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# 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

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

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.