Why would an investor trade a variance swap over a volatility swap? Is it simply related to the leverage involved in a Var (i.e. sigma-squared) or is there something else to it?
Var and vol swaps are very similar products, with the leverage (convexity) being the biggest theoretical difference, yes.
In the actual market however they are very different. After the 2008 debacle var swaps in the single stock space are not too common, whereas single stock vol swaps are regularly quoted. One interesting perspective is trading one versus the other to get clean exposure to convexity.
Derman et al has a long note on this from 1999. Variance swaps are actually the more natural choice. It has nothing to do with leverage. From the linked article:
Although options market participants talk of volatility, it is variance, or volatility squared, that has more fundamental theoretical significance. This is so because the correct way to value a swap is to value the portfolio that replicates it, and the swap that can be replicated most reliably (by portfolios of options of varying strikes, as we show later) is a variance swap.
A little further down in the same article, he discusses how volatility swaps are actually a derivative on variance swaps:
Since variance can be replicated relatively simply, it is useful to regard volatility as the square root of variance. From this point of view, volatility is itself a square-root derivative contract on variance. Thus, a volatility swap can be dynamically hedged by trading the underlying variance swap, and its value depends on the volatility of the underlying variance – that is, on the volatility of volatility.
As you know both var swap & vol swap are traded on vol. The difference comes in convexity. Although variance swap payoffs are linear with variance they are convex with volatility. Because of the convexity, a variance swap will always outperform a contract linear in volatility of the same strike. This convexity is the reason that variance swaps strikes trade above at-the-money volatility.
In case of large swing in volatility, var swap will give far better result than vol swap.
(a) From the dealers perspective, single stock vol swap is much easier to hedge. (b) From the clients perspective, vol swap is nearly impossible to unwind with anyone but the original dealer (c) Var>Vol spread trades in the IBD market all the time and is the only truly liquid product with vol of realized vol exposure (options on realized var don't really trade these days).