# emaxdrawdown

Compute expected maximum drawdown for Brownian motion

## Syntax

```EDD = emaxdrawdown(Mu, Sigma, T)
```

## Arguments

 `Mu` Scalar. Drift term of a Brownian motion with drift. `Sigma` Scalar. Diffusion term of a Brownian motion with drift. `T` A time period of interest or a vector of times.

## Description

`EDD = emaxdrawdown(Mu, Sigma, T)` computes the expected maximum drawdown for a Brownian motion for each time period in `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.

The output argument `ExpDrawdown` is computed using an interpolation method. Values are accurate to a fraction of a basis point. Maximum drawdown is nonnegative since it is the change from a peak to a trough.

 Note   To compare the actual results from `maxdrawdown` with the expected results of `emaxdrawdown`, set the `Format` input argument of `maxdrawdown` to either of the nondefault values (`'arithmetic'` or `'geometric'`). These are the only two formats `emaxdrawdown` supports.

## References

Malik Magdon-Ismail, Amir F. Atiya, Amrit Pratap, and Yaser S. Abu-Mostafa, "On the Maximum Drawdown of a Brownian Motion," Journal of Applied Probability, Volume 41, Number 1, March 2004, pp. 147–161.