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

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.

Keywords:

• Black–Scholes implied volatility
• Volatility smile
• Volatility skew
• Small-time asymptotics
• Asymptotic expansion
• Fractional Brownian motion
• Stochastic volatility model
• Local volatility model
• Rough fractional stochastic volatility model

Acknowledgements

The author is grateful to Martin Forde, Jim Gatheral, Mathieu Rosenbaum and the two anonymous referees for their very helpful comments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science under KAKENHI [grant number 24684006].