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.
