Optimizing Multiple Stock Trading Rules using Genetic Algorithms

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1 Optimizing Multiple Stock Traing Rules using Genetic Algorithms Ariano Simões, Rui Neves, Nuno Horta Instituto as Telecomunicações, Instituto Superior Técnico Av. Rovisco Pais, Lisboa, Portugal. ABSTRACT This paper stuies investments strategies for stock markets base on a combination of multiple Technical Inicators rules. The strategies generate are compare with the Buy an Hol, an with the single Technical Inicators. The experiments show that the combine strategies outperform the results of each strategy iniviually, an also the results of the Buy an Hol for 3 Major Stock Inexes, between 2004 an General Terms Algorithms, Economics, Experimentation, Performance. Keywors Optimization, Technical Analysis, Evolutionary Algorithms, Financial Analysis, Moving Average.. INTRODUCTION The stuy of profitable traing rules in the stock market constitutes a wiely known problematic in financial markets. Although the existence of those rules still generate great controversy for many economists an acaemics [6]. On the other han, investor, traers, an other stakeholers of financial an investment firms, with large experience in the stock market, claim that it is possible to have excessive returns (compare with the Buy an Hol) using algorithmic traing [][4]. One investment technique commonly use is Technical Analysis, which forecasts the price of stocks base only on the price of the stock an the volume trae in the past. Momentum strategies base on the continuation in the evolution of a stock price on their recent history [0][8], have prove to be consistently more profitable than the inexes where those stocks were inclue. The founation of Technical Analysis is the Dow Theory, written by Charles Dow, founer of Wall Street Journal where the main ieas of the Dow Theory where publishe, in the en of the XIX century [][3] The main iea of this Theory is that stock markets move accoring to trens. These trens are more important with the longer the time-frame they ha been active, an can overlap. This means that in a large uptren small ownsizes of short term can occur, but the tren is not over until strong signals of reversal occur. In this paper we will concentrate on ientifying meium to long term trens. Although a theoretical explanation of why these meiums to long term trens occur is not the focus of this paper, several causes can be pointe out. The first an probably the more important is the economical cycles theory, which states that the economy has long perios (months or years) of growth followe by perios of ecline or stagnation. With this in min an knowing that the stock market reflects not only the current performance of companies, but also the expectations about the future it is possible to ientify a correlation between economic cycles an stock market cycles. Aitionally to the economic cycles are other factors that can contribute to the price fluctuation like share buyback, which mainly occurs when companies generate large profits (it woul be a o iea for a company in financial nees, to allocate money to stock repurchase), which implies that share buy backs are one most of the time in a uptren economical cycle, when the company generate excess profits. Other important factor is the money flow of the stock market (especially in countries like the U.S.A. where most citizens have investments in the stock market). The effect of the money flow can be foun even in companies or mutual funs with goo performance, as the aversion to risk increases to the public in the beginning of the Bear Markets, people ten to reirect their capital to more secure investments like bons, eposits (or even commoities as occurre in this Bear market). This ecapitalization of the stock market (an because the number of shares is the same) leas the stocks own following the simply rule of supply an eman: supply increase (as people try to sell) an eman stagnate or even ecrease (people change to ifferent markets). Genetic Algorithms are optimization techniques base on the principles of natural evolution. In [7] is provie a formal stuy of this subject. This paper presents a genetic algorithm for optimizing Technical Inicators parameters in orer to maximize returns. Other GAs have been previously use to optimize technical inicators parameters, in particular [7] an to evelop investment strategies base on technical inicators [] [8] [9] [20] [2]. In this sense, we propose the use of a GA to obtain the set of inicators an their parameters, which shoul be use to

2 preict a aily market value. Initially we have applie GAs to fin the more suitable parameters for the SMAC, MAD an RSI inicators. After that we combine several strategies, so that a buy or short-sell signal is only mae when the majority of the strategies agree, again a GA is use to optimize the Technical Inicators Parameters of all the strategies use. The next section will iscuss the relate work on the Genetic Algorithms an various traing strategies currently use in Technical Analyses. Section 3 explains the system architecture an the investment strategies use in this paper, the markets an years use to test those strategies. Also in this section the overall escription of the GA is shown, an the fitness, selection, crossover an mutation functions use. In section 4 the results are presente an a highlight of the most relevant results is mae. In section 5 the conclusions of this stuy are shown. 2. RELATED WORK One of the most use an olest strategies to ientify trens is the crossing of Moving Averages. This strategy consists of having two Moving Averages, one of long term, an other of meium term. A buying signal is generate when the Meium Moving Average crosses up the Long Moving Average, whenever the cross is ownwars a selling signal is generate. This strategy has been stuie by [2] an by []. This stuies conclue that from 90 to 2000 the Crossing of the Moving Average perform better than the Buy an Hol strategy, except for the perio from 980 to 2000 where the market exhibite a regular uptren, an no excess profits where possible as reporte in [5]. More complete stuies of other Technical Inicators has been mae, like the one in [3] who stuies the profitability of 76 Technical Inicators with robust results for some inicators. Many papers have been recently publishe on the use of GAs to optimize technical inicators like [7], which use GAs to optimize the parameter of a single Technical Inicator, the MACD (Moving Average Convergence- Divergence) with 3 parameters, an an extra parameter for the history winow size. Another solution base also on optimizing Technical Inicators parameters is the one use in [], where the chromosome is compose by the MACD, RSI an history winow size, also a comparison between single an multiobjective is mae. Besies GAs others optimization techniques has been applie to this area of stuy, like neural networks in [2], where the neural network uses for the inputs the price, volume, interest rate an foreign exchange rate. Also other more unexplore approaches like pattern recognition as been trie in [5] which explores a more visual approach to Technical Analysis. Other technical information has been stuie. The influence of volume as a preicting tool was stuie in [4] [6], the inicator is base in the suen increase of the volume to generate a buy signal. This stuy concentrates in the optimization of technical traing rules which has not been yet teste with GAs, like the SMAC an MAD strategies, an also, combines these two strategies in one chromosome trying to achieve better an soli returns than with the solo strategies. 3. METHODOLOGY The propose system consists on a Genetic Algorithm couple with a market return evaluation moule base on the return of the strategies in ifferent markets in specific time-frames. 3. SYSTEM ARCHITECTURE Figure System Overall Architecture. The complete process can be summarize as: The user starts by specifying the markets to analyze an next chooses the Technical Inicators use in the strategy. Finally, the user chooses the train an test perio. Afterwars, the Genetic Algorithm Kernel runs several number of times, optimizing the parameters of the strategy for the markets an training perio chosen. Finally for each run of the GA, its return on the test perio is calculate. Detail info is shown to the user isplaying the optimize strategy an the return for each market in the test an in the training perio. 3.. MODULES DESCRIPTION This section presents the overall escription of each moule an their main responsibilities. Technical Inicators: This moule is responsible for the creation an management of the technical inicators use by all the strategies. This unit calculates the value of the technical inicators for a specifie inex an time perio an stores

3 it s calculation for later reuse. This moule is also responsible for calculating the strategies ecisions (if it shoul buy, short-sell or be out of the market). Train an Testing Perios: The Time Perio moule controls the time components of the Stock Inexes, in this unit the user can specify which time perios the Genetic Algorithm will use for optimization, an which time perio shoul be use for test, an its configuration (continuous, sliing winow, an others.) Stock Market Inexes: This moule is responsible for loaing the stock market inexes from the source (a.csv file) an giving access to the ata to the other parts, the store information inclues the close value, the open, high, low an the ate. Market Evaluation: In this block it is calculate the return an other metrics for evaluating the investment strategy (like the Sharpe, number of traes execute, ROI, an others.). The results can be evaluate for several types of metrics, yearly or monthly an with simple or compoun average. Genetic Algorithm: The Genetic Algorithm Moule is the most important because it is the one who oes the core functions of the system. This moule uses ata from all the other moules to calculate the perfect strategy with the Technical Inicators specifie by the user for the specifie markets, in the training perio. The crossover is a one-point crossover, an parents are chosen base on a roulette-wheel selection. Optimize Strategy: Finally this moule is responsible for showing the user the result of the optimization. Besie the best strategy obtaine, it also shows results from various runs of the Genetic Algorithm, so the user can test the average results an robustness of the solution. For each strategy it shows the return in the test an training perio, the yearly return an the Sharpe. 3.2 TRAIN AND TEST DATA SET The time perio chosen for training was from January 993 to 3 December 2003, eleven years of aily ata. This time perio was chosen for two main reasons. The first one is that the time perio shoul be big enough to be statically relevant an to avoi any kin of bias ue to a small sample perio. Seconly, the market ata shoul be similar in nature to the markets where the system is going to be applie. With the constant changes in the stock market in the last years, like online traing, algorithmic traing, high volume traing, an with the increase in the spee an amount of exchange information an short elays for new information to reach an change markets evolution, early an mi 20th century ata may be meaningless to current moels to preict stock markets behavior. The testing perio was from January 2004 to 3 December 2009, six years of testing. This perio was chosen to test the GAs in an almost real situation, simulating that the investor ha run the training in 993 to 2003, an applie these strategies until the present. Also, the fact that the markets ha been very stressful an that this has been a very ifficult perio for all the operators in the market, meaning that fining a successful strategy in this type of market is not an easy task. The markets teste where the S&P500 (USA), FTSE00 (Englan) an DAX30 (Germany). They represent the main inexes of the main evelope economies. These are markets that behave in a stable an orerly fashion for long perios. They also inclue several big companies in ifferent sectors which gives an extra stability to them. They react mainly to company profits an major economic events. They also have high volume of transactions an are ifficult to manipulate ue to high stanars of regulation an size. 3.3 TECHNICAL INDICATORS For the strategies use the Simple Moving Average will be applie, which can be calculate using the following expression (): SMA n ( ) = n t= n+ P( t) Where n is the time perio (in ays), is the ay where the moving average is calculate, P(t) is the value of the Inex at ay t. An example of this inicator for a SMA of 200 ays is presente in Figure 2. () Figure 2 - Evaluation of the SMA(200) from 2000 to 200 in the S&P 500. The first strategy to be teste was the Simple Moving Averages Crossover (SMAC) which is compose by two Moving Averages (MA) with ifferent time perios. One of

4 the MA is a long term MA, an the other is a short term MA. A buying signal is generate whenever the short term MA crosses over the long term MA, an a sell signal is generate whenever the short term MA crosses uner the long term MA. Following this strategy the investor will buy (or maintain) the Inex whenever Eq. (2) is higher than zero, an will short sell whenever Eq. (2) will be lower than zero. SMAC s, l ( ) = P( t) P( t) s t= s+ l t= l+ (2) Where l is the time perio use for long term, s the time perio for short term, an P(t) the value of the Inex at ay t. An example of this inicator for a SMAC of 200 an 50 ays is presente in Figure 3. Figure 3 - Evaluation of the SMAC(200, 50) from 2000 to 200 in the S&P 500. Another inicator that will be use in this paper is the Moving Average Derivate (MAD). It is an extene version of the MA Change escribe in []. In the original version it is calculate by subtracting e value of the current MA with the value of the MA in the previous ay. In mathematics this is simply the secant to the MA curve in the last two ays. In this way the Derivate of the MA can be calculate base on the efinition of Secant of the MA (Eq. 3). Where n is the time perio use to calculate the MA an g is the istance between the two ays to calculate the secant (the original strategy consists of a fixe g with value ). MAD n, g ( ) = t= n+ P( t) ng t= n+ P( t g) (3) Figure 4 - Evaluation of the MAD(200, 50) from 2000 to 200 in the S&P 500. In this way the value of the MAD reflects the current value of the Inex. As mentione the strategy consists of buying when the MAD is larger than zero an short sell when it is less than zero. The strategy introuce in this paper is the MAD (Moving Average Derivate) an consists on having only one MA. The iea behin this strategy is to buy the Inex when the Derivate of the MA is positive (meaning that the Inex will go up), an short sell when the Derivate is negative. An example of the calculation of this Inicator with the parameters, 200 for the long Moving Average, an 50 for the gap, can be seen in Figure 4, where is shown the evolution of the S&P 500 from 2000 to 200 an the respective values of the MAD. This inicator gives a buy orer when the MAD crosses the zero in an ascening slope an a sell orer when it crosses the zero in a escening slope. Other inicator use was the Relative Strength Inex (RSI). The RSI inicator is a momentum oscillator use to compare the magnitue of a stock s recent gains to the magnitue of its recent losses, in orer to etermine overbought or oversol conitions. The formula use on its calculation is: 00 RSI n ( ) = 00 Ups ( n) + Downs ( n) Where n is the time perio (in ays), is the ay where the inicator is calculate. Ups is the sum of gains over the n perio an Downs is the sum of losses over the n perio. When calculate, the RSI line forms a signal between 0 an 00, which specifies etermine overbought or oversol conitions when its value is above or below specific levels. (4)

5 An example of the graphical representation of the RSI 4, with buy signal on level 30, an sell signal on level 70, is shown on Figure 5. Figure 5 - Evaluation of the RSI 4 from 2006 to 2009 in the S&P PARAMETERS OF TECHNICAL INDICATORS After efining the strategies it is necessary to efine the parameters to use both in the SMAC, MAD, an RSI strategies. As the strategies base on Moving Averages have two parameters, with similar meanings: The first parameter is similar to both strategies, the time perio of the long term MA. The secon parameter in one strategy is the time perio of a short term MA an in the other strategy is the istance between the two points use to calculate the secant. Both this parameters shoul be a meium term perios. The RSI strategy use has 5 parameters. The first one is the perio of the RSI, the secon one is the level for buying long positions, the thir one is the sell level to exit this positions. The fourth an fifth parameters refer to the shortselling strategy, the fourth is the level for enter a shortselling position, an the last parameter is the level for exiting short-selling position. 3.4 GENETIC ALGORITHM KERNEL 3.4. GENETIC ENCODING The chromosome create must represent the Technical Inicators use, in this way the SMAC chromosome is represente by two genes, one for the shortest MA other for the longest MA in ays (natural numbers), the interval of this values is between an 250 (this value is above the largely use MA for long term analysis: 200 ays). The same rule applies to the MAD chromosome, where one of the parameters is the gap an the other the number of ays of the MA. The RSI is represente with five genes, all being natural numbers, one for each parameter. In the next table is represente the maximum an minimum level allowe for each gene: Table - Limits for the parameters of the Technical Inicators. Gene Minimum Maximum SMAC Longest MA SMAC Shortest MA 200 MAD Longest MA 250 MAD Gap 250 RSI Perio 0 4 RSI Buy Entry Level 0 00 RSI Buy Exit Level 0 00 RSI Short-Sell Entry Level 0 00 RSI Short-Sell Exit Level 0 00 Aitionally to that an since the Chromosome will have several Inicators it s necessary to have a weight mechanism in the chromosome. To tackle this problem with each Technical Inicator in the Chromosome is associate a weight between 0 an 5. An each chromosome will also have a weight require to trae, between an the maximum weight possible (number of Technical Inicators in the chromosome times 5). To calculate the final ecision in each ay, each rule ecision ( for a buy ecision, 0 for no ecision, an - for shortsell) will be multiplie for the weight associate with that rule. If the total some of the ecisions is higher than the Weight Require to Trae, a buy signal is generate, an if the sum is lower than the negative value of the Weight Require to Trae, a short-sell signal is generate. In Table 2 it is shown a representation of a possible chromosome with two strategies, the SMAC an the MAD: Table 2 - An example of a Chromosome WTT W SMAC W MAD Chromosome The table also shows the Weight to Negotiate (WTT), an the Weigth associate with each one of the strategies FEATURES OF THE GA The Genetic Algorithm use for the optimization uses a stanar optimization proceure. The selection of iniviuals for crossover is chosen base on a roulette wheel selection (but only the best half of the population enters the selection process), an the probability of being chosen is equal to the ratio: iniviual fitness function /

6 Sum of fitness of all l iniviuals. Each iniviual can be chosen any number of times for crossover (the only exception is that an iniviual cannot be chosen to crossover with himself). The crossover is a one-point crossover, each breaing generates the two possible istinct chilren an inclues them in the population. In the chromosome ome of only one inicator (SMAC, MAD or RSI) the chilren are create by swapping the long an shortest MA ay. In the multiple strategies chromosome the chilren are create by ranomly selecting a point in the mile of the chromosome an swapping the genes (of the two parents) to the left an to the right of the crossover point. The fitness function use was also being analyze on this research, so several research functions where teste, an their escription an their results can be foun on the next chapter. 4. RESULTS 4. METRICS USED On Investment: This is the most basic metric use for evaluating investment strategies, an it calculates how much money you earn, for each unit of money investe, the formula to calculate the ROI is in (5). Vf - Vi ROI = V Drawown: This metric is use to calculate risk an it measures the maximum lost (in percentage) that the strategy has suffere over time. Sharpe : The Sharpe is a measure that was create by Nobel Prize William Sharpe, to measure the rewar-to-variability ratio of a traing strategy [9]. This measure allow to compare two strategies with ifferent returns, an see if the aitional return of one strategy is ue to applying a more risky strategy, or to a smarter investment strategy. The Sharpe formula is (6): R Sharpe = σ i R f Where R is the average return of the strategy, R f is the risk free rate (normally the rate of the US treasuries security). An σ is the stanar eviation of the strategy. The risk free rate must be on a treasury security with the same timeframe that the investment strategy, since we are consiering 6 years, the more suitable security is the 5 year Treasury Note. A secure investment woul be buying a 5 year Treasury Note on 2004 an with a 3.36% yiel. (6) (5) Sortino : Sortino is similar to the Sharpe, because it s also a rewar/risk racio. The main ifference is it only penalizes the negative returns an not the positive but isperse results as the Sharpe oes. The Sortino is calculate like the Sharpe, but instea of the Stanar Deviation it uses the Downsie Risk. The ownsie risk is the eviation of the values that are below some threshol (for example, below 0%). 4.2 COMPARISON OF RESULTS All the results presente in this chapter were base on 50 runs mae for each strategy, the histograms present the results of the 50 runs, an the values presente in the tables are the average value of the 50 runs. First Test The Impact of the Fitness function: The first approach was to try several ifferent fitness functions, the fitness functions chosen where base on the metrics presente above, an the main goal was to ientify fitness functions that coul fin less risky solutions, even if they ha lower returns. The four fitness functions use were: 2 x Drawown Sharpe Sortino In Figure 6 we can see the Histogram for the test perio of the Annualize s for the 50 runs for each fitness function. Figure 6 Histogram of the Annualize s for the 4 Fitness Functions. In this figure we can see that the functions have ientical results an can be approximate to Gauss istribution. In Table 3 we can see the Evaluation Metrics of the 4 Fitness Functions uring the test perio. For each evaluation metric (line of the table) the best result is highlighte in bol.

7 We can see that the results of all strategies are not very significant ifferences, for example, average returns ranging between 8.2% an 9.%, an the average Sharpe ratio between 0.52 an 0.6. Table 3 Evaluation Metrics for the 4 fitness functions teste. Evaluation Metric Av. Annualize Average Drawown Average Sharpe Average Sortino Rent. - 2x DD Sharpe 9.0% 9.00% 8.50% 8.20% 27.90% 27.20% 29.0% 27.90% In the next picture we see the evolution of overall return, where once again we can see that the final returns are very similar. Figure 7 Evolution of overall return for the 4 fitness functions in the test perio (2004 to 2009). Sortino Secon Test - Chromosome with only one Technical Inicator: Several tests were mae, with ifferent chromosome configurations. This test has a chromosome with only one traing rule. The three traing rules presente before (MAD, SMAC an RSI) were use an the Histogram comparing these strategies is on Figure 8: As can be seen from the figure, chromosomes with MAD an SMAC strategies take clear avantage over the strategy base on RSI. Since the MAD an SMAC have a curve similar to the normal istribution with mean aroun 8% to 9% an the strategy base on RSI has the majority of the executions with returns of less than zero. In the next table are the evaluation metrics for these strategies. Again, for each evaluation metric (line of the table) the best result is highlighte in bol. Table 4 Evaluation Metrics for the MAD, SMAC an RSI. Evaluation Metric Av. Annualize MAD 8.80% Average Drawown 27.30% Average Sharpe Average Sortino SMAC RSI 6.20% -2.40% 3.90% 9.20% % The results of this table confirms what ha alreay been foun in histogram, MAD an SMAC have positive results, an the MAD is clearly the best strategy, because it has the best result in most metrics. The fact that the strategy RSI has rawown s better than the others is ue to the fact that the strategy is very little time on the market (as it is a losing strategy, the optimization performe by the GA is to negotiate the least possible) an therefore limit the losses. Thir Test Chromosome with the same two strategies (but ifferent parameters): In the next test the chromosomes where moifie in orer to ouble the strategies presente in the previous example. In this way the chromosomes teste ha the following strategies: MAD, MAD, SMAC, SMAC, an RSI, RSI. The histogram of these chromosomes results are in Figure 9. Figure 8 - Histogram of the Annualize s of the three original Technical Inicators.. Figure 9 Histogram of the Annualize s of the chromosomes with ouble traing rules.

8 Regaring the previous test, it can be seen that the strategies SMAC an MAD improve their results by reucing elements with the worst returns, which are much less than in the previous test. As for the RSI it remaine with poor performance. In Table 5 are the evaluation metrics for these strategies. Table 5 Evaluation Metrics for the ouble strategies chromosomes. Evaluation Metric Av. Annualize MAD, MAD SMAC, SMAC 8.90% 8.20% Average Drawown 27.20% 27.40% 24.00% Average Sharpe Average Sortino RSI, RSI -3.00% Again we conclue that the strategies MAD an SMAC get much better results that the RSI. MAD strategy has some avantage by having the best performance on most metrics. In comparison with this table an the previous: The Average get s better, whit the MAD going from 8.4% to 8.9% an the SMAC going from 6.2% to 8.2%. Regaring the Sharpe, MAD remains practically (0.6 vs. 0.60), an SMAC rises from 0.29 to We can conclue that oubling the strategies has a positive effect on returns, with almost no impact on risk, the average rawown oes not increase, an the Sharpe an Sortino o not ecrease. Last Test Chromosome with 3 combine strategies an comparison with the Buy & Hol): In this test the best strategies from the previous tests have been combine. The strategies base on MAD an SMAC are combine in to a single chromosome, in orer to achieve even better results. In this way, the strategies create are a combination of 3 Technical Inicators, one from the first test, an two repeate strategies from the secon test. These strategies are the SMAC, SMAC, MAD (or simply 2xSMAC, MAD ) an the MAD, MAD, SMAC (or 2xMAD, SMAC ). These strategies are then compare with the Buy & Hol to see if they can beat the market. In Figure 0 are the annualize nualize returns of these two strategies an the Buy & Hol, both strategies beat the Buy & Hol. As they have the majority of the executions in intervals with big return than the Buy & Hol. Figure 0 Histogram of the Annualize s of the new strategies an Buy & Hol. In Table 6 we can see the Evaluation Metrics for the 2xSMAC, MAD, 2xMAD, SMAC an Buy & Hol strategies. Table 6 Evaluation Metrics for the three strategies. Evaluation Metric Av. Annualize 2xSMAC, MAD 9.0% Average Drawown 26.80% Average Sharpe Average Sortino xMAD, SMAC Buy & Hol 9.0% 3.40% 27.70% 56.80% In this table we can see that the 2xMAD, SMAC is the best strategy because it has the best results in almost all metrics. Although the 2xSMAC, MAD is not far away in most metrics, an has a lower rawown than the other strategy (once more the ifference is tiny). In comparison with the Buy & Hol, once again, both strategies emonstrate clear avantages, in all metrics. Not only the returns are better, but the risk associate metrics are also better. Meaning that these strategies obtain better gains that the Buy & Hol without increase the strategies risk. Figure Evolution of the three strategies overall return from 2004 to In Figure we can see the evolution of the strategies in the test perio. We can see that both strategies cannot

9 obtain relevant gains in the Bull Market (2004 to the en of 2007) compare to the Buy & Hol. In fact, the 2xMAD, SMAC use the Buy & Hol strategy uring that perio, staying all the time in the market. But when the Bear Market of 2008 arrives, the optimize strategies can etect the change of the tren in time, an profit by starting short- -selling positions. 5. CONCLUSIONS This ocument presente the use of Genetic Algorithms to optimize the parameters of various Technical Inicators an with them create various traing strategies. The results obtain showe that this strategies beat significantly the Buy an Hol (the 2xMAD. SMAC strategy ha an average of 9.% against the 3.4% of the Buy an Hol), once more proving the valiity of Technical Analysis. The use of the compose chromosomes has also shown better results than the use of any of the inicators iniviually. 6. BIBLIOGRAPHY [] Boas-Sagi, D. J., Fernánez, P., Hialgo, J. I., Soltero, F. J., an Risco-Martín, J. L. Multiobjective optimization of technical market inicators. In Proceeings of the GECCO '09 Annual Conference Companion on Genetic an Evolutionary Computation Conference, Montreal, Canaa, 2009, [2] Brock, William & Lakonishok, Josef & LeBaron, Blake, 992. Simple Technical Traing Rules an the Stochastic Properties of Stock s. Journal of Finance, American Finance Association, vol. 47(5), pp [3] Canegrati, E., A Non-Ranom Walk own Canary Wharf, in MPRA Paper. 2008, University Library of Munich. [4] Chan, E. P., Quantitative Traing, John Wiley an Sons, New Jersey, [5] Ellis, Craig A. & Parbery, Simon A., Is smarter better? A comparison of aaptive, an simple moving average traing strategies. Research in International Business an Finance, Elsevier, vol. 9(3), 2005, [6] Fama, Eugene F., 998. Market efficiency, long-term returns, an behavioral finance. Journal of Financial Economics, Elsevier, vol. 49(3), pp [7] Fernánez-Blanco, P., et al., Technical market inicators optimization using evolutionary algorithms. in Proceeings of the 2008 GECCO conference companion on Genetic an evolutionary computation Atlanta, USA, [8] Gorgulho, A., Neves, R. an Horta N., et al., Using GAs to Balance Technical Inicators on StockPicking for Financial Portfolio Composition. in Proceeings of the 2009 GECCO conference companion on Genetic an evolutionary computation Montreal, Canaa, [9] Hassan, G. an Clack C., "Robustness of Multiple Objective GP Stock-Picking in Unstable Financial Markets," in Genetic an Evolutionary Computation Conference, Montreal, Canaa, 2009, [0] Jegaeesh, N., an Titman, S. s to buying winners an selling losers: Implica-tions for stock market efficiency, Journal of Finance 48, 993, [] Kaufman, P. J., New Traing Systems An Methos, John Wiley & Sons Inc., San Francisco, CA [2] Kimoto, T. an Asakawa, K., Stock Market Preiction System with Moular Neural Networks, IJCNN International Joint Conference on Neural Networks, vol., 990, -6. [3] Kirkpatrick, C. D. an Dahlquist R. D., Technical Analysis, FT press [4] Leigh, W., Moani N. an Hightower, R., A computational implementation of stock charting: abrupt volume increase as signal for movement in New York Stock Exchange Composite Inex, Decision Support Systems, Volume 37, Issue 4, September 2004, pp [5] Leigh, W., Frohlich, C. J., Hornik, S., Purvis, R. L. an Roberts, T. L., Traing With a Stock Chart Heuristic, IEEE Transactions on systems, man, an cybernetics Part A: Systems an Humans, vol. 38, no., january 2008 [6] Leigh, W. an Purvis, R., Implementation an valiation of an opportunistic stock market timing heuristic: One-ay share volume spike as buy signal, Expert Systems with Applications, Volume 35, Issue 4, November 2008, [7] Michalewicz, 994. Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, 2n eition, 994 [8] Rey, D. M. an Schmi, M. M., "Feasible momentum strategies: Evience from the Swiss stock market," Financial Markets Portfolio Management, vol. 2, 2007, [9] Sharpe, W. F, The Sharpe. Journal of Portfolio Management 2 () 994, [20] Wagman, L, "Stock Portfolio Evaluation: An Application of Genetic-Programming-Base Technical Analysis," Genetic Algorithms an Genetic Programming at Stanfor, 2003, [2] Yan, W. an Clack, C., Evolving Robust GP Solutions for Hege Fun Stock Selection in Emerging Markets, in Proceeings of the 2007 GECCO conference companion on Genetic an evolutionary computation. 2007, Lonon, UK,

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