The general structure of market
data
the statistical analysis of trading data, and stochastic
modelling using it, is an ongoing area of research in finance because,
despite intensive study, a comprehensive
understanding of the
structure of capital markets exchange trading data remains elusive. For
an overview of this issue, Chapter 5 of Sonification
and
Information provides a more formal introduction, including more
detail of the techniques illustrated here.
As markettime unfolds, the price of a traded security
‘wanders’ from
value to value. This is known mathematically as a stochastic process,
or a random function. Amongst the simplest types of such stochastic
processes is a random walk, called so because it may be thought of as a
simplified model for an individual walking on a line who at each point
of time either takes one step to the right with probability p or one
step to the left with probability (1–p). Because the domain over which
this function is defined is a time interval, and prices move discretely
(step from one value to another), this random function is also known as
a discrete time series and securities trading datasets can thus be
classified as discrete multidimensional time series whose
momenttomoment behaviour is analogous to geometric Brownian motion.
Two principal concerns are ways to accurately describe the way prices
are distributed statistically and whether or not, and to what extent,
autocorrelation exists and can be detected, even preempted, as market
prices evolve. Understanding the accuracy of the description of price
distribution is important for the risk analysis of various trading
instruments in the longer term, and understanding the inherent
autocorrelation is important in attempts to predict future, especially
catastrophic, events.
A little history: The Efficient Market Hypothesis
In
1900, Louis Bachelier, using methods that had been created for
analysing gambling, conjectured that price fluctuations in day–to–day
exchange–traded securities were independent random variables. He
provided little empirical evidence to support the assumption, yet his
work, which also included techniques for analysing the value of
government bonds and displaying optionsrelated strategies, is today
considered seminal. Bachelier’s thesis was revolutionary, but largely
ignored and forgotten and it was Albert Einstein's independent
description that brought the solution for Brownian motion to the
attention of physicists.
The unexpected crashes of stock markets in 1929, and the depression
that followed, stimulated mathematicians to attempt a better
understanding of market action through statistical analysis. In 1962
Paul Cootner published an anthology of quantitative analysis that
became the basis for what is known as the Efficient Market Hypothesis,
which Eugene Fama later formalised in 1965. Simply stated, the
Efficient Market Hypothesis assumes that the market price of a trading
security reflects all that is known about it; the difference in price
from one point in time to another, simply reflecting any new
information about the security as it becomes known. After digesting the
new information together with an assessment of the risks involved, the
collective consciousness that is the market finds an equilibrium price.
This is called the random walk version of Efficient Market Hypothesis.
A random walk is a sufficient but not a necessary condition for market
efficiency. That is, whilst market efficiency does not necessarily
imply a random walk (it may be some other process), a random walk does
imply market efficiency. Such a random walk is called Brownian
motion. If Bachelier’s conjecture is correct, as is assumed by
the Efficient Market Hypothesis, prices would exhibit no
autocorrelation and a statistical analysis would reveal a
normallydistributed, or Gaussian, probability density function (PDF).
Mandelbrot ups the ante
As a result of the discrepancy between this theoretical perspective and
his analysis of the way certain speculative prices, such as those of
cotton, moved, Benoit Mandelbrot became dissatisfied with this
simplified model. Mandelbrot is one of the significant contributors in
the field and his technical monograph summarises a unique and
influential perspective built over many decades. His investigations
showed that real markets exhibit much larger variability, as well as
greater leptokurtosis and skewness than a normal distribution. He
realised that the market process could be better described by a Lévy
flight.
In addition to their distribution, markets also exhibit momentary
autocorrelations. Modern econometrics and financial engineering place
considerable import on understanding such phenomena because the
increased likelihood of extreme events, both positive and negative,
indicates greater market volatility than if the markets are normally
distributed, which, in turn, would impact on risk assessment, options
pricing and portfolio theory in general.
The principal statistical techniques used in market analysis and
simulation today, use Bayesian partitioning, variable length Markov
chains and Monte Carlo simulations, and Generalized AutoRegressive
Conditional Heteroskedasticity (GARCH) modelling. There seems nothing
intrinsic in these techniques to limit the application of sonification
to them and preliminary discussions indicated that some researchers in
the field are interested in experimenting with such techniques to
assist in the comprehension of their abstraction.
