This paper derives a higher-order asymptotic expansion of the conditional characteristic function of an Itô semimartingale increment over a shrinking time interval, under a very general setting that allows for rough volatility and jumps of arbitrary activity. The spot characteristics, including volatility and jump size functions, can exhibit local behavior like fractional Brownian motion, and the model encompasses many popular rough volatility models. The expansion shows how different features—drift, diffusion, jumps, and roughness—contribute at different orders, and importantly, that contributions from jumps and volatility roughness can be of the same asymptotic order but differ in their dependence on the characteristic function exponent, enabling their separation.

As an application, the authors develop a nonparametric estimator of the Hurst parameter of the diffusive volatility process using portfolios of short-dated options. The estimator exploits the decay rate of the argument of the conditional characteristic function as the time horizon shrinks. When the underlying asset price contains jumps, a bias correction is proposed by differencing characteristic functions over different short horizons. Numerical experiments illustrate the accuracy of the expansions and the performance of the estimator. The results have implications for volatility estimation in rough settings, where ignoring higher-order terms can lead to significant errors.