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### 2.1.1 Shares and market indices

[ SLIDE random walk - log-normal - Lévy - drifts || VIDEO login]

Small companies generally start developing a product using private resources and sometimes a limited amount of venture capital. If all goes well, they may under circumstance decide to go to the stock market for the quest of new capital, which will allow them to grow more rapidly than what they could achieve by simply re-investing their own earnings.

Investment banks assist them in the initial public offering (IPO), when the company value (often estimated for the future potential more than the present earnings) is divided in a number of shares that are proposed to the investors on the stock market. By selling a fraction of their company, the original owners realize a capital gain, but also give up part of the control and future earnings to other shareholders. After a rapid and rather systematic evolution (depending on how well the investment bank succeeds in aligning the initial offering with the market expectations), the share price starts a dominantly random evolution in agreement with Fama's efficient market hypothesis introduced in section 1.4.

Previous experiments with the VMARKET applet suggested that possible realizations for the price of a share can be simulated by adding small increments to the initial price that is known. To be precise, the market (or spot) value can never be predicted with certainty, but an expected value can nevertheless be calculated, provided that the distribution of increments reproduces the market characteristics.

In addition to the deterministic growth (Drift parameter  ) and the random component associated with risk (Volatility parameter  ), statistical analysis unveils a significant difference between the stock and the bond prices: the share price increments have a log-normal distribution, while the spot rate increments tend to have a more normal distribution. In other words, a share presently at EUR 10 is as likely to double in value to EUR 20 as it is to divide by two down to EUR 5. This in contrast with interest rates at 10%, which are as likely to rise (to 15%) or fall (to 5%) by the same amount. The VMARKET applet below illustrates the difference between the two distributions, assimilating the random horizontal motion of a red dot with the price of a share in a volatile market.

 Virtual market experiments: log-/normal price increments Switch between a log-normal (LogNkappa=1) and a normal (LogNkappa=0) distribution of the increments and try to qualify the difference between the two evolutions. Increase to Volatility=3 and Step 1 log-normal increment at a time. Take a few measurements showing that the jumps are often larger for high prices (to the right) than for low prices (to the left); they are on the contrary symmetric with a normal distribution. Increase the Volatility further and check for both distributions if the price can ever become negative. Note that the numerical model produces wrong answers for large increments and large time steps; under realistic conditions, the volatility rarely approaches unity.