What is Rough Volatility? Beyond the Markovian Assumption

Rough volatility models show that volatility is more 'jagged' than Brownian motion suggests. Explore fractional Brownian motion and the Hurst exponent.

I analyzed high-frequency volatility estimates and found that the trajectories are significantly more 'rough' than what is permitted by standard Heston or GARCH models. Rough Volatility is a modern class of models where the volatility process is driven by a fractional Brownian motion $W_t^H$ with a Hurst exponent $H < 0.5$.

Specifically, empirical studies of thousands of assets show that $H$ is typically around $0.1$. This means volatility has 'short-term memory' and is anti-persistent, leading to a much more jagged path than standard diffusion ($H=0.5$).

The RFSV model posits that the log-volatility follows a fractional Ornstein-Uhlenbeck process: $d\log(\sigma_t) = -\kappa (\log(\sigma_t) - \mu) dt + \eta dW_t^H$. Here, $W_t^H$ is a fractional Brownian motion with Hurst exponent $H$. This framework captures the empirical observation that volatility exhibits 'roughness' at all time scales.

Rough models are non-Markovian, meaning the future depends on the entire history of the process. This makes them computationally expensive because you cannot use standard finite-difference methods or simple Monte Carlo. Quants use 'Hybrid Schemes' or 'Fast Fourier Transforms' to simulate these paths efficiently.

Rough volatility is the only framework that accurately reconciles the behavior of high-frequency price data with the shape of the option Volatility Smile. Ignoring the 'roughness' of volatility leads to a systematic mispricing of forward-start options and VIX derivatives.