4.  How to model stock (asset) prices

It is not much of a complication to include a measure of the cost of borrowing money,
namely to introduce a risk-free interest rate, say "r".  Another complication of more
significance is how to anticipate possible asset price swings.  This is where the notion
of a "random walk" enters.  The most popular notion of how asset prices move
is that their prices are a solution to a certain stochastic differential equation.  This
equation then has a few consequences, namely that the asset price fluctuates randomly
but nevertheless has an expected mean growth rate and an expected volatility.

The usual, simplest, model for asset prices, S, is given by:

    dS = S (μ dt  +  σ  dX)

i.e.

"relative change in S"  =  μ dt  +  σ  dX

This should seem reasonable: the relative or PERCENTAGE change in S in a short time dt  is reasonably modeled by a mean growth rate, μ, and a random component, given by σ dX.  Think of dX as a standardized but random change, and so σ measures how susceptible S is to the random vagaries.  

Simulation will require using "pseudo-random numbers".  Such numbers are not so easy to come by.  Let's view the result of RANDU: (using our applet).

RANDU was the IBM mainframe's built-in FORTRAN random number generator
(an example of a standard "linear-congruential" pseudo-random number generator).
We will now try to proceed to the web to view the following: take
3 consecutive pseudo-random numbers between 0 and 1, call them x, y and z, and plot them as a point in .  Then take the next three consecutive numbers, and plot them as another point in .

Visit
    http://www.tamu-commerce.edu/coas/math/FACULTY/KREMIN/kremin.html
(third "bulleted"-item from the bottom).
  
  Lesson from that site: it's not that easy to come up with a good set of
  pseudo-random numbers for simulation purposes.  Nevertheless, we shall go on.