Self-organization and Finite Size Effects in Agent Models for Financial Markets

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Self-organization and Finite Size Effects in
Agent Models for Financial Markets
CCSS ETH Zurich June 8 - 12, 2009
Luciano Pietronero Collaborators
Valentina Alfi, Matthieu Cristelli and Andrea
Zaccaria Institute of Complex Systems, CNR
and University of Rome La Sapienza,
Italy Centro Fermi, Via Panisperna, Roma (WEB
page http//pil.phys.uniroma1.it)
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• SUMMARY
• Workable ABM, clear math and properties
• New elements N variable, Stylized Facts due
• to Finite Size Effects, Self-organization
• Approximate scaling, no strict universality
• effective exponents depend on situation
• Liquidity crises Order Book Model for finite
  liquidity
• ABM in the Global Network

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• ABM perspective
• The market seems to evolve spontaneously
  towards
• states with intrinsic instability which then
  collapse
• or explode triggered by minor or irrelevant
  perturbations
• Importance of social interactions (herding)
  effects
• especially in situations of uncertainity with
  respect to
• the fundamentals of economics (fear, panic,
  euphoria)
• Breaking of the cause-effect relation and of
  the
• traditional economic principles
• Relation to Critical phenomena and SOC in
  physics

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Physics, Complexity,
Socio-Economics Physics try to discover the
laws of nature Economics are there laws to be
discovered? evolutive elements, adaptivity, the
whole society is involved Complexity new vision
and possible point of contact
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MODELS AND BASIC PROBLEMS
INTERDISCIPLINARY APPLICATIONS
Ising (1911) Scaling, Criticality (64 - 70) and
RG Group (gt72) Percolation (70-80) Glasses
Spin Glasses etc.(gt74) Deterministic Chaos
(78) Fractal Geometry (80-90) Polymers and Soft
Matter Dynamical Systems and Turbulence Fractal
Growth Physical Models DLA/DBM
(82-84) Selforganized Criticality Sandpile
(87) Granular Systems (90) Minority Game
(97) Rare Events Complex Networks (gt2000)
Condensed Matter problems Phase
Transitions Magnetic Systems Bio-inspired
Problems Astrophysics Geophysics Information
Theory Optimization Economics and Finance Social
Sciences (Random Walk, Bachelier 1900) Agent
Based Models (very many) Apply old Models
or develop New Models? Universality?
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Universality?
In Nature all trees are alike but not
identical. Similarity and common basic structure
but no strict universality. Exponents can
therefore depend on specific situations richness
to be explored.
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• Basic Stylized Facts (Universal?)
• Arbitrage -- Random Walk (BS)
• Fat tails, Volatility Clustering etc.
• AND ALSO
• Non stationarity
• Self-organization
• Global Network

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ABM model with moving average-based strategies
(V. Alfi, L.P., A. Zaccaria 2008)
(Linear dynamics to start more stable and easy
to treat)
N players
NF fundamentalists
NC chartists
At each time step, each agent can change its
strategy with probabilities
Price formation (price change related to excess
demand liquidity problem)
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Switching between F and C strategies
Origin of the Finite size effects
F
C
fluctuations
Kirman 1993 AlfaranoLux 2006
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Too fast fluctuations
Nc
Intermittency OK (Stylized Facts)
N50
Too low fluctuations
N500
NB For N diverging fluctuations are suppressed.
Therefore Stylized Facts correspond to finite
size effects
N5000
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Asymmetric case Basically Fundamentalists with
bubbles due to Chartists (not quite realistic in
these times)
If the transition probabilities are symmetric the
equilibrium distribution is bimodal or unimodal
depending on the parameters
With asymmetric transition probabilities the
scenario is richer
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TENDENCY TO FUNDAMENTALISM INSTITUTIONAL
INVESTORS IN QUIET TIMES
bimodal region
relative number of chartists
For large value of N chartists are suppressed
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N50
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N500
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N5000
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N dependence of price fluctuations Switching
effect between Fundamentalists and Chartists
N500
N 50
N 5000
Changes of opinion are too fast
Too stable and dominated by fundamentalists
Intermittent behavior OK
Puzzle Interesting fluctuations appear at finite
N and disappear for infinite N unlike Critical
Phenomena in Physics
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N1 M10 b510-4 K0.05 B1 g0.1 s1
(Pf 0)
NB even a single agent can show some
intermittency
Bursts of price fluctuations appear
spontaneously and are clearly due to Chartists
dynamics (Possibility of analytical studies)
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N100 M10 b110-3 K0.002 B1 g0.1 s1
N100
NB All the parametrs are now in full control BUT
fine tuning is always necessary
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Autocorrelation functions of returns and square
returns NB SF arise from Finite Size Effects
Probability density function of price-returns
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(red N50 black N500 green N5000)
NB largest peaks Correspond to the Intermediate
case. It takes some stability in the C state to
develop a bubble
ABM results for the Self-organized state
Real data from NYSE stock
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(red N50 black N500 green N5000)
NB Black (N500) is the only case leading to
stylized facts
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• Basic criterion for Self-Organization
• Agents decide whether trading (or not)
  depending
• on the price movements they observe
• Stable prices Less trading
• Large action (price movements) More trading
• (Euphoria, bubbles, panic, crashes)
• Caution some agents may prefer a stable market
  and be
• scared by fluctuations. This would require an
  analysis of different
• time scales and, in any case, these agents
  certainly do not
• produce the Stylized Facts

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Each agent calculate the price-volatility on the
previous T steps
On the basis of the calculated volatility each
agent has a probability to enter/leave the market
if the volatility is above/under a certain
threshold
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Self-Organization in action Different starting N
(50, 500, 3000) evolve and finally converge to
the Quasi-critical state (N500) which
corresponds to the Stylized Facts
N2
N
N1
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Linear dynamics N 500 Hetherogeneity with
respect to their time horizon Volatility
clustering is decreased because the behavior is
less coherent
Apparent power law behavior but no fundamental
critical phenomenon
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Limiting cases Only chartists and only
Fundamentalists Volatility Clustering disappears
for both limits
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Diffusion properties for the two limiting
cases and for the mixed one (red)
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Simple approximation superposition of only C and
only F limiting cases. Volatility Clustering is
not reproduced. Population dynamics is important
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Multiplicative dynamics autocorrelations
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Multiplicative dynamics Extreme sensitivity to
parameter region. Slightly different parameters
lead to very different Fat Tails
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Comparison between linear and multiplicative
dynamics Fat Tails are usually larger for the
Multiplicative case
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Multiplicative case Volatility fluctuations for
different values of N General behavior is similar
to the linear case
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Multiplicative case Self-Organized-Intermittency
(relatively slow convergence from large N)
N
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Power laws and universality?

• Herding naturally leads to population
  switching (i.e. F vs C)
• For a given N and a single time horizon this
  leads to a
• characteristic time scale.
• Distribution of trading horizons leads to
  many time scales

• Nonstationarity key element for the
  Self-organization
• traders may decide NOT to play or to
• play variable amounts of shares (volume)
• Switching situation Finite Size Effects

• Strong deviations from gaussian behavior but
  not necessarily
• critical with universal power laws.
• NB. Different opinions about data analysis
• HE Stanley R. Cont J. Kertesz D.Sornette C.
  Tsallis

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Fat Tail effective exponents as a function of
model propertiesSituations with more Chartists
lead to larger Tails
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ABM ideal
ABM real
Subprime
General Motors
Lehman
Recent Crisis Pf???
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External fluctuations affect all properties but
mostly the estimate of the Fundamental Price
Different Situation
if
Fundamentalists are discouraged
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ABM Environment
BASIC ANSATZ
Fundamentalists dominate in the long run (?)
But in a complete model this may require
evolution and adaptation for all possible
instabilities
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Back to Liquidity problem Liquidity seems more
important than volume or news for price
changes Microscopic model for the order book
finite liquidity (crisis). This should be
included in a realistic ABM
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Order Book ABM
In a typical Agent-Based Model (ABM) the price
evolution is a coarse-grained clearing/adjustment
mechanism that does not take into account the
liquidity of the market
real markets
Therefore we need to investigate the microscopic
mechanisms for price formation in order to find
ß(g)
Order book model
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Order Book in a nutshell
incoming limit order
ask
ask
limit order
bid
bid
incoming market order
market order
V
?
?p
?q
spread s(t)?
spread s(t)?
p
p
best bid b(t)?
best ask a(t)?
best bid b(t)?
best ask a(t)?
p(t)?
a buy limit order can be placed in the interval
-8,a(t)
limit order bid può cadere tra -8,a(t)
a sell limit order can be placed in the interval
b(t),8
limit order ask può cadere tra b(t),8
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Order Book regimes
We can identify two different regimes in the
order book dynamics

• Very liquid market
• Small price variations
• Behavior similar to a continuous system
• Illiquid market discreteness
• Large price variations
• The discreteness of the system is crucial

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Price Impact Function
?
?p
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Liquid market
bid
ask
bid
market order
market order
V
?p
spread s(t)?
p
p
p(t)?
b(t)?
a(t)?
best bid b(t)?
best ask a(t)?
a(t1)?
p(t1)?
p(t1)-p(t) fraction of a tick
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Illiquid market
bid
ask
bid
market order
market order
V
?p
spread s(t)?
p(t)?
p
p
a(t)?
b(t)?
best bid b(t)?
best ask a(t)?
p(t1)?
a(t1)?
p(t1)-p(t) several ticks
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Price Impact Function
The Price Impact Function (PIF) can be considered
as the response function of a stock, that is ...
If an agent submit a virtual market order of
volume ? at time t, what will be the average
price change at time tt?
The Price Impact Function of real market is a
concave function with respect to the order volume
Markets are not in a linear response regime
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Price Impact Surface
We want to study the role played by
liquidity/granularity in price response but the
normal PIF is calculated averaging on order book
configurations with different liquidity/granularit
y
We define the Price Impact Surface (PIS) which is
instead a function of volume and
liquidity/granularity
where g is a measure of liquidity/granularity
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Price Impact Surface - 1
Model result for t400 and k4
is concave as one measured in real order book
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Price Impact Surface - 2
If we rescale the PIS with the average impact
function that is proportional to
we observe a quasi-collapse in a unique curve
(in particular for small values of the order
volume)
Relation to ABM Small N leads to more Sparse
orders and more granularity. N dependent price
formation
Therefore the PIS can be approximately factorized
as
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Stylized Facts of Order Book
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Summary

• Price movement leads to increase of effective
  action (N). Multiplicative cascade (avalanche
  like SOC, sandpile, absence of cause-effect
  relation) fat tails and volatility clustering
• The reason that price returns have much less
  correlations depends on the fact that they are
  functions of many more parameters (arbitrage).
• The specific structure of the fluctuations is due
  to the competition between stability and
  instability which is controlled by the rates of
  the changes of opinion
• Stylized Facts seem to correspond to finite size
  effects in N and in time. Conceptual and
  practical implications.
• Self-organization in a quasi-critical state
  arises from the agents strategies with respect
  to price movements
• Origin of price variations no volume, no news
  (?)
• Liquidity crisis. Order book model to generalize
  
• the equation of price formation. Granularity of
  the O.B.

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Work in progress

• Reinterpretation of different exponents in data
  analysis
• richness of non-universality
• Sequence of correlated orders and anomalous
• response function (D.Farmer J.P. Bouchaud)
• In our ABM we set Pf const. This corresponds to
  a basically stable situation which is not very
  realistic in these days
• Possible integration of the ABM with the overall
• financial network (Gallegati, Stiglitz et al)
• Empirical analysis of Herding
• Identification of market sentiment (F or C)
• Different time scales for agents (apparent power
  law?)

For refs see EPJB Special issue on Complexity
(2009) Ed. by F. Schweitzer
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