Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Herding, minority game, market clearing and
efficient markets in a simple spin model
framework
Ladislav Kristoufek & Miloslav Vosvrda
Institute of Information Theory and Automation
Czech Academy of Sciences
&
Institute of Economic Studies
Faculty of Social Sciences
Charles University
Czech Republic
Workshop of the Econophysics Network 2017
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Outline
1
Introduction and motivation
2
Ising model in finance
Ising model basics
Connection to financial markets
3
Simulations setting and results
4
Discussion
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Agent-based modeling (ABM) in economics and
finance
ABMs have attracted much attention in economics and
finance in recent years as they describe reality better than
the traditional simplified models.
Perfectly rational representative agent is substituted by a
boundedly rational agent.
Decisions are usually based on simple heuristics rather
than utility maximization.
The resulting systems are then driven endogenously.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Agent-based modeling (ABM) in economics and
finance
In finance, ABM foundations were laid by Brock & Hommes (1997,
1998) – strategy-switching agents and possible bifurcation dynamics.
Early papers of Lux & Marchesi (1999) and Kaizoji (2000) introduced a
possibility of generating returns-like series from simple models based
on interactions between multiple agents.
This avenue was further expanded by Bornholdt (2001) and his
adjustment of the Ising model for financial markets. This model:
combines the traditional 2D-Ising model and the minor game dynamics;
mimics the most important financial stylized facts – volatility clustering, fat
tails and slowly decaying autocorrelation function of absolute returns
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Motivation
Since then, various generalizations and expansions of the Bornholdt
approach have been introduced. These are nicely summarized in the
review by Sornette (2014). There are three interesting (and important)
outcomes:
The ability of models to recover the stylized facts is often very sensitive to
the parameter choice.
Majority of papers focus primarily on retrieving the stylized facts and touch
the interpretation of parametric values only on surface.
Majority of papers are not able of outperform the original Bornholdt model
in the sense of the stylized facts coverage.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Our contribution
We inspect the implications of the financial Ising model towards capital
market efficiency. Specifically:
We focus on the model parameters and how they influence returns
dynamics in the optics of efficient market hypothesis (EMH).
Instead of asking “What combination of parameters yields returns and
volatility mimicking the stylized facts?”, we ask “What combination of
parameters yields returns consistent with the efficient market
hypothesis (EMH)?”
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Efficient market hypothesis
Efficient market hypothesis (EMH) based on independent
works of Fama and Samuelson has been a cornerstone of
finance and financial economics for decades.
Simply put, EMH yields fair markets, i.e. prices reflect all
available information (weak, semi-strong and strong
versions of EMH).
From econometric perspective, EMH implies that financial
returns should:
not be serially correlated (both martingale and random walk
versions of EMH);
follow Gaussian distribution (random walk version of EMH)
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Outline
1
Introduction and motivation
2
Ising model in finance
Ising model basics
Connection to financial markets
3
Simulations setting and results
4
Discussion
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Basics of Ising model (1)
The Ising model is a ferromagnetic model in statistical
mechanics consisting of discrete integer variables, called
spins, organized in a lattice. The spins can only take
values −1 and +1.
The spins are organized in a squared lattice of k × k
dimension. Let si (t) denote the spin value on the i-th
position in the lattice at time t ∈ {0, . . . , T }, and
i ∈ {1, . . . , k 2 }.
The total magnetization of the system at time t is given as
2
k
1 X
M(t) = 2
sj (t).
k
(1)
j=1
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Basics of Ising model (2)
The intermediary between total magnetization and
neighbouring forces is the local field hi . It is defined as
2
hi (t) =
k
X
Jij si (t) − αsi (t) |M(t)| ,
(2)
j=1
where α > 0 is the global coupling constant, Jij = 1 for
close neighbours of si and Jij = 0 otherwise.
The local field hi summarizes the two possibly
counteracting influences on spin i – local and global.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Basics of Ising model (3)
In our simulations, neighbours of si (t) are four sites
adjacent to si (t) on both horizontal and vertical axis –
specifically, periodic boundary conditions forming a torus.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Basics of Ising model (4)
The spins in the lattice are updated according to the
following formulae (heat-bath dynamics):
1
1 + exp(−2βhi (t))
1−p
si (t + 1) = +1
with p =
(3)
si (t + 1) = −1
with
(4)
where p is the probability of a spin flip and β is a
parameter.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Outline
1
Introduction and motivation
2
Ising model in finance
Ising model basics
Connection to financial markets
3
Simulations setting and results
4
Discussion
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Ising model basics
Connection to financial markets
Connection to financial markets
Spins can be treated as traders on the market.
+ or − magnetization can be seen as an intention to buy or sell,
respectively.
Local magnetization effects can be seen as herding effect on the
markets.
Global magnetization effects can be seen as total market mood and is
reminiscent of a classic minority game.
The total magnetization can be viewed as the general mood or tendency
of the whole market and is computed as a simple average of all spins.
Financial returns are defined as
r∆t (t) = M(t) − M(t − ∆t)
(5)
and we consider ∆t = 1 in the rest of the presentation so that
r (t) ≡ r1 (t)
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Simulations setting
2-dimensional 25 × 25 lattice – 625 agents.
Parameter α is varied between 0 and 15 with a step of 1.
Parameter β is varied between 0 and 4 with a step of 0.25.
A randomized matrix is used at time t = 0, i.e. si (0) = 1
with p = 0.5 for all i.
For each setting, we perform 100 simulations with time
series length T = 1000.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Testing procedure
Test Gaussian distribution with the Jarque-Bera test.
Test serial correlation with the Ljung-Box test.
Critical level is set to 90%.
We are interested in a proportion of simulations where
normality and/or no serial correlation are rejected.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Results – no autocorrelation
"NO AUTOCORRELATION"- MODEL I
"NO AUTOCORRELATION" - MODEL I
4
3.75
3.5
100%
3.25
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Alpha
3
4
3.5
0.25
0
0%
2
0.5
10%
2.5
0.75
20%
1
1
30%
1.5
1.25
40%
0.5
Rejection rate
0.75
0
1.5
50%
12
1.75
60%
15
0
Beta
6
2
70%
9
2.25
80%
0
2.5
3
2.75
Rejection rate
90%
3
Beta
Alpha
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Results – Gaussian distribution
"GAUSSIAN" - MODEL I
"GAUSSIAN"- MODEL I
4
3.75
3.5
100%
3.25
0
10%
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Alpha
L. Kristoufek & M. Vosvrda
3
6
Alpha
3.5
0.25
0
1
4
0%
0.5
2
0.75
20%
2.5
1
30%
1
1.25
40%
1.5
Rejection rate
1.5
50%
0.5
1.75
60%
12
Beta
15
0
2
70%
9
2.25
80%
0
2.5
3
2.75
Rejection rate
90%
3
Beta
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Summary of the results – no autocorrelation
The plane is practically split into two, separated by β = 0.5.
Note that the value is very close to βC ≈ 0.441.
Strongly non-linear dependence between β and the
rejection rate. Local minima are neither β = 0 nor
β → +∞, but β = 0.25 and β = 1.25.
When local field plays no role, i.e. β = 0, EMH is still
rejected in practically all cases.
Monotone increasing relationship between α and the
rejection rate. We are thus closest to EMH for α = 0.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Summary of the results – Gaussian distribution
Rejection rate attains low levels only for inverse
temperatures below βC .
Above βC , rejection rates jump quickly upwards.
Results only mildly depend on α even though they form a
weak U-shape.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Summary of the results – put together
The model is able to generate serially uncorrelated and
normally distributed returns only for a rather narrow range
of parameters.
Interestingly, the serially uncorrelated returns are found
also for β < βC which is a new finding not discussed in the
literature which usually focuses only on β > βC . The model
dynamics for the inverse temperatures below the critical
value is thus not as uninteresting as usually claimed.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Interpretation and discussion – “mainstream”
Focusing on the martingale version of EMH, i.e. only on
the no autocorrelation assumption, we find two rejection
rates minima – {α, β} = {0, 0.25} and {α, β} = {0, 1.25}.
If we stick with the classical interpretation of α and β as the
intensities of the global and the local coupling, respectively,
we can argue that the efficient market is found for no global
coupling but some local coupling.
This is well in hand with an intuitive feeling that markets
would not work if they were completely random, which
would be the case for {α, β} = {0, 0} when the agents
make their decisions on the 50-50 basis.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Interpretation and discussion – “alternative”
Interpreting β as the intensity of the local coupling is rather far-fetched
(it is part of the heat-bath dynamics of the system but not of the local
field itself).
It makes much more sense to take α as a weight of how much more
important the global coupling is compared to the local one:
The higher the α parameter, the more influence the global coupling has.
If α = 0, the dynamics is solely driven by the local coupling (herding).
If α 1, the dynamics is driven solely by the global coupling (minority
game behavior).
Such interpretations are not much different from the ones made using the
standard interpretation of α and β. However, we are able to make such
claims using only one of the parameters.
Some level of local interactions is thus needed for the efficient market.
2
(reminder) hi (t) =
k
X
Jij si (t) − αsi (t) |M(t)|
j=1
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Interpretation and discussion – “alternative”
To look deeper into the interpretation of β, we use the idea presented in
McCauley (2009) who discusses market efficiency in the sense of
market clearing, i.e. clearing of supply and demand, and its connection
to entropy of the market. We refer to this as the technical efficiency of
the market.
If the market clears perfectly, it is technically efficient.
In practice, 100% efficiency is impossible. However, EMH assumes exactly
this.
Such level of efficiency suggests that there is no energy loss in the system
and as such, the entropy of the system does not increase.
If there is energy coming into the system (i.e. agents take actions), it is
only possible to have no change in entropy if the temperature of the
system approaches infinity. This yields zero inverse temperature β.
For positive inverse temperatures, the system entropy increases and it is
not technically efficient.
This gives us a new interpretation of the β parameter in the financial Ising
model.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Interpretation and discussion – “alternative”
The results clearly show that the markets are not efficient in the EMH
perspective for β = 0 which is parallel to the perfect market clearing.
At least some market frictions are necessary for the market to be
efficient.
Note that such claim does not go completely against the notion of the
market efficiency as laid down by Fama (1970) who states three
sufficient conditions for efficient markets – no transaction costs, all
available information freely available to all agents, and all agents agree
on implications of such information and future distributions of the traded
assets. However, these are sufficient and not necessary conditions. As
specifically noted by Fama himself, such assumptions do not reflect the
real financial markets.
Our results suggest that not only the frictions do not always go against
efficiency, but they mainly suggest that frictions are needed for the
market to be efficient in the EMH sense. To reach the efficient market,
there need to be frictions.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets
Introduction and motivation
Ising model in finance
Simulations setting and results
Discussion
Time for questions.
L. Kristoufek & M. Vosvrda
Herding, minority game, market clearing and efficient markets