Random Walk Theory

The random walk theory (RWT) or the theory of the symmetrical odyssey is a theory that mathematically describes the course of market prices over time (especially share prices and other security prices ). It is also called random flight statistics . The term random walk or symmetrical odyssey is a direct consequence of the market efficiency hypothesis .

description

According to the random walk theory, a course can also be understood as a signal and analyzed and modeled according to the teachings of signal theory .

One possible model would be, for example

${\displaystyle S(t)=T(t)+P(t)+U(t)}$

It stands for the signal, i.e. the course, the drift component, the periodic component and an independent noise component . ${\displaystyle S(t)}$ ${\displaystyle T(t)}$ ${\displaystyle P(t)}$ ${\displaystyle U(t)}$

The drift component and the periodic component are combined to form the trend , which can be described by moving averages. It is equal to the information input function due to the instantaneous manifestation of all information, i.e. H. the real information content of the course. This is a random function , as there is no way of predicting the future course.

The threshold here is synonymous with the independent noise component. In the random walk theory it is assumed to be without information. A Brownian motion is postulated here. ${\displaystyle U(t)}$

Criticism of the random walk theory

Signal analysis using time series analysis of indices such as the DAX or the Dow Jones Industrial Average shows that the threshold is not white noise .

Time series analysis of the threshold

The threshold is not normally distributed , but has so-called " fat distribution ends ", i.e. H. there is leptokurtosis . Furthermore, it does not have a quasi-constant amplitude : there are large amplitude fluctuations of the threshold, which form so-called volatility clusters . The threshold is a function of the noise with heteroscedasticity .

A good approximation of the threshold is given by the GARCH models . However, this only applies to the past; the forecasting skills are not particularly good.

Comparison with general approaches

According to Otto Loistl, ARMA models according to Box-Jenkins have best-fit approximation approaches for most DAX values ​​that do not correspond to the random walk theory, since these approaches do not have vanishing ones. ${\displaystyle ARMA(p,d,q)\cdot (P,D,Q)}$ ${\displaystyle p,q}$

Other approaches

As an alternative to the random walk theory, the course can be approximated with Markov chains, i.e. the approach of a function with complete forgetfulness.

literature

• Otto Loistl: capital market theory . Oldenbourg Verlag, Munich 1994, ISBN 3-486-22968-0 .