Hi,I'm trying to get market values of volatility swaps using P. Carr et R. Lee's paper. My problem is about the definition of the payoff. If we start to price a variance swap, for me, it doesn't matter to define the realized variance as the sum of the squared log returns or the sum of squared log returns minus the mean of log returns since the difference is just a European option: ln(S_T/S_0)^2. However, I cannot figure out how to adapt Carr & Lee results if we consider a volatility swap and it seems that the standard definition of the payoff of the volatility swap uses the centralized version. Thanks in advance for your comments.Z

You can't. Isn't life wonderful?

The only reason why variance swaps remain popular despite their "counter-intuitiveness" (variance is not intuitive to traders as opposed to volatility so you may think that only vol swaps would be traded), is that they can be statically replicated by a strip of vanillas as shown in the paper you mention. This approach (widely used altough it has some shortcomings) is definitely not applicable to vol swaps, so these are generally priced with stoch vol models.

the Carr lee result does apply to vol swaps. how practical it is, is a different question.

Vol swaps have vol-of-var exposure which can not be statically replicated with vanillas. You can still hedge them dynamically with volgamma of OTM & ITM vanillas, but it is painful and you can bleed a lot on bid-offer.

figaro,I agree on the dynamic hedge. You use OTM and ITM vanillas because you have a stoch vol model? The reasons why you don't use varswap modeling (Consistent variance curve models by H. Buehler, for example) comes from the ln(S_T/S_0)^2 term in the payoff of the volswap? Am I right?

What you are hedging is volgamma (2nd deriv wrt vol), and volgamma of atm options is close to 0 - hence you need to use itm & otm to get enough volgamma for the hedge. There is no problem hedging with varswaps instead of vanillas, but this is where liquidity & bid-offer costs come in - bid-offer for such options is higher than atm vanillas. That is the advantage of varswaps over volswaps - you only pay bid-offer once.The problem with using stoch vol is that there is no liquid options market to hedge exposure to vol-of-vol (or vol-of-var, whichever you prefer to think of), so your hedging strategy will inevitably end up taking a long or a short position in it. In practice, you would offset this exposure with another exotic trade on the same underlier, but again you come up against bid-offer...Stick with varswaps.

In Remark 3.8, there's a simple lower bound for the vol swap if the 1st and 2nd moments of the integrated variance are knownhttp://www.maths.bris.ac.uk/~mamsf/Calibrating ... rde.pdfC&L showed that the vol swap rate < variance swap rate, and that (in the uncorrelated setting) VolSwapRate> ATM volatility, using Jensen's inequality, and derive a back of the envelope estimate for the VolswapRateM

< and > should be <= and >=

QuoteOriginally posted by: FordeIn Remark 3.8, there's a simple lower bound for the vol swap if the 1st and 2nd moments of the integrated variance are knownThat is the key "if". If you know something about vol of var (e.g. the total variance of var, as in your paper), you know something about vol swap forwards. If you know everything about vol of var, you know everything about vol swap forwards. If you know nothing about vol of var (e.g. you only know vanilla prices), you know nothing about vol swap forwards.