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Towards Natural-Language Reasoning Agent-Based Artificial ... Towards Natural-Language Reasoning Agent-Based Artificial ... Presentation Transcript

  • 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.
  • 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.
  • Assumptions
    • Neoclassical Financial Market Models:
      • Rational Expectation
      • Deductive Reasoning
    • This Model:
      • Bounded Rationality
      • Inductive Reasoning Process
      • Fuzzy Notion
    SFASM
  • 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.
  • 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.
  • The Model (Market Environment)
    • Two Assets:
    • Payoff Units
    • Stock d ~ AR(1)* N
    • Risk-free Bond r ~ Fixed Infinite
    • *The current dividend, d t , is announced and becomes public information at the start of time period t .
  • The Model (Market Environment)
    • N Agents:
      • Utility Function (CARA):
      • U i,t ( W i,t ) = -exp(-  W i,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
    • 1. At time t, the dividend, d t , realizes.
    • 2. Forecast :
      • using the recently best performance rule base
    • 3. Submit demand function:
    Market Flow
  • Market Flow (cont.)
    • 4. The market declares a price p t 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
  • Expectation
    • The forecast equation hypothesis used is:
    • where a and b are forecast parameters.
  • Decision Flow Crisp Conditions Fuzzy Decisions Crisp Decisions Fuzzy Notions defuzzify fuzzify Inside Thinking Outside Environment
  • 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’’.
  • 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
  • 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
  • Membership Function for Descriptor low moderately-low moderately-high high
  • Membership Function for forecast parameter ‘ a ’ low moderately-low moderately-high high
  • Membership Function for forecast parameter ‘ b ’ low moderately-low moderately-high high
  • Fuzzy Condition-Action Rule
    • In general, we can write a rule as:
      • [x 1 , x 2 , x 3 , x 4 , x 5 | y 1 , y 2 ], where x 1 , x 2 , x 3 , x 4 , x 5  {0, 1, 2, 3, 4} and y 1 , y 2  {1, 2, 3, 4}.
    • We would interpret the rule
    • [x 1 , x 2 , x 3 , x 4 , x 5 | y 1 , y 2 ] as:
      • ``If p * r/d is x 1 and p/MA(5) is x 2 and p/MA(10) is x 3 and p/MA(100) is x 4 and p/MA(500) is x 5 , then a is y 1 and b is y 2 ’’
  • 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.
  • Rule Base (an example)
  • 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.
  • Example
    • Consider a simple fuzzy rule base with the following four rules.
    • 1 st rule:
    • If 0.5p/MA(5) is low then a is moderately high and b is moderately high .
    • 2 nd rule:
    • If 0.5p/MA(5) is moderately low then a is low and b is high .
    • 3 rd rule:
    • If 0.5p/MA(5) is high then a is moderately low and b is moderately low .
    • 4 th rule:
    • If 0.5p/MA(5) is moderately high then a is high and b is low .
  • 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.
  • Response of 1st rule (example)
  • Response of 2nd rule (example)
  • Response of 3rd rule (example)
  • Response of 4th rule (example)
  • Summary
    • Rule Membership Decisions
    • 1 st Rule 0
    • 2 nd Rule 0.5
    • 3 rd Rule 0
    • 4 th Rule 0.5
    a is moderately high b is moderately high . a is low b is h igh . a is moderately low b is moderately low . a is high b is low .
  • Defuzzify of Forecast Parameters ‘ a ’ and ‘ b ’
  • 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.
  • 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%.
  • 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.
  • The Market Experiments Linn & Tay (2001b)
    • Experiments:
      • Experiment 1 (slow learning)
      • Experiment 2 (fast learning)
    • Benchmarks :
      • Disney and IBM stocks
  • Experiments Parameters
  • 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.
    • 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
  • 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)
    • 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)
    • Other Non-linearity
      • exists other nonlinear dependent for each ARMA-TARCH fitted residual series (BDS test)
  • 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.
  • Conclusions (cont.)
    • The framework offers an alternative perspective on capital markets that extends beyond the traditional paradigms.