What are the limitations of brownian motion in its applications to finance?
So where to begin? Continuity is a big thing as it fails to take into account jumps, the Gaussian assumption is another big one. However, looking deeper into it stationarity is a huge problem as it applies to financial time series.
However, it does an OK job at simulation stuff in the long-run.
From this paper:
The geometric Brownian motion model implies that the series of ﬁrst differences of the log prices must be uncorrelated. But for the S&P 500 as a whole, observed over several decades, daily from 1 July 1962 to 29 Dec 1995, there are in fact small but statistically signiﬁcant correlations in the differences of the logs at short time lags.