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 ...
No. Here we want to consider yet another distribution of the form:
(1)       f(x) = A e-b SQRT( 1+k(x-m)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 right-most picture, in Figure 1. >And that 20 is the area under the f-curve? 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(x-m)2) = 1+k(x-m)2/2     (approx.)
so we can write f(x) as:
f(x) = A e-b(1+k(x-m)2/2) = Ae-b e-bk(x-m)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)(x-m)2/s2

In order to match these characteristics near the Mean, we set
bk = 1/s2   or   b = 1/ks2

That gives:
(2)       f(x) = A e-1/ks2 SQRT( 1+k(x-m)2 )

Since we also want = 1   we choose A so that
A e-1/ks2 SQRT( 1+k(x-m)2 )dx = 1

That makes A a function of k, so we'll write:
 (3)       A(k) = 1 e-1/ks2 SQRT( 1+k(x-m)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(x-m)2/2)
 where     A(k) = 1 e-1/ks2 SQRT( 1+k(x-m)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 k-values 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 k-value, 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 k-values, 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(x-m)2 )dx ?
I do a sum, like so, for a bunch of j's: Σ e-1/ks2SQRT(1+k(xj-m)2)Δx
>Huh?