# Why not Volatility Swaps?

A common complaint about variance swaps is that they pay-off based on realised variance and not simply realised volatility. Why are there not volatility swaps, products which payout linearly on the difference between implied and realised volatility? If these are more easily understandable why are they not regularly traded?

Whilst volatility can be seen as more of an intuitive measure (being a standard deviation it is measured in the same units as the underlying), variance is in some sense more fundamental – especially because it is additive. For example, the variance of the sum of two independent distributions is just the sum of each of their variances. This linearity of variance makes calculating variance swap mark-to-markets and forwards particularly simple.

As pointed out, the exposure of delta-hedged options to volatility, after accounting for the gamma, is actually an exposure to the difference between implied and realised volatility squared. In this sense, a variance swap mirrors a kind of ideal delta-hedged option whose gamma remains constant. Furthermore, variance swaps are relatively easy to replicate. Once the replicating portfolio of options has been put in place, only delta-hedging is required. No further buying or selling of options is necessary.

All this explains why variance swaps are attractive instruments to trade, but still does not explain why volatility swaps are not also frequently traded.

The main theoretical difficulty with volatility swaps is that they cannot be statically replicated through options. A replicating portfolio must dynamically trade options, and make relatively strong assumptions about the underlying volatility process – in particular about the volatility of volatility. This makes any replication process model dependent, and therefore much more prone to errors than the theoretically robust variance swap replication.

In fact the convention of quoting variance swap notionals in vega, rather than variance, amounts can be seen as an attempt to treat variance swaps like volatility swaps. The vega notional represents the average P&L for a 1 vega change in volatility, but with the convexity meaning that longs will profit by more if volatility increases, and lose by less if volatility decreases. Thus for small changes in volatility, where the effect of the variance swap convexity is relatively limited, variance swaps (measured in vega notional) locally approximate volatility swaps.

Volatility swaps can then be thought of as variance swaps without the convexity. The discount of the volatility swaps to variance swaps should therefore reflect the value of this convexity, which in turn is determined by the volatility of volatility. Seen in this light, the ability to calculate a fair price for volatility swaps, requires not only a price for the volatility (variance), but also a price for the volatility of volatility – i.e. the means of valuing options on volatility.

Since it is variance which arises naturally from delta-hedging options, in terms of volatility products, variance should be thought of as the true underlying. In this framework, volatility swaps are naturally thought of as derivatives of variance – paying the square-root of the variance swap contract.

In fact we could dynamically trade a long variance swap (buying more as volatility decreases and selling as volatility increases) to hedge out this convexity bias.

Similar to delta-hedging an option the P&L made from the resulting buy low – sell high strategy (for variance) will lead to a P&L based on the volatility of volatility: the larger this volatility-of-volatility, the bigger the discount of the dynamically replicated volatility swap to the variance swap.

However, besides issues arising from the transaction costs of dynamically trading variance swaps, estimating this volatility-of-volatility contribution from the dynamic hedging of the variance swap is problematic and model-dependent, making the volatility swap contracts difficult to price.

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