Chueh-Yung Tsao 1

Agent-Based Artificial Stock Markets:

Towards Natural-Language Reasoning Artificial Adaptive Agents (4)

• Linn & Tay (2001a). ``Fuzzy Inductive Reasoning, Expectation Formation and the Behavior of Security Prices,’’ JEDC.

• Linn & Tay (2001b). ``Fuzzy Inductive Reasoning and Nonlinear Dependence in Security Returns: Results from Artificial Stock Market Environment,’’ working paper.

Chueh-Yung Tsao 2

Motivations

• Some might question whether it is reasonable to assume that traders are capable of handling a large number of rules.

• The previous study on artificial stock market have reported that some statistical properties of simulated returns do not match the real returns.

Chueh-Yung Tsao 3

Assumptions

• Neoclassical Financial Market Models:– Rational Expectation– Deductive Reasoning

• This Model:– Bounded Rationality– Inductive Reasoning Process– Fuzzy Notion

SFASM

Chueh-Yung Tsao 4

Inductive Reasoning Process

• Two-step Process– Possibility-elaboration

Creating a spectrum of plausible hypotheses based on our experience and the information available.

– Possibility-reductionThese hypotheses are tested to see how well they

connect the existing incomplete premises to explain the data observed. Reliable hypotheses will be retained ; unreliable ones will be dropped and ultimately replaced with new ones.

Chueh-Yung Tsao 5

Fuzzy Notion

• Literature Supports: – Smithson (1987), Smithson and Oden (1999)

• Some Reasons: – Justifying the assumption that agents are able to

process and compare hundreds of different rules simultaneously when making choices.

Chueh-Yung Tsao 6

The Model (Market Environment)

• Two Assets:

Payoff Units

Stock d ~ AR(1)* N

Risk-free Bond r ~ Fixed Infinite

*The current dividend, dt, is announced and becomes public information at the start of time period t.

ttt dddd ερ +−+= − )( 1

Chueh-Yung Tsao 7

The Model (Market Environment)

• N Agents:– Utility Function (CARA):

Ui,t(Wi,t) = -exp(-λWi,t )

(homogeneous, time-independent, time-additive, state-independent, and zero time-preference utility function)

– Expectation: heterogeneously– Decision: share holdings of stock– Object: maximizing subjective expected utility of next

period wealth

Chueh-Yung Tsao 8

1. At time t, the dividend, dt, realizes.

2. Forecast :– using the recently best performance rule base

3. Submit demand function:

][ 11, ++ + ttti dpE

2,

11,, ˆ

)1(][ˆ)(

ti

ttttitti

rpdpEpx

σλ+−+

= ++

Market Flow

Chueh-Yung Tsao 9

Market Flow (cont.)

4. The market declares a price pt that will clear the market:– tatonement process

5. Evaluate the forecasting error for each rule base:

6. Update rule bases every k periods: – Using GAs

2,,1

2,,1

2,, )]()[()1( ttjitttjitjit dpEdpee +−++−= −− θθ

Chueh-Yung Tsao 10

Expectation

• The forecast equation hypothesis used is:

where a and b are forecast parameters.

bdpadpE ttttt ++=+ ++ )()( 11

Chueh-Yung Tsao 11

Decision Flow

CrispConditions

FuzzyDecisions

CrispDecisions

FuzzyNotions

defuzzify

fuzzifyInside

ThinkingOutside

Environment

Chueh-Yung Tsao 12

Fuzzy Condition-Action Rule

• The format of a rule is:– ``If specific conditions are satisfied then the

values of the forecast equation parameters are defined in a relative sense’’.

– e.g. ``If {price/fundamental value} is low, then a is low and b is high’’.

Chueh-Yung Tsao 13

Fuzzy Condition-Action Rule

• Five market descriptors (five information bits) are used for the conditional part of a rule:– p*r/d, p/MA(5), p/MA(10), p/MA(100),

p/MA(500)

• Two forecast parameters (two forecast bits) are used for the conditional part of a rule:– a & b

Chueh-Yung Tsao 14

Fuzzy Condition-Action Rule

• We present fuzzy information about a variable with the codes: 1 2 3 4 0

low moderately-low moderately-high high absence

• We present fuzzy information about a parameter with the codes: 1 2 3 4

low moderately-low moderately-high high

Chueh-Yung Tsao 15

Membership Function for Descriptor

lowmoderately-low moderately-high

high

Chueh-Yung Tsao 16

Membership Function for forecast parameter ‘a’

low

moderately-low moderately-high

high

Chueh-Yung Tsao 17

Membership Function for forecast parameter ‘b’

low

moderately-low moderately-high

high

Chueh-Yung Tsao 18

Fuzzy Condition-Action Rule

• In general, we can write a rule as:– [x1, x2, x3, x4, x5| y1, y2], where x1, x2, x3, x4, x5 ∈

{0, 1, 2, 3, 4} and y1, y2 ∈ {1, 2, 3, 4}.

• We would interpret the rule

[x1, x2, x3, x4, x5| y1, y2] as:

– ``If p*r/d is x1 and p/MA(5) is x2 and p/MA(10) is x3 and p/MA(100) is x4 and p/MA(500) is x5, then a is y1 and b is y2’’

Chueh-Yung Tsao 19

Rule Base• Single fuzzy rule can not specify the

remaining contingencies. Therefore, three additional rules are required to form a complete set of beliefs.

• Fore this reason, each rule base contains four fuzzy rules.

• At any given moment, agents may entertain up to five different market hypothesis rule bases.

Chueh-Yung Tsao 20

Rule Base (an example)

Chueh-Yung Tsao 21

Defuzzify of Fuzzy Decisions

• We employ the centroid method , which is sometimes called the center of area

,method to translate the fuzzy decisions into specific values for a a and b.

Chueh-Yung Tsao 22

Example

Consider a simple fuzzy rule base with the following four rules.

1st rule: If 0.5p/MA(5) is low then a is moderately high and b is moderately high.

2nd rule: If 0.5p/MA(5) is moderately low then a is low and b is high.

3rd rule:If 0.5p/MA(5) is high then a is moderately low and b is moderately low.

4th rule:If 0.5p/MA(5) is moderately high then a is high and b is low.

Chueh-Yung Tsao 23

Example (cont.)

• Now suppose that the current state in the market is given by p = 100, d = 10, and MA(5) = 100.

• This gives us, 0.5p/MA(5) = 0.5.

Chueh-Yung Tsao 24

Response of 1st rule (example)

Chueh-Yung Tsao 25

Response of 2nd rule (example)

Chueh-Yung Tsao 26

Response of 3rd rule (example)

Chueh-Yung Tsao 27

Response of 4th rule (example)

Chueh-Yung Tsao 28

Summary

Rule Membership Decisions

• 1st Rule 0

• 2nd Rule 0.5

• 3rd Rule 0

• 4th Rule 0.5

a is mode rate ly high b is mode rate ly high. a is low b is high. a is mode rate ly low b is mode rate ly low . a is high b is low .

Chueh-Yung Tsao 29

Defuzzify of Forecast ‘Parameters a’ ‘and b’

Chueh-Yung Tsao 30

Genetic Algorithms

• GAs are applied to retain the reliable rule bases, drop the unreliable rule bases, and create new rule bases.

• The fitness measure of a rule base is calculated as follows:

where β is constant and s is the specificity of the rule base.

sef jitjit β−−= 2,,,,

Chueh-Yung Tsao 31

The Market Experiments Linn & Tay (2001a)

• Experiment 1 (slow learning)– k = 1000– Using best rule base with probability 1.

• Experiment 2 (fast learning)– k = 200– Using best rule base with probability 1.

• Experiment 3 (fast learning with doubt)– k = 200– Using best rule base with probability 99.9%.

Chueh-Yung Tsao 32

Why we introduce ‘a state of doubt’ to catch the actual figure of kurtosis?• Although during the first few hundred of time steps,

kurtosis is always rather large ( because of initialized randomly and trying to figure out how to coordinate), once agents have identified rule bases that seem to work well, excess kurtosis decrease rapidly.

• From that point on, it is extremely difficult to generate further excess kurtosis without exogenous perturbation, because it is difficult to break the coordination among agents.

• We suspect the large kurtosis observed in actual returns series may have originated from such exogenous events as rumors or earnings surprises.

Chueh-Yung Tsao 33

The Market ExperimentsLinn & Tay (2001b)

• Experiments: – Experiment 1 (slow learning)– Experiment 2 (fast learning)

• Benchmarks:

– Disney and IBM stocks

Chueh-Yung Tsao 34

Experiments Parameters

Chueh-Yung Tsao 35

Results (Linn & Tay (2001a))

• The results of this model are similar to those of LeBaron et al. (1999) in which their model is based upon a crisp but numerous rules.

• A modification of the model, i.e., fast learning with ‘doubt’, is shown to produce return kurtosis measures that are more in line with actual data.

Chueh-Yung Tsao 36

• It is found that the market moves in and out of various states of efficiency. Moreover, when learning occur slowly, the market can approach the efficiency of a REE

Chueh-Yung Tsao 37

Results (Linn & Tay (2001b))

• Normality: – rejects normality for each series (Jarque-Bera

test)

• Linearity: – exists linear dependent for each series (Ljung-

Box Q test)– does not exist any linear dependent for each

ARMA fitted residual series (Ljung-Box Q test)

Chueh-Yung Tsao 38

• Non-linearity: – exists nonlinear dependent for each ARMA

fitted residual series (using both correlation dimension and BDS test methods)

• ARCH Effect: – exists ARCH behavior for each ARMA fitted

residual series (Ljung-Box Q test and LM test)– does not exist any ARCH effect for each

ARMA-TARCH fitted residual series (Ljung-Box Q test and LM test)

– exists other nonlinear dependent for each ARMA-TARCH fitted residual series (BDS test)

Chueh-Yung Tsao 39

• Other Non-linearity– exists other nonlinear dependent for each

ARMA-TARCH fitted residual series (BDS test)

Chueh-Yung Tsao 40

Conclusions

• These two papers begin by presenting an alternative model of decision-making behavior, genetic-fuzzy classifier system, in capital markets where the environment that investors operate in is ill-defined.

• The results indicate that the model proposed in this paper can account for the presence of nonlinear effects observed in real markets.

Chueh-Yung Tsao 41

Conclusions (cont.)

• The framework offers an alternative perspective on capital markets that extends beyond the traditional paradigms.