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An Algorithmic Information-Theoretic Approach to the Behaviour of Financial Markets

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Using frequency distributions of daily closing price time series of several financial market indexes, we investigate whether the bias away from an equiprobable sequence distribution found in the data, predicted by algorithmic information theory, may account for some of the deviation of financial markets from log-normal, and if so for how much of said deviation and over what sequence lengths. We do so by comparing the distributions of binary sequences from actual time series of financial markets and series built up from purely algorithmic means. Our discussion is a starting point for a further investigation of the market as a rule-based system with an 'algorithmic' component, despite its apparent randomness, and the use of the theory of algorithmic probability with new tools that can be applied to the study of the market price phenomenon. The main discussion is cast in terms of assumptions common to areas of economics in agreement with an algorithmic view of the market.
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doi: 10.1111/j.1467-6419.2010.00666.x
AN ALGORITHMIC INFORMATION-
THEORETIC APPROACH TO THE
BEHAVIOUR OF FINANCIAL MARKETS
Hector Zenil
IHPST, Universit´
e de Paris 1 (Panth´
eon-Sorbonne)
Laboratoire d’Informatique Fondamentale de Lille (USTL)
Jean-Paul Delahaye
Laboratoire d’Informatique Fondamentale de Lille (USTL)
Abstract. Using frequency distributions of daily closing price time series of
several financial market indices, we investigate whether the bias away from an
equiprobable sequence distribution found in the data, predicted by algorithmic
information theory, may account for some of the deviation of financial markets
from log-normal, and if so for how much of said deviation and over what sequence
lengths. We do so by comparing the distributions of binary sequences from actual
time series of financial markets and series built up from purely algorithmic means.
Our discussion is a starting point for a further investigation of the market as a rule-
based system with an algorithmic component, despite its apparent randomness,
and the use of the theory of algorithmic probability with new tools that can be
applied to the study of the market price phenomenon. The main discussion is
cast in terms of assumptions common to areas of economics in agreement with
an algorithmic view of the market.
Keywords. Algorithmic complexity; Algorithmic probability; Closing price move-
ments; Computable economics; Experimental economics; Financial markets;
Information content; Stock market
1. Introduction
One of the main assumptions regarding price modelling for option pricing is
that stock prices in the market behave as stochastic processes, that is, that price
movements are log-normally distributed. Unlike classical probability, algorithmic
probability theory has the distinct advantage that it can be used to calculate the like-
lihood of certain events occurring based on their information content. We investigate
whether the theory of algorithmic information may account for some of the devia-
tion from log-normal of the data of price movements accumulating in a power-law
distribution.
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432 ZENIL AND DELAHAYE
We think that the power-law distribution may be an indicator of an information-
content phenomenon underlying the market, and consequently that departures
from log-normality can, given the accumulation of simple rule-based processes
– a manifestation of hidden structural complexity – be accounted for by Levin’s
universal distribution, which is compatible with the distribution of the empirical
data. If this is true, algorithmic probability could supply a powerful set of tools
that can be applied to the study of market behaviour. Levin’s distribution reinforces
what has been empirically observed, viz. that some events are more likely than
others, that events are not independent of each other and that their distribution
depends on their information content. Levin’s distribution is not a typical probability
distribution inasmuch as it has internal structure placing the elements according to
their structure specifying their exact place in the distribution, unlike other typical
probability distributions that may indicate where some elements accumulate without
specifying the particular elements themselves.
The methodological discipline of considering markets as algorithmic is one facet
of the algorithmic approach to economics laid out in (Velupillai, 2000). The focus
is not exclusively on the institution of the market, but also on agents (of every
sort), and on the behavioural underpinnings of agents (rational or otherwise) and
markets (competitive or not, etc.).
We will show that the algorithmic view of the market as an alternative
interpretation of the deviation from log-normal behaviour of prices in financial
markets is also compatible with some common assumptions in classical models
of market behaviour, with the added advantage that it points to the iteration of
algorithmic processes as a possible cause of the discrepancies between the data and
stochastic models.
We think that the study of frequency distributions and the application of
algorithmic probability could constitute a tool for estimating and eventually
understanding the information assimilation process in the market, making it possible
to characterise the information content of prices.
The paper is organised as follows: In Section 2 a simplified overview of the
basics of the stochastic approach to the behaviour of financial markets is introduced,
followed by a section discussing the apparent randomness of the market. In section
4, the theoretic-algorithmic approach we are proposing herein is presented, preceded
by a short introduction to the theory of algorithmic information and followed
by a description of the hypothesis testing methodology 5.4. In Section 6.2.1,
tables of frequency distributions of price direction sequences for five different
stock markets are compared to equiprobable (normal independent) sequences of
length 3 and 4 to length 10 and to the output frequency distributions produced
by algorithmic means. The alternative hypothesis, that is that the market has an
algorithmic component and that algorithmic probability may account for some
of the deviation of price movements from log-normality is tested, followed by
a backtesting Section 6.2 before introducing further considerations in Section
7 regarding common assumptions in economics. The paper ends with a short
section that summarises conclusions and provides suggestions for further work in
Section 8.
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ALGORITHMIC INFORMATION APPROACH TO THE MARKETS 433
2. The Traditional Stochastic Approach
When events are (random) independent of each other they accumulate in a normal
(Gaussian) distribution. Stock price movements are for the most part considered to
behave independently of each other. The random-walk like evolution of the stock
market has motivated the use of Brownian motion for modelling price movements.
Brownian motion and financial modelling have been historically tied together
(Cont and Tankov, 2003), ever since Bachelier (1900) proposed to model the price
Stof an asset on the Paris stock market in terms of a random process of Brownian
motion Wtapplied to the original price S0. Thus St=S0+σWt.
The process Sis sometimes called a log (or geometric)Brownian motion.Data
of price changes from the actual markets are actually too peaked to be related to
samples from normal populations. One can get a more convoluted model based on
this process introducing or restricting the amount of randomness in the model so
that it can be adjusted to some extent to account for some of the deviation of the
empirical data to the supposedly log-normality.
A practical assumption in the study of financial markets is that the forces behind
the market have a strong stochastic nature (see Figure 1 of a normal distribution
and how a simulated market data may be forced to fit it). The idea stems from the
main assumption that market fluctuations can be described by classical probability
theory. The multiplicative version of Bachelier’s model led to the commonly used
Black–Scholes model, where the log-price Stfollows a random walk St=S0exp
[σt+σWt].
The kind of distribution in which price changes actually accumulate (see Figure
2) is a power law in which high-frequency events are followed by low-frequency
events, with the short and very quick transition between them characterised by
asymptotic behaviour. Perturbations accumulate and are more frequent than if
normally distributed, as happens in the actual market, where price movements
accumulate in long-tailed distributions. Such a distribution often points to specific
kinds of mechanisms, and can often indicate a deep connection with other,
seemingly unrelated systems.
Figure 1. In a Normal Distribution any Event Is More or Less Like any Other.
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Figure 2. Events in a Long-Tailed (Power-Law) Distribution Indicate that Certain Days
Are Not Like any Others.
0 500 1000 1500 2000 2500 3000
–100
–50
0
50
Figure 3. Simulated Brownian Walks Using a CA, an RNG and only one True Segment
of Daily Closing Prices of the DJI.
As found by Mandelbrot (Mandelbrot, 1963) in the 1960s; prices do not follow
a normal distribution; suggesting as it seems to be the case that some unexpected
events happen more frequently than predicted by the Brownian motion model (see
Figure 3). On the right one walk was generated by taking the central column of
a rule 30 cellular automaton (CA), another walk by using the RandomInteger[]
random number function built in Mathematica. Only one is an actual sequence of
price movements for 3 000 closing daily prices of the Dow Jones Index (DJI).
3. Apparent Randomness in Financial Markets
The most obvious feature of essentially all financial markets is the apparent
randomness with which prices tend to fluctuate. Nevertheless, the very idea of
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chance in financial markets clashes with our intuitive sense of the processes
regulating the market. All processes involved seem deterministic. Traders do not
only follow hunches but act in accordance with specific rules, and even when
they do appear to act on intuition, their decisions are not random but instead
follow from the best of their knowledge of the internal and external state of the
market. For example, traders copy other traders, or take the same decisions that
have previously worked, sometimes reacting against information and sometimes
acting in accordance with it. Furthermore, nowadays a greater percentage of the
trading volume is handled electronically, by computing systems (conveniently
called algorithmic trading) rather than by humans. Computing systems are used
for entering trading orders, for deciding on aspects of an order such as the timing,
price and quantity, all of which cannot but be algorithmic by definition.
Algorithmic however, does not necessarily mean predictable. Several types
of irreducibility, from non-computability to intractability to unpredictability, are
entailed in most non-trivial questions about financial markets, as shown with clear
examples in (Velupillai, 2000) and (Wolfram, 2002).
In (Wolfram, 2002) Wolfram asks whether the market generates its own
randomness, starting from deterministic and purely algorithmic rules. Wolfram
points out that the fact that apparent randomness seems to emerge even in very
short timescales suggests that the randomness (or a source of it) that one sees
in the market is likely to be the consequence of internal dynamics rather than of
external factors. In economists’ jargon, prices are determined by endogenous effects
peculiar to the inner workings of the markets themselves, rather than (solely) by
the exogenous effects of outside events.
Wolfram points out that pure speculation, where trading occurs without the
possibility of any significant external input, often leads to situations in which prices
tend to show more, rather than less, random-looking fluctuations. He also suggests
that there is no better way to find the causes of this apparent randomness than
by performing an almost step-by-step simulation, with little chance of beating the
time it takes for the phenomenon to unfold – the timescales of real world markets
being simply too fast to beat. It is important to note that the intrinsic generation of
complexity proves the stochastic notion to be a convenient assumption about the
market, but not an inherent or essential one.
Economists may argue that the question is irrelevant for practical purposes. They
are interested in decomposing time series into a non-predictable and a presumably
predictable signal in which they have an interest, what is traditionally called a trend.
Whether one, both or none of the two signals is deterministic may be considered
irrelevant as long as there is a part that is random-looking, hence most likely
unpredictable and consequently worth leaving out.
What Wolfram’s simplified model shows, based on simple rules, is that despite
being so simple and completely deterministic, these models are capable of
generating great complexity and exhibit (the lack of) patterns similar to the apparent
randomness found in the price movements phenomenon in financial markets.
Whether one can get the kind of crashes in which financial markets seem to cyclicly
fall into depends on whether the generating rule is capable of producing them from
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Figure 4. Patterns out of Nothing: Random Walk by 1000 Data Points Generated Using
the Mathematica Pseudo-Random Number Generator Based on a Deterministic Cellular
Automaton.
time to time. Economists dispute whether crashes reflect the intrinsic instability of
the market, or whether they are triggered by external events. In a model, in Lamper
et al. (2002) for example, sudden large changes are internally generated suggesting
large changes are more predictable – both in magnitude and in direction as the
result of various interactions between agents. If Wolfram’s intrinsic randomness is
what leads the market one may think one could then easily predict its behaviour
if this were the case, but as suggested by Wolfram’s Principle of Computational
Equivalence it is reasonable to expect that the overall collective behaviour of the
market would look complicated to us, as if it were random, hence quite difficult to
predict despite being or having a large deterministic component.
Wolfram’s Principle of Computational Irreducibility (Wolfram, 2002) says that
the only way to determine the answer to a computationally irreducible question is
to perform the computation. According to Wolfram, it follows from his Principle of
Computational Equivalence (PCE) that ‘almost all processes that are not obviously
simple can be viewed as computations of equivalent sophistication: when a system
reaches a threshold of computational sophistication often reached by non-trivial
systems, the system will be computationally irreducible’’.
Wolfram’s proposal for modelling market prices would have a simple programme
generating the randomness that occurs intrinsically (see Figure 4). A plausible, if
simple and idealised behaviour is shown in the aggregate to produce intrinsically
random behaviour similar to that seen in price changes. In Figure 4, one can see
that even in some of the simplest possible rule-based systems, structures emerge
from a random-looking initial configuration with low information content. Trends
and cycles are to be found amidst apparent randomness.
An example of a simple model of the market as shown in (Wolfram, 2002),
where each cell of a cellular automaton corresponds to an entity buying or selling
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Figure 5. On the Top, the Rule 90 Instruction Table. On the Left the Evolution of Rule 90
from a Random Initial Condition for 100 Steps. On the Right the Total Differences at
Every Step between Black and White Cells.
at each step (see Figure 5). The behaviour of a given cell is determined by the
behaviour of its two neighbours on the step before according to a rule. The plot
on the left gives as a rough analog of a market price differences of the total
numbers of black and white cells at successive steps. A rule like rule 90 is additive,
hence reversible, which means that it does not destroy any information and has
‘memory’ unlike the random walk model. Yet, due to its random looking behaviour,
it is not trivial shortcut the computation or foresee any successive step. There
is some randomness in the initial condition of the cellular automaton rule that
comes from outside the model, but the subsequent evolution of the system is fully
deterministic. The way the series plot is calculated is written in Mathematica as
follows: Accumulate[Total/@(CA/.{0→−1})] with CA the output evolution of
rule 90 after 100 steps.
4. An Information-Theoretic Approach
From the point of view of cryptanalysis, the algorithmic view based on frequency
analysis presented herein may be taken as a hacker approach to the financial
market. While the goal is clearly to find a sort of password unveiling the rules
governing the price changes, what we claim is that the password may not be
immune to a frequency analysis attack, because it is not the result of a true random
process but rather the consequence of the application of a set of (mostly simple)
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rules. Yet that doesn’t mean one can crack the market once and for all, since for
our system to find the said password it would have to outperform the unfolding
processes affecting the market, which, as Wolfram’s PCE suggests, would require
at least the same computational sophistication as the market itself, with at least
one variable modelling the information being assimilated into prices by the market
at any given moment. In other words, the market password is partially safe not
because of the complexity of the password itself but because it reacts to the cracking
method.
Whichever kind of financial instrument one looks at, the sequences of prices
at successive times show some overall trends and varying amounts of apparent
randomness. However, despite the fact that there is no contingent necessity of true
randomness behind the market, it can certainly look that way to anyone ignoring
the generative processes, anyone unable to see what other, non-random signals may
be driving market movements.
von Mises’ approach to the definition of a random sequence, which seemed at
the time of its formulation to be quite problematic, contained some of the basics of
the modern approach adopted by Per Martin-L¨
of (L¨
of, 1966). It is during this time
that the Keynesian (Keynes, 1936) kind of induction may have been resorted to as
a starting point for Solomonoff’s seminal work (Solomonoff, 1964) on algorithmic
probability.
Martin-L¨
of gave the first suitable definition of a random sequence. Intuitively, an
algorithmically random sequence (or random sequence) is an infinite sequence of
binary digits that appears random to any algorithm. This contrasts with the idea of
randomness in probability. In that theory, no particular element of a sample space
can be said to be random. Martin-L¨
of’s randomness has since been shown to admit
several equivalent characterisations in terms of compression, statistical tests and
gambling strategies.
The predictive aim of economics is actually profoundly related to the concept of
predicting and betting. Imagine a random walk that goes up, down, left or right by
one, with each step having the same probability. If the expected time at which the
walk ends is finite, predicting that the expected stop position is equal to the initial
position, it is called a martingale. This is because the chances of going up, down,
left or right, are the same, so that one ends up close to one’s starting position,
if not exactly at that position. In economics, this can be translated into a trader’s
experience. The conditional expected assets of a trader are equal to his present
assets if a sequence of events is truly random.
Schnorr (1971) provided another equivalent definition in terms of martingales
(Downey and Hirschfeldt, 2010). The martingale characterisation of randomness
says that no betting strategy implementable by any computer (even in the weak
sense of constructive strategies, which are not necessarily computable) can make
money betting on a random sequence. In a true random memory-less market, no
betting strategy can improve the expected winnings, nor can any option cover the
risks in the long term.
Over the last few decades, several systems have shifted towards ever greater
levels of complexity and information density. The result has been a shift towards
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Figure 6. By Extracting a Normal Distribution from the Market Distribution, the
Long-Tail Events Are Isolated.
Paretian outcomes, particularly within any event that contains a high percentage of
informational content.1
Departures from normality could be accounted for by the algorithmic component
acting in the market, as is consonant with some empirical observations and common
assumptions in economics, such as rule-based markets and agents.
If market price differences accumulated in a normal distribution, a rounding
would produce sequences of 0 differences only. The mean and the standard
deviation of the market distribution are used to create a normal distribution, which
is then subtracted from the market distribution. Rounding by the normal distribution
cover, the elements in the tail are extracted as shown in Figure 6.
4.1 Algorithmic Complexity
At the core of algorithmic information theory (AIT) is the concept of algorithmic
complexity,2a measure of the quantity of information contained in a string of digits.
The algorithmic complexity of a string is defined as the length (Kolmogorov, 1965;
Chaitin, 2001) of the shortest algorithm that, when provided as input to a universal
Turing machine or idealised simple computer, generates the string. A string has
maximal algorithmic complexity if the shortest algorithm able to generate it is not
significantly shorter than the string itself, perhaps allowing for a fixed additive
constant. The difference in length between a string and the shortest algorithm able
to generate it is the string’s degree of compressibility. A string of low complexity
is therefore highly compressible, as the information that it contains can be encoded
in an algorithm much shorter than the string itself. By contrast, a string of maximal
complexity is incompressible. Such a string constitutes its own shortest description:
there is no more economical way of communicating the information that it contains
than by transmitting the string in its entirety. In AIT a string is algorithmically
random if it is incompressible.
Algorithmic complexity is inversely related to the degree of regularity of a string.
Any pattern in a string constitutes redundancy: it enables one portion of the string to
be recovered from another, allowing a more concise description. Therefore highly
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regular strings have low algorithmic complexity, whereas strings that exhibit little
or no pattern have high complexity.
The algorithmic complexity KU(s)ofastringswith respect to a universal Turing
machine Uis defined as the binary length of the shortest programme pthat produces
as output the string s.
KU(s)=min{| p|,U(p)=s)
Algorithmic complexity conveys the intuition that a random string should be
incompressible: no programme shorter than the size of the string produces the
string.
Even though Kis uncomputable as a function, meaning that there is no effective
procedure (algorithm) to calculate it, one can use the theory of algorithmic
probability to obtain exact evaluations of K(s) for strings sshort enough for which
the halting problem can be solved for a finite number of cases due the size (and
simplicity) of the Turing machines involved.
4.2 Algorithmic Probability
What traders often end up doing in turbulent price periods is to leave aside the
‘any day is like any other normal day’ rule, and fall back on their intuition, which
leads to their unwittingly following a model we believe to be better fitted to reality
and hence to be preferred at all times, not just in times of turbulence.
Intuition is based on weighting past experience, with experience that is closer
in time being more relevant. This is very close to the concept of algorithmic
probability and the way it has been used (and was originally intended to be used
(Solomonoff, 1964)) in some academic circles as a theory of universal inductive
inference (Hutter, 2007).
Algorithmic probability assigns to objects an aprioriprobability that is in some
sense universal (Kirchherr and Li, 1997). This aprioridistribution has theoretical
applications in a number of areas, including inductive inference theory and the time
complexity analysis of algorithms. Its main drawback is that it is not computable
and thus can only be approximated in practise.
The concept of algorithmic probability was first developed by Solomonoff (1964)
and formalised by Levin (1977). Consider an unknown process producing a binary
string of length kbits. If the process is uniformly random, the probability of
producing a particular string sis exactly 2k, the same as for any other string of
length k. Intuitively, however, one feels that there should be a difference between
a string that can be recognised and distinguished, and the vast majority of strings
that are indistinguishable to us as regards whether the underlying process is truly
random.
Assume one tosses a fair coin 20 three times and gets the following outcomes:
00000000000000000000
01100101110101001011
11101001100100101101
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The first outcome would be very unlikely because one would expect a pattern-less
outcome from a fair coin toss, one that resembles the second and third outcomes. In
fact, it would be far more likely that a simple deterministic algorithmic process has
generated this string. The same could be said for the market: one usually expects
to see few if any patterns in its main indicators, mostly for the reasons set forth in
Section 3. Algorithmic complexity captures this expectation of patternlessness by
defining what a random-looking string looks like. On the other hand, algorithmic
probability predicts that random-looking outputs are the exception rather than the
rule when the generating process is algorithmic.
There is a measure which describes the expected output of an abstract machine
when running a random programme. A process that produces a string swith a
programme pwhen executed on a universal Turing machine Uhas probability
m(s). As pis itself a binary string, m(s) can be defined as being the probability that
the output of a universal Turing machine3Uis swhen provided with a sequence
of fair coin flip inputs interpreted as a programme.
m(s)=U(p)=s2−|p|=2K(s)+O(1)
That is, the sum over all the programmes pfor which the universal Turing machine
Uoutputs the string sand halts.
Levin’s universal distribution is so called because, despite being uncomputable,
it has the remarkable property (proven by Leonid Levin himself) that among
all the lower semi-computable semi-measures, it dominates every other.4This
makes Levin’s universal distribution the optimal prior distribution when no
other information about the data is available, and the ultimate optimal predictor
(Solomonoff’s original motivation (Solomonoff, 1964) was actually to capture the
notion of learning by inference) when assuming the process to be algorithmic (or
more precisely, carried out by a universal Turing machine). Hence the adjective
‘universal’.
The algorithmic probability of a string is uncomputable. One way to calculate
the algorithmic probability of a string is to calculate the universal distribution by
running a large set of abstract machines producing an output distribution, as we
did in (Delahaye and Zenil, 2007).
5. The Study of the Real Time Series versus the Simulation
of an Algorithmic Market
The aim of this work is to study the direction and eventually the magnitude of time
series of real financial markets. To that mean, we first develop a codification
procedure translating financial series into binary digit sequences. Despite the
convenience and simplicity of the procedure, the translation captures several
important features of the actual behaviour of prices in financial markets. At the
right level, a simplification of finite data into a binary language is always possible.
Each observation measuring one or more parameters (e.g. price, trade name) is an
enumeration of independent distinguishable values, a sequence of discrete values
translatable into binary terms.5
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5.1 From AIT Back to the Behaviour of Financial Markets
Different market theorists will have different ideas about the likely pattern of 0’s
and 1’s that can be expected from a sequence of price movements. Random walk
believers would favour random-looking sequences in principle. Other analysts may
be more inclined to believe that patterned-looking sequences can be spotted in the
market, and may attempt to describe and exploit these patterns, eventually deleting
them.
In an early anticipation of an application of AIT to the financial market (Mansilla,
2001), it was reported that the information content of price movements and mag-
nitudes seem to drastically vary when measured right before crashes compared to
periods where no financial turbulence is observed. As described in Mansilla (2001),
this means that sequences corresponding to critical periods show a qualitative
difference compared to the sequences corresponding to periods of stability (hence
prone to be modelled by the traditional stochastic models) when the information
content of the market is very low (and when looks random as carrying no
information). In Mansilla (2001), the concept of conditional algorithmic complexity
is used to measure these differences in the time series of price movements in two
different financial markets (NASDAQ and the Mexican IPC), here we use a different
algorithmic tool, that is the concept of algorithmic probability.
We will analyse the complexity of a sequence sof encoded price movements,
as described in Section 6.2.1 We will see whether this distribution approaches
one produced artificially – by means of algorithmic processes – in order to
conjecture the algorithmic forces at play in the market, rather than simply assume
a pervasive randomness. Exploitable or not, we think that price movements may
have an algorithmic component, even if some of this complexity is disguised behind
apparent randomness.
According to Levin’s distribution, in a world of computable processes, patterns
which result from simple processes are relatively likely, while patterns that can
only be produced by very complex processes are relatively unlikely. Algorithmic
probability would predict, for example, that consecutive runs of the same
magnitude, that is, runs of pronounced falls and rises, and runs of alternative
regular magnitudes have greater probability than random-looking changes. If one
fails to discern the same simplicity in the market as is to be observed in certain
other real world data sources (Zenil and Delahaye, 2010), it is likely due to the
dynamics of the stock market, where the exploitation of any regularity to make
a profit results in the deletion of that regularity. Yet these regularities may drive
the market and may be detected upon closer examination. For example, according
to the classical theory, based on the average movement on a random walk, the
probability of strong crashes is nil or very low. Yet in actuality they occur in cycles
over and over.
What is different in economics is the nature of the dynamics some of the data
are subject to, as discussed in Section 3, which underscores the fact that patterns
are quickly erased by economic activity itself, in the search for an economic
equilibrium.
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Assuming an algorithmic hypothesis, that is that there is a rule-based – as
opposed to a purely stochastic – component in the market, one could apply the
tools of the theory of algorithmic information, just as assuming random distributions
led to the application of the traditional machinery of probability theory.
If this algorithmic hypothesis is true, the theory says that Levin’s distribution is
the optimal predictor. In other words, one could run a large number of machines
to simulate the market, and m, the algorithmic probability based on Levin’s
universal distribution would provide accurate insights into the particular direction
and magnitude of a price based on the fact that the market has a rule-based
element. The correlation found in the experiments described in the next section
6.2.1 suggests that Levin’s distribution may turn out to be a way to calculate and
approximate this potentially algorithmic component in the market.
5.2 Unveiling the Machinery
When observing a certain phenomenon, its outcome fcan be seen as the result of
a process P. One can then ask what the probability distribution of Pgenerating f
looks like. A probability distribution of a process is a description of the relative
number of times each possible outcome occurs in a number of trials.
In a world of computable processes, Levin’s semi-measure (a.k.a universal
distribution) establishes that patterns which result from simple processes (short
programmes) are likely, while patterns produced by complicated processes (long
programmes) are relatively unlikely. Unlike other probability measures, Levin’s
semi-measure (denoted by m) is not only a probability distribution establishing that
there are some objects that have a certain probability of occurring according to said
distribution, it is also a distribution specifying the order of the particular elements
in terms of their individual information content.
Figure 7 suggests that by looking at the behaviour of one market, the behaviour
of the others may be predicted. But this cannot normally be managed quickly
enough for the information to be of any actual use (in fact the very intention of
succeeding in one market by using information from another may be the cause
rather than the consequence of the correlation).
In the context of economics, if we accept the algorithmic hypothesis (that price
changes are algorithmic, not random), mwould provide the algorithmic probability
of a certain price change happening, given the history of the price. An anticipation
of the use of this prior distribution as an inductive theory in economics is to be found
in Velupillai (2000). But following that model would require us to calculate the
prior distribution m, which we know is uncomputable. We proceed by approaching
mexperimentally in order to show what the distribution of an algorithmic market
would look like, and eventually use it in an inductive framework.
Once mis approximated, it can be compared to the distribution of the outcome
in the real world (i.e. the empirical data on stock market price movements). If
the output of a process approaches a certain probability distribution, one accepts,
within a reasonable degree of statistical certainty, that the generating process is
of the nature suggested by the distribution. If it is observed that an outcome s
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Figure 7. Series of Daily Closing Prices of Five of the Largest Stock Markets from
01/01/2000 to January 01/01/2010. The Best Sequence Length Correlation Suggests that
Markets Catch Up with Each Other (Assimilate Each Others’ Information) in about 7–10
Days on Average.
occurs with probability m(s), and the data distribution approaches m, one would
be persuaded, within the same degree of certainty, to accept a uniform distribution
as the footprint of a random process (where events are independent of each other),
that is, that the source of the data is suggested by the distribution m.
What Levin’s distribution implies is that most rules are simple because they
produce simple patterned strings, which algorithmically complex rules are unlikely
to do. A simple rule, in terms of algorithmic probability, is the average kind of
rule producing a highly frequent string, which according to algorithmic probability
has a low random complexity (or high organised complexity), and therefore looks
patterned. This is the opposite of what a complex rule would be, when defined
in the same terms – it produces a pattern-less output, and is hence random-
looking.
The outcomes of simple rules have short descriptions because they are less
algorithmically complex, means that in some sense simple and short are connected,
yet large rules may also be simple despite not being short in relative terms. And
the outcomes of the application of simple rules tend to accumulate exponentially
faster than the outcomes of complicated rules. This causes some events to occur
more often than others and therefore to be dependent on each other. Those events
happening more frequently will tend to drastically outperform other events and will
do so by quickly beginning to follow the irregular pattern and doing so closely for
a while.
The first task is therefore to produce a distribution by purely algorithmic means
using abstract computing machines6– by running abstract computational devices
like Turing machines and cellular automata (CA).
It is not only in times of great volatility that one can see that markets are
correlated to each other (see Figure 7). This correlation means that, as may
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be expected, markets systematically react to each other. One can determine the
information assimilation process time by looking at the correlations of sequences
of daily closing prices of different lengths for five of the largest European and U.S.
stock markets. It is evident that they react neither immediately nor after an interval
of several days. As suggested by the table in Section 6.1.2, over a period of 20
years, from January 1990 to January 2010, the average assimilation time is about a
week to a week and a half. For one thing, the level of confidence of the correlation
confirms that even if some events may be seen as randomly produced, the reactions
of the markets follow each other and hence are neither independent of each other
nor completely random. The correlation matrix 6.1.2 exhibits the Spearman rank
correlation coefficients, followed by the number of elements compared (number of
sequence lengths found in one or another market), underlining the significance of
the correlation between them.
5.3 Binary Encoding of the Market Direction of Prices
Market variables have three main indicators. The first is whether there is a
significant price change (i.e. larger than, say, the random walk expectation), the
second is the direction of a price change (rising or falling), and third, there is its
magnitude. We will focus our first attempt on the direction of price changes, since
this may be the most valuable of the three (after all whether one is likely to make
or lose money is the first concern, before one ponders the magnitude of the possible
gain or loss).
In order to verify that the market carries the algorithmic signal, the information
content of the non-binary tuples can be collapsed to binary tuples. One can encode
the price change in a single bit by ‘normalizing’ the string values, with the values of
the entries themselves losing direction and magnitude but capturing price changes.
Prices are subject to such strong forces (interests) in the market that it would be
naive to think that they could remain the same to any decimal fraction of precision,
even if they were meant to remain the same for a period of time. In order to spot
the algorithmic behaviour one has to provide some stability to the data by getting
rid of precisely the kind of minor fluctuations that a random walk may predict. If
not, one notices strong biases disguising the real patterns. For example, periods of
price fluctuations would appear less likely than they are in reality if one allows
decimal fluctuations to count as much as any other fluctuation.7
This is because from one day to another the odds that prices will remain exactly
the same up to the highest precision is extremely unlikely, due to the extreme forces
and time exposure they are subject to. Even though it may seem that at such a
level of precision one or more decimal digits could represent a real price change,
for most practical purposes there is no qualitative change when prices close to the
hundreds, if not actually in the hundreds, are involved. Rounding the price changes
to a decimal fraction provides some stability, for example, by improving the place
of the 0ntuple towards the top, because as we will see, it turns out to be quite
well placed at the top once the decimal fluctuations have been gotten rid of. One
can actually think of this as using the Brownian motion expectation to get rid of
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meaningless changes. In other words, the Brownian movement is stationary in our
analysis, being the predictable (easy) component of the minor fluctuations, while
the actual objects of study are the wilder dynamics of larger fluctuations. We have
changed the focus from what analysts are used to considering as the signal, to
noise, and recast what they consider noise as the algorithmic footprint in empirical
data.
5.3.1 Detecting Rising versus Falling
As might be expected given the apparent randomness of the market and the
dynamics to which it is subject, it would be quite difficult to discern clear patterns
of rises versus falls in market prices.
The asymmetry in the rising and falling ratio is explained by the pattern deletion
dynamic. While slight rises and falls have the same likelihood of occurring, making
us think they follow a kind of random walk bounded by the average movement of
Brownian motion, significant rises are quickly deleted by people taking advantage
of them, while strong drops quickly worsen because of people trying to sell rather
than buy at a bargain. This being so, one expects to see longer sequences of
drops than of rises, but actually one sees the contrary, which suggests that periods
of optimism are actually longer than periods of pessimism, though periods of
pessimism are stronger in terms of price variation. In Mansilla (2001) it was
reported that the information content of price movements and magnitudes seem
to drastically vary when measured to intervals of high volatility (particularly
right before crashes) compared to periods where no financial turbulence is
observed.
We construct a binary series associated to each real time series of financial
markets as follows. Let {pt}be the original time series of daily closing prices of
a financial market for a period of time t. Then each element bi{bn}, the time
series of price differences, with n=t1, is calculated as follows:
bi=1pi+1 >pi
0pi+1 pi
The frequency of binary tuples of short lengths will be compared to the frequency
of binary tuples of length the same length obtained by running abstract machines
(deterministic Turing machines and one-dimensional CA).
5.4 Calculating the Algorithmic Time Series
Constructing Levin’s distribution mfrom abstract machines is therefore necessary
in order to strengthen the algorithmic hypothesis. In order to make a meaningful
comparison with what can be observed in a purely rule-governed market, we will
construct from the ground up an experimental distribution by running algorithmic
machines (Turing machines and CA). An abstract machine consists of a definition
in terms of input, output and the set of allowable operations used to turn the input
into the output. They are of course algorithmic by nature (or by definition).
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The Turing machine model represents the basic framework underlying many
concepts in computer science, including the definition of algorithmic complexity.
In this paper we use the output frequency distribution produced by running a finite,
yet very large set of Turing machines with empty input as a means to approximate
m. There are 11 019 960 576 four-state two-symbol Turing machines (for which the
halting condition is known, thanks to the Busy Beaver game). The conception of
the experiment and further details are provided in Delahaye and Zenil, (2007) and
Zenil and Delahaye (2010). These Turing machines produced an output, from which
a frequency distribution of tuples was calculated and compared to the actual time
series from the market data produced by the stock markets encoded as described in
5.3.1. A forthcoming paper provides more technical details (Delahaye and Zenil,
2011).
Also, a frequency distribution from a sample of 10 000 four-colour totalistic CA
was built from a total of 1 048 575 possible four-colour totalistic CA, each rule
starting from an random initial configuration of 10–20 black and white cells, and
running for 100 steps (hence arbitrarily halted). Four-colour totalistic CA produce
four-symbol sequences. However only the binary were taken into consideration,
with the purpose of building a binary frequency distribution. The choice of this
CA space was dictated by the fact that the smallest CA space is too small, and
the next smallest space too large to extract a significant enough sample from it.
Having chosen a sample of four-colour totalistic CA, the particular rules sample
was randomly generated.
For all the experiments, stock market data sets (of daily closing prices) used
covered the same period of time: from January 1990 to January 2010. The stock
market code names can be readily connected with their full index names or symbols.
Spearman’s rank coefficient was the statistical measure of the correlation between
rankings of elements in the frequency distributions. As is known, if there are no
repeated data values, a perfect Spearman correlation of +1or1 occurs when
each of the variables is a perfect monotone function of the other. While 1 indicates
perfect correlation (i.e. in the exact order), 1 indicates perfect negative correlation
(i.e. perfect inverse order).
6. Experiments and Results
It is known that for any finite series of a sequence of integer values, there is a
family of countable infinite computable functions that fit the sequence over its
length. Most of them will be terribly complicated, and any attempt to find the
simplest accounting for the sequence will face uncomputability. Yet using the tools
of AIT, one can find the simplest function for short sequences by deterministic
means, testing a set of increasingly complex functions starting from the simplest
and proceeding until the desired sequence assuring the simplest fit is arrived at.
Even if the likelihood of remaining close to the continuation of the series remains
low, one can recompute the sequence and find good approximations for short
periods of time.
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The following experiment uses the concept of algorithmic complexity to find the
best fit in terms of the simplest model fitting the data, assuming the source to be
algorithmic (rule-based).
6.1 Rankings of Price Variations
In the attempt to capture the price change phenomenon in the stock market, we
have encoded price changes in binary strings. The following are tables capturing
the rise and fall of prices and their occurrence in a ranking classification.
6.1.1 Carrying an Algorithmic Signal
Some regularities and symmetries in the sequence distribution of price directions in
the market may be accounted for by an algorithmic signal, but they also follow from
a normal distribution. For example, the symmetry between the left- and right-hand
sides of the Gaussian curve with zero skewness means that reverted and inverted
strings (i.e. consecutive runs of events of prices going up or down) will show
similar frequency values, which is also in keeping with the intuitive expectation of
the preservation of complexity invariant to certain basic transformations (a sequence
of events 1111... are equally likely to occur as 0000..., or 0101... and 1010...).
This means that one would expect nconsecutive runs of rising prices to be
as likely as nconsecutive runs of falling prices. Some symmetries, however, are
broken in particular scenarios. In the stock market for example, it is very well
known that sequences of drastic falls are common from time to time, but never
sequences of drastic price increases, certainly not increases of the same magnitude
as the worst price drops. And we witnessed such a phenomenon in the distributions
from the Dow Jones Index (DJI). Some other symmetries may be accounted for by
business cycles engaged in the search for economic equilibrium.
6.1.2 Correlation Matrices
With algorithmic probability in hand, one may predict that alternations and
consecutive events of the same type and magnitude are more likely, because
they may be algorithmically more simple. One may, for example, expect to see
symmetrical events occurring more often, with reversion and complementation
occurring together in groups (i.e. a string 10noccurring together with 0n1and
the like). In the long-term, business cycles and economic equilibria may also be
explained in information theoretic terms, because for each run of events there are
the two complexity-preserving symmetries, reversion and complementation, that
always follow their counterpart sequences (the unreversed and complement of the
complement), producing a cyclic type of behaviour.
The correlations shown in Table 2 indicate what is already assumed in looking
for cycles and trends, viz. that these underlying cycles and trends in the markets
are more prominent when deleting Brownian noise. As shown later, this may be
an indication that the tail of the distribution has a stronger correlation than the
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elements covered by the normal curve, as could be inferred from the definition of
a random walk (i.e. that random walks are not correlated at all).
The entries in each comparison table (Table 1 to 7) consist of the Spearman
coefficient followed by the number of elements compared. Both determine the
level of confidence and are therefore essential for estimating the correlation. Rows
compare different stock markets over different sequence lengths of daily closing
prices, represented by the columns. It is also worth noting when the comparisons,
such as in Table 2, displayed no negative correlations.
6.2 Backtesting
Applying the same methodology over a period of a decade, from 1980 to 1990 (old
market), to three major stock markets for which we had data for the said period of
time, similar correlations were found across the board, from weak to moderately
weak – though the trend was always towards positive correlations.
6.2.1 Old Market versus CA Distribution
The distributions indicate that price changes are unlikely to rise by more than
a few points for more than a few days, while greater losses usually occur
together and over longer periods. The most common sequences of changes are
alternations. It is worth noticing that sequences are grouped together by reversion
and complementation relative to their frequency, whereas traditional probability
would have them occur in no particular order and with roughly the same frequency
values.
Tables 8 and 9 illustrate the kind of frequency distributions from the stock
markets (in this case for the DJI) over tuples of length 3 with which distributions
from the market data were compared with and its statistical correlation evaluated
section between four other stock markets and over larger periods of time up to 10
closing daily prices.
6.3 Algorithmic Inference of Rounded Price Directions
Once with the tuples distributions calculated and a correlation found, one can apply
Solomonoff’s (Solomonoff, 1964) concept algorithmic inference. Let’s say that by
looking two days behind of daily closing prices one sees two consecutive losses.
The algorithmic inference will say that with probability 0.129 the third day will
be a loss again. In fact, as we now know, algorithmic probability will suggest
that with higher probability the next day will only repeat the last values of any
run of 1’s or 0’s and the empirical distribution from the market will tell us that
runs of 1 s (gains) are more likely than consecutive losses (before the rounding
process deleting the smallest price movements) without taking into account their
magnitude (as empirically known, losses are greater than gains, but gains are more
sustainable), but runs of consecutive 0’s (losses) will be close or even more likely
than consecutive losses after the rounding process precisely because gains are
smaller in magnitude.
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Table 1. Stock Market versus Stock Market.
Market vs. Market 4 5 6 7 8 9 10
CAC40 vs. DAX 0.059|16 0.18|32 0.070|62 0.37|109 0.48|119 0.62|87 0.73|55
CAC40 vs. DJIA 0.31|16 0.25|32 0.014|62 0.59|109 0.27|124 0.34|95 0.82|51
CAC40 vs. FTSE350 0.16|16 0.019|32 0.18|63 0.15|108 0.59|114 0.72|94 0.73|62
CAC40 vs. NASDAQ 0.30|16 0.43|32 0.056|63 0.16|111 0.36|119 0.32|88 0.69|49
CAC40 vs. SP500 0.14|16 0.45|31 0.085|56 0.18|91 0.16|96 0.49|73 0.84|45
DAX vs. DJIA 0.10|16 0.14|32 0.13|62 0.37|110 0.56|129 0.84|86 0.82|58
DAX vs. FTSE350 0.12|16 0.029|32 0.12|63 0.0016|106 0.54|118 0.81|89 0.80|56
DAX vs . NASDAQ 0.36|16 0.35|32 0.080|62 0.014|110 0.64|126 0.55|96 0.98|48
DAX vs. SP500 0.38|16 0.062|31 0.20|56 0.11|88 0.11|94 0.43|76 0.63|49
DJIA vs. FTSE350 0.35|16 0.13|32 0.022|63 0.29|107 0.57|129 0.76|99 0.86|56
DJIA vs. NASDAQ 0.17|16 0.13|32 0.0077|62 0.079|112 0.70|129 0.57|111 0.69|64
DJIA vs. SP500 0.038|16 0.32|31 0.052|55 0.14|89 0.37|103 0.32|86 0.60|59
FTSE350 vs. NASDAQ 0.36|16 0.38|32 0.041|63 0.54|108 0.68|126 0.57|107 0.66|51
FTSE350 vs. SP500 0.50|16 0.50|31 0.12|56 0.11|92 0.25|101 0.26|96 0.29|66
NASDAQ vs. SP500 0.70|16 0.42|31 0.20|56 0.024|91 0.41|111 0.23|102 0.42|61
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Table 2. In Contrast, when the Markets are Compared to Random Price Movements, Which Accumulate in a Normal Curve, They
Exhibit no Correlation or Only a Very Weak Correlation, as Shown in Section 6.1.2.
Marketvs.Market 4 5 678910
CAC40 vs. DAX 0.58|11 0.58|15 0.55|19 0.37|26 0.59|29 0.55|31 0.60|28
CAC40 vs. DJIA 0.82|11 0.28|15 0.28|21 0.29|24 0.52|29 0.51|32 0.33|36
CAC40 vs. FTSE350 0.89|11 0.089|15 0.41|20 0.17|22 0.59|30 0.28|28 0.30|34
CAC40 vs. NASDAQ 0.69|11 0.27|14 0.28|18 0.44|23 0.30|30 0.17|34 0.61|30
CAC40 vs. SP500 0.85|11 0.32|15 0.49|20 0.55|24 0.42|33 0.35|35 0.34|36
DAX vs. DJIA 0.76|11 0.45|16 0.56|20 0.35|26 0.34|28 0.25|35 0.24|33
DAX vs. FTSE350 0.61|11 0.30|16 0.58|19 0.14|25 0.30|29 0.34|30 0.21|31
DAX vs . NASDAQ 0.40|11 0.27|16 0.36|18 0.75|25 0.28|29 0.28|35 0.50|33
DAX vs. SP500 0.14|12 0.36|17 0.72|20 0.64|28 0.42|31 0.52|34 0.51|32
DJIA vs. FTSE350 0.71|11 0.30|16 0.63|20 0.71|22 0.21|28 0.28|31 0.35|33
DJIA vs. NASDAQ 0.58|11 0.52|15 0.33|19 0.58|23 0.46|29 0.49|37 0.51|35
DJIA vs. SP500 0.70|11 0.20|16 0.45|21 0.29|26 0.35|32 0.37|36 0.55|36
FTSE350 vs. NASDAQ 0.73|11 0.57|15 0.70|17 0.48|23 0.62|28 0.34|33 0.075|35
FTSE350 vs. SP500 0.66|11 0.65|16 0.56|19 0.18|24 0.64|32 0.32|32 0.52|38
NASDAQ vs. SP500 0.57|11 0.37|15 0.41|18 0.32|24 0.30|34 0.19|35 0.35|40
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Table 3. When Random Price Movements Are Compared to Rounded Prices of the Market (Denoted by ‘r. Market’ to Avoid
Accumulation in the Normal Curve) the Correlation Coefficient is too Weak, Possessing no Significance at all. This may Indicate that
it is the Prices Behaving and Accumulating in the Normal Curve that Effectively Lead the Overall Correlation.
r. Market vs. Random 4 5 6 7 8 9 10
DJIA vs. random 0.050|16 0.080|31 0.078|61 0.065|96 0.34|130 0.18|120 0.53|85
SP500 vs. random 0.21|16 0.066|30 0.045|54 0.16|81 0.10|99 0.29|87 0.32|57
NASDAQ vs. random 0.12|16 0.095|31 0.11|60 0.14|99 0.041|122 0.29|106 0.57|68
FTSE350 vs. random 0.16|16 0.052|31 0.15|61 0.14|95 0.30|122 0.50|111 0.37|77
CAC40 vs. random 0.32|16 0.15|31 0.13|60 0.16|99 0.19|119 0.45|109 0.36|78
DAX vs. random 0.33|16 0.023|31 0.20|60 0.14|95 0.26|129 0.31|104 0.31|77
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Table 4. Comparison between the Daily Stock Market Sequences versus an Hypothesised Log-Normal Accumulation of Price
Directions.
Market vs. Random 4 5678910
DJIA vs. random 0.21|12 0.15|17 0.19|23 0.033|28 0.066|33 0.31|29 0.64|15
SP500 vs. random 0.47|12 0.098|17 0.20|25 0.32|31 0.20|38 0.41|29 0.32|20
NASDAQ vs. random 0.55|11 0.13|16 0.093|20 0.18|26 0.015|37 0.30|35 0.38|25
FTSE350 vs. random 0.25|11 0.24|16 0.053|22 0.050|24 0.11|31 0.25|23 0.49|13
CAC40 vs. random 0.12|11 0.14|15 0.095|22 0.23|26 0.18|30 0.36|23 0.44|14
DAX vs. random 0.15|12 0.067|18 0.12|24 0.029|31 0.31|32 0.27|27 0.59|15
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Table 5. The Comparison to TM Revealed Day Lengths Better Correlated than other, Although their Significance Remained Weak
and Unstable, with a Tendency, however, to Positive Correlations.
Market vs. TM 5 6 7 8 9 10
DJIA vs. TM 0.42|16 0.20|21 0.42|24 0.021|35 0.072|36 0.20|47
SP500 vs. TM 0.48|18 0.30|24 0.070|32 0.32|39 0.26|47 0.40|55
NASDAQ vs. TM 0.67|17 0.058|25 0.021|32 0.26|42 0.076|49 0.17|57
FTSE350 vs. TM 0.30|17 0.39|22 0.14|29 0.43|36 0.013|41 0.038|55
CAC40 vs. TM 0.49|17 0.026|25 0.41|32 0.0056|38 0.22|47 0.082|56
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Table 6. When Compared to the Distribution from Cellular Automata, the Correlation was Greater. Each Column had Pairs of Score
means: (0.09, 16), (0.17, 32), (0.09, 64), (0.042, 116), (0.096, 144), (0.18, 119), (0.41, 67) for 4 to 10 Days, for which the Last 2 (9
and 10 days Long) Have Significant Levels of Correlation According to their Critical Values and the Number of Elements Compared.
Market vs. CA 4 5 6 7 8 9 10
DJIA vs. CA 0.14|16 0.28|32 0.084|63 0.049|116 0.10|148 0.35|111 0.51|59
SP500 vs. CA 0.16|16 0.094|32 0.0081|64 0.11|116 0.088|140 0.17|117 0.40|64
NASDAQ vs. CA 0.065|16 0.25|32 0.19|63 0.098|116 0.095|148 0.065|131 0.36|65
FTSE350 vs. CA 0.16|16 0.15|32 0.12|64 0.013|120 0.0028|146 0.049|124 0.42|76
CAC40 vs. CA 0.035|16 0.36|32 0.21|64 0.064|114 0.20|138 0.25|114 0.33|70
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Table 7. Comparison Matrix of Frequency Distributions of Daily Price Directions of Three Stock Markets from 1980 to 1990.
Old Market vs. CA 4 5 6 7 8 9 10
DJIA vs. CA 0.33|10 0.068|16 0.51|21 0.15|28 0.13|31 0.12|32 0.25|29
SP500 vs. CA 0.044|13 0.35|19 0.028|24 0.33|33 0.45|33 0.00022|30 0.37|34
NASDAQ vs. CA 0.45|10 0.20|17 0.27|24 0.16|30 0.057|31 0.11|34 0.087|32
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Table 8. 3-Tuples Distribution from the DJI Price Difference Time Series for the Past
80 Years. 1 Means that a Price Rose, 0 that it Fell or Remained the Same as Described
in the Construction of the Binary Sequence bnas Described in Section 5.3.1 and
Partitioned in 3-Tuples for this Example. By Rounding to the Nearest Multiple of .4
(i.e. Dismissing Decimal Fraction Price Variations of this Order) Some More Stable
Patterns Start to Emerge.
Tuple Prob.
000 0.139
001 0.130
111 0.129
011 0.129
100 0.129
110 0.123
101 0.110
010 0.110
Table 9. 3-Tuples from the Output Distribution Produced by Running all 4-state
2-Symbol Turing Machines Starting from an Empty Tape First on a Background of 0’s
and then Running it Again on a Background of 1’s to Avoid Asymmetries Due to the
Machine Formalism Convention.
Tuple Prob.
000 0.00508
111 0.00508
001 0.00488
011 0.00488
100 0.00488
110 0.00468
010 0.00483
101 0.00463
To calculate the algorithmic probability of a price direction biof the next
closing price by looking nconsecutive daily rounded prices behind, is given
by
P(bi)=m(bin...bi)
That is, the algorithmic probability of the string constituted by the nconsecutive
price directions of the days before followed by the possible outcome to be estimated,
with mLevin’s semi-measure described in equation (6.3). It is worth noting however
that the inference power of this approach is limited by the correlation found between
the market’s long tails and the distributions calculated by means of exhaustive
computation. Tables with distributions for several tuple lengths and probability
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values will be available in Delahaye and Zenil (2011), so that one can empirically
calculate mby means of the results of exhaustive computation.
For example, the algorithmic model predicts a greater incidence of simple
signatures as trends under the noise modelled by the Brownian motion model,
such as signatures 000... of price stability. It also predicts that random-looking
signatures of higher volatility will occur more if they are already occurring, a
signature in unstable times where the Brownian motion no longer works in these
kind of events outside the main Bell curve.
7. Further Considerations
7.1 Rule-Based Agents
For sound reasons, economists are used to standardising their discussions by starting
out from certain basic assumptions. One common assumption in economics is
that actors in the market are decision makers, often referred to as rational agents.
According to this view, rational agents make choices by assessing possible outcomes
and assigning a utility to each in order to make a decision. In this rational choice
model, all decisions are arrived at by a rational process of weighing costs against
benefits, and not randomly.
An agent in economics or a player in game theory is an actor capable of decision
making. The idea is that the agent initiates actions, given the available information,
and tries to maximise his or her chances of success (traditionally their personal
or collective utilities), whatever the ultimate goal may be. The algorithm that
each agent takes may be non-deterministic, which means that the agent may
make decisions based on probabilities, not that at any stage of the process a
necessarily truly random choice is made. It actually doesn’t matter much whether
their actions may be perceived as mistaken, or their utility questioned. What is
important is that agents follow rules, or if any chance is involved there is another
large part in it not random at all (specially when one takes into consideration the
way algorithmic trading is done). The operative assumption is that the individual
has the cognitive ability to weigh every choice he/she makes, as opposed to taking
decisions stochastically. This is particularly true when there is nothing else but
computers making the decisions.
This view, wherein each actor can be viewed as a kind of automaton following
his or her own particular rules, does not run counter to the stance we adopt here,
and it is in perfect agreement with the algorithmic approach presented herein (and
one can expect the market to get more algorithmic as more automatization is
involved). On the contrary, what we claim is that if this assumption is made, then
the machinery of the theory of computation can be applied, particularly the theory
of algorithmic information (AIT). Hence market data can be said to fall within the
scope of algorithmic probability.
Our approach is also compatible with the emergent field of behavioural
economics (Camerer et al., 2003), provided the set of cognitive biases remain
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ALGORITHMIC INFORMATION APPROACH TO THE MARKETS 459
grounded in rules. Rules followed by emotional (or non-rational) traders can be as
simple as imitating behaviour, repeating from past experience, acting out of fear,
taking advice from others or following certain strategy. All these are algorithmic
in nature in that they are rule based, despite their apparent idiosyncrasy (assuming
there are no real clairvoyants with true metaphysical powers). Even though they
may look random, what we claim, on the basis of algorithmic probability and
Levin’s distribution, is that most of these behaviours follow simple rules. It is the
accumulation of simple rules rather than the exceptional complicated ones which
actually generate trends.
If the market turns out to be based on simple rules and driven by its intrinsic
complexity rather than by the action of truly random external events, the choice
or application of rational theory would be quite irrelevant. In either case, our
approach remains consistent and relevant. Both the rational and, to a large extent,
the behavioural agent assumptions imply that what we are proposing here is that
algorithmic complexity can be directly applied to the field of market behaviour,
and that our model comes armed with a natural toolkit for analysing the market,
viz. algorithmic probability.
7.2 The Problem of Over-Fitting
When looking at a set of data following a distribution, one can claim, in statistical
terms, that the source generating the data is of the nature that the distribution
suggests. Such is the case when a set of data follows a model, where depending
on certain variables, one can say with some degree of certitude that the process
generating the data follows the model.
It seems to be well known and largely accepted among economists that one can
basically fit anything to anything else, and that this has shaped most of the research
in the field, producing a sophisticated toolkit dictating how to achieve this fit as
well as how much of a fit is necessary for particular purposes, even though such a
fit may have no relevance either to the data or to particular forecasting needs, being
merely designed to produce an instrument with limited scope fulfilling a specific
purpose.
However, a common problem is the problem of over-fitting, that is, a false
model that may fit perfectly with an observed phenomenon. A statistical comparison
cannot actually be used to categorically prove or disprove a difference or similarity,
only to favour one hypothesis over another.
To mention one of the arbitrary parameters that we might have taken, there is
the chosen rounding. We found it interesting that the distributions from the stock
markets were sometimes unstable to the rounding process of prices. Rounding to
the closest .4 was the threshold found to allow the distribution to stabilise. This
instability may suggest that there are two different kinds of forces acting, one
producing very small and likely negligible price movements (in agreement to the
random walk expectation), and other producing the kind of qualitative changes in
the direction of prices that we were interested in. In any case this simply results
in the method only being able to predict changes of the order of magnitude of the
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rounding proceeding from the opposite direction, assuming that the data are not
random, unlike the stochastic models.
Algorithmic probability rests upon two main principles: the principle of multiple
explanations, which states that one should keep all hypotheses that are consistent
with the data, and a second principle known as Occam’s razor, which states
that when inferring causes, entities should not be multiplied beyond necessity,
or, alternatively, that among all hypotheses consistent with the observations, the
simplest should be favoured. As for the choice of an aprioridistribution over a
hypothesis, this amounts to assigning simpler hypotheses a higher probability and
more complex ones a lower probability. So this is where the concept of algorithmic
complexity comes into play.
As proven by Levin and Solomonoff, the algorithmic probability measure (the
universal distribution) will outperform any other, unless other information is
available that helps to foresee the outcome, in which case an additional variable
could be added to the model to account for this information. But since we’ve been
suggesting that information will propagate fast enough even though the market
is not stochastic in nature, deleting the patterns and making them unpredictable,
any additional assumption only complicates the model. In other words, Levin’s
universal distribution is optimal over all non-random distributions (Levin, 1973),
in the sense that the algorithmic model is by itself the simplest model fitting
the data when these data are produced by a process (as opposed to being
randomly generated). The model is itself ill-suited to an excess of parameters
argument because it basically assumes only that the market is governed by
rules.
As proven by Solomonoff and Levin, any other model will simply overlook some
of the terms of the algorithmic probability sum. So rather than being more precise,
any other model will differ from algorithmic probability in that it will necessarily
end up overlooking part of the data. In other words, there is no better model taking
into account the data than algorithmic probability. As Solomonoff has claimed, one
can’t do any better. Algorithmic inference is a time-limited optimisation problem,
and algorithmic probability accounts for it simply.
8. Conclusions and Further Work
When looking at a large-enough set of data following a distribution, one can in
statistical terms safely assume that the source generating the data is of the nature
that the distribution suggests. Such is the case when a set of data follows a normal
distribution, where depending on certain statistical variables, one can, for example,
say with a high degree of certitude that the process generating the data is of
a random nature. If there is an algorithmic component in the empirical data of
price movements in financial markets, as might be suggested by the distribution of
price movements, AIT may account for the deviation from log-normality as argued
herein. In the words of Velupillai (Velupillai, 2000) – quoting Clower (Clower,
1994) talking about Putnam’s approach to a theory of induction (Putnam, 1975,
1990) – This may help ground ‘economics as an inductive science’ again.
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ALGORITHMIC INFORMATION APPROACH TO THE MARKETS 461
One may well ask whether a theory which assumes that price movements follow
an algorithmic trend ought not to be tested in the field to see whether it outperforms
the current model. The truth is that the algorithmic hypothesis would easily
outperform the current model, because it would account for recurrent periods of
instability. The current theory, with its emphasis on short-term profits, is inclined to
overlook these, for reasons that are probably outside the scope of scientific inquiry.
In our understanding, the profits attributed to the standard current model are not
really owed to the model as such, but rather to the mechanisms devised to control
the risk-taking inspired by the overconfidence that the model generates.
In a strict sense, this paper describes the ultimate possible numerical simulation
of the market when no further information about it is known (or cannot be known
in practise) assuming no other (neither efficient markets nor general equilibrium)
but actors following a set of rules and therefore to behave algorithmically at least
at some extent hence potentially modelled by algorithmic probability.
The simulation may turn out to be of limited predictive value – for looking no
more than a few days ahead and modelling weak signals – due to the deleting
patterns phenomenon (i.e. the time during which the market assimilates new
information). More experiments remain to be done which carefully encode and take
into consideration other variables, such as the magnitude of prices, for example,
looking at consecutive runs of gains or loses.
Acknowledgments
Hector Zenil wishes to thank Vela Velupillai and Stefano Zambelli for their kind invitation
to take part in the workshop on Nonlinearity, Complexity and Randomness at the
Economics Department of the University of Trento, and for their useful comments.
Jason Cawley, Fred Meinberg, Bernard Franc¸ ois and Raymond Aschheim provided helpful
suggestions, for which many thanks. And to Ricardo Mansilla for pointing us out to his
own work and kindly provided some data sets. Any misconceptions remain of course the
sole responsibility of the authors
Notes
1. For example, if one plots the frequency rank of words contained in a large corpus
of text data versus the number of occurrences or actual frequencies, Zipf showed
that one obtains a power-law distribution.
2. Also known as programme-size complexity, or Kolmogorov complexity.
3. A universal Turing machine is an abstraction of a general-purpose computer.
Essentially, as proven by Alan Turing, a universal computer can simulate any other
computer on an arbitrary input by reading both the description of the computer to
be simulated and the input thereof from its own tape.
4. Since it is based on the Turing machine model, from which the adjective universal
derives, the claim depends on the Church–Turing thesis.
5. Seeing it as a binary sequence may seem an oversimplification of the concept of
a natural process and its outcome, but the performance of a physical experiment
always yields data written as a sequence of individual observations sampling certain
phenomena.
6. One would actually need to think of one-way non-erasing Turing machines to
produce a suitable distribution analogous to what could be expected from a sequence
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of events that have an unrewritable past and a unique direction (the future), but the
final result shouldn’t be that different according to the theory.
7. The same practise is common in time series decomposition analysis, where the
sole focus of interest is the average movement, in order that trends, cycles or other
potential regularities may be discerned.
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Preface.- Acknowledgments.- Introduction.- I. Background.- Preliminaries.- Computability Theory.- Kolmogorov Complexity of Finite Strings.- Relating Plain and Prefix-Free Complexity.- Effective Reals.- II. Randomness of Sets.- Martin-Lof Randomness.- Other Notions of Effective Randomness.- Algorithmic Randomness and Turing Reducibility.- III. Relative Randomness.- Measures of Relative Randomness.- The Quantity of K- and Other Degrees.- Randomness-Theoretic Weakness.- Lowness for Other Randomness Notions.- Effective Hausdorff Dimension.- IV. Further Topics.- Omega as an Operator.- Complexity of C.E. Sets.- References.- Index.
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To measure the impact of Bayesian reasoning, this paper investigates the occurrence of two words, "Bayes" and "Bayesian," since 1970 in journal articles in a variety of disciplines, with a focus on economics and statistics. The growth in statistics is documented, but the growth in economics is largely confined to economic theory/mathematical economics rather than econometrics.
Book
This book was originally published by Macmillan in 1936. It was voted the top Academic Book that Shaped Modern Britain by Academic Book Week (UK) in 2017, and in 2011 was placed on Time Magazine's top 100 non-fiction books written in English since 1923. Reissued with a fresh Introduction by the Nobel-prize winner Paul Krugman and a new Afterword by Keynes’ biographer Robert Skidelsky, this important work is made available to a new generation. The General Theory of Employment, Interest and Money transformed economics and changed the face of modern macroeconomics. Keynes’ argument is based on the idea that the level of employment is not determined by the price of labour, but by the spending of money. It gave way to an entirely new approach where employment, inflation and the market economy are concerned. Highly provocative at its time of publication, this book and Keynes’ theories continue to remain the subject of much support and praise, criticism and debate. Economists at any stage in their career will enjoy revisiting this treatise and observing the relevance of Keynes’ work in today’s contemporary climate.
Article
A new approach to the understanding of complex behavior of financial markets index using tools from thermodynamics and statistical physics is developed. Physical complexity, a quantity rooted in the Kolmogorov–Chaitin theory is applied to binary sequences built up from real time series of financial markets indexes. The study is based on NASDAQ and Mexican IPC data. Different behaviors of this quantity are shown when applied to the intervals of series placed before crashes and to intervals when no financial turbulence is observed. The connection between our results and the efficient market hypothesis is discussed.