SURPRISE 95

BrownianMotion AndThe EconomicWorld- ArticleOne


Contents

Project
Introduction
Applications of Brownian Motion to Market Analysis
Conclusion
Sources and References

Project

Brownian motion is a sophisticated random number generator, based on a process discovered in plants(R. Brown; 1827) This continuous random motion of solid microscopic particles when suspended in a fluid medium is due to the consequence of continuous bombardment by atoms and molecules. It has a wide range of applications, including modeling noise in images, generating fractals, growth of crystals and stock market simulation.In this porject, brownian motion is firstly introduced based on the research work by Albert Einstein and a number of its real-life applications are then briefly explained. In this first article, my project partner will concentrate on introducing Brownian motion generally and I will look into some applications of it in the financial world.

Introduction

The explanation of Brownian motion, given by Einstein in 1905 and based on the kinetic-molecular conception of matter, is considered one of the fundamental pillars supporting atomism in its victorious struggle against phenomenological physics in the early years of this century. Discovered by Robert Brown in 1827, of the continuous movement of small particles suspended in a fluid, it did not arouse interest for a long time, until at the close of the century when Guoy's conviction and research (that Brownian motion constituted a clear demonstration of the existence of molecules in continuous motion) brounght it to the attention of the Physics world. However, all nineteenth-century research remained at the qualitative level and yet it was able to clarify some general characteristics of the phenomenon: the completely irregular, unceasing, motion of the particles is not produced by external causes, and it does not depend on the nature of the particles but only on their size. It was only in 1905 when a quantitative analysis was brought about, where Einstein succeeded in stating the mathematical laws governing the movements of particles on the basis of the principles of the kinetic-molecular theory. He derived a formula whereby the path described by a molecule on the average is not proportional to time, but proportional to the square root of time. This follows from the fact that the path described during two consecutive unit time-intervals are not always to be added, but just as frequently have to be subtracted.

This article will deal mainly with the following field :
Stock Market Analysis

Appliactions of Brownian Motion to Market Analysis


Background Information

Around 1900, Louis Bachelier first proposed that financial markets follow a 'random walk' which can be modeled by standard probability calculus. In the simplest terms, a "random walk" is essentially a Brownian motion where the previous change in the value of a variable is unrelated to future or past changes.

Brownian motion has desirable mathematical characteristics, where statistics can be estimated with great precision, and probabilities can be calculated, and hence scientists and analysts often turn to such an independant process when faced with the analysis of a multidimensional process of unknown origin (ie. the stock market). The Brownian motion theory and Random Walk model are widely applied to the modelling of markets, and the insight that speculation can be modeled by probabilities extends from Bachelier and continues to this day.


Brownian Motion in the Stock Market

In the middle of this century, work done by M.F.M Osborne showed that the logarithms of common-stock prices, and the value of money, can be regarded as an ensemble of decisions in statistical equilibrium, and that this ensemble of logarithms of prices, each varying with time, has a close analogy with the ensemble of coordinates of a large number of molecules. Using a probability distribution function and the prices of the same random stock choice at random times, he was able to derive a steady state distribution function, which is precisely the probability distribution for a particle in Brownian motion. A similar distribution holds for the value of money, measured approximately by stock market indices. Sufficient, but not necessary conditions to derive this distribution quantitatively are given by the conditions of trading, and the Weber-Fechner law. (The Weber-Fechner law states that equal ratios of physical stimulus, for example, sound frequency in vibrations/sec, correspond to equal intervals of subjective sensation, such as pitch. The value of a subjective sensation, like absolute position in physical space, is not measurable, but changes or differences in sensation are, since by experiment they can be equated, and reproduced, thus fulfilling the criteria of measurability).

A consequence of the distribution function is that the expectation values for price itself increases , with increasing time intervals 't', at a rate of 3 to 5 percent per year, with increasing fluctuation, or dispersion, of Price.This secular increase has nothing to do with long-term inflation, or the growth of assets in a capitalistic economy, since the expected reciprocal of price, or number of shares purchasable in the future, per dollar, increases with time in an identical fashion.

Thus, it was shown in his paper that prices in the market did vary in a similar fashion to molecules in Brownian motion. In another paper presented around the same period, it was also found that there is definite evidence of periodic in time structure (of the prices in Brownian motion) corresponding to intervals of a day, week, quarter and year : these being simply the cycles of human attention span.


With compounding evidence and widespread acceptance that Brownian motion exists in market structures, many researches and studies have since taken place, revolving and evolving around this theory.

For example, a statistical analysis of the New York Stock Exchange composite index to show that Levy Processes do exist in it was carried out by R.N.Mantegna, and he showed that the daily variations of the of the price index are distributed on a 'Levy' stable probability distribution, and that the spectral density of the price index is close to one expected for a Brownian motion.


Optimal Dynamic Trading with Leverage Constraints

Specific investigations, for example, like finding a solution for the optimal dynamic trading strategy of an investor who faces leverage constraints (ie. a limitation on his ability to borrow for the purpose of investing in a risky asset), makes use of the assumption that the value of the risky asset follows a geometric Brownian motion, and this assumption allows for a quantitative analysis to be possible , and hence arriving at explicit solutions to optimal portfolio problems containing leverage and minimum portfolio return constraints.


Investment, Uncertainty, and Price Stabilization Schemes

Another application of Einstein's theory is seen in the paper done by William Smith, who uses the method of regulated Brownian motion to analyze the effects of price stabilization schemes on investment when demand is uncertain. He investigates the behavior of investment when price is random, but subject to an exogenous ceiling, and with the aid of the mathematics of regulated Brownian motion, demonstrated that price controls mitigate the response of investment to changes in price, even when controls are not binding. The conclusions developed would be applicable to any economic situation involving smooth costs of adjustments of stocks when prices are uncertain but subject to government control (ie. rent controls, hiring/firing decisions in the presence of a minimum wage).


A Brownian Motion Model for Decision Making

The Brownian model was also made use of by L.Romanow to develop a model for a decision making process in which action is taken when a threshold criterion level is reached. The model was developed with reference to career mobility, and it provides an explanation of an important feature of promotion processes in internal labour markets. The model assumes continuous observation of behavior (of employees) and that the only route for leaving a job is by promotion. This suggests that the important mechanisms in the process are the basic evaluation procedure -- rating which includes a random component (Brownian motion theory), and the dicision rule -- promote when an estimated average reaches a criterion level. The model was able to provide substantive qualitative results and hence is of good use to the 'real' world in decision making policies.

Conclusion

The applications of this "random theory" are indeed far and wide, and in this particular area in the economists' realm, random events must occur in order to foster innovation. If we know exactly what was to come, we would stop experimenting. We would stop learning. Thus, Brownian motion and the random walk hypothesis offer us a way to understand how markets and economies function. There are no guarantees, as yet, that they will make it easier for us to make money, however, we will be better able to develop strategies and assess risks using them.


Sources

L.Bachelier, Theorie de la Speculation (Gauthier-Villars, Paris, 1900)
P.H.Cootner,Ed.,The Random Character of Stock Market Prices (MIT Press,1964)
R.N.Mantegna, Physica A179 232 (1991)
Allyn L.Romanow,Journal of Mathematical Sociology,1984,Vol.10, pp 1-28
W.T.Smith,Journal of Econ Dynamics and Control 18,(1994),pp 561-579
S.J.Grossman, Journal of financial and quantitative analysis,(1992),Vol27,pp151-172
E.E.Peters, Fractal Market Analysis,J.Wiley and Sons, 1994.
A.Einstein, The Brownian Movement, 1956.
BLPES Library,LSE
Bath Information and Data Services(BIDS)
Internet Newsgroups
Imperial College and Science Museum Libraries


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