Why VIX Futures Lie Between Volatility and Variance Swap Bounds

# VIX future's lower and upper bounds


# VIX future's lower and upper bounds












A Tale of Two Indices, by Carr and Wu (Jrl. of Derivatives, Spring 2006)

As per the above paper of Carr and Wu (page 24 and 25), the price of a VIX future has for lower bound the fair strike of a forward-starting Vol swap, and for higher bound the square root of the fair strike of a forward-starting var swap.

While I understand the maths behind (Jensen's inequality), what would be the intuitive explanation of this result?

## Answer by mountshoutcap (score 2)

https://quant.stackexchange.com/a/77763

Brief premise.

> volatility swap: you are essentially agreeing to trade future realized volatility. payout depends on the difference between the realized volatility of the underlying asset and the volatility level agreed upon (the strike)

> variance swap: you are agreeing to trade the square of future realized volatility. again the payout depends on the difference between the realized and the strike variance.

> Jensen's inequality: given a convex function, the function of the expectation is less than or equal to the expectation of the function. if I had to explain it to my grandpa, the average of squared numbers is always higher than the square of their average. but we know that already.

> Volatility (VIX for our case) futures (thanks @nbbo2 for commenting): you're agreeing to buy/sell the future level of vol at a predetermined price on a specified future date. The payout is based on the difference between the VIX future's price when you enter the contract and the actual level of volatility at the contract's maturity.

to put it "practical" terms, from a market perspective, arbitrageurs between these instruments would not allow the VIX futures to drift too far away from these bounds. if they were price below the lower bound (below the vol swap strike), the arbitrage wizards could profit by going long the future and short the swap. on the other hand, if VIX futures were to break the upper bound (square root of variance swap strike), the arbitrageurs could go short the future and long the swap.

Hope this helps.

## Answer by Achrbot (score 1)

https://quant.stackexchange.com/a/77772

As you mention, the bounds are due to a convexity correction because of the square root function.

I think the most intuitive way to understand them, are by considering the VIX future and forward starting volatility swap as derivatives on variance swaps.

Let $K_{t}^{\tau}$ denote the variance swap rate for $\tau = 30$ days. Under some technical assumptions, as outlined in the paper you linked, the VIX future $F_t^T$ with maturity $T$ is a forward price on $\sqrt{K_T^\tau}$, and hence its value is given by \begin{align} F_t^T = \mathbb{E} \left[\sqrt{K_T^\tau} \vert \mathcal{F}_t \right]. \end{align}

Now, because the square root is a concave function, the expectation of the sqrt is lower than the sqrt of the expectation. This is because high values of $K_t$ contribute relatively more to $\mathbb{E} \left[K_T^\tau \vert \mathcal{F}_t \right]$ than to $\mathbb{E} \left[\sqrt{K_T^\tau} \vert \mathcal{F}_t \right]$ (Jensen's inequality).

Because of this, we get $$ F_t^T \leq \sqrt{\mathbb{E} \left[K_T^\tau \vert \mathcal{F}_t \right]}, $$ and we recognize that the term under the square root, is the forward starting variance swap rates. Essentially the same argument holds for the lower bound.