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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...
...two new terms, one of which modifies theta and the other modifies vanna, but I think the effect should be small most of the time.
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