motivated by e-mail from Ron Mc.
Predictions: Part II ... a continuation of Part I
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.
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:
We also match the value and slope at the place where we changed from one to the other.
- When x is close to m, f(x) ≈ B e-|x - m|2/2 ... like the Normal distribution
- 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
- And the necessary condition for a probability distribution: f(x) dx = 1
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.
Note that F(m) = 1/2
... so the probability of getting less than the Mean m is 50%.
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, 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.
>That's the pudding?
We'll do this:
See Figure 1?
- 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.
The Density distributions (Actual, Laplace and Normal) look like this:
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.
>Where's the spreadsheet?
It looks like this:
Click on picture to download spreadsheet.
There's also an Explain sheet:
>And your conclusion?
My conclusion? I think it's great fun!!
>What's that options stuff?
If you predict the stock price, say 30 days into the future, then you can estimate the value of an option. That's in the spreadsheet, too.
>I assume you use this to predict future prices, eh?
Uh ... actually, I use this.