Brownian Motion Path and Maximum Drawdown

We illustrate the evolution of a geometric Brownian motion simulating a daily stock return series. The jagged blue line is the cumulative return of the daily return series. The red line is the maximum drawdown to date of the series. Adjusting the mean and standard deviation sliders demonstrates how the cumulative return and maximum drawdown change with respect to these parameters for a given underlying set of random shocks. Adjusting the "new random case" slider allows you to see different random cases to get a sense of how variable a return series can be for a single set of parameters. Note that in general the greater the ratio of mean to standard deviation, the smoother the return evolution is and the smaller the maximum drawdown is.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


A geometric Brownian motion is the de facto standard model for stock price evolution. It is locally represented by the simple stochastic differential equation
where is the stock price, μ is the "drift" parameter and σ is the standard deviation. The equation says that over independent time increments of size Δt, the stock price's fractional change is normally distributed with mean μΔt and standard deviation σ. The ratio of μ and σ is sometimes called the information ratio of the price evolution and it is tied to the maximum drawdown of the series.
This Demonstration allows the user to choose annualized values of μ and σ (in percentage) and then shows the cumulative returns of the random series and its maximum drawdown series. The maximum drawdown at any point in time is the largest peak to trough change in the series. The plot shows that the maximum drawdown is tightly linked to the information ratio of the series.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+