Main Content

Compute expected maximum drawdown for Brownian motion

computes the expected maximum drawdown for a Brownian motion for each time period in
`ExpDrawdown`

= emaxdrawdown(`Mu`

,`Sigma`

,`T`

)`T`

using the following equation:

$$dX\left(t\right)=\mu dt+\sigma dW\left(t\right).$$

If the Brownian motion is geometric with the stochastic differential equation

$$dS\left(t\right)={\mu}_{0}S\left(t\right)dt+{\sigma}_{0}S\left(t\right)dW\left(t\right)$$

then use Ito's lemma with *X*(*t*) = log(*S*(*t*)) such that

$$\begin{array}{c}\mu ={\mu}_{0}-0.5{\sigma}_{0}{}^{2},\\ \sigma ={\sigma}_{0}\end{array}$$

converts it to the form used here.

[1] Malik, M. I., Amir F. Atiya,
Amrit Pratap, and Yaser S. Abu-Mostafa. “On the Maximum Drawdown of a Brownian
Motion.” *Journal of Applied Probability.* Vol. 41, Number
1, March 2004, pp. 147–161.