Genetic Learning and the Stylized Facts of Foreign Exchange Markets

1. 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@icu.ac.jp, http://www.bwl.uni-kiel.de/vwlinstitute/gwrp/german/team/lux.html

2. 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

3. returns = ln(pt) – ln(pt-1)

4. 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. > 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

5. Fat Tails (Leptokurtosis) of the Distribution of Returns

6. 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

7. 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) -> GA learning in K-W Model

8. 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 -> 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)

9. 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 • -> 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)

10. Agents‘ Optimization Problem max U(c(t), c(t+1)) subject to: Strategic choice variables:

11. 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))

12. Selection of equilibrum? Out-of-equilibrium dynamics? • Learning of agents via genetic algorithms: • each agent‘s 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

13. Genetic Algorithms: Binary Coding A chromosome (agent) is represented by a binary bit-string: From binary to real numbers: aj{0,1}

14. 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

15. 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

16. Example with realistic time series properties of returns: Binary coded GAs: 50 agents, pmut = 0.01, w1 = 10, w2 = 4, U = c(t)c(t+1) Question: sensitivity with respect to genetic algorithm parameters and number of agents

17. Is this result robust?-> variation of pmut, NWhat causes the dynamics?-> analysis of large economy limit

18. Log of exchange rate

19. Constant mutation probability pmut = 0.01, varying population size Varying mutation probability, constant population size N= 60 N 20 100 200 1000 2000 4000 10000 Median  2.64 3.17 3.54 3.15 2.74 1.92 1.82 pmut 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 Median  2.16 2.81 3.25 3.51 3.81 4.11 4.25 4.42 4.46 4.53 Table 1: Variation of Tail Index Estimate from Binary Coded GAs Estimation of  in Prob(ret. > x) x- via maximum likelihood. Median values from 100 (25) replications with 2,000 observations each

20. Varying mutation probability, constant population size N= 60 Constant mutation probability pmut = 0.01, varying population size absolute 0.36 0.42 0.43 0.38 0.35 0.29 0.28 0.22 0.22 0.18 absolute 0.33 0.40 0.35 0.16 0.07 0.07 0.04 pmut 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 N 20 100 200 1000 2000 4000 10000 raw -0.03 -0.05 -0.12 -0.17 -0.25 -0.30 -0.37 -0.43 -0.46 -0.51 raw -0.02 -0.06 -0.18 -0.56 -0.61 -0.48 -0.40 Table 2: Variation of Index of Fractional Differentiation d from Binary Coded GAs Estimation of d in E[rt 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).

21. Varying mutation probability, constant population size N= 60 Constant mutation probability pmut = 0.01, varying population size Rej. of  > 1 (=1) 0.56 (0.48) 0.72 (0.72) 0.96 (0.92) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) N 20 100 200 1000 2000 4000 10000 Median  0.98 0.99 0.98 0.91 0.86 0.81 0.56 Rej. of  > 1 (=1)  0.54 (0.47) 0.64 (0.59) 0.95 (0.91) 0.99 (0.99) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (1.00) pmut 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 Median  0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.93 0.92 0.91 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).

22. Log of exchange rate pmut = 0.01 pmut = 0.05

23. Influence of number of agents: real coding, N = 200

24. Influence of number of agents: real coding, N = 2,000

25. Influence of number of agents: real coding, N = 20,000

26. First and second moments

27. The Large Economy Limit GA learning leads to gradual adjustment of choice parameters towards momentary optimum: with U = c(t) c(t+1) -> cyclic dynamics between corner equilibria f = 0 and f = 1

28. ci fi Indifference curves for f(t+1) = 0.55, f(t) = 0.5 (left) and f(t+1) = 0.5, f(t) = 0.55 (right)

29. A snapshot of the evolution of the population

30. 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 ( < 1000) are needed to get realistic time series • with large number of agents: inherent randomness of the artificial economy gets lost -> measurable macroeconomic quantities (pi(t), e(t)) become deterministic quantities • local instability develops into persistent oscillations: cyclic learning dynamics