A Model for the
Stock Price Behaviour

What are Shares or Stocks?

A share is a portion of the ownership of a company. The holders of the shares (also known as shareholders) have a stake in the assets and profits of the company, proportional to the fraction of the shares they won.

For example, if a company has 100 shares outstanding and a person owns 10 of them, then he/she owns 10% of the company. Most of the company shares give the shareholders the right to vote.

Trading shares is not limited to just big players like Investment Banks and Fund Managers. Millions of ordinary people buy and sell shares everyday. Many use online share brokers, who will buy and sell shares on behalf of the customers. Most of the high street banks provide this facility for a nominal fee to their customers.

The self invested personal pension (SIPP) scheme entirely relies on one's ability to buy and sell shares at the right time and for the right price to build their pension fund. The schemes like this one are increasingly relevant in the current economic environment, so an understanding of how a given stock market index or a stock price behave might respond to other economic indicators would be beneficial to anyone who considers SIPP or SIPP-like plans.

Let us take a look at the behaviour of a stock price quantitatively during a normal tading day. Normal trading day is a day when no significant external economic policy meansures were announced.

How Does the Share Price Behave?

The share price of a company is assumed to follow a stochastic process. More specifically the share price S is said to follow the geometric brownian motion (GBM). This means S satisfies the following stochastic differential equation:

Here S is the stock price. is the expected annual rate of return per unit time. is the volatility of the stock price - the standard deviation of the percentage annual return. Z is the Weiner Process. Therefore the change in the stock price is given by:

Here is a random drawing from a standard normal distribution with mean 0 and variance 1. The discrete version of the above equation can be written, after dividing the equation by S, as:

The left-hand-side (lhs) of the above equation is the return provided by the share in a short period of time. The first term on the right-hand-side (rhs) is the expected value of this return, and the second on the rhs is the stochastic component of the return.

The variance of the return is nothing but the variance of the stochastic component - rhs term 2. This is because the rhs term 1 (the mean expected value of the return) is constant. Therefore the return - the change in stock price with respect to the stock price - is normally distributed. That is: