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The implied volatility smirk

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  • Jin Zhang
  • Yi Xiang

Abstract

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility-moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.

Suggested Citation

  • Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
  • Handle: RePEc:taf:quantf:v:8:y:2008:i:3:p:263-284
    DOI: 10.1080/14697680601173444
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    Cited by:

    1. 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.
    2. Sol Kim & Geul Lee, 2017. "Lead–Lag Relationship Between Returns and Implied Moments: Evidence from KOSPI 200 Intraday Options Data," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-20, September.
    3. Elyas Elyasiani & Silvia Muzzioli & Alessio Ruggieri, 2016. "Forecasting and pricing powers of option-implied tree models: Tranquil and volatile market conditions," Department of Economics 0099, University of Modena and Reggio E., Faculty of Economics "Marco Biagi".
    4. J. C. Arismendi & Marcel Prokopczuk, 2016. "A moment-based analytic approximation of the risk-neutral density of American options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 23(6), pages 409-444, November.
    5. Fajardo, José, 2015. "Barrier style contracts under Lévy processes: An alternative approach," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 179-187.
    6. José Fajardo, 2017. "A new factor to explain implied volatility smirk," Applied Economics, Taylor & Francis Journals, vol. 49(40), pages 4026-4034, August.
    7. Huimin Zhao & Jin E. Zhang & Eric C. Chang, 2013. "The Relation between Physical and Risk-neutral Cumulants," International Review of Finance, International Review of Finance Ltd., vol. 13(3), pages 345-381, September.
    8. Jianhui Li & Sebastian A. Gehricke & Jin E. Zhang, 2019. "How do US options traders “smirk” on China? Evidence from FXI options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(11), pages 1450-1470, November.
    9. Haehean Park & Baeho Kim & Hyeongsop Shim, 2019. "A smiling bear in the equity options market and the cross‐section of stock returns," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(11), pages 1360-1382, November.
    10. Szu, Wen-Ming & Wang, Ming-Chun & Yang, Wan-Ru, 2011. "The determinants of exchange settlement practices and the implication of volatility smile: Evidence from the Taiwan Futures Exchange," International Review of Economics & Finance, Elsevier, vol. 20(4), pages 826-838, October.
    11. Kai‐Jiun Chang & Mao‐Wei Hung & Yaw‐Huei Wang & Kuang‐Chieh Yen, 2019. "Volatility information implied in the term structure of VIX," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(1), pages 56-71, January.
    12. Jansen, Jeroen & Das, Sanjiv R. & Fabozzi, Frank J., 2018. "Local volatility and the recovery rate of credit default swaps," Journal of Economic Dynamics and Control, Elsevier, vol. 92(C), pages 1-29.
    13. Yoon, Sun-Joong, 2017. "Time-varying risk aversion and return predictability," International Review of Economics & Finance, Elsevier, vol. 49(C), pages 327-339.
    14. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    15. Jean-Philippe Aguilar & Jan Korbel, 2019. "Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model," Risks, MDPI, Open Access Journal, vol. 7(2), pages 1-14, April.
    16. Xiaolan Jia & Xinfeng Ruan & Jin E. Zhang, 2021. "The implied volatility smirk of commodity options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(1), pages 72-104, January.
    17. Wenqing Bao & ChunLi Chen & Jin E. Zhang, 2013. "Option Pricing with Lie Symmetry Analysis and Similarity Reduction Method," Papers 1311.4074, arXiv.org.

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