# Introduction

Dispersion trading refers to the practice of selling index variance while buying variance of its constituents at the same time.

The reverse strategy can also be employed obviously but is not as popular.

Consider the index as a basket of stocks: The variance of the index is related to that of the individual stocks by: If we hold the realized variances of every component stock constant, the maximum for the index variance is reached when the correlation between all the components is 1. If the correlation between stocks is < 1, the index variance is lower. The more "dispersed" the stocks are, the lower is the index variance.

A measure of dispersion can be defined as: or alternatively .

D = 0 corresponds to the case when there is no dispersion --> all correlations = 1.

Being long dispersion is equivalent to being short correlation, and vice versa.

Another way of expressing this ‘average correlation’ is to assume that the correlation is the same for all distinct pairs of stocks and then rewrite the above equation to solve for this average correlation. Sometimes, it is easier to calculate the less accurate "correlation proxy" for a well-diversified ptf: It can be interpreted as the "average" of all correlations between all pairs of stocks in the index including a stock with itself.

When n is high, it can be seen that: Therefore and Realized correlation = correlation calculated using realized volatilities.

Implied correlation = correlation calculated using  implied volatilities.

Implied volatilies decide the price of traded instruments such as vanilla options and variance swaps.

The success of dispersion trades relies on the fact that statistically  the realized correlation tend to be below the implied correlation.

Historically, if one were long dispersion (short implied correlation), on average, one made more money than the amount lost.

There are many different reasons for this phenomenon, mainly:

- more market demand for index volatility than that of the individual stock.

- correlation jumps when extreme market conditions exist (global recession / market crash) while it stays low in a normal and uneventful market.

To long the volatility of each component stock and short the index volatility, one can either trade vanilla options or variance swaps.

The variance swap provides direct exposure to variance without the unnecessary cost and hassle of hedging against daily stock movements.

One issue in dispersion trade is to decide the relative weight for index and constituents variances. There is no single "correct" relative weight to use.

Example:

- "vega neutral" weights aim to make the sum of constituents vega and index vega zero, so that the trade is hedged  against fluctuations in level of vol.

- "Premium neutral" weights make the initial premium of buying constituents and selling index cancel each other.

In reality, it is impractical to trade all constituents. Often, a selection of names in the index is used --> proxy basket.