Why can a variance swap be perfectly statically hedged whereas a volatility swap requires dynamic hedging?
Possible reference request to the corresponding literature.
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$\begingroup$ Basically the fact that a variance swap can be statically replicated with options of all strikes is a famous and non obvious result, see emanuelderman.com/writing/entry/… and sbossu.com/docs/VarSwaps.pdf. $\endgroup$– Alex CCommented Apr 13, 2016 at 23:25
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$\begingroup$ i have a detailed description of the pricing and hedging in More Mathematical Finance. The vol swap stuff was initially developed by Carr and Lee. $\endgroup$– Mark JoshiCommented Apr 16, 2016 at 10:57
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1$\begingroup$ the variance swap hedging requires dynamic stock hedging. The vol swap requires dynamic option hedging. $\endgroup$– Mark JoshiCommented Apr 16, 2016 at 10:58
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$\begingroup$ Guys, if you write these as answers I can give you the points for your efforts. $\endgroup$– TrajanCommented Apr 16, 2016 at 15:55
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$\begingroup$ @AlexC see the above $\endgroup$– TrajanCommented Apr 16, 2016 at 21:04
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2 Answers
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There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More Mathematical Finance among other places.
It all started with the work of Anthony Neuberger on the Log Contract in 1994, which seemed to be a theoretical result about a non-existent contract. It led to a solution for Var swaps, the famous Derman paper More Than You Ever Wanted To Know About Volatility Swaps 1999 1; see also Bossu's Just What you Need to Know about Variance Swaps 2 for a simpler treatment. Then Peter Carr and Roger Lee wrote Robust Replication of Volatility Derivatives 3in 2009 which addressed Vol Swaps.
To summarize:
A Variance swap can be replicated by a static position in options plus a dynamic position in the underlying. This is a beautiful and very practical result. You have to hold options of all strikes at the given maturity, with holdings inversely proportional to the square of the strike.
A Volatility swap can be replicated with a dynamic position in options. This is not very practical as the transaction costs for continuously buying and selling a large number of options will eat you alive. As a result Vol Swaps are not much traded and Var Swaps are preferred.
Finally, the new method for pricing variance swaps was adopted by the CBOE in 2004 as a way of calculating VIX values. Essentially they compute VIX^2 by this method and then publish the square root of this at frequent intraday intervals. So whenever people look at the VIX they are implicitly relying in this method.
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2$\begingroup$ "It is summarized in Mark Joshi's book More Mathematical Finance" --> where is it summarized in this book ... ? $\endgroup$– OlórinCommented Mar 15, 2017 at 10:11
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Not sure if still relevant for the OP, but as I lack modesty I'd like to say that it is possible, to a good approximation, to hedge varswaps dynamically using 3 delta-hedged options only. This is something I found out relatively recently, and explained in the following note:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4542475
The static portfolio of a continuum of options (and a dynamic position in the underlying) for varswaps replication is beautiful theoretically, but not feasible in practice.
Volswaps is a different matter, the square root really messes things up, and I am not sure there can be an 'easy' hedge for the volswap. As the volswap can be regarded as derivative on the terminal value of a varswap, it can be written as a static strip of options on realized variance (a la Carr-Madan), but not as a static strip of options on the underlying asset.
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1$\begingroup$ nice work, Frido. However, I wonder if there are any liquid indices left in the market for which the market smile is generated by a stochastic volatility model (?) Wouldn't this imply that we can fit these curves with only three parameters which definitely is not the case for almost all indices right now? Did you have the chance to backtest this at some point? $\endgroup$– SI7Commented Aug 22, 2023 at 20:25
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$\begingroup$ @SI7 Great question / remark. The fact that only 3 options were used does not mean it only works for SV models with 3 parameters. The number of options depends on the expansion, not on the number of SV params. I could have continued expanding and included up to fourth moment of realized volatility. In that case I'd have 5 options. I could also have expanded to higher order in correlation (which is much trickier and work in progress). Using three options is let's say the lowest order expansion. Btw, it can also be used if smile is generated by variance gamma model I think. $\endgroup$– FridoCommented Aug 23, 2023 at 5:41