VIX Futures Pricing

Recently I came across a blog called macroption, which seems to carry some interesting materials. 

The blogger specifically explains the difference between the term structure and futures curve of VIX, which can sometimes be confusing. To eliminate the ambiguity, we can use the full notation, with three arguments, to denote forward volatility: $ V(t,S,T) $ (which is similar to the full notation of LIBOR). $ V(t,S,T) $ is, of course, the forward $T-S$ day volatility starting at a future time $S$ as observed at time $t$. With this in mind, the term structure of VIX would be a graph of $V(0,0,T)$ against $T$; while the futures curve of VIX would be a graph of $V(0,S,S+\delta)$ against $S$ ($\delta$ in this case would be 30 days).

Now that we have clarified the notation, let's talk about VIX futures pricing and how convexity plays a role. $VIX^2$ is the fair strike of a variance swap, which can be replicated using vanillas in a model-independent manner (recap here). To express it in terms of expectation, we can write

$$ VIX_0^2 = E^Q_0[V(0,0,30)] $$

However, the VIX index and hence its related derivatives (futures and options) are quoted not in $VIX^2$ terms, but simply in $VIX$ itself (in other words, as we see that VIX is at 14.5, the fair strike of a variance swap is 0.145 * 0.145 = 0.021). If we consider a future contract, in general if the underlying is commodities, FX or stocks we can simply apply the no-arbitrage rule and arrive at the future price using the appropriate funding/dividend/storage/convenience yields. However, these simply don't exist for VIX, which is not a tradable entity. So, we have to rely on

$$ F(0,T) = E^Q_0[\sqrt{VIX_T^2}] \neq \sqrt{E^Q_0[VIX_T^2]}$$

This is where the convexity adjustment comes in (in fact it should be a 'concavity adjustment', as the square root function $\sqrt{x}$ is concave in $x$). We can approximate the future price as

$$ F(0,T) \approx \sqrt{E^Q_0[VIX_T^2]} - \frac{var^Q_0[VIX_T^2]}{8[E^Q_0[VIX_T^2]]^{2/3}}$$

(see, for example, Zhu and Lian,  An Analytical Formula for VIX Futures and its Applications, 2010) The negative sign of the higher order correction term clearly shows concavity.