1. 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
Good afternoon everybody, My name is Yu Chen. I am from University of Tokyo. The title of my talk is …

2. Contents of the Presentation
Concerning the Price Fluctuation of Financial Assets
What are “stylized facts” ?
Why do “stylized facts” occur ?
How to utilize “stylized facts” ?
My talk concersn the price fluctuation of financial assets, in particular those so-called stylized facts. Since most of you in this symposium are familiar with these issues. I will be as brief as possible.

3. What to Observe in the Market?
The Asset Price
The Price-Changes
Return
Log Return
Here is the defition of the log-return.
Nikkei 225 index from 1970 to Nikkei 225 consists of the 225 top-rated, blue-chip Japanese companies listed in the First Section of the Tokyo Stock Exchange.
What Are Stylized Facts?

4. Stylized Facts Observed
The Fat-Tailed Price Distribution
The Volatility Clustering
Here are two stylized facts, the fat-tailed distribution and the volatility clustering, observed in the foreign exchange market.
Calculated from the minute data of Yen-Dollar rate,
What Are Stylized Facts?

5. The Classic Theory for Financial Markets
The Efficient Market Hypothesis (EMH) All information concerning a financial asset is already incorporated into the current price.
The Implications
No way to make risk-free profit
Completely rational traders
Random-walk like price-changes
Independent Price-Changes
Identical PDFs
Gaussian PDF for Price-Changes
These stylized facts are at odds with the classical theory of financial markets. Typically, the efficient market hypothesis says that the distribution should Gaussian and there should be no correlation in price returns, no matter linear or nonlinear.
What Are Stylized Facts?

6. The Fat-Tailed Distribution
The Extraordinarily High Risk
The Universal Existence
The Power-Law Distribution
Levy
Distribution
The Six Sigma Probability
However the stylized facts are universal. These are the well-known results in econophysics.
From Gopikrishna et al.C
What Are Stylized Facts?

7. The Volatility Clustering
Exponential Decay of the Autocorrelation
The Problem of EMH
A Universal Existence
Just another well-known results regarding the volatility clustering of S&P 500.
What Are Stylized Facts?
From Gopikrishna et al.C

8. Available Approaches to Analyze the Market
Standard Finance Theory
The Stochastic Differential Equation
Limitations
Based on wrong assumptions
Awkward to describe traders’ behaviors
Econophysics
Concepts and Theories
Phase transition
Chaos, fractal
Self-organization
Approaches
Replica method
Normalization group method
Agent-based simulations
Evolutionary computations
If we review the recent work in econophysics, we find concepts and theories of statiscal mechanics and complex systems are being applied in the analysis of market. And the available approches are the following. We take the simulation approach.
Why Do Stylized Facts Occur?

9. Simple Agent Model for Complex Price Dynamics
Simplified Description of Traders’ Activities
No way to model even one trader perfectly
Easier to analyze the market mechanism
Collective Fluctuation Recovered to that of Price-Changes
The large scale collective dynamics is often insensitive to small scale behaviors
Succeeded Examples in Physics
Spin model for phase transition
Percolation model for critical phenomena
Lattice gas model for fluid flow
In the simulation of market, we like to use those toy models, since we have some experience in simulation of fluid flow using the so-called lattice gas method. The method is very similar to the toy models of markets. Very simple in micro-dynamics neverthelsss can recover to certain extent the coarse-grained dynamics.
Why Do Stylized Facts Occur

10. The Minority Game (MG) Parameters to Specify a Minority Game WIN
A Multi-Agent, Multi-Strategy, Repeated Game
Those Belonging to the Minority Win the Game
Parameters to Specify a Minority Game
Player number
Number of strategies for each player
Memory of the player +1
WIN -1
These are the basic minority games, I do not waste time here. 00 1 01 -1 10 11 ……
Why Do Stylized Facts Occur?

11. Adaptive Learning in MG
Some Definitions
The th agent’s bid
The total bid (excess demand)
Score all the strategies
The Adaptive Learning of Agents
Also skip.
Only those being minority get positive pay-offs
What Do Stylized Facts Occur?

12. Characteristics of MG Basic Properties Volatility Predictability
Congestion in strategy space
Emergence of Cooperation
Symmetric to Asymmetric Phase Transition
The macroscopic quantities to observe are these two, volatility and predictability. The control parameter ¥alpha is not so clearly defined. Since P can be a measure of the size of strategy pool, I put it as the congestion in strategy space.
Why Do Stylized Facts Occur?

13. Relation between MG and Market Model 1
Time Evolution of an Idealized Market Model
Traders’ actions: strategic investment; position clearing; evaluation of strategies
Asset price / Traders’ capital
Scores of investment strategies
According Marsili’s paper, the relationship between the original MG and a simple market model was clarified. In the market model, traders have three kinds of actions: strategic investment; position-clearing and strategy evaluation. The asset price and traders’ capital are updated every time, but scores of the investment strategies are updated every two time steps.

14. Relation between MG and Market Model 2
Details of the Model
Traders’ order
Clear positions
Price formation
Demand and supply
Balance (D = S*p)
Score the strategies
Capital updates
Score updates
Details of these formulae are given in Marsili’s paper. So I shall not explain here, simply pointing out that a part of traders’ capital change is used in the update of strategies.

15. Relation between MG and Market Model 3
Simplification of the Model
Synchronization of traders
Neglect position-clearing and capital updates
Understanding the MG Market Model
Predict the future excess demand
Fundamentalists’ viewpoint
Chartists’ viewpoint
Start from this simple model, if we further assume the synchronization of traders and neglect the position-clearing, the model can be reduced to the MG. There is also another way to reduce the model to MG without these assumptions. If we neglect the position-clearing and force the traders to evaluate their strategy right after the strategic investment, then traders have to predict the future excess demand. There are two ways to do this, according to Marsili’s paper, one is fundamentalists’ viewpoint and the other is the chartists’ viewpoint. From this, we know that the original MG market consists of pure fundamentalists.

16. Extension of the MG Market Model
Two Kinds of Traders: Producers and Speculators
Speculators have the right not to trade: GCMG
Definitions
Producer
Speculator
The 0-Strategy
Public Information
Threshold for Trade
Log-Return
Trade Volume
Scoring of the Strategies +1 -1
Ofcourse, in the real market, neither traders behave synchronously, nor would all of them be fundamentalists. We adopt a grand canonical MG market model suggested by Challet and Marsili. Characteristics of this model are the inclusion of two kinds of traders and the threshold for trades. Producers have only one strategy and do not have the ability to do adaptive learning. Speculators are like the agents in the original MG. Scores of speculators’ strategies are compared with a constant income represented by ¥epsilon. Only if scores became higher than this, speculators would join the market.
Why Do Stylized Facts Occur?

17. Recovery of Stylized Facts
Fat Tailed Distribution
Volatility Clustering
With certain set of parameters, the stylized facts can be recovered.
Why Do Stylized Facts Occur?

18. Threshold’s Effect on Fat-Tailed Distribution
Without the 0-Strategy
With the 0-Strategy
The mechanism behind the fat-tailed distribution was already explained by Challet and Marsili. There are two factors which influence the distribution form. First is the 0-strategy or the threshold, second is the congestion in strategy space. From the left figure, we see if the 0-strategy is switched off, no matter how we change the number of speculators it sticks to be an exponential distribution. However, in the right figure, if the 0-strateg is on, when there are enough number of speculators the fat-tailed distribution could emerge.
Why Do Stylized Facts Occur?

19. The Effect of Speculation on Fat-Tailed Distribution
Speculation causes larger fluctuations
A phase transition with the increase of speculators
Effects of speculators are clearer in these figures. On the left, the fluctuation of price return (in red) is compared with the active number of speculators (in green). On the right, we can clearly see a phase change in the magnitude of price fluctuations when the number of speculators exceeds a critical value. At the critical vale, the -3 power law emerges.
Where the -3 power
law emerges
Why Do Stylized Facts Occur?

20. Market Mechanism for Fat-Tailed Distribution
The Important Role of Threshold
In Model: the right not to play
In 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 Space
In 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
We can also explain the correspondences between the model and the real market. I skip this.
Why Do Stylized Facts Occur?

21. The Effect of Np/Ns on VC
Autocorrelation of the Volatility
The Correlation Time of Volatility
We investigate the relation between numbers of traders and the correlation time of absolute returns. In the simulation, volatility clustering appears sometimes and disappear some times. As we increase the number of producer and fix the number of speculator, the correlation time in volatility increases. As we increase the number of speculator and fix the number of producer the correlation time decreases. How to explain this? We suggested the following scenarios.
Why Do Stylized Facts Occur?

22. Three Scenarios in the Modeled Market
Production
The Profitable
Chances e2 e5 e3 e4
Consumption
In the market producers produce profitable chances or the predictability of price return, and speculators consume these chances to earn money whenever possible. Depending on the relative number of speculator and producer, there could be three states, namely the chance rich state, the chance depleted state and the chance critical state. Volatility clustering show itself at the chance critical state.
S1: Chance Rich
VC Disappeared
S2: Chance Critical
VC Emerged
S3: Chance Depleted
VC Disappeared
Why Do Stylized Facts Occur?

23. Evidence 1 and 2 The increase of producer increases the predictability
The increase of speculator decreases the predictability
These are evidences for the production of predictability and consumption of it. e1 e2
Why Do Stylized Facts Occur?

24. Evidence 3 and 4 e3 e4 When the number of speculator is
Looking at the change of predictability of price return and the correlation of volatility simultaneously, we can find the following… (in captions of figures)
When the number of speculator is
(relatively) too small, the predictability
becomes high while the correlation low
When the number of speculator is
(relatively) too large, the predictability
is zero while the correlation decreases.
Why Do Stylized Facts Occur?

25. Evidence 5
When the predictability is just depleted , the maximum of correlation time of the absolute returns appears. e5
Correlation time reaches its maximum right after the predictability is depleted
Why Do Stylized Facts Occur?

26. Market Mechanism for Volatility Clustering
The 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 Space
In Model: The maximum of time correlation emerges when the predictability is just depleted
In Real World: The market always seems like efficient
Also, we can relate things in the model to the real market. If this analysis is right, it means that the volatility clustering becomes one of the evidences for the criticality of financial market.
Why Do Stylized Facts Occur?

27. Application of the Minority Game to Predicting Price
The Goal: To predict price-changes in financial markets
Is It Predictable?
Volatility clustering
The predictability
Why Use Minority Game
The simple, built-in adaptive learning
GA-like algorithm easily applicable
We tried to apply the MG to predict the price change. First, to answer the question whether the price is predictable we compares (in the right figure) the predictability of three kinds of time series against the time horizons.
The Predictability of Time Series:
GCMG market model; FX Market (only sign); and a spin model on network for stock market
How to Utilize Stylized Facts?

28. optimization of strategy distribution
The Predicting Method
Financial Market
The Modified Adaptive Learning
Data Generation
Another MG simulation
A random walk
A GCMG Simulation
A Network Stock Market Model Simulation
FX Data ( )
Black Box
complex mechanism
Generate
Real market or artificial market generate time series data as a black box. A part of these data are used to train the predictor minority game. And the trained minority game are used to predict the remained part of data. First four datasets are generated by artificial markets. They are another MG, random walk, GCMG and a network model. The last one is the data generated by real market.
Learning
Predict
Predictor MG
Minority Game
optimization of strategy distribution
How to Utilize Stylized Facts?

29. A Simple Preliminary Test
To Predict Another MG
Comparison with the Prediction of a Random Walk
It seems reasonable that MG’s prediction of MG gives a good result, particularly compared with the meaningless prediction of a random walk. Correlations between the predicted data and those being predicted are shown in the right figure.
How to Utilize Stylized Facts?

30. Two Further Tests To predict the time series generated by a GCMG
To predict the time series generated by a network stock market model
The same comparisons are done for MG predictions of GCMG and network model for stock market.
How to Utilize Stylized Facts?

31. The Reason for the failure
The bad performance in predicting the network market model is due to its poorness in recovering the stylized facts. In particular the volatility clustering is almost indetectable.
The Price-Change Distribution
Autocorrelation of Volatility
How to Utilize Stylized Facts?

32. Prediction of the FX Market
Finally, compared with the prediction of a random walk the prediction of MG to FX data are somewhat meaningful.
How to Utilize Stylized Facts?