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Geometric Brownian motion simulation and extracting mu and sigma


This is my usual way of creating stock simulations
View attachment 33716

But then what confuses me is that when I run simulations using python using the code below

View attachment 33717
mu and sigma extracted for the simple returns are almost always closer to the values I have set.
mu = 0.001
sigma = 0.05

(to give you a brief explanation of the code - I have simulated 100000 paths with 10 steps
for each simulation I store in list - mean of log returns, simple returns and sample standard deviation of log returns, simple returns.
Then after 100000 simulations, I calculated their average)

What am I getting wrong here?
Thank you in advance for your thoughts!
Last edited:


now I think mu should be log return + 1/2 * log sample standard deviation squared
sigma should be the log sample standard deviation