We know the delta of a portfolio of options is simply the sum of deltas of the individual options. But are there any additional known properties about the total delta (or other greeks) of a portfolio of options?
More specifically, how does the total delta of a portfolio change the more options one adds? If there's a random collection of calls and puts on the same underlying but with varying strikes and maturities, shouldn't the likelihood of offsetting deltas increase? Is it possible to apply the central limit theorem here to derive some general rules how the greeks of such a portfolio behave as more options are being added to such a portfolio?