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Journal of Banking & Finance 37 (2013) 578-586 Contents lists available at SciVerse ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf Fuzzy logic, trading uncertainty and technical trading Nikola Gradojevic a,c,, Ramazan Genay b,c a Faculty of Business Administration, Lakehead University, 955 Oliver Road, Thunder Bay, ON, Canada P7B 5E1 Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6 c The Rimini Center for Economic Analysis, Via Patara, 3, 47900 Rimini (RN), Italy b a r t i c l e i n f o Article history: Received 12 August 2012 Accepted 24 September 2012 Available online 4 October 2012 JEL classication: G0 G1 F3 Keywords: Foreign exchange markets Technical trading Uncertainty Fuzzy logic Filtering a b s t r a c t From the market microstructure perspective, technical analysis can be protable when informed traders make systematic mistakes or when uninformed traders have predictable impacts on price. However, chartists face a considerable degree of trading uncertainty because technical indicators such as moving averages are essentially imperfect lters with a nonzero phase shift. Consequently, technical trading may result in erroneous trading recommendations and substantial losses. This paper presents an uncertainty reduction approach based on fuzzy logic that addresses two problems related to the uncertainty embedded in technical trading strategies: market timing and order size. The results of our high-frequency exercises show that 'fuzzy technical indicators' dominate standard moving average technical indicators and lter rules for the Euro-US dollar (EUR-USD) exchange rates, especially on high-volatility days. 2012 Elsevier B.V. All rights reserved. 1. Introduction Technical trading models typically rely on technical indicators constructed from past price and volume information that generate discrete (buy or sell) trading recommendations. Such models are atheoretic and ignore fundamental information about the price; however, they have been shown to result in trading protability. Specically, the success of technical trading violates the weak form of the efcient market hypothesis, which states that past prices should not assist traders in earning unusually high returns. In general, the literature examines the practical value of two types of analysis: charting, which identies geometric patterns in the history of prices, and technical indicators approach, which mechanically applies mathematical trading rules constructed from past and present prices. Studies that nd charting protable include Chang and Osler (1999), Lo et al. (2000) and Savin et al. (2007), whereas evidence for the protability of technical indicators can be found in Neftci (1991), Levich and Thomas (1993), Brock et al. (1992), Neely et al. (1997), Allen and Karjalainen (1999) and Genay (1992). Corresponding author at: Faculty of Business Administration, Lakehead University, 955 Oliver Road, Thunder Bay, ON, Canada P7B 5E1. Tel.: +1 807 343 8419; fax: +1 807 343 8443. E-mail addresses: ngradoje@lakeheadu.ca (N. Gradojevic), rgencay@sfu.ca (R. Genay). 0378-4266/$ - see front matter 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankn.2012.09.012 Some of the above contributions combine technical analysis with other statistical methodologies. For example, Lo et al. (2000) show that charting based on automatic pattern recognition with kernel regressions adds value to the investment process. In a related study, Savin et al. (2007) produce similar results for price patterns. Genay (1992) uses technical indicators to feed an articial neural network model and thereby demonstrates that trading signals produced by such a combination outperform a simple buy-and-hold strategy. Allen and Karjalainen (1999) apply genetic programming to search for ex-ante ''optimal'' trading rules. In summary, several papers present evidence that technical analysis can be informative for the price, although its protability can vary over time, which is in line with the adaptive markets hypothesis (Lo, 2004).1 In light of the market microstructure theory, technical trading may be protable when informed traders make systematic mistakes or when uninformed traders have a predictable impact on price (Harris, 2003). When technical traders reveal and trade on mistakes made by informed traders, they in turn themselves become informed traders. The trading of these so-called information-oriented technical traders corrects prices and improves market efciency. Information-oriented technical trading is, 1 See Irwin and Park (2007) and Menkhoff and Taylor (2007) or, more recently, Neely and Weller (2012) for extensive surveys on the application of technical analysis in the foreign exchange and equity markets. N. Gradojevic, R. Genay / Journal of Banking & Finance 37 (2013) 578-586 however, quite difcult in practice because informed traders correct their mistakes and learn from their past actions. This process diminishes protable information-oriented technical trading opportunities. In contrast to technical trading on the informed traders' actions, sentiment-oriented technical traders exploit predictable price patterns caused by uninformed traders. Such an order-anticipating approach attempts to front-run the uninformed traders and trade before they trade. Sentiment-oriented technical trading can be useful when it correctly anticipates the impacts that uninformed traders will have on prices. In this paper, we extend the activity of sentiment-oriented technical traders (or order anticipators) to 'uncertainty reduction', whereas the uninformed traders are considered to be pure technical traders who employ simple technical indicators. The practice of mechanically applying technical indicators in investment management without any uncertainty considerations could potentially be dangerous. Uncertainties in foreign exchange (FX) and equity markets can originate from, for example, market regime shifts, the impact of large trades on price, short-sale restrictions, incomplete data or behavioral issues. Recently, Lo and Mueller (2010) have argued that the presence of inappropriately identied uncertainties in a quantitative investment strategy can adversely affect risk management efforts. They introduce ve levels of uncertainty: perfect certainty (e.g., direct trading costs), risk (e.g., probability distributions of trading volumes), fully reducible uncertainty (e.g., statistical framework for time series analysis), partially reducible uncertainty (e.g., multiple market regimes) and irreducible uncertainty (e.g., tail risk). Each of these levels is to be addressed with an appropriate set of skills and methodologies. For instance, the application of time series or linear regression analysis to tail events rather than extreme value theory could be disastrous for an investment strategy. In the same vein, utilizing technical indicators in trading while neglecting the uncertainty aspect of such actions would probably result in a series of unnecessarily risky trading recommendations. Our analysis addresses the issue of trading model uncertainty (or trading uncertainty) that belongs to the partially reducible uncertainty domain, as dened above. This situation involves the uncertainty in decision making that arises if there is insufcient knowledge regarding the appropriateness of the trading model. Trading uncertainty represents a renement of the economic uncertainty concept also known as model uncertainty. Model uncertainty generally arises from the potential incorrectness of the choice of the model that is generating the data. In this context, Pesaran and Timmermann (2002) link model uncertainty to an investor facing many competing forecasting models.2 This problem is the reason why predictability and protable opportunities in nancial markets are short-lived (Timmermann and Granger, 2004).3 This paper introduces fuzzy logic as a tool that can help traders to control for the trading uncertainty aspect of employing technical indicators. The information generated by technical indicators is possibly imprecise, incomplete or unreliable. Fuzzy logic by its very nature tolerates uncertainty by dening variables, i.e., technical indicators, as imprecise linguistic terms that cover a broad fuzzy variable range; for example, trading signals can be expressed in a more sophisticated fashion as a 'strong buy', 'hold' or 'very strong sell.' Furthermore, technical indicators such as moving averages are prone to indicating a turning point later than the it actually occurs, as they are essentially imperfect lters 2 This type of uncertainty is also used to denote heteroskedasticity in the conditional variance of asset returns (Easley and O'Hara, 2010). 3 Model averaging techniques are typically used to reduce model uncertainty. Such approaches can be found when model uncertainty is present in the shaping of monetary policy (Cogley et al., 2011), hedge fund pricing (Vrontos et al., 2008) and investment management (Avramov, 2002; Pesaran et al., 2009). 579 with a nonzero phase shift (Genay et al., 2001). Management of uncertainty in such situations is of the utmost importance. Fuzzy logic involves more continuous and conservative decision making than buy or sell recommendations, and it thereby partially reduces trading uncertainty in volatile markets. In addition, fuzzy logic can reduce trading costs by controlling for the order size, whereas pure technical indicators commit all available funds to a trading position. By using fuzzy logic, we attempt to resolve two problems related to the uncertainty embedded in investment strategies based solely on technical trading rules: market timing (''when to trade'') and order size (''how to trade''). Studies on the application of fuzzy logic in nancial economics have been scarce (Bekiros and Georgoutsos, 2007) and usually are considered mostly in tandem with other methodologies such as articial neural networks (Gradojevic, 2007) or reinforced learning of agent-based systems (Bekiros, 2010; Tay and Linn, 2001). Additionally, Bojadziev and Bojadziev (1997) uses fuzzy logic to evaluate a client's risk tolerance based on the annual income and total net worth, whereas Serguieva and Hunter (2004) evaluate the risk associated with investing in 35 UK companies traded on the London Stock Exchange. With regard to technical trading, research efforts have centered on fuzzy logic-assisted charting (Zhou and Dong, 2004) but not on technical indicators. The fact that charting is primarily visual, whereas the technical indicators approach is essentially mathematical, suggests that the latter is more amenable to statistical methodologies such as fuzzy logic. In this paper, our goal is to reduce the trading uncertainty of the standard technical indicators approach by utilizing fuzzy logic technical trading rules that are more robust with respect to errors in decision-making (trading). We directly compare the efcacy of standard technical indicators with that of fuzzy technical indicators for high-frequency (1-min) EUR-USD exchange rates in 2005. Furthermore, we develop ve testable hypotheses that involve the relationship between high-frequency protability and volatility (Hypotheses 1-3) and the ranking of the trading strategies (Hypotheses 4 and 5). We nd that the extension of standard technical trading strategies with the fuzzy control methodology results in improved protability. Our results show that fuzzy technical indicators are particularly useful for reducing trading losses on highly volatile days of the week. On such days, prots from pure technical trading strategies decrease and prots from fuzzy technical trading strategies increase. Overall, higher volatility leads to greater excess returns of fuzzy technical trading strategies relative to pure technical trading strategies. The documented gains in conditional mean returns are, in general, statistically signicant over 50 weeks of 1-min EUR-USD exchange rates, whereas the buy-and-hold strategy performs poorly, irrespective of volatility. Finally, we link our results to market microstructure in the sense that the protability of fuzzy logicbased technical trading might be used to explain the motivation for pursuing sentiment-oriented technical trading strategies. Such strategies are devised by order anticipators who front-run uninformed traders who apply simple technical trading rules. In accordance with Bekiros (2010), we argue that fuzzy control acts as a learning mechanism through which it is possible to better predict turning points and to thus react before uninformed technical traders trade. Hence, we conclude that the protability of sentiment-oriented technical trading is particularly pronounced during high-volatility trading sessions. In Section 2, we provide a brief overview of fuzzy logic, including an illustrative example for the S&P-500 Index. The data are described in Section 3. This section also develops testable economic and ranking hypotheses for the competing trading strategies. The construction of our fuzzy technical indicators and our results are reported in Section 4. We conclude and offer some potential future research avenues in Section 5. 580 N. Gradojevic, R. Genay / Journal of Banking & Finance 37 (2013) 578-586 2. Fuzzy logic fundamentals Fuzzy logic is built upon the notion of fuzzy sets (Zadeh, 1965). Unlike traditional sets (intervals), fuzzy sets allow for the concept of partial membership. This enables discrimination between elements that are relevant to the phenomenon of interest and those of borderline importance that involve imprecision and uncertainty. Information granules, such as ''high speed'', ''signicant risk'' or ''strong sell'', can be processed using fuzzy logic whereby each linguistic term (''high'', ''signicant'' and ''strong'') describes a fuzzy set. A fuzzy set (A) dened in X is represented by its membership function as follows: A:X ? [0, 1]; where A(x) denotes a degree of membership of x in A. Membership functions can be of various types, including triangular, trapezoidal, Gaussian, sigmoidal and polynomial. It should be noted that larger values of a membership function indicate higher degrees of membership.4 Any fuzzy model has three main components: (1) a fuzzy ''rule base'' in the form of a set of ''if-then'' rules (expert knowledge about the model), (2) a fuzzication module that transforms the explanatory variables (inputs) into fuzzy variables and (3) a defuzzication module that converts the conclusion from the fuzzy domain into the dependent variable (output). To design the fuzzy model, information must be gathered on how to construct the rule base. Typically, this information is represented by the expert knowledge about the process or is compiled by studying the historical data. The rules can, for instance, state that ''if the long moving average is hLARGEi and the short moving average is hVERY SMALLi, then the technical trading signal is hSTRONG SELLi.''5 In a fuzzy system, the process of generating the output (trading recommendation) begins with fuzzifying the inputs (components of technical indicators, such as moving averages or lter values) and then executing all of the active rules from the rule base. The process generates fuzzy conclusions about the output variable for each rule. The conclusions of the active rules used in decision making are then aggregated into a fuzzy conclusion about the output variable that captures the inuence of the output membership functions associated with the rules. After defuzzication, a single value output (trading position) is generated. The following example will illustrate fuzzy decision making ('fuzzy technical indicators') and compare it with the standard moving average technical indicators approach. Fig. 1 plots daily closing prices along with the 50-day moving average (MA (50)) for the S&P-500 Index from July 1, 2010 to September 30, 2010. We will concentrate on two occasions when the price penetrated MA (50) from below, thus indicating a buy signal: August 17, 2010 and September 2, 2010. On the rst date, the S&P-500 Index closes at 1092.54, and the MA (50) is 1088.33. On the second date, the corresponding gures are 1090.10 and 1081.26, respectively. The standard moving average technical indicator generates a buy signal on both days and incurs a loss on the rst signal because the price makes an unanticipated drop on August 19. However, the fuzzy moving average technical indicator accounts for the magnitude of discrepancy between the S&P-500 Index value and the MA (50) and generates a 'WEAK BUY' signal (i.e., invest roughly 40% of your current endowment) on August 17. Hence, in contrast to the standard moving average, the fuzzy logic approach does not commit all investment funds to the position. Furthermore, on September 2, 2010, fuzzy logic recognizes the larger discrepancy between the S&P-500 Index value and the MA (50) and generates a 'STRONG BUY' recommendation (i.e., invest roughly 92% of your 4 Gradojevic (2007) and Cox (1992) provide a detailed treatment of fuzzy logic fundamentals. Essentially, fuzzy logic in the form of approximate reasoning has found applications in medicine, engineering, business, economics and meteorology. 5 The design of the fuzzy logic model applied in this paper largely follows Gradojevic (2007). current endowment). The same signal is generated by the standard MA (50) indicator, and as the S&P-500 Index continues to rise, both strategies generate a prot. Our model has two inputs, i.e., MA (50) and daily closing price, which are both fuzzied on the interval [0, 1] into the following ve triangular fuzzy membership functions: ''VERY SMALL,'' ''SMALL,'' ''MEDIUM,'' ''LARGE'' and ''VERY LARGE''. The output is a trading recommendation that is fuzzied on the interval [1, 1] into ve triangular fuzzy membership functions with the following labels: ''STRONG SELL,'' ''WEAK SELL,'' ''HOLD,'' ''WEAK BUY'' and ''STRONG BUY''. The rule base contains 52 = 25 rules that compare all possible combinations of the two inputs and produce the appropriate outputs. Of interest are the fuzzy trading recommendations on August 17, 2010 and September 2, 2010. For the values of the inputs on August 17, the fuzzy system output is 0.409, which corresponds to a 'WEAK BUY' signal. On September 2, the output generated by the fuzzy system is 0.918, which is a 'STRONG BUY' signal. Clearly, this example shows that by producing a more conservative trading signal on August 17, fuzzy control successfully avoided the extent of losses incurred by the standard moving average trading indicator. In our example, MA (50) missed the turning point on August 19, 2010 because of the so-called phase shift of the moving average lter (Genay et al., 2001). An ideal trading lter should retain lower frequencies with lesser weights towards higher frequencies. Such a lter preserves the temporal memory of the data while eliminating excessive higher frequency noise. The fuzzy Gaussian lter has such a capability. However, the MA (50) lter selectively concentrates near zero frequency with a compressed presence at lower frequencies. Such arbitrary frequency selection may omit the temporal memory necessary to identify local trends and turning points. The fuzzy rule base generates a continuous decision surface in the form of mapping from the inputs to the output. It basically accounts for the distance between the inputs and produces a trading signal that identies the exact fraction of the funds that are to be allocated to a position. Here, the distance between the inputs (i.e., between the price and the moving average or the lter) is viewed as a measure of trading uncertainty that increases as the distance decreases. As can be seen in the above example, this additional processing of pure technical indicator signals reduces the losses from missed turning points. Additionally, fuzzy logic cuts trading costs by not committing all available funds to a trading position. Therefore, although both strategies are subject to the same transaction cost, fuzzy control can adjust the trading volume. The remainder of this section describes the relevant aspects of the fuzzy control design employed by this paper. As in Gradojevic (2007), the input membership functions are Gaussian and the output is represented by triangular membership functions. Furthermore, the inference mechanism is the so-called ''Mamdani inference'' (Mamdani and Assilian, 1975), whereas the defuzzication method is the ''centroid of area''. All (long-short) moving average differences and the differences between the lter value and price are normalized and fuzzied on the interval [1, 1]. Gaussian fuzzy membership functions are used, as they produce a relatively smooth input-output mapping. These functions are dened by a mean and standard deviation that are arbitrarily set to slice the variable domain into overlapping Gaussian functions that have the same shape and the highest degree of membership for the mean value. Each input is characterized by nine states, and the following fuzzy sets are assigned: ''VERY NEGATIVE,'' ''NEGATIVE,'' ''MEDIUM NEGATIVE,'' ''WEAK NEGATIVE,'' ''STABLE,'' ''WEAK POSITIVE,'' ''MEDIUM POSITIVE,'' ''POSITIVE'' and ''VERY POSITIVE''. Similarly, the trading recommendation variable is assumed to have nine states represented by linear-shaped functions as follows: ''VERY STRONG SELL,'' ''STRONG SELL,'' ''MEDIUM SELL,'' ''WEAK 581 N. Gradojevic, R. Genay / Journal of Banking & Finance 37 (2013) 578-586 1160 S&P 500 Index MA(50) 1140 17Aug2010 02Sep2010 S&P 500 Index price 1120 1100 1080 1060 1040 1020 Jul 1, 2010 Aug 16, 2010 Sep 30, 2010 Date Fig. 1. Moving average technical indicators for the S&P-500 Index. The data cover the period from July 1, 2010 to September 30, 2010. The daily closing S&P-500 Index prices are plotted with a solid line, whereas the 50-day moving average (MA (50)) is plotted as a dashed line. We consider two dates when a buy signal was generated: August 17, 2010 and September 2, 2010 (marked with circles). SELL,'' ''HOLD,'' ''WEAK BUY,'' ''MEDIUM BUY,'' ''STRONG BUY'' and ''VERY STRONG BUY''. These states uniquely dene the trading strategy in which positive signals are interpreted as long positions and negative signals as short positions.6 3. Data and testable hypotheses 3.1. Data characteristics Our dataset is from the Electronic Broking Services (EBS) (level 1.5) and consists of tick-by-tick FX transaction prices for the EUR/ USD exchange rates spanning January 10 through December 23, 2005 for a total of 50 weeks (250 days). EBS operates as an electronic limit order book and is used for global interdealer spot trading. EBS is dominant for the EUR-USD and USD-JPY currency trading, whereas the GBP-USD currency pair is traded primarily on Reuters. The average daily EUR-USD trading volume (in USD) on EBS in 2003 was between 50 and 70 billion dollars, which was well above that on the NYSE (40 billion dollars). To avoid extreme high-frequency noise and no-activity periods in very small time windows and also to introduce as much uncertainty as possible, we focus on the 1-min frequency. This gives us 1440 observations over each 24-h period for a total of 360,000 data points. The top panel of Fig. 2 displays the 1-min USD/EUR exchange rate, and the bottom panel presents the volatility as the squared 1-min returns. Clearly, the USD appreciated over the data span, from about 1.35 USD/EUR to 1.18 USD/EUR. This trend is followed by several volatility outbursts that are mostly located at support levels. Considering that the USD appreciation trend was strongly reversed in 2006, 2005 can be viewed as a relatively risky year for currency trading. In terms of trading intensity, on average, for the EUR-USD market, there are roughly 8000 buy orders and 6000 sell orders on a given day. Fig. 3 plots the average number of 1-min buy (Panel 6 Increasing the number of states to 11 or 13 or decreasing the number of states to 7 or 5 does not alter our main conclusions about the usefulness of fuzzy control in technical trading. A) and sell (Panel B) orders as well as the total number of trades (Panel C) on each day of the week. As EBS data level 1.5 do not reveal the EUR-USD trading volume, even though the shapes of intraday trading activity curves resemble those from the literature, they should be interpreted with caution. In this paper, we are interested in the day-of-the-week effects and will thus ignore any intraday activity. It appears that, on average, Monday is the day with the lowest trading activity, whereas Fridays exhibit the highest number of trades, which is driven by buy orders (see Panel A). We will show in the following section that our fuzzy control methodology will be the most useful on Fridays, when the price volatility is highest. 3.2. Economic and ranking hypotheses The rst research question of interest concerns the highfrequency relationship between the volatility of foreign exchange returns and technical trading returns. Kho (1996) nds that periods of higher (lower) technical trading returns correspond to high (low) risk premia and volatility. However, Reitz (2006) argues that the information content of a technical trading signal is low when high exchange rate volatility disturbs inference. This mixed evidence leads to the rst testable hypothesis: Hypothesis 1. Higher volatility is associated with lower prots from pure technical trading strategies. Next, we are interested in whether fuzzy technical indicators are more useful than pure technical indicators in periods of high volatility. Therefore, our second testable hypothesis is: Hypothesis 2. Higher volatility is associated with greater prots from fuzzy technical trading strategies. The benchmark trading strategy for both technical indicator approaches is the buy-and-hold strategy. This passive strategy may be protable over longer time periods with low volatility. However, in high frequency data, buying and holding could be risky regardless of volatility. This argument motivates our third testable hypothesis: 582 N. Gradojevic, R. Genay / Journal of Banking & Finance 37 (2013) 578-586 USD/EUR exchange rate USD/EUR exchange rate (Jan. 10 Dec. 23, 2005) 1.35 1.3 1.25 1.2 1.15 0.5 1 1.5 2 2.5 3 USD/EUR exchange rate volatility 1min intervals 5 Volatility (1min squared returns) 5 3.5 3.5 x 10 x 10 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 1min intervals 3 3.5 5 x 10 Fig. 2. Exchange rate and volatility. The top panel plots the 1-min USD/EUR exchange rate and illustrates the appreciation of the USD in 2005. The bottom panel plots the volatility of 1-min squared exchange rate returns. Hypothesis 3. Volatility is not related to prots or losses from the buy-and-hold strategy. slt1 ;l2 mlt1 mlt2 ; In our high-frequency setting, it would also be important to establish ranking hypotheses with respect to the protability of the technical trading-based and the buy-and-hold strategies. First, we would like to compare the two technical trading strategies as follows: where l1 and l2 are the lengths of the short and the long moving averages, respectively. Many possible combinations of moving averages can be used, but this paper will concentrate on (l1, l2) = [(1, 50), (1, 200), (5, 200), (2, 200), (1, 150)], where l1 and l2 are 1-min intervals. In a straightforward application of technical indicators, buy signals are generated when slt1 ;l2 P 0 and sell signals are initiated when slt1 ;l2