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I'm trying to figure out summing lognormals...If you have lognormal stock prices, and create a basket of 100 stocks, the basket price should be lognormal too, right? Yet I hear it said that lognormals sum to normals.What about perfectly correlated lognormals: dX_1 = X_1*dW dX_2 = 2 *X_2*dWThen what will the distribution of (X_1+X_2) look like? It seems like it has to be a lognormal, but how can you calculate it?

The lognormal is closed under addition. Products of lognormalsare lognormal. Sums of lognormals are not lognormal. I doubtsums of lognormals are normal, either.

if you sum enough independent things, it will look normal. Sums of lognormals are not anything nice in general, however.

Thanks MJ and DavidJN. So the S&P should have a more or less normal distribution then? A bit distorted, perhaps, due to some correlation? For perfect correlations, this goes against my intuition for how a good model should behave. I'm going to have to do some simulating to satisfy my curiosity...

for the S&P, the constituents are correlated so the basket need not become normal. it's a definite issue in math finance that we expect both baskets and their constituents to be lognormal

There is quite a lot in the literature about summing lognormals esp. with a view to pricing baskets. You can get quite tight bounds using comonotonicity and conditioning, but no closed forms.