# Comparison of Brownian Motion Expected Drawdown and simulated results

Can anyone tell me whether results as predicted by Brownian Motion for a given mean and std, match what you get by measuring actual drawdown from simulated results over a number of iterations?

• The longer you run the simulation or the higher the number of iterations you run it the closer it should be.
– user7228
Feb 13, 2014 at 15:40
• some years later... I am facing exactly your topic and I am finding inconsistencies between theory and simulation. (quant.stackexchange.com/questions/42031) Did you develop further experience in on this topic? Discussion welcome 🙂 Oct 5, 2018 at 6:03

Its very simple,

One of Brownian Motion (a.k.a. Wiener process in Mathematics) properties is that each increment from s->t is normally distributed with mean = 0 and sd = t-s.

So, if the process that drives your simulated results is ~N(0, t-s) distributed for each increment s->t with 0<=s<=t then yes, your simulated draw downs should match the ones predicted by a Wiener process (one which is driven by a Brownian Motion). Otherwise, it is not.

Take a look at the following paper about the Maximum Drawdown distribution:

On the Maximum Drawdown of a Brownian Motion

The authors end up with an approximative series for the density. It is implemented in the function maxdd of the R-package fBasics. There are convenient functions dmaxdd, pmaxdd and rmaxdd. Calculating the Expected Drawdown should be easy.

Just compare your results with the output of this package (mean, quantiles, etc.) and you should be fine.

Actually, there is no need to "simulate" drawdowns of a brownian motion then - just take random samples with rmaxdd.

When you say "match" or "close" you probably mean that the means converge if sample size increases?

By the law of large numbers, the means of the sampled maximum drawdowns will converge to the expected maximum drawdown (although convergence maybe slow - expecially if the distribution does not have finite variance). Actually, the empirical distributions "approach" the maximum drawdown distribution.

• Thanks you - that is interesting. BUT what I meant by simulate is creating (random generated) an accumulated revenue stream with mean x and std y. Then measure largest drawdown, and repeat several times. Would that number be close to what be generated by maxdd? Jan 15, 2013 at 16:37
• @ManInMoon I edited my post Jan 16, 2013 at 8:42