the Levy distribution
motivated by e-mail from R. Brown.

Suppose that:

  • The price of a stock is P(t), at time t.
  • Suppose, further, that the next price (at time t+1) is randomly distributed.
  • Suppose that the distribution of possible price changes (at time t) has some (as-yet-unknown) distribution.
It has been observed that this "t to t+1" change is not Normally distributed.
It has also been observed that the change may be very large.
It has also been observed that ...
>Okay! Not Normal ... hence abnormal, right?
Patience.
Remember what the garden variety Normal distribution looks like?            
>That's the probability density function and S the Standard Deviation, right?
[N1]
       

Yes, with Mean = 0. Now let's suppose that S, the Standard Deviation, is a function of x, like so:
[L1]   S = x3/2 / c1/2   where c is some parameter.
In fact, this distribution can be generated via MS Excel using:   NORMDIST(x,0,x^(3/2)*c^(1/2),0)
Then that Normal distribution changes its personality ... and looks like so:            
[L2]
and so (Figure 1):            

That distribution has a name and its ...

>Levy!
Yes, after Paul Pierre LÚvy

>So why that strange transformation [L1].
Patience.
However, note that, for x very large, the exponential term in the numerator is very nearly 1.
Hence the Levy distribution decreases like 1/x3/2 ... and that gives fat tails.
Indeed, the Mean of the Levy distribution is ... uh ... infinite.
That's because:  


Figure 1: Levy distribution

Once upon a time I was staring at the stock prices for XOM. They looked like this:    
See? The volatility (or Standard Deviation) changed dramatically.
That characteristic implies that assuming some constant Standard Deviation is way off base.
That leads to something called the Hurst Exponent.
That leads to ...

>But you never finished that Hurst Exponent tutorial!
Uh ... yes ... I mean NO. I get easily distracted.


Figure 2
Anyway, let's assume that the Standad Deviation for the gains increases when the gains are large.
In fact, let's assume that:
[g1]   S = k xα   where k and α are some parameters.
We stock this S into [N1] and get Figure 3:

>I assume you mean you'll stick this S into the Normal formula.
I'm glad to see you're paying attention.

Anyway, if we vary α (as in Fig. 2) we get lots of possible distributions.

Of course, to be a valid density distribution, we'd need to have:  
... which says that the probability that x lies in (0, ∞) is 1.


Figure 3: modified-Levy distribution
That requirement places a restriction on k, once you've picked your favourite value for α.

>And that restriction is ... what?
I have no idea.
Note, however, that the blue curve in Fig. 2 is the same as the blue curve in Fig. 1. (
They got k = c = 1 and α = 3/2.


Checking Levy

Note that, in the above discussion, the values of x are positive.
We assume they represent the gain factors for some stock. That is, if the stock return is 8.2%, then the corresponding gain factor is 1.082.
Then, unless we have returns less than -100%, our gain factors will be positive. (Let's just call 'em "gains", okay?)

Okay, let's first consider the daily gains for that XOM stock that we noted earlier (in Figure 2).
For the past 10 years, the stock performance looked like this:        
We look at the distribution of the 2500 daily stock gains.
They look like this:

Figure 4
The average (or Mean) daily return was about 0.05% so the average "gain" was about 1.0005.

>Is that Levy?
It don't look like Levy, eh?
Let' try some others (over the same time period). Here, we'll use bar graphs ('cause f(x) dx gives the number of gains in some small interval).

>And you like to show the interval, right?
Yes ... and because the charts look more sanitary.

Microsoft

General Electric

Johnson & Johnson
In fact, the cumulative probability looks even more sanitary:

>Cumulative?
That's F(x), the integral of f(x) from 0 to x. For any x, it gives the percentage of gains less than x.    
See the blue dot? It says that 90% of gains are less than 1.025.

>And that means less than 2.5% daily return.
You got it.


Cumulative Distribution