|
|
Cellular Automata for Financial
System and its randomness
|
|
Introduction
In Dr. Wolfram’s book, he
mentioned how Cellular Automata can apply to Financial Market. I think
the most obvious future is how to predict the randomness of all Financial
Markets. One of the most important futures of Cellular Automata is modeling
Financial Markets. With the traditional way, we use mathematics, with its
emphasis on reducing everything to numerical functions and use mathematic
calculations and outcome to simulating market. But the most difficult
thing is how we can predict the randomness by using economic theories or
mathematic functions. Practical experience suggests that particularly on
short timescales much of the randomness is purely a consequence of
internal dynamics in the market, and has little to do with the nature or
value what is being trade. In Dr.
Wolfram’s book, he give us good explain the Mathematica and origin of
randomness. This give us an idea
of why we can not predict the randomness of Financial Market even we
reduce all environment condition that would infect the Financial Market
to minim. In this paper, we will review Dr. Wolfram’s Cellular Automata
and its randomness rules. And apply this into Financial Market using
simplest Investors’
behavior; study the effects of various elements of
investor behavior on market dynamics and asset pricing by using Cellular
Automata.
What are the Cellular Automata
and its randomness rules?
Cellular Automata can discrete dynamical
system with simple construction but complex self-organizing behavior.
This behavior is completely specified in terms of local relations who at
each step each cell computes its new state from that of its close
neighbors. Evidence is presented that all one-dimensional cellular
automata fall into four distinct universality classes. Three classes
exhibit behavior analogous to limit points, limit cycles and chaotic
attractors. The fourth class is probably capable of universal
computation, so that properties of its infinite time behavior are
undecided. The different classes of cellular automaton behavior allow
different levels of prediction of the outcome of cellular automaton
evolution from particular initial states. In the first class, the outcome
of the evolution is determined (with probability 1), independent of the
initial state. In the second class, the value of a particular site at
large times is determined by the initial values of sites in a limited
region. In the third class, a particular site value depends on the values
of an ever-increasing number of initial sites. Random initial values then
lead to chaotic behavior. Nevertheless, given the necessary set of
initial values, it is conjectured that the value of a site in a class 3
cellular automaton may be determined by a simple algorithm. On the other
hands, in class 4 cellular automata, a particular site value may depend
on many initial site values, and may apparently be determined only by an
algorithm equivalent in complexity to explicit simulation of the cellular
automaton evolution. For these cellular automata, no effective prediction
is possible; their behavior may be determined only by explicit
simulation.
Dr. Wolfram’s Simple rule for Financial
Market
In the financial world, most simulation models in economics and
finance assume that investors are rational. However, experimental studies
reveal systematic deviations from rational behavior. How can we determine
the effect of investors' deviations from rational behavior on asset
prices and market dynamics? By using Cellular Automata, we assume investor
behavior and to model it as empirically and experimentally observed. This
can be explained by investors' quasi-rationality. Being able to predict
how people will invest and setting asset prices accordingly is inherently
appealing, and the combination of Cellular Automata and statistical
mechanics can make such modeling possible.
Base on Dr. Wolfram
understands. In the most naïve economic theory, price is a reflection of
value, and the value of an asset is equal to the total of all future
dividends. The prices are in fact determined not by true value, rather by
the best estimates of that value at any time. For Example, the stocks we trade are not
equal to the real asset of company. Instead, stocks are determined by the
company performance, earnings, and investor’s confidence…etc. The
estimates of value are affected by many events that go on in the world. In Financial Market, people try identified
every situation and take that as input into Financial Simulate Model, but
turns out it is still unpredictable. Even we could identify all situations,
and speculate we trade without any significant external input, but the
result is still random fluctuation. It is hard to understand why there
should be any significant fluctuations in prices at all. We ask ourselves
“Is the random fluctuation inevitable?” May be not, here is example: In
negotiation between two parties, we often see that in the beginning, the
prices of two parties will offer are very far from each other. But it is
common to see fairly smooth convergence to a final price that both
parties will satisfy. For large number of parties, we can get same result
simply by construct some algorithms that operate between them. So why in
financial market, which originate as a trade place for multiparty, could
not get such smooth out result? In his book Dr. Wolfram gives us an
example viewing a market as being like simple 1-Dimensional Cellular
Automata. Each cell corresponds to
single trading entity choose to buy or sell at that step. The behavior of
a given cell is determined by looking at the behavior of its two
neighbors on the step before. According his strategy, if both neighbors
buy, the investor will sell. If either neighbor buys, the investor will
buy too. If both neighbors sell, the investor will sell. The Market price
is the running difference of the total numbers of buying and selling at
successive steps.
|
buy
|
|
buy
|
buy
|
|
sell
|
sell
|
|
buy
|
sell
|
|
sell
|
|
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
The plot below gives us a
market price the running difference of the total numbers of buys and
sells.
Although the behavior of the
system running on very simple rule, but the plot above looks in many
respects random.
Cellular Automata for Simple
Financial Market
Base on Dr. Wolfram’s model
to simulating Financial Market as a minimal idealization one which tries
viewing a market as being like a simple 1-D Cellular Automata. We can
come out some strategy more close to real market. The main force that
drives the Financial Market is Consumer Confidence, when Consumer
Confidence is high, the market is in Bull Market, every entity is in buying
stage, and the price of stocks going up. When Consumer Confidence is low,
the market is in Bear market, every entity is in selling stage, and the
price of stocks going down. Base on this we develop Cellular Automata
rules that the color of the cell at a particular step specifies whether
that entity chooses to buy or sell at that step. We can imagine the way
that information flows in a market, for example the cells looking at its
neighbor’s behavior to determine its own activity. So we have:
Bear Market Strategy
What this Cellular Automata
rule reflects in the real market is that if the stock market in bear
market, when consumer confidence is very low. The investor only sees
every one around him selling, so he sells too. Only time he is buying is
when every one else buying. From CA side, the cell look both its
neighbors, if both sides are buy, he buys. If either side is selling or
both sides are selling, he sells.
|
buy
|
|
buy
|
buy
|
|
sell
|
sell
|
|
buy
|
sell
|
|
sell
|
|
|
buy
|
|
|
sell
|
|
|
sell
|
|
|
sell
|
|
The complete system behavior of Bear Market Strategy
The price movement
Bull Market Strategy
What this Cellular Automata
rule reflects in the real market is that if the stock market in bull
market, when consumer confidence is very high. The investor only sees
every one around him buying, so he buys too. Only time he is selling is
when every one else selling. From CA side, the cell look both its
neighbors, if both sides are sell, he sells. If either side is buying or
both sides are buying, he buys.
|
buy
|
|
buy
|
buy
|
|
sell
|
sell
|
|
buy
|
sell
|
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
The complete system behavior of Bull market Strategy
The price movement
Another main force that
infect the investor’s decision is it selves passed experience. Sometimes
the trader’s passed experience gives them an instinct of what they should
do in the Market environment. This together with the behavior of people
around him makes his decision. Apply to Cellular Automata; the cell is
not only looking at its neighbors states, but also look at itself passed state
to determine its future color. So base on the Bear and Bull Strategy we
already have, for example in Bull Market, we can come out the following
new strategies:
|
buy
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
buy
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
sell
|
sell
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
sell
|
buy
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
sell
|
|
|
buy
|
|
|
sell
|
|
|
buy
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
sell
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
|
sell
|
|
|
buy
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
sell
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
|
buy
|
buy
|
buy
|
buy
|
sell
|
sell
|
sell
|
buy
|
buy
|
sell
|
sell
|
sell
|
|
|
buy
|
|
|
buy
|
|
|
buy
|
|
|
sell
|
|
The above, cells’ color is
determine by both its neighbor and it’s previous color. Over all it shows
the buy activity in the market.
Cellular Automata for
Complicate Financial Market
In the real financial world, the
values of socket market are determined by some complexity environment
which different entities apply their own strategy. Instead we have same
strategy apply to every one in above; we give each individual cell inside
Cellular Automat its own strategy. And each individual’s trade strategy
is changing according the environment. It is obviously in real world, if
some one have good trading strategy, soon or late, every one else will
following. It is so simple that term of good strategy is depending on how
much you will make. So the price is determined globally by other people’s
strategy change. So the investor not only looks the trade behavior of its
neighbor but also the trader strategy of its neighbor. We looking in a
timescales of order weeks or months---and in some cases perhaps even
hours or days, the trade strategy can be as a sequence of trade behaviors.
And the entity would be able remember its neighbors trade behavior in
several pervious steps to make it decision. In the Cellular Automata
world, we can use 2-D CA to modeling this activity. The cells starts look
all neighbors around him to make trade decision with its own trade
strategy.
Time---1 step
|
sell
|
sell
|
sell
|
|
sell
|
S/buy
|
buy
|
|
buy
|
buy
|
buy
|
Time---2 steps
|
sell
|
buy
|
sell
|
buy
|
sell
|
|
sell
|
sell
|
sell
|
sell
|
sell
|
|
buy
|
sell
|
S/b
|
buy
|
buy
|
|
sell
|
buy
|
buy
|
buy
|
buy
|
|
buy
|
buy
|
sell
|
buy
|
sell
|
In the above 2-D Cellular
Automata, the distance leave the center cell represent the time of each
step. The closest neighbor to the center cell has more influence to the
center cell’s state than the distanced cell. And cell could compare their strategy’s
“success” with the neighbor cell’s strategy and copy the best strategy.
The origin of Randomness of
Financial Market
In Financial Market, the random
movement in price in sense just reflections of random changes going on in
the outside environment, for example the strategy we describe above. But even
every one have same strategy, it turns out it still looks fairly random. With
simplest rule but still random, so where is this randomness come from? Perhaps
random fluctuations are just an inevitable feature of the way that prices
adjust to their correct values. But certainly we can construct Cellular
Automata rules in class IV that operate between larger numbers of parties
that would also lead to fairly smooth behavior. From Dr. Wolfram’s book,
simple models do not necessarily have simple behavior. For example, in
stock market, the rule is simple, but the outcome is complex or I say exhibit
significant randomness. Why is there randomness in markets in the first
place? Practical experience suggests that particularly on short
timescales much of the randomness that one sees is purely a consequence
of internal dynamics in the market. In Dr. Wolfram’s book, he give use a
good example of intrinsic generation of randomness. In intrinsic
randomness generation, every cell in effect actively contributes to the
randomness. This randomness occurs as a consequence of the dynamics
of the system, even though the initial conditions are simple. In
rule 30 cellular automation, the basic rule for the system is very
simple. The
Cellular Automata is started from a single nonzero site. And
initial condition is also very simple. But the randomness is always generated.
Rule 30
Yet despite the lack of
anything that can reasonably be considered random input, the evolution of
the system nevertheless intrinsically yields behavior which seems in many
respects random. This gives us a good idea how the randomness of
Financial Market come from. In past hundred years, people study the
randomness of Financial Market and hope to be able predict it. But with
respect all we have studied is the randomness from the environment in the
Financial Market. This gives us the question, why after we take all the
factor from outside environment that would generate randomness out, it
still give us the randomness and leading us to unpredictable world. Dr. Wolfram’s rule 30 gives us the best
example, simple rule but out come is random. So are we waste all the time
to try study the randomness of financial market that comes from internal,
which nobody would be able to predict. I think the Financial Systems is
combining mechanism for randomness from the environment and the
randomness from intrinsic generation. In past century or so, people put
all their energy on study of randomness from the environment. After years
develop the best mathematical financial model still can not predict the
real Financial Market. Why? Because people never studied the randomness
from intrinsic generation in financial system, until Dr. Wolfram and his
discovery of Cellular Automata gave us the answer.
CA Financial Model vs. Mathematical Financial Model
What is advantage and disadvantage of Cellular Automata Financial
Model compare to the traditional mathematical financial modeling,
nIn the
Mathematical Financial Model, we have:
Data-àFunction-àOutput-àDecision
nIn
Cellular Automata Financial Model we have:
Entity-àStrategy-àBehavior
The Cellular Automata
Financial Model gives us the complete automation of Financial Market. It
just like a seed in the earth can grow itself. But at this moment we can
not apply Cellular Automata to real market, because Cellular Automata
could not take complicate financial data as input. So it can not simulate
complicate Financial Environment. Also Cellular Automata can not make
decision of buy or sell with time scale and volume. But we know
Mathematical Financial Modeling have been developed in years, and Cellular
Automata Financial Modeling just come out. My idea is to develop a model that combine Mathematical
Function and Cellular Automata by
using Cellular Automata simulating Macro scope environment and using
Mathematical Formula simulating Microenvironment. And take the result of
mathematical modeling as the input of Cellular Automata, so we can see
the result of price movement in large scale.
Conclusions
nCellular
Automata has give us first look of the origin of randomness of Financial
Market
nUse Cellular
Automata Model to simulating Financial market just begin, it will have a
long way to go…
References
"Universality and Complexity in Cellular Automata" - Stephen
Wolfram
“A new Kind of Science”, page 429-432- Stephen Wolfram
“Artificial
Stock Market Simulator by Cellular Automata” -Tomomi TAKASHINA
“Cellular Automat and Computational Finance (99)” -Stern
School of Business,
NYU
"Artificial Life : The Quest for a New Creation" - Steven
Levy
|