As far as I understand, in most of the cases we derive the option valuation assuming that the log-return of the asset is partly driven by its own Brownian motion, and we use Geometric Brownian motion (GBM) for stock option valuation because stock price cannot become negative in this setting. My question is that when we use the GBM for individual stocks, in order to find the price process of a portfolio of stocks (like an index), is it still correct to assume that the portfolio return is also driven by GBM? In other words, while adding arithmetic Brownian motions will still be arithmetic Brownian motion, this is not the case for GBM. Is that correct?
You are right, a weighted average of GBMs is not a GBM, but something else. Unfortunately the resulting process is not known analytically and therefore people still assume a GBM for indexes. (Keep in mind that the real life processes, for both stocks and indexes are not exactly GBM anyway. It is just an approximation. If anything GBM is a better fit for indexes than for individual stocks. Stocks can "jump" a large amount on bad news while indexes are more continuous (though they still can have some jumps, a non-GBM feature)).