Predictions: Part II ... a continuation of Part I
motivated by e-mail from Ron Mc.

In Part I we downloaded a bunch of stock prices from Yahoo and generated a distribution of returns so we could attempt a prediction of future prices.
Although we used actual returns, we might also try to mimic the actual returns with a Laplace Distribution.

>Huh?
Remember when we tried to generate some distribution that gave "fat tails", here?
We started off our distribution so it looked like a Normal Distribution, then switched to an exponential decay.
If the Mean of the returns is m, then it went something like this:

1. When x is close to m, f(x) ≈ B e-|x - m|2/2   ... like the Normal distribution
2. When x is far from m, f(x) ≈ C e-|x - m|   ... a simple exponential decay, so it decreases less dramatically than a Normal distribution
3. And the necessary condition for a probability distribution:   f(x) dx = 1
We also match the value and slope at the place where we changed from one to the other.

Anyway, if we forget about Normal and just assume a simple exponential decay, for all x, it'd be the Laplace Distribution ... so we'll play with that.
the Laplace Distribution:

Density Distribution:       f(x) = (1/2a) e-|x - m |/a       -∞ < x < ∞
 Cumulative Distribution: F(x) = (1/2) e(x - m)/a       -∞ < x ≤ m F(x) = 1 - (1/2)e-(x - m)/a       m ≤ x < ∞

a = StandardDeviation/sqrt(2).

Note that F(m) = 1/2 ... so the probability of getting less than the Mean m is 50%.
Note, too, that the parameter a is related to the standard deviation of returns ... so we can get that from the actual returns.

>Is it any good? Remember; the proof is in the pudding.
Thanks for reminding me, but that's what we're about to investigate.
 We'll do this: Download ten years worth of daily returns for, say GE stock. Determine the cumulative distribution of daily returns. Determine, as well, the Laplace distribution (and, for fun, the Normal distribution) with the same Mean and Standard Deviation as the actual returns. See Figure 1? The Density distributions (Actual, Laplace and Normal) look like this: Figure 1
>That's the pudding?
Patience. We can also play with some predictions.
In Part I we selected a bunch of returns at random and attempted to predict the probability of attaining some future price.
Let's do the same thing with our Laplace distribution.

It looks like this: