Volatility: Rough or Not? A Short Review
It is well-known that the assumption of constant volatility in the Black-Scholes model for pricing financial contracts is wrong and may lead to serious mispricing, especially for any exotic derivative contracts. A classic approach is to use a deterministic local volatility model to take into account the variation both in the time dimension and in the underlying asset price dimension. But the model is still not realistic in terms of forward smile (the implied volatilities of forward starting options). A stochastic volatility component must be added to correct for it. More recently, the concept of rough volatility emerged in many academic papers. Instead of using a classic Brownian motion for the stochastic volatility process, a fractional Brownian motion is used. The idea of using a fractional Brownian motion for financial time-series can be traced back to Mandelbrot, but it is only relatively recently that it has been proposed for the volatility process (and not the stock process).
After many published papers on the subject, Rama Cont and Purba Das recently added their preprint Rough Volatility: Fact or Artefact? on SSRN, raising the question whether using a rough volatility process makes sense from a practical point of view. Indeed, they show that the Hurst index measured is different for the realized volatility (even based on very short sampling intervals) and the instantaneous volatility. The authors make the conjecture that is is all due to the discretization error (a.k.a. the microstructure noise), and that in fact, a standard Brownian motion is compatible with the low Hurst index of the realized volatility.
As mentioned in their conclusion, they are not alone in showing that the rough volatility assumption may not be justified, LCG Rogers has also published a note in 2019 Things we think we know, with the same conclusion.
In a 2019 paper, Masaaki Fukasawa, Tetsuya Takabatake and Rebecca Westphal also attempt to answer the question Is volatility rough?. They present the same problem with the estimation method of Rosenbaum and Gatheral, namely that their measure may actually not measure the true Hurst index, and this could be why it is always close to H=0.1, regardless of the market index. It is shown that for specific models, the Rosenbaum and Gatheral measure is dependent on the period length considered (5 minutes vs. 1 minute lead to different Hurst indices). The authors therefore devise a more robust (and much more complex) way to estimate the Hurst index of time series, which is not so sensitive to the period length considered. They find that the market volatility, measured through the realized variance proxy is rough, with H=0.04, rougher than what Jim Gatheral suggested.
To conclude, the study of Gatheral and Rosenbaum, which motivated the use of rough volatility, was based on “biased” estimates of the Hurst index of the realized volatility of high frequency data as evidenced in the three papers cited above. Clearly then, this justification does not hold. But Fukasawa et al. suggest the volatility is rougher, and it is slightly surprising that Rama Cont and Purba Das do not take more those results into account. They shortly discuss it, but then go on proposing another biased estimate (one which seems to lead to the same results as Rosenbaum and Gatheral). To their credit, Rama Cont and Purba Das show that the microstructure noise is far from IID, while Fukasawa et al. assume it is close to IID in their method. Perhaps their indirect conclusion is that Fukasaza et al. estimates are also biased (i.e. measuring something else - although the results presented in their paper seem solid).
Now, there are also other interesting features of the rough volatility models, such as the variation of the at-the-money skew in time, which mimicks the market implied skews in practice, a feature not easily achieved by stochastic volatility models (this typically requires a multidimensional stochastic volatility process to approach it crudely).