Volatility is rough:
Since the publication of the Volatility is rough paper, research on this topic has flourished. News, events and related papers can be found at Rough Volatility.
Mathieu Rosenbaum and his coauthors show that log-volatility behaves as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale, and that long memory in volatility might be an artifact from the application of classical statistical procedures to assets that follow the rough fractional stochastic volatility (RFSV) model.
So this model not only describes accurately the scaling found in time series, but also gives rise to pricing models that predict the implied volatility surface with remarkable precision. This Python notebook by Jim Gatheral describes in detail both the time series analysis and the pricing of options under the RFSV model:
Comparison of SPX volatility and simulated (RFSV model):
The simulated and actual graphs look very similar; in both there are persistent periods of high volatility alternating with low volatility periods.
H∼0.1 generates very rough looking sample paths (compared with H=1/2 for Brownian motion), therefore the name “rough volatility”.
We can find fractal-type behavior, because the graph of volatility over a small time period looks like the same graph over a much longer time period. This feature of volatility has been investigated both empirically and theoretically.
As above, so below:
Given that rough volatility gives us a better description of the historical returns, it is expected that it will provide us with a better prediction of the future behavior of the returns and volatility; in particular, it should be better at estimating the implied volatility surface. A series of articles (below) show how to adapt the Heston model to the RFSV process, and Mathieu and Omar show in The microstructural foundations of leverage effect and rough volatility that the typical behavior of market participants at the high frequency scale will lead to the leverage effect and rough volatility (the model also uses Hawkes processes).
The success of this model in reconciling microstructure with implied volatility surfaces provides us with an unified framework for pricing and risk management.
- Markovian structure of the Volterra Heston model
- Multi-factor approximation of rough volatility models
- Short-term at-the-money asymptotics under stochastic volatility models
- Perfect hedging in rough Heston models
- The microstructural foundations of leverage effect and rough volatility
- The characteristic function of rough Heston models
- Market impact can only be power law and it implies diffusive prices with rough volatility