(2), pages 1-16, April.
    4. Wong, Hoi Ying & Chan, Chun Man, 2007. "Lookback options and dynamic fund protection under multiscale stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 357-385, May.
    5. 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.
    6. Hamza Guennoun & Antoine Jacquier & Patrick Roome & Fangwei Shi, 2014. "Asymptotic behaviour of the fractional Heston model," Papers 1411.7653, arXiv.org, revised Aug 2017.
    7. Matthew Lorig, 2010. "Time-Changed Fast Mean-Reverting Stochastic Volatility Models," Papers 1010.5203, arXiv.org, revised Apr 2012.
    8. Hillebrand, Eric & Schnabl, Gunther & Ulu, Yasemin, 2009. "Japanese foreign exchange intervention and the yen-to-dollar exchange rate: A simultaneous equations approach using realized volatility," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 19(3), pages 490-505, July.
    9. 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.
    10. Jeonggyu Huh, 2018. "Measuring Systematic Risk with Neural Network Factor Model," Papers 1809.04925, arXiv.org.
    11. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    12. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    13. McAleer, Michael & Medeiros, Marcelo C., 2008. "A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries," Journal of Econometrics, Elsevier, vol. 147(1), pages 104-119, November.
    14. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    15. Jaume Masoliver & Josep Perello, 2006. "Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 423-433.
    16. Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    17. Aït-Sahalia, Yacine & Mancini, Loriano, 2008. "Out of sample forecasts of quadratic variation," Journal of Econometrics, Elsevier, vol. 147(1), pages 17-33, November.
    18. Carlos Fuertes & Andrew Papanicolaou, 2012. "Implied Filtering Densities on Volatility's Hidden State," Papers 1203.6631, arXiv.org, revised Mar 2017.
    19. Oliver Pfante & Nils Bertschinger, 2019. "Information-Theoretic Analysis Of Stochastic Volatility Models," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-21, February.
    20. Jeon, Jaegi & Kim, Geonwoo & Huh, Jeonggyu, 2021. "An asymptotic expansion approach to the valuation of vulnerable options under a multiscale stochastic volatility model," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:1208.5802. 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.