Delta Hedging Choices in Index Dispersion Trades
Summary
The note compares delta choices for hedging short-dated at-the-money options in an index dispersion trade, which pairs long single-stock variance with short index variance. It explains that using Black–Scholes delta with the option’s initial implied volatility gives approximate gamma profit and loss tied to the difference between implied and realized variance, weighted by Black–Scholes gamma. A stochastic-volatility delta instead weights that variance difference using the model’s spot volatility and gamma. Bachelier and minimum-variance deltas would produce other exposures and profit-and-loss profiles.
The answer’s practical point is that for short-dated at-the-money options, gamma profit and loss often dominates, so the choice of delta may have little impact. This is an approximation, not a universal result: other profit-and-loss terms may matter, and the discussion does not provide empirical comparisons or prescribe a particular hedge. It frames the hedge choice around which exposure the trader wants to harvest.
Key ideas
- Delta choice changes the gamma-weighted variance exposure in a dispersion trade.
- Black–Scholes hedging links approximate profit and loss to implied versus realized variance.
- A stochastic-volatility delta uses model spot volatility and model gamma in the approximate exposure.
- For short-dated at-the-money options, gamma effects may dominate differences among hedge deltas.
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Full text
# index dispersion - which delta to hedge
# index dispersion - which delta to hedge
In an index dispersion trade (long single stock variance and short index variance) implemented via straddles or option strips, the options delta will be hedged frequently.
What is common practice on the way to calculate the delta eg. Black Scholes delta (so sticky strike) or sticky delta or model-dependent?
Are there resources describing the impact of those different possibilities?
## Answer by Frido (score 1, accepted)
https://quant.stackexchange.com/a/85265
Good question, in my opinion the answer is: use the hedge vol/delta that gives you what you want to harvest.
For example, suppose the realized dynamics of the underlying asset (be it index or single name) over the interval $[t,t+dt]$ is $$ dS_t = \sigma_R S_t dW_t. $$
Assume you hedge using the BS delta with the initial IV of the option. Then your P/L over $[t,t+dt]$ is approximately $$ \tfrac12 \Gamma^{BS}(IV) S_t^2(IV^2 - \sigma_R^2) dt. $$ So then you harvest the BS Gamma weighted difference between the implied variance and realized variance. (For short dated ATM option you can more or less ignore the other P/L terms.)
Suppose on the other hand you've calibrated an SV model to the skew, and your SV model is $$ dS_t = \sigma_t S_t dW_t $$ and you hedge using the SV delta with spot vol $\sigma_t$. Then your P/L is approximately $$ \tfrac12 \Gamma^{SV}(\sigma_t) S_t^2(\sigma_t^2 - \sigma_R^2) dt. $$
If you use Bachelier delta you'll get a different P/L and using minimum-variance delta will yield yet another P/L. Etc.
However: since we are talking about short-dated ATM options (which you usually use for dispersion trades), and since for short-dated ATM option the other P/L terms aside from Gamma P/L can be more or less ignored we have the situation that for short-dated ATM options it almost doesn't matter which delta you use. Because if it did matter then one choice of delta would be systematically better than others, which in a sense is arbitrage. I believe this is another way to understand the 'trading gist' of Andreasen and Huge's presentation "Wots me delta"Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.