# Is the VIX a Martingale?

Say the S&P500 follows a Gaussian diffusion process, so that: $$VIX^2_{T,t}=\frac{1}{T}\mathrm{E}_t^\mathbb{Q}\left[\int_t^{t+T}\sigma_s^2ds\right]$$ where $T$ is the tenor (assume fixed), $t$ is the current time, $\mathbb{Q}$ is the risk neutral measure, and $\sigma_s$ is the volatility of the diffusion process.

Question: Is the VIX process a Martingale?

Progress: By writing down the Martingale Property, using the Tower Property of conditional expectation, and a change of variables, I get the following: $$\mathrm{E}_t(\text{VIX}_{T,t+\tau}^2)=\frac{1}{T}\mathrm{E}_t^\mathbb{Q}\left[\int_t^{t+T}\sigma_{s+\tau}^2ds\right]$$ It seems I need some assumption on the volatility process $\sigma_s$.
• What do you mean by "a Gaussian diffusion process"? Is $dS = \sigma S dB$ where $B$ is a standard Brownian motion such a process? – Hans Feb 16 '18 at 1:01
• @Hans Yeah! With an additional term for the interest rate: $dS_t/S_t=rdt+\sigma_tdB_t$ – Guilherme Salomé Feb 16 '18 at 2:25
• That is not the conventional nomenclature. You are referring to an Ito process. No, of course this is in general not a martingale. Just take any time dependent integrable function as $\sigma$. – Hans Feb 16 '18 at 6:34