When Brownian Motion Predicts Simulated Maximum Drawdowns
Summary
The document clarifies when drawdowns measured from simulated results should agree with Brownian motion predictions. The key condition is that each process increment over an interval has a normal distribution with zero mean and variance equal to the interval length. If the simulation uses those increments, its drawdown behavior should follow the Wiener process model; otherwise, agreement is not expected.
For maximum drawdown comparisons, the document points to an approximate density series and software functions that provide the distribution and random samples. It suggests comparing sample means, quantiles, and empirical distributions with those implied by the theoretical maximum drawdown distribution. As the sample size grows, sample means converge to the expected value under the law of large numbers, though convergence may be slow. The discussion concerns the model’s assumptions and sampling behavior; it does not provide a numerical comparison or establish that Brownian motion is a suitable model for any particular strategy or return series.
Key ideas
- Brownian motion increments must be normally distributed with zero mean and variance proportional to elapsed time.
- Drawdown predictions from a Wiener process apply only when the simulated process has the required increment behavior.
- Maximum drawdown distributions can be approximated and sampled directly for comparison with simulations.
- Larger samples bring empirical means toward the expected maximum drawdown, but convergence may be slow.
- Comparisons can include both distribution summaries and the full empirical distribution.
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Full text
# Comparison of Brownian Motion Expected Drawdown and simulated results # 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? ## Answer by Matt Wolf (score 1, accepted) https://quant.stackexchange.com/a/7001 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. ## Answer by vanguard2k (score 3) https://quant.stackexchange.com/a/6998 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.
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