When Squared VIX Could Behave Like a Martingale

Summary

The document frames a question about whether VIX is a martingale when the S&P 500 follows a Gaussian diffusion. It defines squared VIX for a fixed tenor as the risk-neutral expectation of average integrated variance over the tenor. It then applies the martingale property, the tower property of conditional expectation, and a change of variables to express the conditional expectation of future squared VIX in terms of volatility over a shifted interval.

The document does not include an answer or establish conditions under which the martingale property holds. Its derivation suggests that the result depends on assumptions about the volatility process, but it does not specify those assumptions or distinguish the behavior of VIX from squared VIX. It is therefore a useful setup for studying conditional variance and martingale questions, rather than a resolved result or a complete pricing argument.

Key ideas

• Squared VIX is framed as the risk-neutral expected average integrated variance over a fixed tenor.
• Testing the martingale property requires comparing current squared VIX with the conditional expectation of its future value.
• The tower property and a change of variables lead to an expression involving volatility over a shifted time interval.
• The document leaves the needed assumptions on the volatility process unspecified and does not resolve the question.

Tags

# Is the VIX a Martingale?


# 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?

Thanks for helping! If you know of any helpful resource that could lead me to the answer please share.

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 $\sigma_s$.

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