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For both time-series, just plot the log returns. You will see that one is not a Random-Walk .. the S&P500 since you will get values that far beyond the normal distribution. Just watch this video by Benoit Mandelbrot (starting at 11min:54sec). Looking at both graphs, your eyes can fool you making you believe that both are generated by Random Walks...


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I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ...


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You have just luckily created 1 path of the random walk by chance that fitted the S&P. You can create another random walk and it will look much different. The efficient markets hypothesis predicts that stock prices behave as random walks, so it is likely that S&P looks similar to that. However, one cannot predict the next step to make a profit, ...



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