In the recent years, banks have sold structured products such as worst-of
options, Everest and Himalayas, resulting in a short correlation exposure. They
have hence become interested in offsetting part of this exposure, namely buying
back correlation. Two ways have been proposed for such a strategy : either pure
correlation swaps or dispersion trades, taking position in an index option and
the opposite position in the components options. These dispersion trades have
been set up using calls, puts, straddles, variance swaps as well as third
generation volatility products. When considering a dispersion trade using
variance swaps, one immediately sees that it gives a correlation exposure.
Empirical analysis have showed that this implied correlation was not equal to
the strike of a correlation swap with the same maturity. The purpose of this
paper is to theoretically explain such a spread. In fact, we prove that the P&L
of a dispersion trade is equal to the sum of the spread between implied and
realised correlation - multiplied by an average variance of the components -
and a volatility part. Furthermore, this volatility part is of second order,
and, more precisely, is of volga order. Thus the observed correlation spread
can be totally explained by the volga of the dispersion trade. This result is
to be reviewed when considering different weighting schemes for the dispersion
trade.