Here's what we want to do: The random variable x represents a possible stock return and we measure deviations of x from the Mean m; that's x-m. We use as units the Standard Deviation of the set of possible returns, namely s. To that end we introduce the variable : u = (x - m) / s We then invent F(u) such that F(u(x)) goes from 0 to 1 as x goes from -∞ to +∞. Calculate f(u(x)) = dF/dx = (dF/du) (du/dx) = (1/s) dF/du and arrange to have f (u(x)) dx = 1. Note that our f(u) and F(u) are really functions of x, eh? Adjust F so that the various moments exist : un f(u) du That means f(u) must decrease VERY rapidly as |u| ∞ Adjust F to match historical data. Adjust F so that f has fat tails. Adjust F so that ...

Figure 1a Figure 1b

>Adjust F so f has fat tails? Well, yes ... compared to, say, the normal distribution. Remember that, for the normal distribution, f(u) decreases as u increases in magnitude. (We call this magnitude |u| and it measures how far x is, from its mean), That decreasing behaviour is like e-u2. See how quickly it decreases compared to, say, e-|u| ? >So you invent F(u) so it decreases like e-|u| ? No, it's f(u) that we want to decrease like e-|u| ... or , at least, less rapidly than e-u2.

Figure 2

>For example?
In Part II we looked at F(u) = (1/2)(1 + tanh(u)) and noted that:

tanh(u) = (eu - e-u) / (eu + e-u) +1 or -1   as u +∞ or -∞         Note that F(u) = (1/2)(1 + tanh(u)) = e2u / ( e2u + 1) = 1 / ( 1 + e-2u) lies between 0 and 1. cosh(u) = (1/2) (eu + e-u)         Note that cosh(u) ≈ (1/2) e|u|   as |u| ∞ f(u) = dF/dx = (1/s) dF/du = (1/2s) (1/ cosh2(u))         Note that f(u) ≈ (2/s) e-2|u|   as |u| ∞ We then adopt: [1]       F(u) = (1/2)(1 + tanh(u)) = e2u / ( e2u + 1) = 1 / ( 1 + e-2u) ... where u = (x - m) / s [2]       f(u) = (1/2s) / cosh2(u) ... where u = (x - m) / s If we know F(u) = k and wish to find u, we solve for u. This involves the inverse function u = F-1(k): [3a]       u = F-1(k) = (1/2) log[ k / (1 - k) ] Since u = (x - m) / s we have (for a given value of F, namely k): [3b]       x = m +(s/2) log[ k / (1 - k) ] ... where 0 < k < 1

>And they look like Figure 1, right?
Right ... and we've got fat tails. Remember? Here we compared weekly returns for GE and a random selection from that same set of returns then a random selection from a normal distribution. The first two had the occasional LARGE return ... but the normal distribution was tranquil in comparison.

>Yeah, I remember that tranquil characterization, but ...
Okay, so here's a similar collection:

>Your tanh distribution is ... how shall I say it? It's obese!

Yah, too many huge returns, more fat than necessary. We can fix that by choosing, in [3b] above:

[3c]       x = m +(s/A) log[ k / (1 - k) ] ... with A = 3.

That'd give (for example) Figure 3a: Figure 3a

Figure 3b

However, changing A is equivalent to changing the standard deviation s, as in Figure 3b.
We need to introduce more parameters, other than m and s, so that ...

>So that the distribution isn't so symmetrical, eh?
Uh ... yes.

Introducing Asymmetry

Note that a symmetrical distribution like Figure 3b can be made asymmetrical by replacing u, as in f(u), by g(u).

>Huh?
I mean, instead of looking at F(u) = 1/(1 + e^-2u) as we did above, we replace u by some invented function g(u).
That'd give: F(g) = 1/(1 + e^-2g) ... and now we fiddle with g(u).

>A picture is worth a thousand ...
Here are some pictures:

Figure 3a

Figure 3b

Figure 3c

>That makes four parameters! Isn't that overkill?
Apparently von Neumann once said:
"With four parameters I can fit an elephant and with five I can make him wiggle his trunk."

>Aha! See? Four is too many! With four you can fit an elephant!
Well, that depends upon what functions you're using.
If I'm allowed to give names to previously unknown functions, then I can fit an elephant with ... uh, a single parameter.
>Yeah, sure. What unknown functions?
These:

 

>Very funny. Just invent a function or two. Why not? Have you heard of Bessel functions or Legendre polynomials or ... >Invented by those guys? Why not? Haven't you heard of Joe InverseSine?   >Okay, but can we get back to distributions? Is g(u) = u + A u2 + B u3 any good ? Figure 4 shows a "best" fit to a bunch of S&P 500 returns. >I assume you can get lots of shapes by changing A and B, right? Yes, A and B and m and s. Check this out.

Figure 4

>So where's the spreadsheet?
Click here or, if that don't work, try to RIGHT-click and Save the target link file.

>Is there some kind of guarantee on the spreadsheet?
Always  

The spreadsheet looks like this