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?
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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. |
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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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