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closing share price (C). Two types of trading days are ofinterest:O[ 1] H[0] δ. In both cases 0 δ 10 is userselectable.Definition: (up-trend day)IV. F UZZY S YSTEM D ESIGNO L 0.1(H L)C H 0.2(H L)(1)Definition: (down-trend day)O H 0.1(H L)C L 0.2(H L)(2)Qualitatively, this simply means for an up day the openingprice is close to the day’s low and the closing price is close tothe day’s high. Similarly, for a down-trend day the openingis at or near the high and the close is near or at the low forthe day. An interesting property of trend days—which keeninvestors can exploit—is the high-low differential tends tobe relatively large.Studies have identified several trading data features thatoften proceed up or down-trend days. These features, described below, are defined in terms of O, C, H and L. Todayis indexed with i 0 and previous days are indexed i 1, 2,and so on. O( 1) denotes the next trading day opening price. NRkWith H[i] and L[i] denoting the high and low for thei-th day, the range is defined as R[i] H[i] L[i].NRk exists if today’s range is less than the ranges forthe previous k 1 days. That is,R[0] min(R[1], . . . , R[k 1])(3)NRk days represent volatility contraction, which oftentimes leads to volatility expansion in the form of widerange days. The greater the number of narrow rangedays, the greater the counter reaction in wide rangingdays. DOJIDOJI indicates that the open and close for the tradingday are within some small percentage (x) of each other.A DOJI means the market reflects temporary priceindecision and often signals a major reversal in themarket. DOJI is a predicate function—i.e., it returns 1(TRUE) or 0 (FALSE). It is defined asDOJI (x) 10 O C x · (H L)otherwise(4)Hook dayA hook day occurs when the price opens outside theprevious day’s range and then proceeds to reverse direction, generally indicating a reaction to temporarilyoverbought or oversold market conditions.There are two versions of a hook day. For the up hookday O[ 1] L[0] δ and for the down hook day318Given stock market historical data, it is always possibleto (deterministically) analyze it and mark where any of thefeatures described in the previous section are present. But amore subtle and challenging problem must be dealt with. Therule-base contains a set of fuzzy rules that predict whetherthe next trading day is likely to be a trend day. Investmentdecisions—i.e., whether to buy, sell or hold—are made basedon these predictions. Fuzzy rules are used because crisp rulesare too restrictive. An example will help fix ideas.The crisp rule “if NR7 then . . .” is true if and only ifthe range during the current day R[0] is less than the rangeduring any of the previous six days R[1], R[2] . . . R[6]. Therule won’t be true if even one of the previous ranges is lessthan R[0]. But suppose the inequality is satisfied for say fiveout of the six days, which makes the antecedent almost true?As another example, Nison [9] asked“How do you decide whether a near-DOJI day(that is, where the open and close are very close,but not exact) should be considered a DOJI? Thisis subjective and there are no rigid rules. . .”Fuzzy reasoning can effectively deal with such uncertainties. Crisp rules, which must give either a ’yes’ or a ’no’answer, cannot handle these situations but fuzzy rules canbecause they can also provide fuzzy answers somewhere inbetween a ’yes’ or ’no’.In our approach stock market data is analyzed to determinehow closely it matches the formal definitions of the featuresdescribed in the previous section. Membership functionsreturn a value between 0 and 1 indicating to what degreefeatures are present. The resultant fuzzy variables are thencollected into fuzzy if-then rules, which constitutes the trading rulebase. The outputs of active rules—i.e., rules whoseantecedent are satisfied—are combined into a fuzzy outputvariable. This variable is defuzzified to produce a crisp valueon the unit interval, which the desirability of buying stockon the next trading day.A. Membership FunctionsFuzzification is the process that maps days (D) ontothe unit interval via a membership function µ(D). Moreprecisely, D represents the number of previous days thata particular feature is satisfied. For instance, for the NR7feature D {0, 1, . . . , 6}. Then µ(0) 0 means the NR7definition was not satisfied during any of the six previousdays, µ(6) 1 means the definition was satisfied duringall six previous days (i.e., NR7 is definitely present) and0 µ(D) 1 means NR7 was satisfied for some D 6days. Trapezoidal membership functions are most appropriatefor the most of the features (see Figure 1). NRk2008 IEEE Symposium on Computational Intelligence and Games (CIG'08)

For this feature the equation is slightly different for eachvalue of k. Let µk (x) denote the membership functionfor NRk. Thenµk (x) 0c (x υmin ) 1x υminυmin x υmaxx υmax(5)with parameter values as shown in Table I.k467c1/21/31/3υmin234υmax467TABLE IPARAMETER VALUES FOR NARROW RANGE FEATURESFig. 2.In the above equation x D η where D k isthe number of days where (3) holds and, with R̃ max1 j k R[j],DOJI membership function with ρ 0.1.where x L[0] δ O[ 1] for an up hook day andx O[ 1] δ H[0] for a down hook day.R̃ R[0]R̃Notice that η increases the membership value forsmaller previous day ranges.η Fig. 3.Fig. 1. Membership functions for the features.B. The Fuzzy Rule-BaseDOJIFor this feature the membership function equation is 1 x/ρ 0 x ρµ(x) (6)0otherwisewhere typically ρ [0.05, 0.30). x represents thepercent difference between O and C and ρ representsthe threshold percentage. Hook DayThe formula for this membership function is 02(x 0.5)µ(x) 1x 211 2 x 0x 0Hook day membership function(7)Unfortunately, just detecting the presence or absence ofa single feature is not a very good trend day predictor. Theproblem is to find combinations of features that make a goodtrend day predictor4 .If there are N total features, then there are N total rulesin the rule-base5 . Consider the ruleif x is NR4 then output is up-trend dayThe semantics of this rule is as follows. The term “xis NR4” means ranges for the current and the previousthree days are computed. x represents how many of thosedays meet the NR4 definition. This crisp data value is theargument for the NR4 membership functions which returns4 Certain real parameter values must also be chosen. For example, thepercentage threshold input value is needed for the DOJI feature.5 There would be 2N total rules if both up-trend and down-trend days arepredicted.2008 IEEE Symposium on Computational Intelligence and Games (CIG'08)319

a number between 0 and 1 to give the degree of membershipfor NR4. The output value is the degree of membership foran up-trend day.A singleton fuzzifier is used for the inputs. The set ofpossible outputs isΛ 0.250.50.751.0These output values represent the likelihood an investorwould buy stock because a given fuzzy rule had fired.The fuzzy rule-base is encoded as a matrix M with onecolumn for every λi Λ and one row for every fuzzyrule [10]. Each rule is of the form “if x is feature then yis an up-trend day”. In this work we investigated 5 featuresso M has 5 rows corresponding to, from top to bottom, thefeatures NR4, NR6, NR7, DOJI, and Up Hook Day (withδ 0.5). There are four columns, one for each value in Λ.A typical matrix might look like 0 0.60.50 0 0.33 0.33 0.33 1.00.9 M 0 0.1 0 0.44 0.50.1 0 0.10.20.7where each mij M is a weight. The first row in the abovematrix is thus interpreted as the investor saying If the currentday is the feature NR4, then I think the likelihood of an uptrend day tomorrow is 0.5 or 0.75 (out of 1) with strengths0.6 and 0.5, respectively. Notice the strengths do not haveto sum to 1.0.Rule processing is straightforward. Given the rule-basematrix M , the vector of fuzzy numbersA (µNR4 , µNR6 , µNR7 , µDOJI , µhook )is created by running the training data through the individualmembership functions. A new A vector is created for eachtrading day. For example, on the 50th training day theranges R(47), R(48) and R(49) are computed and µ4 (x) iscomputed using (5). Similarly the other membership functionvalues are determined using (6) and (7). Then A M Bwherebj max min {ai , mij }1 i 5The fuzzy output vector B is then defuzzified to get a crispoutput value indicating the desirability of buying shares ofstock. A center of average defuzzification was used. That is,PNi 1 bi · λi(8)U PNi 1 λiwhere λi is the i-th singleton from Λ and N is the numberof active rules.The crisp output U [0, 1] indicates the likelihood of anup-trend day or, equivalently, the desirability of purchasingmore stock shares. However, stock is only purchased if thedesirability exceeds a user-selected threshold. We chose 0.8320for this threshold. If the output value is greater than 0.8,then a ‘buy’ signal is generated and how high above 0.8determines the number of shares to buy. Specifically, theamount of stock purchased increased linearly the higher theoutput was above 0.8, subject to sufficient funds in the bankaccount, up to a maximum of 20 shares. Stock was bought atthe opening price and sold at the closing price. All proceedswere deposited into the bank account at the end of eachtrading day.C. The Coevolutionary AlgorithmThe fuzzy rule-base was evolved using a (40 40)evolution strategy. Both recombination and mutation areused as variation operators. Each offspring are created byrandomly choosing two parents. After appending together thecolumns of the M matrix in each parent to form a real vector,standard 1-point crossover, followed by mutation, createsthe offspring. Mutation added a gaussian distributed randomvariable with zero mean and standard deviation σ 0.2 toeach term in the M matrix. The mutation was redone if thecomponent value was less than 0.1 or more than 1.0. For thedefuzzified output any value below 0.5 was automatically setto 0 because a desirability of buying stock with a value thatlow makes no sense. The evolutionary algorithm was run for750 generations.Both the α and the ω rule-bases—corresponding to the αand the ω brokerage firms—evolve independently. Fitness ofa rule-base is determined by tournaments with the competitorrule-bases. Specifically, each individual in the α populationwas provided with 400 and then used its rule-base as aninvestment strategy over 150 consecutive trading days witha start date chosen randomly from the first 850 tradingdays in the training market database. The ω population didthe same over the same 150 trading days. Afterwards eachindividual in both populations has an ending bank accountbalance after selling all shares purchased during the tradingperiod. The bank account balance of each individual in theα (ω) population was compared against 10 randomly chosenindividuals from the ω (α) population. The individual withthe largest bank account records a win.Winning in this game has a positive payoff while losinghas a negative payoff. Every time an individual wins againstan individual in its tournament set, the winner gets 2% ofthe loser’s bank account balance. Conversely, if the individualloses against a member of its tournament set, he has to pay2% of his bank account balance to winner. Hence, individualsget paid or have to pay every time they participate in atournament. (Accounts were not actually credited or debiteduntil after all tournaments were finished.)This payoff approach to conducting tournaments emulatesreal-world trading. Brokerage firms compete for investordollars. The bank account associated with each individualcan be thought of as money given to a brokerage firm byinvestors. Each firm starts out with the same amount andtrades according to the strategy defined by its fuzzy rulebase. Firms with lower bank accounts at the end of thetrading period have poorer strategies and poor strategies2008 IEEE Symposium on Computational Intelligence and Games (CIG'08)

cause investors to move their money elsewhere—i.e., tobetter performing firms.V. E XPERIMENTAL R ESULTSA database of 1600 trading days were available for trainingand testing. The first 1000 days are reserved for trainingwhereas the 600 remaining days are used for validationtesting. Each training epoch consists of 150 consecutivetrading days, with a random starting date and partitioned intothree overlapping 90 day segments. The second segment usedthe last 60 days from the first segment plus 30 new days; thethird segment used the last 30 days of the first segment plus60 new days. The overlaps help mitigate market volatility.Each training epoch started with a 400 bank account.The bank balance at the end of the first 90 day trainingsegment was the beginning bank balance of the second 90day segment. Similarly the third segment beginning balancewas the same as the second segment ending balance. (Bankaccount balances are adjusted as explained in the previoussection.) Testing was conducted in the same way, except the150-day window was chosen from the latter 600 days in thedatabase.Figure 4 shows the results from a typical run on a 90day trading session from the testing portion of the database.Notice the desirability exceeded the 0.8 threshold on only afew days.Fig. 4. Typical trading behavior for the best fuzzy trading rule sets over a90 trading period. Shares were bought whenever the desirability exceeded0.8.Table II shows the bank account balances for the top fiveand bottom five trading strategies after completing the 150days of trading using the training database. Notice for bothfirms the top five strategies had positive returns whereasthe bottom five strategies had negative returns. The positivereturns are considerably higher than the starting balanceof 400. Remember the ending bank account balances arenot due solely to buying stock at a reasonable price. Goodperforming strategies have a higher payoff by taking moneyout of the bank accounts of poorer performing strategies.This zero-sum game format matches how investors in thereal world choose brokerage firms: firms with good strategies attract more investor’s dollars while firms with poorerstrategies drive away investor’s dollars.α Firmω FirmAccount Balances fromTop 5 Strategies ( )548.4, 498.2, 489.6,487.8, 479.6548.9, 548.7, 548.3,547.6, 547.0Account Balances fromBottom 5 Strategies ( )336.1, 335.4, 332.2,331.1, 326.2322.2, 321.2, 319.0,304.5, 300.1TABLE IIBANK ACCOUNT BALANCES FROM TOP AND BOTTOM TRADINGSTRATEGIES FROM TWO BROKERAGE FIRMS . R ESULTS ARE OBTAINEDOVER A 150 TRADING PERIOD WITH AN INITIAL 400 BANK BALANCE .The best α firm fuzzy rule-base strategy was then compared against a simple investment strategy over 90-daytrading periods using the test market data—i.e., the last600 days in the market database, which were not used fortraining. In the simple investment strategy fifty shares ofstock are purchased at the beginning of the trading day andthen sold at the closing price at the end of the trading day.Two trading periods were chosen. The first trading periodbegins on day 1200 while the second trading period beginson day 1400. Figures 5 and 6 compares the simple and fuzzytrading strategies for the first and second trading periods,respectively.Fig. 5. Bank account balances for the top α firm strategy and the simpleinvestment strategy over a 90 day period of test market data starting at the1200th trading day. Both accounts started with a 400 balance.Some interesting observations emerge from a close examination of Figures 5 and 6. The returns from buying andselling stock every day indicates how the market is doing.The market data shows high volatility. Notice the market isgenerally up for several weeks after the 1200th trading daybut reverses dramatically after the 1400th trading day. Moreto the point, the simple investment strategy had a 15.7%return in the first trading period but a -8.2% return during thesecond trading period. Clearly buying stock every day is nota good investment strategy. Conversely, the fuzzy rule-base2008 IEEE Symposium on Computational Intelligence and Games (CIG'08)321

Fig. 6. Bank account balances for the top α firm strategy and the simpleinvestment strategy over a 90 day period of test market data starting at the1400th trading day. Both accounts started with a 400 balance.had a modest 2.5% return during the first trading periodbut no loss during the second trading period. Although thereturns were not as great when the market was up, it didprotect capital when the market was down.VI. F UTURE W ORKIf the two brokerage firms only used their best tradingstrategy, in principle one firm could do some reverse engineering and ultimately deduce the other firms trading strategy(i.e., the fuzzy rules). Notice in Table II that the returnsbetween any of the top two strategies are not significantlydifferent. Hence, a mixed strategy will provide securitywithout sacrificing much of the return.Both brokerage firms used the same membership functions. In practice each brokerage firm would use its owninternally developed membership functions. In this work themembership functions were fixed but in our future experiments we will evolve the membership functions as well.Our evolved trading rule-base only bought stock when themarket data suggested a pending up-trend day. We need toalso evolve a set of down-trend day trading rules. The uptrend day rules would indicate buying opportunities whilethe down-trend days would indicate selling short to helpminimize losses.Finally, our trading strategy relies only on opening andclosing stock prices. Day traders make multiple tradesthroughout the day to maximize returns. Figure 7 shows theprice variation during a 6-hour trading period.322Fig. 7.Stock prices over a single trading day period.Notice that buying the stock at the beginning of the tradingsession and selling the stock after 4 hours would produce ahigher profit than selling or holding based on the openingand closing prices alone, which is what our fuzzy rule baseddecisions on. This suggests fuzzy rules for intra-day tradingwould produce even higher profits. We will investigate thisissue as well.R EFERENCES[1] E. Fama. Efficient ca

open of a predicted trend day and are exited at its close. Four data items were recorded each day: the open share price (O), the high (H) and low (L) for the day and the 3In this work we did not attempt to nd down-trend days, which ould require a different set of strategies. 2008 IEEE Symposium on Computational Intelligence and Games (CIG'08) 317