```-----------------------------------------------------
#                                                   #
#   Chapter 4: Brownian Motion, Wiener Measure      #
#            and the Black-Scholes Model            #
#                                                   #
-----------------------------------------------------

Summary: The returns of liquidly tradable assets like
stocks, stock indices (or futures on them), currencies
and commodities like crude oil, gold and silver have
some typical properties which are common to all of them.
If the returns are normalized, that is, if we subtract
the mean and divide by the standard deviation, their
distribution is close to a normal distribution. This
leads to

S(t_k) = S(t_{k-1}) * (1 + mean + stddev * phi_k )       (1)

with some standard normal distributed random numbers
phi_k and S(t_k) denoting the closing price on day t_k.

If the returns are not considered on a daily basis, but
over some time horizon Delta_t, a statistical analysis
reveals that the mean is actually proportional to
Delta_t and the standard deviation stddev is propor-
tional to sqr(Delta_t). Thus one is lead to

S(t_k) = S(t_{k-1}) * (1 + mean * Delta_t

+ stddev * sqr(Delta_t) * phi_k )       (2)

with t_k = k*Delta_t. Equation (2) is the time discrete
version of the Black Scholes model. The corresponding
product measure of Gaussian densities for the random
numbers phi_k lead in a natural way to the concept of
Brownian motion and Wiener measure.

The geometric Brownian motion arises as the explicit
(and, in discrete time, approximate) solution of the
stochastic difference equation (2). The term "geometric
Brownian motion" and "Black-Scholes model" actually
refer to the same stochastic process.

A data analysis with several plots and histograms for
the returns of SP500, DAX30 and the GE-stock can be
found here:
```
http://hsrm-mathematik.de/WS201516/master/Zeitreihenmodelle/stylized-facts-financial-time-series.html

pdf-file: Chapter 4: Brownian Motion, Wiener Measure and the Black-Scholes Model

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