Why a Weighted Stock Index Is Not Generally a GBM

Summary

The note asks whether an index made from stocks modeled individually as geometric Brownian motions can itself be treated as a GBM for option valuation. It highlights a key distinction: while weighted sums of arithmetic Brownian motions retain that process form, a weighted average of GBMs generally does not. The resulting index process has a more complicated distribution that is not given in a simple analytic form in the discussion.

Despite this mismatch, GBM is often used as a practical approximation for index options. The response suggests it may fit indexes better than individual stocks because single stocks can experience large jumps on company news, while broad indexes tend to move more continuously. This remains an approximation: indexes can also jump, and neither stocks nor indexes exactly follow GBM in real markets. The note offers qualitative reasoning, not empirical tests or a formal model of the weighted process.

Key ideas

• A weighted average of geometric Brownian motions is generally not itself a GBM.
• GBM remains a convenient approximation for pricing index options.
• Broad indexes may fit the GBM assumption better than individual stocks because stock-specific news can cause jumps.
• Indexes can also jump, so GBM is not an exact description of market prices.

Tags

# Using Geometric Brownian Motion for Index Options


# Using Geometric Brownian Motion for Index Options












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?

## Answer by Alex C (score 6, accepted)

https://quant.stackexchange.com/a/41391

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)).

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