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