MODELLING FINANCIAL
CRASHES (AND BUBBLES)
THROUGH THE ISING MODEL
Balaji Harihar
CID – 01511175
Name of supervisor – Dr. William Proud
Name of assessor – Prof. Deeph Chana
Word Count - 5608
Balaji Harihar
Abstract
To model financial crashes and bubbles, the
market has been considered to be bistable, and
the dynamics of the Ising model have been
applied. The main modes of interaction within
the market can be described by thermal-bath
dynamics which tells us how the market
situation can influence the whole world and
also highlights the effect of macroeconomic
factors on this market.
The other mode of interaction is understood by
considering long-range interactions as well as
near-neighbour interactions between agents in
the market. Nowadays, with the facility of
mobile phones and computers, long-distance
trading has become possible and hence its
inclusion into the model.

In the ordered ferromagnetic phase, there are 2
possible orientations of the magnetization and
thus an agent could be in either of the 2 potential
wells (c.f. “bistable model”); one where prices
are going up in general (market growth) and the
other where prices are going down (market
decline), and these orientations are stable.
A crash or bubble will happen when there is a
sudden inversion in the average orientation of
the agents. The paper postulates the cause for
market crashes to be a small informationcarrying signal being enhanced by external
noise.
To test this idea, both numerical and theoretical
analysis is carried out and an important
approximation called mean-field approximation
enables the theoretical analysis to be carried out
with much ease, and not losing too much
information in the process.
This hypothesis turns out to be true in most
cases as market crashes are difficult to
anticipate (hinting towards a weak information
signal) and just before the crash, there is too
much entropy/noise in the market, which
ultimately causes the event to occur.
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Declaration
I declare that the work in this essay titled
“Modelling Financial Crashes (And Bubbles)
Through The Ising Model” has been carried out
by me. All the content in this piece of literature
that has been derived from various sources has
been rightly acknowledged in the list of references
provided.
Content
Introductory chapter – (i) Motivation, aims and
objectives, (ii) Bringing the physics into context
Discussion of methods employed – (i) Overview of both
the analysis, (ii) Theoretical analysis
Results and discussion – (i) A detailed look at the
results…, (ii) …and what do these results imply?
Summary and conclusions
Acknowledgements
References
Introductory Chapter
Motivation, aims and objectives
In a plethora of papers, it has been shown that an
appropriate description of the price of a
commodity over time follows a “complex
stochastic process with non-Gaussian
characteristics” [1-4]. In addition, the background
features of the market like volatility clustering
where the magnitude of change in price directly
correlates to magnitude of change is returns can
be understood by treating it as a collective
phenomenon [5].
Based on this, numerical simulations of a manyagent market can be formed on lines of
percolation theory, Ising model, multiplicative
noise etc. but the reason to choose the Ising model
to depict the behaviour is mainly because it was
used in an infamous paper [6] to study the
Japanese stock market crash of 1997.
The primary focus of this paper is to look at
financial crashes, which are defined as large and
sudden negative returns when market prices are
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generally growing and the exact opposite of what
one would expect in volatility clustering. On the
other hand, a bubble is described as huge positive
price returns on decreasing market prices.
non-interactive factors that affect decision-making
(an example could be the facility of Google
Finance to look at the variation of stock prices in
the market).
Upon some research, the physical concept that
best explains crashes and bubbles (which many
researchers agree upon too) is a phase transition
where the average magnetization of the market is
inverted spontaneously. Most, if not all these
papers [7-10], attribute the cause of crashes to be
a hidden force which is thought to be some piece
of information available to the agents.
Looking at the interaction strength term 𝐴𝑖𝑗 , the
collective response of all the agents is underlined
by it. When 𝐴𝑖𝑗 > 0, agents keep huge faith on the
decision of other agents and go along with
whatever trend is being followed by the market
agents. On the contrary, 𝐴𝑖𝑗 < 0 represents low
faith of an agent on the action of their partners and
the agent rather follows his/her gut feeling.
In this paper, it is postulated that this hidden
force/information signal itself is too weak to cause
the crash, but once it is amplified by external
noise (inflation, interest rates and other
macroeconomic factors being offset) and
fluctuations within the market (various
microeconomic factors being offset),
magnetization inversion is possible through a
process called “stochastic resonance” [11].
The orientation of the agent ‘i’ at an interval ‘t+1’
is updated using Fermi-Dirac statistics (given by
probability ‘p’) to invoke the uniqueness of each
agent and takes the uncertainty of predicting the
agent’s decision by us into account. It is given by
Eq. 2,
Bringing the physics into context
The stock market in this paper is based on the
Ising model format adopted by Kaizoji in Ref. 6
since many features of the market are common to
this model. In our finite lattice (market), there are
‘N’ agents (magnetic dipole moments) with 2
choices of spin orientation σ𝑖 = +1 or -1. When
the orientation is +1, the agent decides to buy
shares of a stock at regular intervals of time ‘t’
(general interval is weekly).
Looking at the formulation of our model, it does
look like an Ising model with the dynamics of a
thermal bath (thermal bath assumes that this
market does not affect the external environment,
and on the other hand, the external environment is
a source of information as well as decisionmaking factors). The resemblance to the Ising
model and its success in understanding society is
supported by the “empirical theory of opinion
formation” [12-14].
Before the orientation of the agent at a time ‘t+1’
is known, it is important to introduce the local
field 𝐼𝑖 (t) which essentially forms the basis for the
environment in which an agent ‘i’ works and
includes the influence of his decision-making
based on interaction with other agents and access
to external information. This local field is given
by Eq. 1,
A useful quantity that can be obtained from the
orientation of each agent is the magnetization,
which is just the weighted average of the sum of
the individual orientations. Denoting the price by
S(t), its dynamics can be understood with the
simple differential equation dS/dt ∝ xS, since
market will be bullish if demand exceeds supply
(and prices increasing in turn which is dS/dt > 0).
If supply exceeds demand, we observe bearish
behaviour and both sides of market behaviour are
described well by the differential equation.
In this equation, ‘z’ gives us the average amount
of non-zero links between agents, 𝐴𝑖𝑗 is the
interaction strength of agent ‘i’ with another agent
‘j’ and ℎ𝑖 (t) is the external field which highlights
Recalling from the principle of volatility
clustering that the price returns are proportional to
the stock price, and in turn to the magnetization,
we obtain the logarithmic equation R(t) =
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log[S(t+1)] – log[S(t)].
This model by itself is very tedious to solve by
hand and is always coded into computers to be
solved numerically. However, a key
approximation, called the ‘mean-field
approximation’ will be helpful in simplifying our
model by taking advantage of the geometry of our
problem, and makes way for a new kind of
analysis to be performed on the model: theoretical
analysis.
In this approximation, we say that 𝐴𝑖𝑗 = A and
ℎ𝑖 (t) = h(t) and we can justify this since the
number of agents and the number of links in turn
are extremely large, and in addition, the
interaction strengths and external fields have a
symmetric distribution about the mean value due
to the randomness of the situation.
Now, we require an expression for x(t+1) under
this approximation such that the crash or bubble
occurs at t = 0 and at times way before or after
that, x(t) is a stable constant value. According to
[15], the most appropriate equation that captures
this behaviour is Eq. 3,
A simple manipulation enables us to write 𝐴𝑖𝑗 as
A + (𝐴𝑖𝑗 – A), and plugging this into Eq. 1, one
part of the interaction term contains (NA/z)x(t)
and this varies proportionally with R(t) (c.f. R(t)
∝ x(t)). This manipulation is very helpful as
that part of the interaction term also shows
how agents react to the price changes, given by
R(t). In a way, it also averages all the reactions
of an agent to the decision of his partners over
the course of time till now.
If we neglect the external field h(t) for now, we
can obtain a good physical picture of how the
model correlates to market behaviour. In the
regime that A < 1 (note A cannot be negative), x =
0 is a steady solution of Eq. 3 while for A>1, we
have a bistable system with two symmetric
solutions 𝑥+ > 0 and 𝑥− < 0.
The x = 0 state is paramagnetic since there is no
preferred orientation of spin, and in economic
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terms, this corresponds to 0 price returns and thus
represents the market being in equilibrium.
Meanwhile the other 2 solutions for A > 1
represent increasing (𝑥+ ) and decreasing (𝑥− )
price returns and can be associated with a growing
and falling market, respectively.
For the purpose of modelling crashes and bubbles
in the mean-field approximation, we will take A >
1 as that provides us with the bistable potential
wells required for the switch of magnetization. A
crash would be an average jump of the market
from the 𝑥+ well to the 𝑥− well while a bubble
would be an average jump from the 𝑥− well to the
𝑥+ well.
The paper by Krawiecki (Ref. 15) takes the
interaction strength to be of the form 𝐴𝑖𝑗 =
A(1+aζ𝑖𝑗 ) with probability P and my analysis will
also be based on this formulation of the strength.
For completeness, 𝐴𝑖𝑗 = 0 with a probability of 1P, although P is taken to be extremely high in the
mean field approximation so that we have a nearly
well interacting model which makes sense in
today’s time with the availability of technology.
ζ𝑖𝑗 is a set of random and unrelated variables
which take values between -0.5 and 0.5 and is the
best possible way to determine the strength of the
links due to randomness of model and
unpredictability of humans.
Now that a vivid picture of the market has been
developed in the absence of the external field, it is
useful to ponder upon how the external field and
information-carrying signal look. As per many
papers [1-15], the most appropriate choice for our
information-carrying signal would be a periodic
wave that causes stochastic resonance; this form
of the signal is chosen only for simplicity and it
does not imply a hidden periodicity involved in
crashes and bubbles.
Moving on to think about the form of the external
field, it was hypothesized that the information
signal being amplified by the noise causes
crashes, and thereby a sum of both these terms
must be present for ℎ𝑖 (t). Along with this, ℎ𝑖 (t)
must also capture the “willingness” to go from
bullish to bearish behaviour (or vice versa), or in
physical terms, there is always a preferred
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orientation associated with ferromagnets where
the energy is lowest. The two parts comprising
this willingness would be the effective strength of
the external field reaching the agent and the
coefficient of reaction for each agent.
Keeping all this in mind, the form of the external
field and information signal are assumed to be Eq.
4,
s(t) is the information signal and ξ(t) describes the
noise within the market which is time dependent
for obvious reasons. The amplitude of s(t), ‘q’, is
too weak to cause a switch in magnetization by
itself but when added to the noise function with a
good enough noise intensity ‘D’, we observe
jumps in both mean-field model as well as full
microscopic model.
Lastly, ‘b’ captures the coefficient of reaction
centred around the 1 while χ𝑖 , just like ζ𝑖𝑗 , is a set
of random unrelated variables in the range (0.5,0.5) and describes the willingness to switch
orientation for an agent.
Discussion of the methods employed
Overview of analysis
The main goal of the analysis, be it numerical or
theoretical, is to investigate how the system
responds to the information signal as a function of
‘D’ with a fixed interaction strength in the mean
field model (A>1) and for different connection
strengths ‘P’ in numerical analysis.
The form of x(t) can be made simpler to work
theoretically by introducing an output function
y(t) = sign(x(t)) where the market crash is
approximated to occur spontaneously and the
market is stable otherwise, away from the time of
crash (assuming the crash only occurs once and
various factors in the market present are too weak
to induce jumps, or in general, the factors causing
bubbles and crashes are randomly and evenly
spread and cancel out the effect of its counterpart,
until our information signal arrives, amplified by
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noise).
The importance of the bistable consideration of
our model is to ensure that when a crash occurs,
its root cause lays in the information signal. The
first set of qualitative analysis will be theoretical
and performed in the mean-field approximation
while numerical analysis will be carried out in
both mean-field and full microscopic model. In
simple words, the output signal measures the
average response of the market to an external
stimulus, making a simple approximation that
there are only two possible states for a spin to take
and that this inversion happens spontaneously.
From the output signal, one can obtain the spectral
density S(ω) by a Fourier transform and in doing
so, we observe a wide noise background and the
graph peaking at odd multiples of 𝜔𝑆 . The
parameter which will be varied with ‘D’ in order
to determine the occurrence of a market crash
under the phenomenon of stochastic resonance is
the Signal to Noise Ratio (SNR) [11]. This ratio is
calculated in the vicinity of the frequency 𝜔𝑠
since we obtain peaks for the spectral density in
that region, and stochastic resonance also occurs if
the spectral density is at a maximum.
In the numerical method, a frequency bandwidth
of 2−12 Hz is used to normalize the SNR because
as per simulations in [11] and [15], they have
obtained the spectral density by averaging the
values from a plethora of series of y(t), each of
which contains 212 points. As the word itself
suggests, the SNR should be a ratio of the residual
signal intensity (perceived intensity to an agent
sans the noise) to the actual noise intensity. The
best way to formulate an equation is to use the
logarithm of the ratio since we are dealing with
many agents, so noise is expected to be
exponential. A suitable way to express the SNR
(in dB) is 10𝑙𝑜𝑔𝑒 [(S(𝜔𝑠 )-𝑆𝑁 (𝜔𝑠 ))/ 𝑆𝑁 (𝜔𝑠 )]
Bringing the physics into context, plenty of papers
have based their simulations on stochastic
resonance to picturize response to an external
stimulus in the Ising domain [16-17], and another
application has been in the “Weidlich model of
opinion formation” [18]. But the difference in
these papers is that the reaction of agents to the
stimuli is measured as a function of thermal
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fluctuations (keeping noise constant). However,
this paper intends to look at the market response
to noise and we thus take this slightly different
approach as compared to the references by
varying ‘D’ and keeping thermal fluctuations.
Lastly, choosing A > 1 ensures ferromagnetism as
spins want to align and we assume that we will
witness rational behaviour from our agents (such
behaviour of aligning with others was observed in
the 1990s when the internet became a huge
sensation, and everybody started investing in
stocks in the internet share market). The
spontaneity and ordering behaviour of the market
makes it apparent that there is high stability, so
spins are far away from critical point.
Theoretical Analysis
This analysis will purely be carried out under the
mean field approximation to evaluate the SNR.
The theory of stochastic resonance outlined in
[11] shall be used. In this theory, for bistable
systems, the SNR can be obtained if the
dependence of the rate of escape from one
symmetric stable state to the other with the noise
is known, denoted by r(D). This dependence on
the escape rate stems from the fact that reaction of
agents (y(t)) (whose Fourier Transform gives the
spectral density) causes a switch in orientation
and a jump from one state to the other, and the
measurable quantity in this jump is the number of
agents switching in a specified amount of time,
which is r(D).
Ferromagnets undergo spontaneous symmetry
breaking when the external field is applied since it
acts as a perturbation to the Hamiltonian of the
system [19]. Due to this, the analysis assumes that
due to the signal, the escape rates now become
asymmetric and time-dependent. This means that
for one half of the cycle of the signal, the escape
rate from, say, the “left state 𝑟+ (D,t)” is greater
than the “right state 𝑟− (D,t)”, and vice versa for
the other half of the signal.
To think of a way to formulate an expression for
our escape rate from the state 𝑟+ (𝐷, 𝑡), we must
remember that the process of noise amplification
is non-additive and thereby only a qualitative
theoretical analysis can be carried out. Various
other factors affect the escape rate, but they are
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miniscule compared to noise, and it will thus be
considered as a first-order effect/perturbation. We
can apply a first-order Taylor expansion to the
escape rate to give Eq. 5,
The 1st order term 𝑟1 (D) is multiplied with the
information signal in order to take the asymmetry
of the overall escape rate into account, and it also
ensures that the escape rate varies in full
synchronization or anti synchronization with the
information signal. 𝑟0 (D) is the symmetric escape
rate which would be the sole escape rate in absence
of the signal.
According to [11], the SNR can now be calculated
from this 1st order approximation, and is given by
Eq. 6,
As seen before, when no external field is applied,
the system lies around the stable states 𝑥+ or 𝑥− .
We have assumed that only the information signal
can cause switch of magnetization and therefore,
except for these large jumps, x(t) oscillates in the
vicinity of the stable states. Keeping this in mind,
for obtaining an estimation of the SNR for h(t) not
equal to 0, a jump of mean orientation is only
possible if stationary point of x(t+1) occurs to the
right of 𝑥+ (crash) or to the left of 𝑥− (bubble).
The problem with this approximation is that it
understates the true value of the probability of jump
since we do not consider the possibility of jumping
under the influence of several noise pulses. With
some simple calculations, we find the condition for
switching to the negative orientation from positive
to be ξ < ξ+ = (-A𝑥+ - q cos(𝜔𝑠 t))/D and the escape
rate 𝑟+ is just the probability distribution function,
written as P(ξ < ξ+ ).
Similarly, the escape rate in the other direction,
from positive to negative, is given by P(ξ > ξ− )
where ξ− is given by (-A𝑥− - q cos(𝜔𝑠 t))/D and the
asymmetric modulation of the rates of escape by
the periodic signal can be clearly seen. As per [11],
in such a situation, the parameters 𝑟0 and 𝑟1 from
Eq. 5 can be written as the given expressions in Eq.
7,
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Results and Discussion
Using this underestimating version of the escape
rate will not suffice, and one way to overcome this
is to somehow combine this method with another
one which overestimates the actual probability of
escape. In light of this, a second train of thought is
developed where we assume the information
carrying signal causes the stable states to disappear
once a jump of magnetization occurs. The reason
this is an overestimation is because at time t+1, the
stable state may reappear due to the external field
being weak in this step, and the jump does not
happen after all.
For our stables states to vanish, the conditions ξ <
ξ+ = (-h0 - q cos(𝜔𝑠 t))/D as well as ξ > ξ− = (h0 q cos(𝜔𝑠 t))/D must be satisfied. The expression for
h0 is given (after some mathematical manipulation
in [11]) by Eq. 8,
Finally, the second set of expressions for 𝑟0 and 𝑟1
is given by Eq. 9,
We know that the signal itself cannot cause
magnetization to flip without the presence of noise,
and this is mathematically expressed by the
condition q < h0 . From this condition, we can
obtain a physical intuition of what h0 means; a
minimum amplitude of the information signal so
that it its amplification with noise can cause a crash
or bubble. If q < h0 , the stable states x+ and x− will
not disappear and neither will the jump of
magnetization occur.
By applying the condition q < h0 , and given that A
> 1, substituting the expressions of Eq. 7 and Eq. 9
into Eq. 6 provides us with curves of SNR against
D with a peak at D > 0 (will be shown in the results
section), so we can anticipate the occurrence of
stochastic resonance in the many-agent model as
well which will be solved numerically and whose
results will be directly reviewed without indulging
too deeply into the computer code, since this is
beyond the scope of this essay.
A detailed look at the results…
Fig. 1. (a) – SNR plotted against D. Parameters are A = 1.5, a =
2, b = 2 and q = 0.2. The solid line plots the theoretical model
using Eq. 6 and 7. The dashed line plots the theoretical model
using Eq. 6 and Eq. 9. Shaded dots show the numerical model
under mean-field approximation while circles represent a full
computational simulation of a market with N = 250 and P = 1.
Fig. 1. (b) – Same parameters used as in (a), but only the full
computational plots are drawn for various probabilities.
Specifically, the squared points represent P = 0.05, diamonds
represent P = 0.25, triangles represent P = 0.5 and circles
represent P = 1. Solid line is line of best fit, just a visual aid for
the reader.
As shown in the graph, all the curves in Fig. 1(a)
peak at a particular value of D > 0 and the results
obtained from numerical method are very similar
to the theoretically postulated model. We have
obtained stochastic resonance in both models as
expected and devoid of the fact that the meanfield theoretical model uses a small number of
agents and a large range of interaction strengths
‘a’ and reaction coefficients ‘b’, the results from
both trains of thought are comparable.
The value of ‘A’ has been taken to be 1.5 (very
deep in the bistable phase) rather than taking it to
be close to unity (as has been done in many papers
including [6]) because for this value of ‘A’, the
magnetization jumps take place due to
fluctuations within the system, which gives
inaccurate responses in the theoretical mean-field
limit for small N (the regime in which this essay
operates) and simulations also take a very long
time.
Moving to Fig. 1(b), it portrays the effect of
changing the interaction strength between agents.
Increasing the strength ‘P’ does lead to the
maxima of the SNR to shift to the left but for
extremely small P (the squared points line), we
observe no stochastic resonance at all, and the
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SNR only keeps decreasing as ‘D’ increases. This
difference in behaviour at low ‘P’ can be
attributed to the order-disorder transition at the
critical probability P𝑐 ≈ 0.01 [20] where for
regions around and below this value, we expect
the system to be in an approximate paramagnetic
state. In this case, a weak signal is enough to
make the agents orient towards a preferred spin
direction, without the need of any external noise.
The ability of paramagnets to orient themselves
quickly is known, and in a financial context, this
implies that people have no way to access
information into the stock market; the very first
piece of information they receive is through the
signal, and thus everybody spontaneously decides
to believe that information and follow that trend.
This kind of situation can be seen in the market of
a newly discovered/invented commodity (like
gold during the Gold Rush or computers in the
1970s where market was disordered due to little
being known about how to market the commodity
[21]).
…And what do these results imply?
Fig. 2 – The graph shows the distribution of normalized
price returns G = (x - < 𝒙 >)/√< 𝒙𝟐 > −< 𝒙 >𝟐 , where
the term < 𝒙 > denotes average over time procured from
computational runs of the mean-field market with the
same parameters as in Fig. 1 and D = 0.4 (near to the value
of D where stochastic resonance occurs); only x > 0 part
is evaluated due to symmetry of Eq. 3 which gives us that
< 𝒙 > = 0.
In our model, we have taken this switching
between magnetizations to be permanent, but this
is not a pragmatic model, as depicted by Fig. 2.
The cumulative distribution of probability ρ(G)
has been plotted against G. As per the study
conducted by [2], ρ(G) must have a maximum at x
= 0 and subsequently, its tail decays according to
a power law. The interpretation of this ‘real’
version of the market is that we expect the market
to remain in equilibrium when we have small
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returns, while the chances of having huge returns
is very low.
But from the mean-field simulation of the market
in Fig. 2, there is a bimodality in the graph (due to
the market being considered bistable), and this
more importantly highlights a high likeliness of a
bubble or crash occurring (large positive or
negative price return). It is important to emphasise
that a bistable form of the stock market was
chosen with a periodic information signal for
obtaining good statistics on the SNR, which in
simple terms measures how the information signal
is enhanced by noise as per most of the papers in
the references.
From Fig. 2, it is also apparent that the agents start
working more synchronously over time as the
graph rises exponentially as returns increase over
time; one way to model this is increasing ‘A’ over
time as long as we are in an ordered (growing or
declining market) regime. As per [6], ordering of
the market occurs in a similar fashion to the way
described in this essay; it goes on to associate this
ordering with the spontaneous market crash
because everybody is placing the same order at
the same time.
The writer of this paper interprets that the market
remains ordered for some time (rather than
immediately crashing) and perhaps the
unprecedented arrival of an insignificant piece of
information rapidly inverts the magnetization due
to amplification by noise, and this bit of
information is the cause of the market crash. Once
the crash or bubble has occurred, the market
generally comes back to the equilibrium state
through decreasing A as time progresses for
example, and the symmetry of the market is
restored too.
Summary and conclusions
In a nutshell, the behaviour of our microscopic
market was encapsulated by the Ising model where
the agents were considered as spins, and the
physics was governed by thermal bath dynamics
and long-range communications with the entire
system being subject to the combined effect of an
external signal and noise.
In order to picture crashes and bubbles, we
assumed that agents had strong interaction
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strengths between them and under this regime, the
market was bistable with one stable state
corresponding to a growing market and the other to
a declining market.
In the model, a crash was defined to be a jump from
the growing market state to the declining market
state (magnetization inversion) due to the arrival of
an information signal. It was also made clear that a
low amplitude signal can also affect the market by
being amplified by noise through a process called
stochastic resonance. This process occurs in both
mean-field and many-agent models and its
dynamics is explained qualitatively through the
adiabatic theory of bistable systems under the
influence of external noise.
The key result from this model is that a supposed
weak stimulus which agents may choose to ignore
can have a large effect on the market by creating
enormous returns if they are enhanced by noise.
This weak information signal is taken to be the
cause of the crash, even though no one anticipates
crashes upon its arrival. For the switch of
magnetization to occur, the market must be far
from equilibrium and there must be a large-scale
ordering of agents just prior to the crash or bubble.
This model does not investigate the effect of a
strong signal on the market because even though
they have the capability of causing crashes, they
are very unlikely since people can read the signal
and act rightly to avoid a financial catastrophe.
Another factor that could cause crashes and is not
studied are external/internal fluctuations, but this is
beyond the scope of the paper to examine since the
cause for the crash/bubble itself is nearly
impossible to find.
Another unavoidable drawback of this model (or
any model in general) is the randomness of the
situation. The reaction coefficients and interaction
strengths were just assigned random numbers
between -0.5 and 0.5 and rather than coming up
with a solution to tackle this, this randomness
opens a new train of thought into thinking about
how alike we are to simple magnetic dipole
moments.
With all the ability to think and make calculated
decisions, a simple spin model possesses the ability
to accurately describe our behaviour. As seen in
this model, some time after the crash, we tend to
8
follow ordering behaviour again, even though this
is the key propellor of crashes/bubbles so this
should intrigue us into thinking how a single atom
by itself is so difficult to understand and predict
(just like a single human), but once many of these
atoms or humans come together, the same Ising
model can perfectly describe the behaviour of both.
Are we ultimately just a pawn in nature’s quest to
achieve what it wants to? Was the introduction of
life in this universe solely to increase its entropy?
Or on the other hand, do non-living things like the
dipole moment in our model also have the ability
to think after all?
How are the subatomic particles working in the
Ising market different from the ones in the human
body? What makes us so different and are we even
any different?
Well, we are ultimately built up of all these
subatomic particles (including the dipole moments)
and our decision-making at the most fundamental
level is left to the brain, which is again a
combination of these atoms, and the ability to think
is just them in action, leading us to make decisions.
This model has made me question the most
fundamental questions on our existence and worth
along with also providing an adept comprehension
of market crashes and bubbles with a touch of
scientific intuition.
Acknowledgements
I would like to thank my supervisor Dr. William G
Proud for providing me with very impactful
insights into my research and posing non-trivial
questions which made me dig deeper into my
research. I would also like to thank Prof. Deeph
Chana for being a wonderful examiner during my
viva and giving extremely helpful feedback. Lastly,
I also owe my gratitude to Charalampos, Rohima,
Isabel, Fatima and Maeve for shedding light on
their research during our weekly meetings and
having fruitful discussions on the topic.
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