Saturday night mid level derivatives trader interview question.

Let's compare two trades:

1) Vega-neutral equity index dispersion trade (long component names on their index weights; short index; same vega) in volatility swap format

2) Vega-neutral equity index dispersion trade in fixed strike options; Black-Scholes delta neutral in index and all names; assume you dynamically manage it to keep vega/gamma on the index weights.

Question: which of these trades will exhibit a higher beta with respect to the direction of the underlying equity index? What factors does this depend on? Can you write down precisely a mathematical expression for the difference in the beta?

People who obviously know the answer to this question, let the kids have a chance! Footnotes below... https://twitter.com/bennpeifert/status/1264542177574584320