# 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.

• Isn't the second integral reminiscent of the Riemann-Liouville representation of fBM except that now the instantaneous volatility and vol of vol is included as a multiplicative factor? I think you should simply look at the expression as defining rough Heston. Jun 29 at 16:27
• See also slides 5 and 8 here: cmap.polytechnique.fr/~euroschoolmathfi18/… Jun 29 at 16:37
• I'm not familiar with rough Heston, however I would recommend having a look at "Pricing under rough volatility" which shows a couple of nice derivations using the integral representation of fBm. Link: papers.ssrn.com/sol3/papers.cfm?abstract_id=2554754 Jul 3 at 1:10