A Simple Model of a Financial Market Based on Herd Behavior

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Genetic Learning and the Stylized Facts of
Foreign Exchange Markets
Thomas Lux University of Kiel Sascha
Schornstein London School of Economics
Seminar talk at International Christian
University, 29 February 2003
Email lux_at_icu.ac.jp, http//www.bwl.uni-kiel.de
/vwlinstitute/gwrp/german/team/lux.html
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Outline

• The Stylized Facts of Exchange Rates and Possible
  Explanations
• The Kareken-Wallace Model
• Genetic Learning in the K-W Model
• Pseudo-Empirical Results on Artificial Economies

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returns ln(pt) ln(pt-1)
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Empirical Background The Stylized Facts

• Random walk property of asset prices and exchange
  rates
• non-rejection of unit root, i.e. st st-1 ?t
• Fat tails of returns
• probability for large returns Prob(ret. gt x) ?
  x-? with ? ? 2,5
• Clusters of volatility
• autocorrelation in all measures of volatility,
  e.g. ret2, abs(ret) etc.
• even long-term dependence E?rt rt-?t ? ?t
  -? ?t 2d-1, d ? 0.35
• Multi-fractality Nonlinear scaling function of
  moments

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Fat Tails (Leptokurtosis) of the Distribution of
Returns
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Volatility clustering in DM/US
Estimates
d 0.29
d 0.24
d 0.07
Temporal dependence E?xt xt-?t ? ?t -? ?t
2d-1
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Stylized Facts as Emergent Phenomena of
Multi-Agent Systems
Interacting Agent Hypothesis dynamics of asset
returns arise endogenously from the trading
process, market interactions magnify and
transform exogenous news into fat tailed returns
with clustered volatility

• References
• Lux/Marchesi (1999,2000), Alfarano/Lux (2001)
• Gaunersdorfer/Hommes (2000)
• Iori (2002), Bornholdt (2001)
• Arifovic/Gencay (2001) -gt GA learning in K-W
  Model

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Fat Tails and Volatility Clustering from
Indeterminacy of Equilibrium

• Stability of the equilibrium depends on the
  distribution of strategies among traders
• However, in the neighborhood of the equilibrium
  all strategies have the same pay-off -gt the
  fraction of agents with certain strategies
  follows a random walk
• Stochastic dynamics of strategy choice repeatedly
  drive the system beyond its stability threshold
  onset of severe, but short-lived fluctuations.

Examples Lux/Marchesi, Bouchaud, Kareken-Wallace
economy with GA learning (Arifovic/Gencay)
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The Open Economy OLG Model (Kareken/Wallace
Economy)

• Two generations, two countries, agents live for
  two periods
• Two assets money holdings in home and foreign
  currency
• No production, given endowments, one homogeneous
  good
• -gt no international trade, only capital
  movements, young agents save and decide about
  capital allocation, spend their savings when old
• Flexible exchange rates
• Identical agents (identical utility function)

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Agents Optimization Problem
max U(c(t), c(t1))
subject to Strategic choice
variables
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Prices
money supply , i 1, 2, ..., N agents
Equilibria Consequences
(1) equilibrium exchange rate is
indeterminate, e ?(0, ?) (2) equilibrium
portfolio composition is indeterminate, f ?0,
1 (3) equilibrium consumption from
maximization of U(c(t), w1 w2 c(t))
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Selection of equilibrum? Out-of-equilibrium
dynamics?

• Learning of agents via genetic algorithms
• each agents choice variables are encoded in a
  chromosome
• after lifespan of each generation (2 periods), a
  new generation is formed via genetic operations
• (i) reproduction according to fitness (utility)
• (ii) crossover recombination of genetic
  material
• (iii) mutation
• (iv) election new chromosomes replace existing
  ones only if at least as fit as parents

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Genetic Algorithms Binary Coding

• A chromosome (agent) is represented by a binary
  bit-string
• From binary to real numbers

aj ?0,1
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• Genetic Operations
• reproduction random selection with
  probabilities depending
• on utility Ui/?Ui (other possibilities
  rank-dependet selection, elitist selection etc.)
• crossover exchange of genetic material
• parents
• off-spring
• mutation flip bits with probability pmut
• election accept off-spring only if at least as
  fit as one of its parents

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Genetic Algorithms Real Coding
A chromosome (agent) is represented by a pair
ci(t), fi(t) Crossover uniform random draw
in the range between c1 and c2 (f1 and
f2) Mutation modification of ci (fi) using
Normal random draws with small variance
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Example with realistic time series properties of
returns Binary coded GAs 50 agents, pmut
0.01, w1 10, w2 4, U c(t)c(t1)
Question sensitivity with respect to genetic
algorithm parameters and number of agents
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Is this result robust?-gt variation of pmut,
NWhat causes the dynamics?-gt analysis of
large economy limit
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Log of exchange rate
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Table 1 Variation of Tail Index Estimate
from Binary Coded GAs
Estimation of ? in Prob(ret. gt x) ? x-? via
maximum likelihood. Median values from 100 (25)
replications with 2,000 observations each
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Table 2 Variation of Index of Fractional
Differentiation d from Binary Coded GAs
Estimation of d in Ert rt-?t ?t 2d-1 via
periodogram regression for both raw and absolute
returns (median values over 100 or 25 Monte Carlo
runs with 2,000 observations each).
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Table 3 Results of Unit-Root Tests for Binary
Coded GAs
Estimates of ? and rejection prob. for ? 1 in
st ? st-1 ?t , s log exchange rate (100 and
25 Monte Carlo runs, respectively).
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Log of exchange rate
pmut 0.01
pmut 0.05
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Influence of number of agents real coding, N
200
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Influence of number of agents real coding, N
2,000
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Influence of number of agents real coding, N
20,000
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First and second moments
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The Large Economy Limit
GA learning leads to gradual adjustment
of choice parameters towards momentary
optimum with U c(t)
c(t1) -gt cyclic dynamics between corner
equilibria f 0 and f 1
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ci
fi
Indifference curves for f(t1) 0.55, f(t) 0.5
(left) and f(t1) 0.5, f(t) 0.55 (right)
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A snapshot of the evolution of the population
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Results

• Kareken-Wallace economy with genetic learning
  can generate realistic time series with even
  numerically accurate features
• Underlying mechanism continuum of equilibria
  which are unstable under learning
• a small probability of mutation and a small
  number of agents ( lt 1000) are needed to get
  realistic time series
• with large number of agents inherent randomness
  of the artificial economy gets lost -gt
  measurable macroeconomic quantities (pi(t), e(t))
  become deterministic quantities
• local instability develops into persistent
  oscillations cyclic learning dynamics