Why can a variance swap be perfectly statically hedged whereas a volatility swap requires dynamic hedging?

Possible reference request to the corresponding literature.

  • $\begingroup$ 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 C Apr 13 '16 at 23:25
  • $\begingroup$ 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 Joshi Apr 16 '16 at 10:57
  • $\begingroup$ the variance swap hedging requires dynamic stock hedging. The vol swap requires dynamic option hedging. $\endgroup$ – Mark Joshi Apr 16 '16 at 10:58
  • $\begingroup$ Guys, if you write these as answers I can give you the points for your efforts. $\endgroup$ – Permian Apr 16 '16 at 15:55
  • $\begingroup$ @AlexC see the above $\endgroup$ – Permian Apr 16 '16 at 21:04

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.

  • 2
    $\begingroup$ "It is summarized in Mark Joshi's book More Mathematical Finance" --> where is it summarized in this book ... ? $\endgroup$ – ujsgeyrr1f0d0d0r0h1h0j0j_juj Mar 15 '17 at 10:11

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