Agent-Based Artificial Stock Markets: Towards Natural-Language Reasoning Artificial Adaptive Agents (4)

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Agent-Based Artificial Stock MarketsTowards
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.

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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.

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Assumptions

• Neoclassical Financial Market Models
• Rational Expectation
• Deductive Reasoning
• This Model
• Bounded Rationality
• Inductive Reasoning Process
• Fuzzy Notion

SFASM
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Inductive Reasoning Process

• Two-step Process
• Possibility-elaboration
• Creating a spectrum of plausible hypotheses based
  on our experience and the information available.
• Possibility-reduction
• These 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.

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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.

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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.

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The Model (Market Environment)

• N Agents
• Utility Function (CARA)
• Uit(Wit) -exp(-Wit )
• (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

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Market Flow

• 1. At time t the dividend dt realizes.
• 2. Forecast
• using the recently best performance rule base
• 3. Submit demand function

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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

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Expectation

• The forecast equation hypothesis used is
• where a and b are forecast parameters.

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Decision Flow
Crisp Conditions
Fuzzy Notions
fuzzify
Inside Thinking
Outside Environment
Fuzzy Decisions
Crisp Decisions
defuzzify
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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.

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Fuzzy Condition-Action Rule

• Five market descriptors (five information bits)
  are used for the conditional part of a rule
• pr/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

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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

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Membership Function for Descriptor
low
high
moderately-low
moderately-high
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Membership Function for forecast parameter a
low
high
moderately-low
moderately-high
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Membership Function for forecast parameter b
low
high
moderately-low
moderately-high
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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 pr/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

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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.

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Rule Base (an example)
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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.

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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.

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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.

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Response of 1st rule (example)
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Response of 2nd rule (example)
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Response of 3rd rule (example)
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Response of 4th rule (example)
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Summary

• Rule Membership
  Decisions
• 1st Rule 0
• 2nd Rule 0.5
• 3rd Rule 0
• 4th Rule 0.5

a is moderately high b is moderately high.
a is low b is high.
a is moderately low b is moderately low.
a is high b is low.
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Defuzzify of Forecast Parameters a and b
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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 is constant and s is the specificity
  of the rule base.

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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.

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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.

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The Market ExperimentsLinn Tay (2001b)

• Experiments
• Experiment 1 (slow learning)
• Experiment 2 (fast learning)
• Benchmarks
• Disney and IBM stocks

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Experiments Parameters
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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.

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• 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

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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)

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• 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)

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• Other Non-linearity
• exists other nonlinear dependent for each
  ARMA-TARCH fitted residual series (BDS test)

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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.

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Conclusions (cont.)

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