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Why Geometric Brownian Motion Cannot Produce Price Jumps

Article Quant Q&A · Author: MikeHeimlich
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Summary

The document explains the distinction between extreme price changes and jumps in a stochastic price path. A lognormal distribution can assign low probability to very large price changes over a time interval, but that does not mean the model’s path jumps instantaneously between values. Under geometric Brownian motion, paths are continuous: if price moves from one level to another, it passes through the intervening levels, even when the overall change is unusually large.

A jump process differs because it can move directly between separated values over an instant, leaving a discontinuity in the sample path. The example contrasts a move from 10 to 100 under the two kinds of process. This is a conceptual explanation rather than an empirical comparison or a guide to estimating jump models; it does not specify how to choose a jump process or assess whether real market prices exhibit jumps.

Key ideas

  • Geometric Brownian motion has continuous sample paths, so it cannot represent instantaneous price jumps.
  • A rare, large price change over an interval is not the same as a discontinuity in the path.
  • A jump process can move between separated price levels without traversing the values between them.
  • The distinction concerns path behavior, not simply the probability assigned to extreme outcomes.

Tags

StatisticsVolatility
Full text
# Geometric Brownian Motion unable to model / predict jumps


# Geometric Brownian Motion unable to model / predict jumps












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 by Holden (score 3, accepted)

https://quant.stackexchange.com/a/45329

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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