Compute expected maximum drawdown for Brownian motion
If the Brownian motion is geometric with the stochastic differential equation
then use Ito's lemma with X(t) = log(S(t)) such that
converts it to the form used here.
This example shows how to use log-return moments of a fund to compute the expected maximum drawdown (
EMaxDD) and then compare it with the realized maximum drawdown (
load FundMarketCash logReturns = log(TestData(2:end,:) ./ TestData(1:end - 1,:)); Mu = mean(logReturns(:,1)); Sigma = std(logReturns(:,1),1); T = size(logReturns,1); MaxDD = maxdrawdown(TestData(:,1),'geometric')
MaxDD = 0.1813
EMaxDD = emaxdrawdown(Mu, Sigma, T)
EMaxDD = 0.1545
The drawdown observed in this time period is above the expected maximum drawdown. There is no contradiction here. The expected maximum drawdown is not an upper bound on the maximum losses from a peak, but an estimate of their average, based on a geometric Brownian motion assumption.
Mu— Drift term of a Brownian motion with drift
Drift term of a Brownian motion with drift., specified as a scalar numeric.
Sigma— Diffusion term of a Brownian motion with drift
Diffusion term of a Brownian motion with drift, specified as a scalar numeric.
T— A time period of interest
A time period of interest, specified as a scalar numeric or vector.
ExpDrawdown— Expected maximum drawdown
Expected maximum drawdown, returned as a numeric.
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.
To compare the actual results from
the expected results of
emaxdrawdown, set the
Format input argument of
either of the nondefault values (
'geometric'). These are the only two formats
 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.