# Weighting dispersion

While a dispersion trade always involves a short index volatility position and a long single stock volatility position, there are different strategies for calculating the ratio of the two trade legs.

• Vega-weighted In a vega-weighted dispersion, the index vega is equal to the sum of the single-stock vega. If both index and single-stock vega rise one volatility point, the two legs cancel and the trade neither suffers a loss or reveals a profit.
The easiest weighting to understand is a vega-weighted dispersion, which by definition has zero vega (as the vega of the short index and long single-stock legs are identical). A vega-weighted dispersion is, however, short gamma and short theta (ie, have to pay theta).
• Theta- (or correlation-) weighted Theta weighting means the vega multiplied by √variance (or volatility for volatility swaps) is equal on both legs. This means there is a smaller single-stock vega leg than for vega weighting (as single-stock volatility is larger than index volatility, so it must have a smaller vega for vega × volatility to be equal). Under theta-weighted dispersion, if all securities have zero volatility, the theta of both the long and short legs cancels (and total theta is therefore zero). Theta weighting can be thought of as correlation-weighted (as correlation ≈ index var / average single stock var = ratio of single-stock vega to index vega). If volatility rises 1% (relative move) the two legs cancel and the dispersion breaks even.
• Theta-weighted dispersion needs a smaller long single-stock leg than the index leg (as reducing the long position reduces theta paid on the long single-stock leg to that of the theta earned on the short index leg). As the long single-stock leg is smaller, a theta-weighted dispersion is very short gamma (as it has less gamma than vega-weighted, and vega-weighted is short gamma).
• Gamma-weighted Gamma weighting is the least common of the three types of dispersion. As gamma is proportional to vega/vol, then the vega/vol of both legs must be equal. As single-stock vol is larger than index vol, there is a larger single-stock vega leg than for vega-weighted.
Gamma-weighted dispersion needs a larger long single-stock leg than the index leg (as increasing the long position increases the gamma to that of the short index gamma). As the long single-stock leg is larger, the theta paid is higher than that for vega-weighted.

#### Weighting Goals

• Theta-weighted dispersion is best weighting for almost pure correlation exposure The sole factor that determines if theta-weighted dispersion makes a profit or loss is the difference between realised and implied correlation. For timing entry points for theta weighted dispersion, we believe investors should look at the implied correlation of an index (as theta-weighted dispersion returns are driven by correlation). Note that theta-weighted dispersion breaks even if single stock and index implied moves by the same percentage amount (eg, index vol of 20%, single-stock vol of 25% and both rise 50% to 30% and 37.5%, respectively).
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• Vega-weighted dispersion gives hedged exposure to mispricing of correlation When a dispersion trade is vega-weighted, it can be thought of as being the sum of a theta weighted dispersion (which gives correlation exposure), plus a long single-stock volatility position. This volatility exposure can be thought of as a hedge against the short correlation position (as volatility and correlation are correlated); hence, a vega-weighted dispersion gives greater exposure to the mispricing of correlation. When looking at the optimal entry point for vega-weighted dispersion, it is better to look at the difference between average singlestock volatility and index volatility (as this applies an equal weight to both legs, like in a vega weighted dispersion). Note that vega-weighted dispersion breaks even if single stock and index implied moves by the same absolute amount (eg, index vol of 20%, single-stock vol of 25% and both rise ten volatility points to 30% and 35%, respectively).
• Gamma-weighted dispersion is rare, and not recommended While gamma weighting might appear mathematically to be a suitable weighting for dispersion, in practice it is rarely used. It seems difficult to justify a weighting scheme where more single-stock vega is bought than index (as single stocks have a higher implied than index and, hence, should move more). We include the details of this weighting scheme for completeness, but do not recommend it.
Empirically, the difference between single-stock and index volatility (ie, vega-weighted dispersion) is not correlated to volatility, which supports our view of vega-weighted dispersion being the best.