Comparing Vega, Theta, and Gamma Weighting in Index Dispersion Trades
Summary
The document outlines three ways to size the long single-stock variance leg against the short index variance leg in a dispersion trade. Vega weighting equates index vega with the sum of constituent-stock vegas. Theta weighting instead matches vega multiplied by the square root of variance, described as volatility for volatility swaps. The text says this makes the single-stock vega exposure smaller than under vega weighting when stock volatility exceeds index volatility, and relates the weighting to correlation exposure.
Gamma weighting matches vega divided by volatility across the two sides, based on gamma’s stated proportionality to that ratio. The document characterizes it as less common and says it implies a larger single-stock vega leg than vega weighting under the same volatility comparison. It poses a question about deriving theta neutrality and calculating the short-leg adjustment, but does not supply a derivation or a complete sizing formula. The descriptions are conceptual and omit contract specifications, portfolio aggregation details, and implementation effects.
Key ideas
- Vega weighting matches index vega to the aggregate vega of the single stocks.
- Theta weighting matches vega multiplied by volatility across the two legs.
- The document associates theta weighting with correlation exposure and relative volatility moves.
- Gamma weighting matches vega divided by volatility and is described as less common.
- No full derivation or contract-specific sizing procedure is provided.
Tags
Full text
# index dispersion trades theta/vega/gamma weighted # index dispersion trades theta/vega/gamma weighted When implementing an index dispersion trade (long single stock variance and short index variance e.g. via variance swaps), 3 common ways to weigh the long leg vs the short leg: theta /vega/gamma weighted. Theta weighted: by definition, weigh the 2 legs such that the theta paid on the long leg is compensated by the theta earned on the short one. One can often read (e.g. here) that theta-weighted is equivalent to having Vega * square root of the variance strike equal on both legs - how to prove that being theta flat leads to this? Assuming we fix the notional on the long leg, and need to adjust the notional on the short leg, what exactly is that weight to apply on the short leg to be theta weighted? Extract from link above: - 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. - 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 sin gle 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. - 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.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.