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Short-time at-the-money skew and rough fractional volatility

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  • Masaaki Fukasawa

Abstract

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.

Suggested Citation

  • Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    • Handle: RePEc:taf:quantf:v:17:y:2017:i:2:p:189-198
    DOI: 10.1080/14697688.2016.1197410
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    References listed on IDEAS

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    1. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    2. Alexey Medvedev & Olivier Scaillet, 2007. "Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 427-459.
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