Rough Heston Volatility and Fractional Brownian Motion

Summary

The document asks how the rough Heston volatility equation relates to empirical evidence that changes in log volatility behave like increments of fractional Brownian motion with a Hurst parameter below one half. It presents the rough Heston variance as a Volterra integral equation: past mean reversion and Brownian shocks contribute through a power-law kernel, with the shock term scaled by the square root of current variance.

The author notes that fractional Brownian motion has a representation using stochastic integrals driven by Brownian motion, and asks how that connects to the rough Heston specification. This highlights the distinction between observed rough behavior in log volatility and the model’s equation for variance. The document contains no derivation or answer, so it does not establish that the two processes are identical; it is a question about their relationship and modeling interpretation rather than evidence for a particular calibration or trading strategy.

Key ideas

• Empirical rough volatility is described through increments of log volatility resembling fractional Brownian motion with low Hurst parameter.
• The rough Heston model specifies variance using a Volterra kernel applied to mean reversion and Brownian shocks.
• The stochastic integral representation of fractional Brownian motion motivates the question of how the model captures roughness.
• The document does not derive an equivalence between the model variance and fractional Brownian motion.

Tags

# 71387


# If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows












From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with Hurst parameter $H< \frac{1}{2}$.

Next, the volatility in the rough Heston model is given as

$$ V_{t}=V_{0}+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \lambda\left(\theta-V_{s}\right) d s+\frac{\zeta}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \sqrt{V_{s}} d W_{s}. (*)$$

I am aware that fBM can be represented in the Mandelbrot-van Ness representation as stochastic integrals with Brownian motion as the integrators.

My question is, what is the connection between $(*)$ and fractional Brownian motion $B^{H}$, because at the moment I do not see it.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.