As the work of Mandelbrot pretty definitively shows[1], you should nearly always include a fat tail on both sides of your distribution.
I am sure that there exist better prediction methods (say, that provide "X% confidence" intervals that are narrower than those of the Brownian model but still contain the predicted value at least X% of the time in the long run). I am a complete amateur in this field.
Indeed, as you say, a fat tail is evident in the distribution of the one-step increments Z(i+1) - Z(i). However, both theory and data tell me that a Gaussian distribution, with linearly increasing variance, works well for larger steps Z(i+n) - Z(i) when n is ~20 hours or more. See this plot:
This is a series of crude histograms of the quantity R(i,n) = (Z(i+n) - Z(i))/sqrt(n), sampled from the reference datafile (Bitstamp 2013-09-01 to 2014-01-30, 1 hour intervals) with various values of i, tallied separately for each stride n. The "n" axis runs across the middle of the plot, with n=1 at the upper left edge and increasing down to the right. The "R" axis is perpendicular to it, with R=0 in the middle. The histogram for each n is normalized to unit sum.
The long fat tails and narrower peak of the distribution for small n is quite evident. However, as n increases, the tails and sharp peak seem to disappear, as one would expect from the Law of Large Numbers.
Actually, I am not really sure about the tails because the number of samples in each histogram also gets smaller when n increases. Perhaps the following plot is more convincing:
In this plot, the horizontal axis is the stride n (hours), and the vertical axis is the log increment D(i,n) = Z(i+n) - Z(i) (not divided by sqrt(n)).
The light brown crosses on each vertical line are a sample of differences D(i,n) with same value of n and various values of i. Each green dot is the mean of the sample increments D(i,n) with same n. Each red dot is the standard deviation of those increments, computed assuming that their expected value is 0 (rather than the empirical mean shown by the green dot). The histogram-like lines are the 2.5% and 97.5% percentiles of those samples.
The orange curve is the deviation sigma(n) = C*sqrt(n) of D(i,n) predicted by the log-Brownian model. The purple curves are the ±2*sigma(n) bounds above and below D=0.
According to this plot, for n ~15 hours or more, the empirical deviation (red dots) is very close to the model C*sqrt(n) (orange line). For n ~30 hours or more, the curves ± 2*C*sqrt(n) folow the empirical 2.5%-97.5% percentiles as accurately as one could expect.
The model clearly fails for smaller n; in particular, over the span of 5 hours, large swings occur more often than would be expected in a log-Brownian model. Therefore, I was too confident in my prediction for tomorrow; but with a bit of luck, I should be safe for the rest of the month.
The slightly ascending green line means that there is a consistent increasing trend in that sample. That trend also manifests itself in the growing gap between the empirical 2.5% percentile and the -2*sigma(n) curve. However, if I had used only the last 2 months of data for the analysis, instead of the last 5, the trend would have been decreasing - and stronger. That is one of the reasons why I did not include a trend term.
I have looked for correlations between successive increments (Z(i) - Z(i-1) and Z(i+1)-Z(i)), but did not see anything clearly significant. If there is such a correlation, it must be very subtle, and should be quickly "forgotten" after a few time steps.
Note that real stock prices are influenced by "real world" factors such as demand for the product, raw material prices, etc. Those factors vary according to their own nature in various time scales, that range from decades to hours. Maybe it is those factors that provide the long-term correlations characteristic of fractal signals?
In contrast, Bitcoin's price is almost entirely set by speculation; while external news may trigger changes, they do not directly determine the magnitude of those changes. ("How many billion dollars were subtracted from Bitcoin's future usage in e-commerce payments because of Shrem's arrest?")
So perhaps Bitcoin's price is indeed better described by a log-Brownian model than by a fractal process...
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