Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in.
I am a mathematician who knows nothing about finance. I heard from a popular source that a something called the Black-Scholes equation is used to model the prices of options. Out of curiosity, I turned to Wikipedia to learn about the model. I was shocked to learn that it assumes that the log of the price of an asset follows a Brownian motion with drift (and then the asset price itself is said to follow a "geometric" Brownian motion). Why, I wondered, should that be a good model? I can understand that asset prices have to be unpredictable or else smart traders would be able to beat the market by predicting them, but there would seem to be many unpredictable alternatives to geometric Brownian motion.
I have found one source that addresses my question, the following book chapter: http://www.probabilityandfinance.com/chapters/chap9.pdf and an argument it alludes to in chapter 11 of the same book. The analysis here looks very interesting, and I am curious if it is generally accepted in the finance community. I have not studied it enough to understand how realistic its assumptions are, however. It apparently depends on a "continuous time" assumption that seems like it might not be very realistic given that real markets move in response to discrete news events such as earnings announcements.