market mechanism analysis using minority game market models y. chen, t. kohashi, m. komiya, y....

Market Mechanism Analysis Using Minority Game Market Models

Y. Chen, T. Kohashi, M. Komiya, Y. Hashimoto and H. Ohashi

Department of Quantum Engineering and Systems Science

Graduate School of Engineering, The University of Tokyo

Contents of the PresentationConcerning the Price Fluctuation of

Financial AssetsWhat are “stylized facts” ?Why do “stylized facts” occur ?How to utilize “stylized facts” ?

What to Observe in the Market?The Asset PriceThe Price-Changes

Return

Log Return

What Are Stylized Facts?

Nikkei 225 index from 1970 to 2002. Nikkei 225 consists of the 225 top-rated, blue-chip Japanese companies listed in the First Section of the Tokyo Stock Exchange.

x[t]

R

t ,t−Δt ≡(x[t] −x[t−Δt])

x[t−Δt]

rt ,t−Δt ≡lnx[t] −lnx[t−Δt]

*if (x[t]−x[t−Δt]) = x[t−Δt] then rt,t−Δt ≈Rt,t−Δt

Stylized Facts ObservedThe Fat-Tailed Price

DistributionThe Volatility

Clustering


p(rt), r

t≡rt,t−1 f [r

t]g[r

t+τ ] , f [x] =g[x] = x

Calculated from the minute data of Yen-Dollar rate, 2006.8-2007.1.

The Classic Theory for Financial MarketsThe Efficient Market

Hypothesis (EMH)All information concerning a financial asset is already incorporated into the current price.

The ImplicationsNo way to make risk-

free profitCompletely rational

tradersRandom-walk like

price-changes

Independent Price-Changes

Identical PDFs

Gaussian PDF for Price-Changes


p(r) =1

2ps2

Ê

ËÁÁÁÁ

ˆ

¯˜̃˜̃

1/ 2

exp -r - r( )

2

2s2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜

f [r

tm]g[r

tn] =0

p(r

tm) =p(rtn ) =p(r)

The Fat-Tailed Distribution

The Extraordinarily High Risk

The Universal Existence

The Power-Law Distribution


p>[r ]: r - a,a = - 3

The Six Sigma Probability

Levy Distribution

From Gopikrishna et al.C

The Volatility Clustering

Exponential Decay of the Autocorrelation

The Problem of EMHA Universal Existence


rt◊r

t+tµ exp(- t / t

0)

rt

◊rt+t

µ t - 0.3

From Gopikrishna et al.C

Available Approaches to Analyze the MarketStandard Finance

TheoryThe Stochastic

Differential Equation

Limitations Based on wrong

assumptions Awkward to describe

traders’ behaviors

EconophysicsConcepts and Theories

Phase transition Chaos, fractal Self-organization

Approaches Replica method Normalization group

method Agent-based simulations Evolutionary

computations

Why Do Stylized Facts Occur?

dx

x=σdX + μdt

Simple Agent Model for Complex Price DynamicsSimplified Description of Traders’ Activities

No way to model even one trader perfectlyEasier to analyze the market mechanism

Collective Fluctuation Recovered to that of Price-ChangesThe large scale collective dynamics is often

insensitive to small scale behaviorsSucceeded Examples in Physics

Spin model for phase transitionPercolation model for critical phenomenaLattice gas model for fluid flow

Why Do Stylized Facts Occur

The Minority Game (MG)

A Multi-Agent, Multi-Strategy, Repeated Game

Those Belonging to the Minority Win the Game

Parameters to Specify a Minority GamePlayer numberNumber of strategies for

each playerMemory of the player


+1

WIN

-1+1

+1

+1-1+

1

WIN

N

S m

…… i S

00 1

01 -1

10 -1

11 1

m =2 ai,sμ(t)

Adaptive Learning in MGSome Definitions

The th agent’s bidThe total bid (excess demand)

Score all the strategies

The Adaptive Learning of Agents

What Do Stylized Facts Occur?

b

i(t) =ai,si (t)

μ(t)

A(t) = bi (t)

i=1

N

∑

U i,s(t +1) =U i,s(t)−ai,s

μ(t)A(t)

s

i(t +1) =argmax

sU i,s(t)

s =1,2,...,S

μ ∈ 0,1,...,P−1{ }

P =2m

Only those being minority get positive pay-offs

i

Characteristics of MGBasic Properties

Volatility

Predictability

Congestion in strategy space

Emergence of CooperationSymmetric to Asymmetric

Phase Transition


σ 2 = A2 =

1P

A2 |μμ=0

P−1

∑

H = A

2=1P

A|μ2

μ=0

P−1

∑

α ≡

PN

Relation between MG and Market Model 1Time Evolution of an Idealized Market Model

Traders’ actions: strategic investment; position clearing; evaluation of strategies

Asset price / Traders’ capitalScores of investment strategies

p[t] C[t]

u[t]

t t +1 t + 2 t + 3 t + 4p[t +1] p[t + 2] p[t + 3] p[t + 4] p[t + 5]

C[t +1] C[t + 2] C[t + 3] C[t + 4] C[t + 5]

u[t +1] u[t + 3]u[t −1]

Relation between MG and Market Model 2Details of the Model

Traders’ orderClear positionsPrice formation

Demand and supply Balance (D = S*p)

Score the strategies Capital updates Score updates

a

i[t]=ai,s*

μ[t] =±1

a

i[t +1] =−ai[t]

(N + A[t]) / 2 (cash); (N - A[t]) / 2p[t] (stock) A[t]= ai[t]i=1

N∑

p[t +1] =p[t](N + A[t]) / (N −A[t])

C

i[t + 2] =Ci[t] +{ p[t+ 2] −p[t]}S[t] +Q[t+1]R[t]

ai[t]A[t+1]N

u

i[t +1] =ui[t−1] + ai[t]A[t+1]

Relation between MG and Market Model 3Simplification of the Model

Synchronization of tradersNeglect position-clearing and capital updates

Understanding the MG Market ModelPredict the future excess demand

Fundamentalists’ viewpointChartists’ viewpoint

A[t +1] =−A[t]; p[t+ 2] =p[t]

u

i[t +1] =ui[t] −ai[t]A[t]

Ei[A[t +1] |t] =−φiA[t]; φi ; −

A[t]A[t+1]A2[t]

φi > 0

φi < 0

Extension of the MG Market Model

Two Kinds of Traders: Producers and Speculators

Speculators have the right not to trade: GCMG

DefinitionsProducer SpeculatorThe 0-StrategyPublic InformationThreshold for TradeLog-ReturnTrade Volume

Scoring of the Strategies


+1

-10 +

1

+1

-10 μ(t)ε

U i,s(t +1) =U i,s(t)−ai,s

μ(t)A(t)−εδs,0

N p, S =1

N s, S >1

rt=A(t) / λ

N p+ Ns

act

ai,0μ(t) =0

Recovery of Stylized Facts

Fat Tailed Distribution

Volatility Clustering


N p=200,Ns =400,S=2,m=4,ε =0.01

0.3τ −3( / )r σ −

Threshold’s Effect on Fat-Tailed DistributionWithout the 0-

StrategyWith the 0-Strategy


N p=500,Ns =100−2000,S=2,m=4,ε =−∞,0.01

Speculation causes larger fluctuations

The Effect of Speculation on Fat-Tailed Distribution

A phase transition with the increase of speculators


Where the -3 power law emerges

N p=200,Ns =400,S=2,m=4,ε =0.01

N p=900,S=2,m=4,ε =0.01

Market Mechanism for Fat-Tailed DistributionThe Important Role of Threshold

In Model: the right not to playIn Real World: the Watch-And-Wait behavior of

traders for various reasons Other examples: Sand-Pile in SOC, Latent Heat ….

The Congestion in the Strategy SpaceIn Model: the self-incurred phase transition

caused by the similarity of agents’ strategies and the optimization of the strategies through the adaptive learning

In Real World: Traders often have similar strategies or they learn the similar strategies


The Effect of Np/Ns on VC

Autocorrelation of the Volatility

The Correlation Time of Volatility


N

p=100,200,Ns =200−400,

m=5,S=2,ε =−∞ m =5,S=2,ε =−∞

Ns=200

Np =0→ 200

Np=200

Ns =200→ 400


The ProfitableChances

Production

Consumption

N p

N s

S1: Chance RichVC Disappeared

S2: Chance CriticalVC Emerged

S3: Chance DepletedVC Disappeared

Three Scenarios in the Modeled Market

Evidence 1 and 2The increase of producer

increases the predictability

The increase of speculator decreases the predictability


m =6,S=2,ε =−∞

2000 400

s

p

NN

== →

200200 500

p

s

NN

== →


When the number of speculator is(relatively) too small, the predictabilitybecomes high while the correlation low

When the number of speculator is(relatively) too large, the predictabilityis zero while the correlation decreases.

Evidence 3 and 4


Correlation time reaches its maximum right after the predictability is depleted

Np=200

Ns =200→ 500m=6,S=2ε =−∞

Evidence 5

Market Mechanism for Volatility ClusteringThe Interaction between Different Types of Agents

In Model: Producers generates the predictability (profitable chance); Speculators competitively consume it

In Real World Producers: Irrational Traders; Speculators: Rational Traders The intelligent takes advantage of the others in financial market

The Criticality of Phase Transition in Information SpaceIn Model: The maximum of time correlation emerges when

the predictability is just depletedIn Real World: The market always seems like efficient


Application of the Minority Game to Predicting PriceThe Goal: To predict

price-changes in financial markets

Is It Predictable?Volatility clusteringThe predictability

Why Use Minority GameThe simple, built-in

adaptive learningGA-like algorithm easily

applicable

How to Utilize Stylized Facts?

The Predictability of Time Series: GCMG market model; FX Market (only sign); and a spin model on network for stock market

The Predicting Method

Minority Gameoptimization of

strategy distribution

Predictor MG


Predict

Generate

Learning

Black Boxcomplex mechanism

Financial Market

The Modified Adaptive Learning

Data Generation Another MG simulationA random walkA GCMG SimulationA Network Stock Market

Model SimulationFX Data (2006.8-2007.1)

A[t]→ A* [t]

u

i[t +1] =ui[t] −ai[t]A

* [t]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 2 4 6 8 10 12 14 16 18 20

Financial Market

Artificial Market

A Simple Preliminary TestTo Predict Another

MGComparison with the

Prediction of aRandom Walk


N =N* =501,m=m* =4,TL =128

N * =501,m* =4N =501,m=5,TL =128

Two Further TestsTo predict the time

series generated by a GCMG

To predict the time series generated by a network stock market model


The Reason for the failure


The Price-Change Distribution Autocorrelation of Volatility


Prediction of the FX Market

market mechanism analysis using minority game market models y. chen, t. kohashi, m. komiya, y....

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