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The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model

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  • Jaehyuk Choi
  • Lixin Wu

Abstract

This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black-Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error in a way similar to the control variate method because the CEV model is the zero vol-of-vol limit of the SABR model. Moreover, the CEV volatility approximation yields a finite value at a zero strike and thus conveniently leads to a small-time asymptotics for the mass at zero. The numerical results compare favorably with the BS volatility approximations in terms of the approximation accuracy, small-strike volatility asymptotics, and no-arbitrage region.

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  • Jaehyuk Choi & Lixin Wu, 2019. "The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model," Papers 1911.13123, arXiv.org, revised Jun 2021.
    • Handle: RePEc:arx:papers:1911.13123
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    1. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-219, March.
    2. Archil Gulisashvili & Blanka Horvath & Antoine Jacquier, 2018. "Mass at zero in the uncorrelated SABR model and implied volatility asymptotics," Quantitative Finance, Taylor & Francis Journals, vol. 18(10), pages 1753-1765, October.
    3. Bin Chen & Cornelis W. Oosterlee & Hans Van Der Weide, 2012. "A Low-Bias Simulation Scheme For The Sabr Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 1-37.
    4. Jaehyuk Choi & Chenru Liu & Byoung Ki Seo, 2019. "Hyperbolic normal stochastic volatility model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 39(2), pages 186-204, February.
    5. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    6. Nian Yang & Xiangwei Wan, 2018. "The survival probability of the SABR model: asymptotics and application," Quantitative Finance, Taylor & Francis Journals, vol. 18(10), pages 1767-1779, October.
    7. Jaehyuk Choi & Lixin Wu, 2021. "A note on the option price and ‘Mass at zero in the uncorrelated SABR model and implied volatility asymptotics’," Quantitative Finance, Taylor & Francis Journals, vol. 21(7), pages 1083-1086, July.
    8. Yang, Nian & Chen, Nan & Liu, Yanchu & Wan, Xiangwei, 2017. "Approximate arbitrage-free option pricing under the SABR model," Journal of Economic Dynamics and Control, Elsevier, vol. 83(C), pages 198-214.
    9. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    10. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    11. Ning Cai & Yingda Song & Nan Chen, 2017. "Exact Simulation of the SABR Model," Operations Research, INFORMS, vol. 65(4), pages 931-951, August.
    12. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    13. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    14. Nan Chen & Nian Yang, 2019. "The principle of not feeling the boundary for the SABR model," Quantitative Finance, Taylor & Francis Journals, vol. 19(3), pages 427-436, March.
    15. Manuela Larguinho & José Carlos Dias & Carlos A. Braumann, 2013. "On the computation of option prices and Greeks under the CEV model," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 907-917, May.
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    Cited by:

    1. Jaehyuk Choi & Byoung Ki Seo, 2023. "Option pricing under the normal SABR model with Gaussian quadratures," Papers 2301.02797, arXiv.org.
    2. Jaehyuk Choi & Lixin Wu, 2021. "A note on the option price and ‘Mass at zero in the uncorrelated SABR model and implied volatility asymptotics’," Quantitative Finance, Taylor & Francis Journals, vol. 21(7), pages 1083-1086, July.
    3. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2022. "A Black–Scholes user's guide to the Bachelier model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 959-980, May.

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