macrocephalopod

Nice question, here's my take (warning: not an options expert). P&L of your option can be approximated as P&L = theta * dt + 0.5 * gamma * ds^2 + vanna * ds * dv + volga * dv^2 (where s is spot, v is implied vol)https://twitter.com/bennpeifert/status/1352069470903181313 …

I ignored delta as assumed hedged, rho because it's small, and vega because I'll come back to it later. The covariance between spot, vol is in the vanna term so to isolate it we want a portfolio that is long vanna, flat volga and gamma.

The 25d call is long vanna, volga and gamma. The 25d put is long volga and gamma, short vanna. The ATM is long gamma and flat vanna/volga. So the portfolio we want is long 25d call, short 25d put and a small amount of ATM to neutralize the gamma.

This portfolio just has theta and vanna, so the implied covariance of s, v is (-1) * theta * dt / vanna -- note vanna, theta will both be positive if put vol > call vol, so the covariance will be negative.

The implied beta is implied covariance / implied variance of the stock. Implied variance is v^s * s^2 * dt, so the implied beta is (-1) * theta / (s^2 * v^2 * vanna) -- note that the dt's cancel (as they should, don't want the answer to depend on time period)

That's the implied beta for a 1 point move in the underlying. Need to multiply by the underlying price to get the implied beta (in vol points) for a 1% move in the underlying.