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

  • Jean-Pierre Fouque
  • Matthew Lorig
  • Ronnie Sircar
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    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.

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    File URL: http://arxiv.org/pdf/1208.5802
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    Paper provided by arXiv.org in its series Papers with number 1208.5802.

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    Date of creation: Aug 2012
    Date of revision: Apr 2015
    Handle: RePEc:arx:papers:1208.5802
    Contact details of provider: Web page: http://arxiv.org/

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    1. Masaaki Fukasawa, 2011. "Asymptotic analysis for stochastic volatility: martingale expansion," Finance and Stochastics, Springer, vol. 15(4), pages 635-654, December.
    2. Chernov, Mikhail & Gallant, A. Ronald & Ghysels, Eric & Tauchen, George, 2002. "Alternative Models for Stock Price Dynamic," Working Papers 02-03, Duke University, Department of Economics.
    3. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    4. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    5. 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.
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