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

TL;DR

This paper develops a fractional stochastic volatility model consistent with the observed power law of implied volatility skew at short maturities, introducing a new asymptotic expansion approach.

Contribution

It constructs a dynamically consistent fractional volatility model with Hurst parameter less than half and derives a novel asymptotic expansion for implied volatility as maturity approaches zero.

Findings

Fractional Brownian motion with H<0.5 models short-term skew.

Asymptotic expansion valid for general models.

Standard local-stochastic volatility models are not dynamically consistent.

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.