11

Indirect Estimation of the Indirect Estimation of the Parameters of Agent Based Parameters of Agent Based Models of Financial MarketsModels of Financial Markets

Peter WinkerPeter Winker

Manfred GilliManfred Gilli

22

OutlineOutline

Background Information.Background Information. Introduction.Introduction. Method.Method. Result.Result. Conclusion of the Paper.Conclusion of the Paper. Further Improvement.Further Improvement.

33

Standard ModelsStandard Models

AssumptionsAssumptions Agents are fully rational.Agents are fully rational. Markets are efficient.Markets are efficient.

44

Rational BehaviourRational Behaviour

Agent is rational ifAgent is rational if He is aware of his alternatives.He is aware of his alternatives. Form expectations about any unknowns.Form expectations about any unknowns. Has clear preferences.Has clear preferences. Chooses his action deliberately after Chooses his action deliberately after

some process of optimization.some process of optimization. Taking into account their knowledge or Taking into account their knowledge or

expectations of other decision makers’ expectations of other decision makers’ behaviour.behaviour.

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Efficient Market HypothesisEfficient Market Hypothesis

All market participants receive and All market participants receive and act on all relevant information as act on all relevant information as soon as it is available.soon as it is available.

Perfect information within the Perfect information within the market.market.

Cannot “beat the market”.Cannot “beat the market”.

66

Agent Based ModelsAgent Based Models

Agents to be heterogenous.Agents to be heterogenous. Agents with limited rational Agents with limited rational

behaviour.behaviour. Market does not need to be efficient.Market does not need to be efficient. Interaction between agents.Interaction between agents.

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ParametersParameters

Not directly observable.Not directly observable. Compare with empirical data.Compare with empirical data. DM/US-$ exchange rate.DM/US-$ exchange rate.

88

Characteristic of DM/US-$Characteristic of DM/US-$

Daily Returns DM/US-$Daily Returns DM/US-$

99

Characteristic of DM/US-$Characteristic of DM/US-$

Excess kurtosis.Excess kurtosis.

Volatility varies over time.Volatility varies over time. AR(1) process with ARCH(1) effect.AR(1) process with ARCH(1) effect. where where

4XXE

ttt rr 110 2

110 ttV

1010

Model (Kirman 1990)Model (Kirman 1990)

Two prevalent views of the world.Two prevalent views of the world. Each agent holds one view.Each agent holds one view. NN agents. agents. State: number of agents, State: number of agents, kk, for first view., for first view. Two agents, A and B, meet at random.Two agents, A and B, meet at random. P(A’s view P(A’s view →→ B’s view) = B’s view) = .. P(A changed his view independently) = . P(A changed his view independently) = . 1

1111

Model (Kirman 1990)Model (Kirman 1990)

If , large shares of first type of If , large shares of first type of agents and second type of agents, agents and second type of agents, respectively, with high probability. respectively, with high probability.

111,

N

k

N

kNkkP

1

11,N

kN

N

kkkP

1

1

N

1212

Fundamentalist / ChartistsFundamentalist / Chartists

There are two types of agents:There are two types of agents: Fundamentalist:Fundamentalist:

Chartist:Chartist:

ttf SSvSE 1

11 tttc SSSE

1313

AdvantagesAdvantages

Complicated non-stationary Complicated non-stationary dynamics.dynamics.

Non-fundamentalist behaviour.Non-fundamentalist behaviour.

1414

SimulationSimulation

Objective function.Objective function. estimated ARCH(1)-effect.estimated ARCH(1)-effect. empirical kurtosis.empirical kurtosis. and mean values from 1000 and mean values from 1000

simulations.simulations. First and last 10% results deleted.First and last 10% results deleted.

empempdd kkf 11

emp1

empdk

dk 1

1515

Monte Carlo SimulationMonte Carlo Simulation

200 Monte Carlo simulation200 Monte Carlo simulation

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Monte Carlo SimulationMonte Carlo Simulation

10000 Monte Carlo simulation10000 Monte Carlo simulation

1717

Monte Carlo SimulationMonte Carlo Simulation

10000 Monte Carlo simulation10000 Monte Carlo simulation

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Threshold AcceptingThreshold Accepting

Initial:Initial: Choose threshold sequence Choose threshold sequence

, set, set

and generate an initial .and generate an initial . Step 1:Step 1: Choose some .Choose some . Step 2:Step 2: Calculate .Calculate . Step 3:Step 3: If , set .If , set . Step 4:Step 4: If , set and go to 1.If , set and go to 1.

Otherwise, output .Otherwise, output .

max,,0, IiTi cx0i

cn xNx cn xfxff

iTf nc xx

maxIi 1iicx

1919

SimulationSimulation

2020

SimulationSimulation

2121

ResultResult

Optimal values are andOptimal values are and

, market is better , market is better characterized by switching moods of characterized by switching moods of the investors than by assuming that the investors than by assuming that the mix of fundamentalists and the mix of fundamentalists and chartists remains rather stable over chartists remains rather stable over time.time.

0008571.03250.0

1

1

N

2222

ConclusionConclusion

Agent based models can replicate Agent based models can replicate empirical data of the financial empirical data of the financial markets.markets.

Parameters may be difficult to Parameters may be difficult to estimate.estimate.

Indirect method can be used.Indirect method can be used. Optimization heuristic may need to be Optimization heuristic may need to be

used.used.

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Further ImprovementFurther Improvement

First and last 10% simulation results First and last 10% simulation results removed. Too much?removed. Too much?

Number of parameters to be Number of parameters to be estimated.estimated.

Only two types of agents?Only two types of agents?

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ReferenceReference

Fama, E.F. 1970, “Efficient capital markets: a review of theory and Fama, E.F. 1970, “Efficient capital markets: a review of theory and empirical work”, empirical work”, Journal of FinanceJournal of Finance, V25, Issue 2, p383-417., V25, Issue 2, p383-417.

Gilli, M., Winker, P. 2003, “A global optimization heuristic for Gilli, M., Winker, P. 2003, “A global optimization heuristic for estimating agent based models”, estimating agent based models”, Computational Statistics & Data Computational Statistics & Data AnalysisAnalysis, 42, p299-312., 42, p299-312.

Kirman, A. 1990, “Epidemics of opinion and speculative bubbles in Kirman, A. 1990, “Epidemics of opinion and speculative bubbles in financial markets”, in Taylor M.P.(eds), financial markets”, in Taylor M.P.(eds), Money and financial Money and financial marketsmarkets, Basil Blackwell Ltd, Oxford, p354-368., Basil Blackwell Ltd, Oxford, p354-368.

Tsay, R.S. 2002, Tsay, R.S. 2002, Analysis of financial time seriesAnalysis of financial time series, John Wiley & , John Wiley & Sons, Inc.Sons, Inc.

Winker, P. 2001, “Application of the optimization heuristic Winker, P. 2001, “Application of the optimization heuristic threshold accepting in statistics”.threshold accepting in statistics”.