Frequently Asked Questions In Quantitative Finance

Chapter 2: FAQs 177

If you know the implied volatilities for the individual
stocks and for the index option then you can back out
an implied correlation, amounting to an ‘average’ across
all stocks:

σI^2 −

∑N

i= 1 w

2

iσ

2

i

∑N

i= 1

∑N

i =j= 1 wiwjρijσiσj

.

Dispersion trading can be interpreted as a view on this
implied correlation versus one’s own forecast of where
this correlation ought to be, perhaps based on historical
analysis.

The competing effects in a dispersion trade are

• gamma profits versus time decay on each of the long
  equity options


• gamma losses versus time decay (the latter a source
  of profit) on the short index options


• the amount of correlation across the individual
  equities

In the example above we had half of the equities increas-
ing in value, and half decreasing. If they each moved
more than their respective implied volatilities would
suggest then each would make a profit. For each stock
this profit would depend on the option’s gamma and the
implied volatility, and would be parabolic in the stock
move. The index would hardly move and the profit there
would also be related to the index option’s gamma.
Such a scenario would amount to there being an aver-
age correlation of zero and the index volatility being
very small.

But if all stocks were to move in the same direction the
profit from the individual stock options would be the
same but this profit would be swamped by the gamma
loss on the index options. This corresponds to a correla-
tion of one across all stocks and a large index volatility.