Fuzzy Inductive Reasoning and the Behavior of Security Prices

1. Fuzzy inductive reasoning, expectation formation and the behavior of security pricesNicholas S.P. Tay , Scott C. Linn • Introduction • Expectation formation and market created uncertainty • Models • Experiments • Results • Conclusions 報告人 黃雅琪

2. Introduction (1) This paper extends the Santa Fe Artificial Stock Market Model (SFASM) in two important directions: • First, some might question whether it is reasonable to assume that traders are capable of handling a large number of rules. We demonstrate that similar results can be obtained even after severely limiting the reasoning process.(by fuzzy reasoning) • Second, the kurtosis of SFASM simulated stock returns is too small as compared to real data. We demonstrate that with a minor modification to how traders go about deciding which of their prediction rules to rely on , we can produce return kurtosis that is comparable to that of actual returns data.

3. Introduction (2) • The actual market environment, not like Neoclassical financial market models, is usually ill-defined, where the ability to exercise deductive breaks down. • We conjecture that individual reasoning in an ill-defined setting can be described as an inductive process in which individuals exhibit limitations in their ability to process and condense information. The objectives of this paper are to first, model an inductive reasoning process and second, to investigate its implications for aggregate market behavior and in particular for the behavior ofsecurity prices.

4. Expectation formation and market created uncertainty Market created uncertainty and volatility Fuzzy reasoning and induction

5. Market created uncertainty and volatility (1) • No one knows for certain what essential ingredients are necessary for explaining the obtuse behavior seen in security prices. A certain type of market created uncertainty may be a potential explanation. • We mean the uncertainty created as a result of the interactions among heterogeneous market participants who must learn to form their expectations in a market environment inherently ill-defined.

6. Market created uncertainty and volatility (2) • The environment is ill-defined because for a market participant to logically deduce his expectations,he needs to know what others are expecting. But since every other market participants needs to do also, there is circularity in the reasoning process. (beauty contest problem) • The result is that no one will be able to logically deduce his expectation. • Consequently, each market participant will by necessity form his price expectations based on a subjective forecast of the explanations of all other market participants. As a result, the market can develop a life of its own and respond in ways not correlated with movements in fundamental values.

7. Market created uncertainty and volatility (3) • In Arthur’s words, if one believes that others believe the price will increase, he will revise his expectations to anticipate upward-moving prices；if he believes others believe a reversion to lower values, he will revise his expectation downward. • Keynes regarded security prices as ‘the outcome of the mass psychology of a large number of ignorant individuals’, with professional speculators mostly trying to outguess the future moods of irrational traders,and thereby reinforcing asset price bubbles. • Dreman (1982) maintains that ‘the thinking of the group’heavily influences the forecast of future events of individual investor,including professionals.

8. Market created uncertainty and volatility (4) These uncertainties create two types of risk for potential arbitrageurs. • Identification risk-It is difficult for arbitrageurs to exploit noise traders because they can never be sure that any observed price movement was driven by noise, which creates profit opportunities, or by news that the market knows but they have not yet learned. • Noise trader risk(future resale price risk)-The price may systematically move away from its fundamental value because of noise trader activity.So, an investor who knows, even with certainty, that an asset is overvalued will still take only a limited short position because noise traders may push prices even further away from their fundamental values.

9. Market created uncertainty and volatility (5) • Another type of risk, fundamental risk (inherent in the market), although not due to the market uncertainty,can also limit arbitrage, because even if noise trader do not move away from fundamental values, changes in the fundamentals might move the price against the investor. • Altogether, these problems limit arbitrage activity,and in turn impair the market’s natural tendency to return itself to a focus on fundamentals. Arbitrage plays an ‘error-correction’ role in the market, bringing asset prices into alignment with their fundamental values.

10. The implication of this discussion • In rational expectations, by fiat we eliminate the market created uncertainty outlined above. • We allow the agents in our model the opportunity to form their expectations based on individual subjective evaluations, thus restoring the potential for market created uncertainty to emerge.

11. Expectations formation~Fuzzy reasoning and induction Fuzzy reasoning • Market participants typically have very little time to decipher the immense amount of information from the stock market. • Some psychologists have argued that our ability to process information efficiently is the outcome of applying fuzzy logic as part of our thought process. • This would entail the individual compressing information into a few fuzzy notions that in turn are more efficiently handled,and then reasoning as if by the application of fuzzy logic. • Smithson (1987) also finds the evidence that some categories of human thought are fuzzy,and the mathematical operations of fuzzy sets as prescribed by fuzzy set theory a realistic description of how humans manipulate fuzzy concepts.

12. Induction (1) • In real life, we often have to draw conclusions based on incomplete information .In this instances, logical deduction fails because the information we have in hand leaves gaps in our reasoning. • Induction is a mean for finding the best available answers to questions that transcend the information at hand. • Nonetheless, induction should not be taken as mere guesswork, and we must find the answers both sensible and defensible.

13. Induction (2) • Inductive reasoning follows a two-step process: 1.Possibilityelaboration-Creating a spectrum of plausible alternatives based on our experience and the information available. 2. Possibilityreduction-These alternatives are tested to see how well they connect the existing incomplete premises to explain the data observed. • The alternative offering the best fit connection is then accepted as a viable explanation.

14. How can induction be implemented in a security market model ? • Each individual in the market continually creates a multitude of ‘market hypotheses’.(step.1) • These hypotheses, which represent the individuals’ subjective expectational models of what moves the market price and dividend, are then simultaneously tested for their predictive ability. • The hypotheses identified as reliable will be retained and acted upon in buying and selling decisions；unreliable ones dropped (step.2), and ultimately replaced with new ones. ( Then repeatedly learning and adapting)

15. Modeling the expectations formation process The expectations formation process described above can be modeled by letting each individual form his expectations using his own personal genetic-fuzzy classifier system. • Each genetic-fuzzy classifier system containsa set of conditional forecast rules (subjective ‘market hypotheses’) that guide decision making. • Inside the CS is a GA responsible for generating new rules, testing all existing rules, and weeding out bad rules.

16. Models • The market environment • The sequence of events • Modeling the formation of expectations -Components of the reasoning model -The specification of forecasting rules -Coding market conditions -Identifying expectation model parameters -Fuzzy rule bases as market hypotheses -An example -Forecasting accuracy and fitness values -Recapitulation

17. The market environment Ⅰ • The basic framework is similar to SFASM. • Two tradable assets in the market: A stock that pays an uncertain dividend dt ,and there are N units of the risky stock. A risk-free bond that pays a constant r and is in infinite supply. • The dividend is driven by an exogenous stochastic process as following, which no agent knows: ~ AR (1) εt ~ iid Gaussian (0,σε2)

18. The market environment Ⅱ • N heterogeneous agents’ CARA utility function: U(W)= -exp(-λW) • Agents are heterogeneous in terms of their individual expectations. • Each agent is initially endowed with one share per agent. • At each date, upon observing the information, agents decide what their desired holdings of each of the two assets should be by maximizing subjective expected utility of next period wealth. ( Solving a sequence of single period problem over an infinite horizon. ~ Myopic)

19. The market environment Ⅲ • The maximizing problem would be Assuming that agent i’s prediction at time t of the next period’s price and dividend are normally distributed with mean and variance then agent i’s demand for shares of the risky assets is given by: • Since total demand must equal the total number of shares issued,for the market to clear, we must have

20. The sequence of events • The current dividend dt is announced at the start of time period t. • Agents then form their expectations based on all current information about the state of the market ( including the historical ) • Once their price expectations are established, agents use Eq.(2) to calculate their desired holdings of the two assets. • The market then clears at pt.( by market clearing condition)；the sequence of events is then repeated and pt+1determined. • At each step, agents learn about the effectiveness of the genetic-fuzzy classifier they relied on, and the unreliable classifiers are weeded out to make room for classifiers with new and perhaps better rules.

21. Components of the reasoning model Modeling the formation of expectations We have argued that individuals apply fuzzy logic and induction to the formation of expectations, so we characterize individual reasoning as the product of the application of a genetic-fuzzy CS, which based in part on the design of the CS originally developed by Holland (1986) . • Three essential components of CS:A set of conditional action rules, a credit allocation system for assessing the predictive capability of any rule and a GA by which rules evolve. • Our genetic-fuzzy CS replace the conventional rules in Holland’s classifier with fuzzy rules. These fuzzy rules also involve a condition-action format but they differ form conventional rules in that fuzzy terms rather than precise terms now describe the conditions and actions.

22. The specification of forecasting rules (1) • The forecast equation hypothesis is ( Proof) Where a & b are the forecast parameters obtained from the activated rule. • As SFASM, the linear forecasting model shown above is optimal when, a) agents believe that prices are a linear function of dividends. b) a homogeneous rational expectations equilibrium obtains. While we place no such restrictions on the system, a linear forecasting model serves as an approximation for the structure likely to evolve over time.

23. A proof of this assertion (1) • Recall that,dividend process is , and demand for the security • Assuming that agents conjecture that price is a linear fn. of the dividend,that is . • Then, ……..(a)

24. A proof of this assertion (2) • Since all the agents are equally risk averse, each agent must hold the same number of shares.We substitute the the one-period ahead forecast into demand fn.,then we obtain: …………………………..(b) • The LHS is a constant, so the RHS cannot exhibit any dependence on time.Therefore, dt must vanish.This lead to .Substituting into (b) Substituting f & e into (a),we obtain: Compare this to

25. The specification of forecasting rules (2) • We use five information bits to specify the conditions in a rule, and two bits to represent the forecast parameter a & b.(Totally, seven bits.) • The format of a rule when fuzzy rules prevail, would therefore be ‘If specific conditions are satisfied then the values of the forecast equation parameters are defined in a relative sense’.One example is ‘If {price/fundamental value} is low, then a is low and b is high’.

26. Coding market conditions (1) • The five information bits used to specify the conditions represent five market descriptors. • The five market descriptors are [p*r/d, p/MA(5), p/MA(10), p/MA(100), and p/MA(500)], where MA(n) denotes an n-period moving average of prices. • The first bit reflects the current price in relation to the current dividend and it indicates whether the stock is above or below the ‘fundamental’ value at the current time；bits 2-5, are ‘technical’ bits which indicate whether the price history exhibits a trend or similar characteristic.

27. Market descriptors are transformed into fuzzy information sets by first defining a range of possibilities for each information bit. And second, by specifying the number and types of fuzzy sets to use for each of these market variables. First,we set the range for each of these variables to [0,1]. Second, we assume that each descriptors has the possibility of falling into four alternative states: ‘low’, ‘moderately low’, ‘moderately high’ and ‘high’. Coding market conditions (2)

28. Coding market conditions (3) • We let the possible states of each market descriptor be represented by a set of four membership functions associated with a specific shape. • We represent fuzzy information with the codes 1,2,3,4 for low, moderately low, moderately high,and high.A ‘0’ is used to record the absence of a fuzzy set.(like # before) • Example:If the condition part of the rule is coded as [01302],this correspond to a state in which the market price is less than MA(5) but some what greater than MA(10) and is slightly less than MA(500). low Moderately high Moderately low high Fig1.Fuzzy sets of the market descriptors

29. Identifying expectation model parameters (1) Now we turn to the modeling of the forecast part of the rule. • We allow the possible states of each forecast parameter to be represented by four fuzzy sets which labeled respectively (with the shapes indicated), ‘low’, ‘moderately low’, ‘moderately high’, and ‘high’. • The universe of discourse for a & b are set to [0.7,1.2]&[-10,19], respectively.The shapes and locations:

30. Identifying expectation model parameters (2) • An example of the forecast part of the rule is the string [2 4],which would indicate that the forecast parameter a is ‘moderately low’ and b is ‘high’. • 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}. (for example,[1 0 3 0 2 ∣2 4 ]) • We would interpret the rule [x1,x2,x3,x4,x5∣y1,y2] as: If p*r/d is x1 、p/MA(5) is x2、p/MA(10) is x3、 p/MA(100) is x4、andp/MA(500) is x5 , then a is y1 and b is y2 ...

31. Fuzzy rule bases as market hypotheses (1) • A genetic-fuzzy classifier contains a set of fuzzy rules that jointly determine what the price expectation should be for a given market state. • Each rule base (a set of rules) represents a tentative hypothesis about the market and reflects a ‘complete’belief. The rule ‘If p*r/d is high, then a is low and b is high’ itself does not make much sense as an hypothesis.Three addition rules (specifying what a&b should be for the case p*r/d is low,moderately low and moderately high)are required to form a complete set of beliefs. • For this reason, each rule base contains four fuzzy rules.

32. Fuzzy rule bases as market hypotheses (2) • Fig.4 shows an example of a rule base. • In order to keep the model manageable yet maintain the spirit of competing rule bases, we allow each agent to work in parallel with five rule bases. • Hence, each agent may derive several different price expectations at any given time,and a agent will acts on the one that has recently proven to be the most accurate. (A modification will be specified in the following page)

33. Fuzzy rule bases as market hypotheses (3) • In a separate experiment, we allow agents to sometimes select the rule base to use in a probabilistic manner. (Triggered by a random event) • Specific event the polarization of negative attitudes (group polarization phenomena) • We think of these negative attitudes as ‘doubts’(with a very low probability to occur) about whether the perceived best rule base is actually best, and agent tie the probability of selecting a rule to its relative forecast accuracy. This modification is a key extension of the SFASM.It produces return kurtosis measures more in line with observed data than that of SFASM,while simultaneously generating return and volume behavior that are otherwise similar to actual data. trigger

34. An example (1) • Consider a simple fuzzy rule base with the following four rules. If 0.5 p/MA(5) is low then a is moderately high and b is moderately high. If 0.5 p/MA(5) is moderately low then a is low and b is high. If 0.5 p/MA(5) is high then a is moderately low and b is moderately low. If 0.5 p/MA(5) is moderately high then a is high and b is low. • Suppose that the current states is given by p=100,d=10,and MA(5)=100. 0.5 p/MA(5)=0.5 • Since 0.5 is outside 1st & 3rd rules’ domain, the forecast parameters will also have zero membership value.

35. An example (2) • Only the 2nd and 4th rules contribute to the resultant fuzzy sets for a & b. • We employ the centroid method to translate the resultant information into specific values for a & b. • We obtain 0.95 and 4.5 for a & b.(Fig.6,8,9,10) • Substituting these forecast parameter into the linear forecasting model, give us the forecast for the next period price and dividend : E(p+d)=0.95(100+10)+4.5=109

36. Fig.6

37. Fig.8

38. Fig.9-10

39. Forecasting accuracy and fitness values (1) • We measure that accuracy for a rule base by the inverse of e2t,i,j. • Define • The variable e2t,i,j is used for several purpose: -In each period the agents refer to e2t,i,j when deciding which rule base to rely on. -It isused by agent i as a proxy for the forecast variance σ2t,i,j . -It isused to compute what we will label a fitness measure. (The fitness measure is used to guide the selection of rule bases for ‘crossover’ and ‘mutation’ in the GA.)

40. Forecasting accuracy and fitness values (2) • Agents revise their rule bases by GA on average every k periods. • We specify the GA as following: -Selection:guided by fitness measure. Mutation:by mutating the values in the rule base array. Crossover:combining part if one rule base array with the complementary part of another. • In general, rule bases that fit the data well, will more likely to produce whereas less fit bases will have a higher probability of being eliminated.

41. Forecasting accuracy and fitness values (3) βis a constant(a cost per unit) S is the specificity of the rule base(eg.fig.4) • The fitness measure of a rule base is calculated as follows: It imposes higher costs on rules leading to larger squared forecast errors and employing greater specificity. • β is introduced to penalize specificity.The purpose is to discourage agents from carrying redundant bits because of limiting ability to store and process information. • The net effect of this is to insure that a bit is used only if agents genuinely find it useful in predictions and in doing so introduces a weak drift toward configurations containing only zeros.

42. Recapitulation (1) • dt announced at start of time period t. • Based on all current information , the five market descriptor [p*r/d, p/MA(5), p/MA(10), p/MA(100), and p/MA(500)]are computed and the forecast models parameters identified. • Agents form Ei,t[pt+1, dt+1]by using a&b from the rule base most accurate.( Except pt undetermined…Ei,t[pt+1,dt+1]=a(pt+dt)+b) ) • On the other side, In their desired share holdings eq., only ptis undetermined. The market cleaning condition determine pt. ,and by substituting pt into e2t,i,j,we get the accuracy of the rule base by the inverse of e2t,i,j..

43. Recapitulation (2) • Learning in the model happens at two different levels. -On the surface, learning happensrapidly as agents experiment with different rule bases and over time discover which rule bases are accurate and worth acting upon and which should be ignored. -At a deeper level, learning occurs at a slower pace as the GA discards unreliable rule bases to make room for new ones.The new, untested rule base will not cause disruptions because they will be acted upon only if they prove to be accurate.

44. Experiments • In these experiments, the primary control parameter is the learning frequency constant k. • Recall that agents use inductive reasoning which basically amounts to formulating tentative hypotheses (rule base), and testing these hypotheses repeatedly against observed data. • Under such a scheme, the learning frequency will play a key role in determining the structure of the rule bases and how well agents are able to coordinate their price expectations.(The reason:next page)

45. The reason why k plays a key role • When learning frequency is high, agents revise their belief frequently.Then, -They will not have adequate time to fully explore whether their market hypotheses are consistent with those belonging to other agents. -Their hypotheses are more likely to be influenced by transient behavior of market variables. Difficult to converge on an equilibrium price expectation. • In contrast, when learning frequency is low, agents are more likely to converge on an equilibrium price expectation.

46. The model’s parameters We conduct three sets of experiments~ Table1:common parameters In the 1st and 2nd experiments, k is equal to 200&1000, respectively. In both experiments, agents form their forecasts using the most accurate base. In the 3rd experiment, k is equal to 200.We assume that there is a 0.1% probability that an agent will decide to select the rule base to act upon in a probabilistic fashion. ( Then, all the remaining agents follow him.)

47. Other features of the experiment • In 3rd experiment,when a state of doubt arises, the probability that agent i will select his rule base j is then linked to the relative forecast accuracy of the rule base by ranking the five rule bases from 0 to 2, in increments of 0.5, and compute the selection probability for each rule base as: ~Unique for each agent • We follow the LeBaron (1999) in refer to the two cases (k=200&1000)as fast learning and slow learning.(asynchronous) • We began with a random initial configuration of rules；then we simulate for 100,000 period to allow any asymptotic behavior to emerge.Subsequently, starting with the configuration attained at t=100,000 we simulated an additional 10,000 periods to generate data for the statistical analysis.We repeated the simulation 10 times to facilitate the analysis of regularities.

48. Results • Simulation results form our experiments show that the model is able to generate behaviors similar to many of the regularities observed in real financial markets. Asset price and returns Trading volume Market efficiency

49. Asset price and returns (1) • Fig 11-13 present snapshots of observed price behavior.

50. Asset price and returns (2) • In these three graphs, the market price is consistently below the REE price.