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

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