Many people would say that the price of a stock some time in the future cannot be predicted. And indeed, if there is some formula which says that the price tomorrow will be R20 and one day later R24, everybody will tomorrow buy their shares until the price rises to R24 (ignoring the effect of interest); what will happen is that the price formula make a wrong prediction for tomorrow's price! So mathematical models for stock price movements are not very useful for an investor who deals in stocks only. Formulae for predicting prices is science fiction.

However, there is a reason to try to model the stock market anyway. For instance, there are contracts traded everywhere in the world of the following type: you have an option so that at the end of the year you can buy an ABSA share for R100. That is, at the end of the year, if you want, you can buy the share at R100 (irrespective of what its price is then), but you are not forced to. How much should you be willing to pay for this contract? Clearly, if the price of the share is more than R100 at the end of the year, this contract will be worth something to you (namely, the difference between the value of the share and R100). But without a mathematical model we cannot say much more about what we should pay today to have this option at the end of the year. We need a description of the " random" behaviour of the stock, even though this model will be unable to predict the specific price that will be seen tomorrow. It will therefore be a "statistical" model.

One model, which is not too bad, is to say that tomorrow's price can be obtained from that of today in the following way. A coin is flipped somewhere in moneyland. Tomorrow´s price is then today´s price plus x if the coin happened to land on heads; and we subtract x instead if the coin landed on tails. (Here x is some fixed number chosen beforehand.) The price of the day after tomorrow is obtained in the same way from tomorrow´s price, and so we continue.

This is called the ¨random walk¨. If one plots (simulated) graphs of a process like this, it resembles that of stock prices we see in newspapers. However, there is a seriously wrong implication which one only needs a little common sense to discover...

In this project we will consider various simple models and make our own conclusions about them. We will also look at statistical ways of testing such models to climb out of the ivory tower and test it against what is actually observed in the market. In addition we will write computer programs needed to handle the numerics and data processing involved in this.
 

• Mr. Gusti van Zyl