Rough volatility: fact or artefact?

Abstract

We investigate the statistical evidence for the use of 'rough' fractional processes with Hurst exponent H < 0.5 for the modeling of volatility of financial assets, using a model-free approach. We introduce a non-parametric method for estimating the roughness of a function based on discrete sample, using the concept of normalized p-th variation along a sequence of partitions. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional processes. We then apply this method to estimate the roughness of realized volatility signals based on high-frequency observations. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Comparison of roughness estimates for realized and instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility always exhibits 'rough' behaviour with an apparent Hurst index H < 0.5. These results suggest that the origin of the roughness observed in realized volatility time-series lies in the microstructure noise rather than the volatility process itself.

Probability Density of Lognormal Fractional SABR Model

“…The main observation in Gatheral et al (2018) is that in using the square root of the realized variance as a proxy for the instantaneous volatility, the logarithm of the volatility process behaves like a fractional Brownian motion in almost any time scale of frequency. The Hurst exponent H inferred from the time series data is less than a half; indeed, H ≈ 0.1, see also Cont and Das (2022) and Rogers (2019). This observation of a small Hurst exponent in the volatility process analyzes the model as more technical and challenging from a stochastic analysis point of view.…”

“…Somewhat on the contrary, in a recent study in Gatheral et al (2018), the Hurst exponent H is estimated as being less than a half; thereby indicating antipersistency as opposed to the persistency of the volatility process. For a more detailed and in-depth consideration of this issue, we refer interested readers to the discussions in Cont and Das (2022) and Rogers (2019). It is also worth mentioning that generalizations of the Heston model to the fractional version have been considered in El Euch and Rosenbaum (2019) and Guennoun et al (2018).…”

The fractional volatility model and rough volatility

“…Hence, if the volatility is driven by fractional Brownian motion, a Hurst index smaller than 1 2 would seem to contradict the market long-range dependent volatility (volatility clustering) [11] [12]. The way the realized volatility is measured at high frequency has however been criticized by several authors [13] [14] [15] suggesting that the origin of the roughness lies in the microstructure noise [16] rather than on the actual volatility process.…”