Geometric Brownian Motion unable to model / predict jumps

Question

In my finance course, we were talking about the flaws of modelling Stock Prices with the geometric Brownian Motion. According to my professor:

"Since the geometric Brownian Motion has continous time sample path, it does not allow for jumps in its values when rare events occur"

I fail to really understand this explication for the no-jump problem. If we take a look at the log normal distribution of the prices, then there should be the small possibility of an "extreme change", resp. a very rare event occurring and therefore experiencing a jump?

If anyone has a more detailed explanation than just the quote above, I would be very thankful

Answer 1

Assume that the value of the sample path of the geometric Brownian motion equals $10$ at time $t_0$ and equals 100 at time $t_0 + \Delta t$. For the value to change from $10$ to $100$, the path should necessarily go over all the values between $10$ and $100$ (possibly with fluctuations) during the intermediate time $\Delta t$; it cannot jump directly from $10$ to $100$.

On the other hand, if we consider a process with jumps, then it is possible for it to be $10$ at time $t_0$ and jump to $100$ after a very small (infinitesimal) amount of time $\Delta t$. This jump will create a gap on the plot of the sample path and may indicate some rare event.