macrocephalopod

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.

I ignored vega above since we don't generally have a view on the direction of implied vol -- *except* that we will move along the vol surface as time passes and as the stock price changes. Accounting for this would introduce...

Agree but the premise of the question is that we’re *not* in a flat smile world (otherwise the implied beta would just be 0!)