Motivations

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-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. 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. Chueh-Yung Tsao

7. The Model (Market Environment) • N Agents: • Utility Function (CARA): Ui,t(Wi,t) = -exp(-Wi,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. Market Flow 1. At time t, the dividend, dt, realizes. 2. Forecast : • using the recently best performance rule base 3. Submit demand function: 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 Chueh-Yung Tsao

10. Expectation • The forecast equation hypothesis used is: where a and b are forecast parameters. Chueh-Yung Tsao

11. Decision Flow Crisp Conditions Fuzzy Notions fuzzify Inside Thinking Outside Environment Fuzzy Decisions Crisp Decisions defuzzify 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: 1 234 0 lowmoderately-lowmoderately-highhighabsence • We present fuzzy information about a parameter with the codes: 1 234 lowmoderately-lowmoderately-highhigh Chueh-Yung Tsao

15. Membership Function for Descriptor low high moderately-low moderately-high Chueh-Yung Tsao

16. Membership Function for forecast parameter ‘a’ low high moderately-low moderately-high Chueh-Yung Tsao

17. Membership Function for forecast parameter ‘b’ low high moderately-low moderately-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 RuleMembershipDecisions • 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. 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  is constant and s is the specificity of the rule base. 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. Chueh-Yung Tsao