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

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

  1. 1. Agent-Based Artificial Stock Markets: Towards Natural-Language Reasoning Artificial Adaptive Agents (4) <ul><li>Linn & Tay (2001a). ``Fuzzy Inductive Reasoning, Expectation Formation and the Behavior of Security Prices,’’ JEDC. </li></ul><ul><li>Linn & Tay (2001b). ``Fuzzy Inductive Reasoning and Nonlinear Dependence in Security Returns: Results from Artificial Stock Market Environment,’’ working paper. </li></ul>
  2. 2. Motivations <ul><li>Some might question whether it is reasonable to assume that traders are capable of handling a large number of rules. </li></ul><ul><li>The previous study on artificial stock market have reported that some statistical properties of simulated returns do not match the real returns. </li></ul>
  3. 3. Assumptions <ul><li>Neoclassical Financial Market Models: </li></ul><ul><ul><li>Rational Expectation </li></ul></ul><ul><ul><li>Deductive Reasoning </li></ul></ul><ul><li>This Model: </li></ul><ul><ul><li>Bounded Rationality </li></ul></ul><ul><ul><li>Inductive Reasoning Process </li></ul></ul><ul><ul><li>Fuzzy Notion </li></ul></ul>SFASM
  4. 4. Inductive Reasoning Process <ul><li>Two-step Process </li></ul><ul><ul><li>Possibility-elaboration </li></ul></ul><ul><ul><ul><li>Creating a spectrum of plausible hypotheses based on our experience and the information available. </li></ul></ul></ul><ul><ul><li>Possibility-reduction </li></ul></ul><ul><ul><ul><li>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. </li></ul></ul></ul>
  5. 5. Fuzzy Notion <ul><li>Literature Supports: </li></ul><ul><ul><li>Smithson (1987), Smithson and Oden (1999) </li></ul></ul><ul><li>Some Reasons: </li></ul><ul><ul><li>Justifying the assumption that agents are able to process and compare hundreds of different rules simultaneously when making choices. </li></ul></ul>
  6. 6. The Model (Market Environment) <ul><li>Two Assets: </li></ul><ul><li>Payoff Units </li></ul><ul><li>Stock d ~ AR(1)* N </li></ul><ul><li>Risk-free Bond r ~ Fixed Infinite </li></ul><ul><li>*The current dividend, d t , is announced and becomes public information at the start of time period t . </li></ul>
  7. 7. The Model (Market Environment) <ul><li>N Agents: </li></ul><ul><ul><li>Utility Function (CARA): </li></ul></ul><ul><ul><li>U i,t ( W i,t ) = -exp(-  W i,t ) </li></ul></ul><ul><ul><li>(homogeneous, time-independent, time-additive, state-independent, and zero time-preference utility function) </li></ul></ul><ul><ul><li>Expectation: heterogeneously </li></ul></ul><ul><ul><li>Decision: share holdings of stock </li></ul></ul><ul><ul><li>Object: maximizing subjective expected utility of next period wealth </li></ul></ul>
  8. 8. <ul><li>1. At time t, the dividend, d t , realizes. </li></ul><ul><li>2. Forecast : </li></ul><ul><ul><li>using the recently best performance rule base </li></ul></ul><ul><li>3. Submit demand function: </li></ul>Market Flow
  9. 9. Market Flow (cont.) <ul><li>4. The market declares a price p t that will clear the market: </li></ul><ul><ul><li>tatonement process </li></ul></ul><ul><li>5. Evaluate the forecasting error for each rule base: </li></ul><ul><li>6. Update rule bases every k periods: </li></ul><ul><ul><li>Using GAs </li></ul></ul>
  10. 10. Expectation <ul><li>The forecast equation hypothesis used is: </li></ul><ul><li>where a and b are forecast parameters. </li></ul>
  11. 11. Decision Flow Crisp Conditions Fuzzy Decisions Crisp Decisions Fuzzy Notions defuzzify fuzzify Inside Thinking Outside Environment
  12. 12. Fuzzy Condition-Action Rule <ul><li>The format of a rule is: </li></ul><ul><ul><li>``If specific conditions are satisfied then the values of the forecast equation parameters are defined in a relative sense’’. </li></ul></ul><ul><ul><li>e.g. ``If {price/fundamental value} is low, then a is low and b is high’’. </li></ul></ul>
  13. 13. Fuzzy Condition-Action Rule <ul><li>Five market descriptors (five information bits) are used for the conditional part of a rule: </li></ul><ul><ul><li>p * r/d, p/MA(5), p/MA(10), p/MA(100), p/MA(500) </li></ul></ul><ul><li>Two forecast parameters (two forecast bits) are used for the conditional part of a rule: </li></ul><ul><ul><li>a & b </li></ul></ul>
  14. 14. Fuzzy Condition-Action Rule <ul><li>We present fuzzy information about a variable with the codes: </li></ul><ul><ul><li>1 2 3 4 0 </li></ul></ul><ul><ul><li>low moderately-low moderately-high high absence </li></ul></ul><ul><li>We present fuzzy information about a parameter with the codes: </li></ul><ul><ul><li>1 2 3 4 </li></ul></ul><ul><ul><li>low moderately-low moderately-high high </li></ul></ul>
  15. 15. Membership Function for Descriptor low moderately-low moderately-high high
  16. 16. Membership Function for forecast parameter ‘ a ’ low moderately-low moderately-high high
  17. 17. Membership Function for forecast parameter ‘ b ’ low moderately-low moderately-high high
  18. 18. Fuzzy Condition-Action Rule <ul><li>In general, we can write a rule as: </li></ul><ul><ul><li>[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}. </li></ul></ul><ul><li>We would interpret the rule </li></ul><ul><li>[x 1 , x 2 , x 3 , x 4 , x 5 | y 1 , y 2 ] as: </li></ul><ul><ul><li>``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 ’’ </li></ul></ul>
  19. 19. Rule Base <ul><li>Single fuzzy rule can not specify the remaining contingencies. Therefore, three additional rules are required to form a complete set of beliefs. </li></ul><ul><li>Fore this reason, each rule base contains four fuzzy rules. </li></ul><ul><li>At any given moment, agents may entertain up to five different market hypothesis rule bases. </li></ul>
  20. 20. Rule Base (an example)
  21. 21. Defuzzify of Fuzzy Decisions <ul><li>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. </li></ul>
  22. 22. Example <ul><li>Consider a simple fuzzy rule base with the following four rules. </li></ul><ul><li>1 st rule: </li></ul><ul><li>If 0.5p/MA(5) is low then a is moderately high and b is moderately high . </li></ul><ul><li>2 nd rule: </li></ul><ul><li>If 0.5p/MA(5) is moderately low then a is low and b is high . </li></ul><ul><li>3 rd rule: </li></ul><ul><li>If 0.5p/MA(5) is high then a is moderately low and b is moderately low . </li></ul><ul><li>4 th rule: </li></ul><ul><li>If 0.5p/MA(5) is moderately high then a is high and b is low . </li></ul>
  23. 23. Example (cont.) <ul><li>Now suppose that the current state in the market is given by p = 100, d = 10, and MA(5) = 100. </li></ul><ul><li>This gives us, 0.5p/MA(5) = 0.5. </li></ul>
  24. 24. Response of 1st rule (example)
  25. 25. Response of 2nd rule (example)
  26. 26. Response of 3rd rule (example)
  27. 27. Response of 4th rule (example)
  28. 28. Summary <ul><li>Rule Membership Decisions </li></ul><ul><li>1 st Rule 0 </li></ul><ul><li>2 nd Rule 0.5 </li></ul><ul><li>3 rd Rule 0 </li></ul><ul><li>4 th Rule 0.5 </li></ul>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 .
  29. 29. Defuzzify of Forecast Parameters ‘ a ’ and ‘ b ’
  30. 30. Genetic Algorithms <ul><li>GAs are applied to retain the reliable rule bases, drop the unreliable rule bases, and create new rule bases. </li></ul><ul><li>The fitness measure of a rule base is calculated as follows: </li></ul><ul><li>where  is constant and s is the specificity of the rule base. </li></ul>
  31. 31. The Market Experiments Linn & Tay (2001a) <ul><li>Experiment 1 (slow learning) </li></ul><ul><ul><li>k = 1000 </li></ul></ul><ul><ul><li>Using best rule base with probability 1. </li></ul></ul><ul><li>Experiment 2 (fast learning) </li></ul><ul><ul><li>k = 200 </li></ul></ul><ul><ul><li>Using best rule base with probability 1. </li></ul></ul><ul><li>Experiment 3 (fast learning with doubt) </li></ul><ul><ul><li>k = 200 </li></ul></ul><ul><ul><li>Using best rule base with probability 99.9%. </li></ul></ul>
  32. 32. Why we introduce ‘a state of doubt’ to catch the actual figure of kurtosis? <ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>We suspect the large kurtosis observed in actual returns series may have originated from such exogenous events as rumors or earnings surprises. </li></ul>
  33. 33. The Market Experiments Linn & Tay (2001b) <ul><li>Experiments: </li></ul><ul><ul><li>Experiment 1 (slow learning) </li></ul></ul><ul><ul><li>Experiment 2 (fast learning) </li></ul></ul><ul><li>Benchmarks : </li></ul><ul><ul><li>Disney and IBM stocks </li></ul></ul>
  34. 34. Experiments Parameters
  35. 35. Results (Linn & Tay (2001a)) <ul><li>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. </li></ul><ul><li>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. </li></ul>
  36. 36. <ul><li>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 </li></ul>
  37. 37. Results (Linn & Tay (2001b)) <ul><li>Normality: </li></ul><ul><ul><li>rejects normality for each series (Jarque-Bera test) </li></ul></ul><ul><li>Linearity: </li></ul><ul><ul><li>exists linear dependent for each series (Ljung-Box Q test) </li></ul></ul><ul><ul><li>does not exist any linear dependent for each ARMA fitted residual series (Ljung-Box Q test) </li></ul></ul>
  38. 38. <ul><li>Non-linearity: </li></ul><ul><ul><li>exists nonlinear dependent for each ARMA fitted residual series (using both correlation dimension and BDS test methods) </li></ul></ul><ul><li>ARCH Effect: </li></ul><ul><ul><li>exists ARCH behavior for each ARMA fitted residual series (Ljung-Box Q test and LM test) </li></ul></ul><ul><ul><li>does not exist any ARCH effect for each ARMA-TARCH fitted residual series (Ljung-Box Q test and LM test) </li></ul></ul><ul><ul><li>exists other nonlinear dependent for each ARMA-TARCH fitted residual series (BDS test) </li></ul></ul>
  39. 39. <ul><li>Other Non-linearity </li></ul><ul><ul><li>exists other nonlinear dependent for each ARMA-TARCH fitted residual series (BDS test) </li></ul></ul>
  40. 40. Conclusions <ul><li>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. </li></ul><ul><li>The results indicate that the model proposed in this paper can account for the presence of nonlinear effects observed in real markets. </li></ul>
  41. 41. Conclusions (cont.) <ul><li>The framework offers an alternative perspective on capital markets that extends beyond the traditional paradigms. </li></ul>

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