2.1.1 Shares and market indices

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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.

In the next chapter, we will study more carefully how the increment dS describing the random component of an asset price S is proportional to a the stochastic increment dX

where the increment dX can be evaluated as a random number drawn from a normal distribution with zero mean (1.4#eq.5) to simulate relatively large time steps longer than one month. Very little programming experience is needed to understand how the log-normal walk (the position or the price) of a single particle has been implemented in

double mu = runData.getParamValue("Drift"); double sigma = runData.getParamValue("Volatility"); for(int j=0; j<numberOfRealisations; j++){ currentState[j][0] += currentState[j][0] * ( mu*timeStep + random.nextGaussian()*sigma*Math.sqrt(timeStep) ); }For those who are not familiar with Java or C++ coding conventions, note that

`x+=1.0`

increments the variable `x`

by the real value one
and `j++`

increments the variable j by the integer value one after
using it for evaluation.
Choosing the input parameter `Drift=0.0`

in the applet also sets
`mu=0.0`

, so that the normally distributed random number
obtained from the function `random.nextGaussian()`

is here simply scaled by the current position to simulate a log-normal
distribution of the increments without drift.
For time steps shorter than one month, the random component of asset price increments is sometimes modeled with a Lévy stable symmetrical distribution

The probability with an index =1.40 have been used in Ref.[15] to first calculate the scaling factor =0.00375 and then reproduce nearly three orders of magnitude of the ``fat tails'' of a

At least some investors have to believe that the price of a share will
rise more rapidly than the return they can earn overnight from a deposit
at a spot rate, which carries little or no risk at all.
A *risk premium* in the range 3-8% is
usually added to the 0-2% spot rate to account for a positive drift
proportional to the share value (e.g. `Drift=SpotRate+0.04`).

A fixed dividend payment
per share is often made each year after
the shareholder's assembly; for simplicity, this can be modeled as a
continuous dividend yield
in the range 1-4% and contributes
with a negative drift to the spot price, since the payments reduce
the total value of the company (e.g. `Drift=SpotRate-Dividend`,
see exercise 2.02).

Before we conclude this short introduction on the modeling of the prices on
the stock market, simply note that any combination of shares can be used to
form a weighted average called *market index*: the most famous such as the
Dow Jones
and
NASDAQ 100
in the US,
the FTSE 100
(Financial Times Stock Exchange, pronounce ``footsee'') in the UK and the
Nikkei 225
in
Japan combine the largest and most prestigious companies (so called ``blue
chips'' - the name stems from the game of poker where the blue chip have
the highest value) in a country and provide a measure of the economic growth
of a nation.

**SYLLABUS** ** Previous:** 2.1 The stock market
**Up:** 2.1 The stock market
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