The Distribution of a Geometric Brownian Motion Price Change

Summary

The document asks for the distribution of the change in an asset price over a future interval when the price follows geometric Brownian motion. The response emphasizes that the two endpoint prices belong to the same process and are therefore dependent, so treating them as independent lognormal variables would not answer the question. It also notes the general fact that sums, and hence differences, of lognormal variables are not themselves generally lognormal, although approximations can sometimes be used.

The answer offers no derivation or explicit distribution for the price change. Its useful contribution is a caution against assuming either independence of the endpoints or a lognormal distribution for their difference. As a result, the question remains unresolved in the source. Any practical application would need to derive the distribution while accounting for the dependence induced by the shared process; the note does not provide parameter-based formulas, approximations, or empirical evidence.

Key ideas

• Prices at two times in one geometric Brownian motion are dependent.
• Differences of lognormal variables are not generally lognormal.
• Approximations of lognormal sums or differences may be possible in some cases.
• The response identifies the dependence issue but does not derive the requested distribution.

Tags

# If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?


# If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?












Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say $$ \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \end{equation} $$

My question is what is the distribution of $S(t+h)-S(t)$ where $h>0$?

I think this is a standard textbook question but I didn't find anything relevant to it yet. If it's duplicated question please refer me to the existed one. I'm working on it at the same time. Any help will be appreciated!

## Answer by Fokko (score 3, accepted)

https://quant.stackexchange.com/a/44608

interesting question, as this problem is quite famous re stock prices I think. So I did some research on it and found this: https://stats.stackexchange.com/questions/238529/the-sum-of-independent-lognormal-random-variables-appears-lognormal

Seems to be very complex and the sum (and therefore the difference) of two lognormals is (generally) not lognormal. However in some cases it is approximated as lognormal.

Anyway, your question is more specific as S(t+h) and S(t) are supposed to be not independent, as it is the same stochastic process, but time shifted. So I guess, it goes more complex as it already is.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.