another Distribution ... continued from
here 
Once upon a time we talked about a distribution of returns with Mean = m and Volatility = s, but one that'd give fatter tails and ...
>Is that your Mystery Distribution?
No. Here we want to consider yet another distribution of the form:
(1) f(x) = A e^{b SQRT( 1+k(xm)2 )}
where we need to choose the numbers A, b and k appropriately. In particular, to mimic the behaviour of some stock returns.
The cumulative distribution is determined by the area beneath the f(x) curve and denoted by:
F(x) = f(u) du
>Huh?
The probability that x will be less than, for example, 5% is F(5) = 20% as shown on the rightmost picture, in Figure 1.
>And that 20 is the area under the fcurve?
Yeah, the area to the left of x = 5%.
 Figure 1

The probability that x is less than infinity is clearly 100% so we need to choose A, b and k so that:
F(∞) = = 1
>Is it equal to "1" or "100%"?
They're the same: 1 means 100%.
Now we'll search for appropriate values for A, b and k.
Note that, when z is close to 0, SQRT(1+z) = (1+z)^{1/2} = 1+z/2 ... approximately.
So, when x is close to its Mean m, SQRT(1+k(xm)^{2}) = 1+k(xm)^{2}/2 (approx.)
so we can write f(x) as:
f(x) = A e^{b(1+k(xm)2/2)} = Ae^{b} e^{bk(xm)2/2} (approx.)
This is the form we'd expect since the Normal distribution has this form, namely:
f(x)_{normal} = 1/SQRT(2π) e^{(1/2)(xm)2/s2}
In order to match these characteristics near the Mean, we set
bk = 1/s^{2} or b = 1/ks^{2}
That gives:
(2) f(x) = A e^{1/ks2 SQRT( 1+k(xm)2 )}
Since we also want = 1 we choose A so that
A e^{1/ks2 SQRT( 1+k(xm)2 )}dx = 1
That makes A a function of k, so we'll write:
(3) A(k) = 
1
e^{1/ks2 SQRT( 1+k(xm)2 )}dx

>But A depends upon the Mean and Standard Deviation as well, right?
Yes, but we know the numbers m and s ... assuming we're trying to mimic the distribution of returns for some particular stock.
>Altogether now?
Yes. Altogether now:
f(x) = A(k) e^{(1/ks2)SQRT(1+k(xm)2/2)}
where A(k) = 
1
e^{1/ks2 SQRT( 1+k(xm)2 )}dx




>So what's k?
Like I said, we choose it so as to mimic the distribution of returns for some particular stock.
>Well, you're gonna have fun evaluating A, eh?
Not on a spreadsheet. Here's what we'll do:
 We'll download daily stock prices for, say GE, and calculate the daily returns.
 Then we'll plot the distribution of these returns, like Figure 2.
 We'll look at the Normal and Lognormal distributions with the same Mean and Standard Deviation as the GE returns. There's little difference, for daily returns.
 We'll note that the peak needs to be taller and we need fatter tails.
 We'll then try out our "other distribution" with various kvalues to see if we can improve upon this.
>Yeah, so?
Okay, if we consider the distribution of GE returns (as per Figure 2) and pick a nice kvalue, we can get Figure 2A, below.
 Figure 2

Figure 2A

>Why k=10?
Actually (it surprised me!), the chart of f(x) is relatively insensitive to your choice of k.
Figure 2B

>That peak may be relatively insensitive, but the tails ...
Aah, yes, the tails. I guess that's because we tried to match the actual distribution near the Mean  and that's near the peak.
But the tails, they're ... uh ...
>Fat!
Well, fatter, for large kvalues, but the peak drops a bit. So we have to compromise between matching the peak and the tails.
Here's a few more:
Figure 2C
>Okay, but how do you evaluate A(k)?
You mean: e^{1/ks2 SQRT( 1+k(xm)2 )}dx ?
I do a sum, like so, for a bunch of j's:
Σ e^{1/ks2SQRT(1+k(xjm)2)}Δx
>Huh?
Don't worry about it.
Remember when we noted that actual returns can sometimes be extreme
... far more (or less!) than one would expect from a Normal or Lognormal distributions.
(See this chart where the upper chart with the wild returns is real
whereas the lower is fictitious, based upon a Normal distribution with the same Mean and Volatility.)
In Figure 3 (top chart) we have the daily returns for MSFT (for the past ten years).
If the distribution were Normal (for example) then, as it turns out, we'd expect a return outside the range 8% to 8% once or twice. In fact, there were ...
>I count 6!
Yes, me too. Six in this 10year period.
On the other hand, if we try to mimic the MSFT returns using our OTHER distribution we can get lots of returns outside (8%, 8%).
>And the match? I mean the actual distribution vs ...
I know what you mean ...
 Figure 3

>Why k = 200? You just pick a k which works, eh?
Yes, of course. What would you suggest?
>I'd suggest ... uh, picking a k which works. So, where's the spreadsheet?
I'm thinking about it ...
to continue
