# High Frequency Trading using Fuzzy Momentum Analysis

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2 Proceedings of the World Congress on Engineering 2 Vol I WCE 2, June 3 - July 2, 2, London, U.K. system uses fuzzy reasoning to analyse the current market conditions according to which a certain equity price is currently moving. The membership functions were later modified using ANFIS. This new approach in the optimisation of membership functions led to an improvement of the results. II. FUZZY INFERENCE There are many types of fuzzy inference systems that have been proposed in literature. However, the most common is the Sugeno model [], which makes use of a list of rules in the form of if-then rules to produce an output for each rule. Rule outputs consist of the linear combination of the input variables as well as a constant term; the final output is the weighted average of each rule s output. The rule base in the Sugeno model has rules of the form:, = + +. (), = + +. (2) Where and are predefined membership functions, and are membership values, and, and are the consequent parameters. When we calculate the equation of first-order Sugeno [], the degree of membership variable of in membership function of are multiplied by the degree of membership variable of and in membership function, and the product is deemed a linear regression weight ( ). Finally, the weighted average and is deemed the final output, which is calculated as follows: = + + (3) In the case of designing a fuzzy system for financial modelling, as is the case in this paper, one should opt to use the Mamdani model [7], which is based on linguistic variables and linguistic output. A fuzzy inference system is a rule-based fuzzy system that can be seen as an associative memory and is comprised of five stages:. An input stage, this consists of measuring what the input would be 2. Database stage which defines membership functions of the fuzzy sets used in the fuzzy rules. 3. A Processing stage that determines what action needs to be carried out based on fuzzy 'IF-THEN' rules. 4. An averaging stage that determines the centre of mass for all system conditions. 5. Finally an output stage that is a crisp control decision or an output. An extension to Fuzzy Inference Systems is the Adaptive Neuro Fuzzy Inference System (ANFIS) which can be used for optimising the membership functions by combining fuzzy reasoning combined with the pattern recognition capability of neural networks [6]. The ANFIS approach learns the rules and membership functions from data [6]. It represents an adaptive network of nodes and directional links with associated learning rules. It is called adaptive because some, or all, of the nodes have parameters which affect the output of the node. These networks identify and learn relationships between inputs and outputs. III. FUZZY LOGIC MOMENTUM ANALYSIS SYSTEM (FULMAS) Creating a fuzzy inference system to detect momentum is a complex task. The identification of various market conditions has been a topic subject to various theories [5] and suggestions. This paper proposes a fuzzy inference system that categorises the market conditions into seven categories based on price movement, using the current volume to determine the participation rates (PR) of the trading system each time. The participation rate is the percentage of volume available in the market that the system will buy or sell at each trade. Let denote the current price, the previous price and is an up/down fluctuating counter that goes positive or negative according to the movement of the price. Whenever the price goes up, it adds, and when the price goes down, it subtracts. This indicator is used to identify market conditions for amount of ticks (where one tick is one price observation), where if the market is moving strongly upwards, it will be detected by having more ones than - or zeros. This can be explained in the equation =, (3) where is the amount of elapsed time that we want to detect the momentum for. The first step in designing the FUzzy Logic Momentum Analysis System, FULMAS, involves defining the market conditions that the fuzzy system has to identify. In this study, the following seven market conditions are used to cover all possible movements of the price series:. Rallying 2. Strong up 3. Slightly up 4. Average 5. Slightly down 6. Strong down 7. Crashing These conditions are considered linguistic values that the fuzzy logic system will use, and they will be responsible for determining the current state in the price formation process and its momentum. As momentum builds, the system considers the previous amount of ticks and performs an inference procedure by adding all the up or down directional movements of the current price to the previous price in order to determine whether the general trend has been up or down after x points. The momentum is detected as displayed in the pseudo-code in Listing. BEGIN P = Price; K = ; Start data feed (i); REPEAT if P(i) > P(i-) then k(i) = k + ; else if P(i) < P(i-) then k(i) = k - ; else then k(i) = ; END Listing : Calculating momentum ISBN: ISSN: (Print); ISSN: (Online) WCE 2

3 Proceedings of the World Congress on Engineering 2 Vol I WCE 2, June 3 - July 2, 2, London, U.K. For example, if we want to detect the momentum of the last x= price observations (ticks), we add all the up, down fluctuations and then input the resulting crisp value to the fuzzy system, which would lie somewhere in the membership functions, as shown in the upper panel in Figure. The choice of triangular membership functions was made after using the expert based method due to their mathematical simplicity. Triangular shapes require three parameters and are made of a closed interval and a kernel comprised of a singleton. This simplifies the choice of placing the membership functions. The expert merely has to choose the central value and the curve slope on either side. The complimentary characteristics of neural networks and fuzzy inference systems have been recognised and the methodologies have been combined to create neuro-fuzzy techniques. Indeed, earlier work by [] described an artificial neural network with processing elements that could handle fuzzy logic and probabilistic information, although the preliminary results were less than satisfactory. In this study, we use ANFIS from another perspective, i.e. to optimise the membership functions already created in the previous section. This is performed by feeding the ANFIS system both the training data and the desired output, and tuning the ANFIS in order to reach the target result by modifying the membership functions. In other words, at each instance, ANFIS is fed the results currently obtained from the fuzzy system in the previous section, and then a set of 'target' prices or data is inserted into ANFIS. This target price will be an optimal price that is far better than the current one (a cheaper price if on buy mode or a higher price if in sell mode). The system runs and modifies the membership functions in each epoch in order to get as close to the optimal price as possible. We also test the approach on another type of membership functions, which is the bell-shaped membership function (see Figure 2, upper panel). Figure and 2 show both applied membership functions before (upper panel) and after (lower panel) ANFIS was initialised to reach optimal 'target' results. Comparing the results of both optimised membership functions, an improvement in the original system was discovered. The optimised triangular membership functions have also outperformed the optimised bell-shaped membership functions; this confirms the experts opinion mentioned above concerning the choice of the triangular membership functions. The same procedure is applied for calculating the linguistic variable volatility, where the linguistic values. Very High 2. High 3. Medium 4. Low 5. Very Low The fuzzy logic system considers both market momentum and volatility. It generates the rules and then takes a decision based on the amount of market participation. The combined approach is summarised in Figure 3. D e g r e e o f m e m b e r s h ip D e g re e o f m e m b e r s h ip D e g r e e o f m e m b e r s h ip D e g re e o f m e m b e rs h ip Initial MFs Initial MFs input2 Final MFs input2 Figure : Triangular membership functions optimised using ANFIS input Final MFs input Figure 2: Bell-shaped membership functions optimised using ANFIS Figure 3: Extracting fuzzy rules from both volatility and momentum ISBN: ISSN: (Print); ISSN: (Online) WCE 2

6 Proceedings of the World Congress on Engineering 2 Vol I WCE 2, June 3 - July 2, 2, London, U.K. Figure 7: Optimised FULMAS outperformance to SVS system for buying NOK Figure 8: Optimised FULMAS outperformance to SVS system for selling NOK VI. CONCLUSION There are still relatively few studies comparing artificial intelligence approaches and classical models such as time series theory. However, recent research continues to suggest that adopting artificial intelligence techniques for the technical analysis of financial systems will yield positive results. With several researchers now questioning the Efficient Market Hypothesis [4], particularly in light of the recent global financial crisis, artificial intelligence approaches such as time series that can improve analysis will likely continue to hold the interest of researchers and investors alike. The problem of order execution is a very complicated one. To be able to provide the best price, an execution system must dynamically change the participation rates at each instance in order to cater to price changes that are driven by momentum and volatility. This study has introduced a system that utilises fuzzy logic in order to justify the current market condition that is produced by the accumulation of momentum. The proposed FULMAS is a fuzzy logic momentum analysis system that outperforms the traditional SVS used in the industry, which is often based on executing orders dependent on the weighted average of the current volume. Results of the implemented system have been displayed and compared against the traditional system. The system reveals that the AI enhanced approach increases the average profitability on orders on both the buy and sell sides Figure 9: Optimised FULMAS outperformance to SVS system for buying VOD REFERENCES [] Boehmer, E (25). Dimensions of execution quality: Recent evidence for U.S. equity markets. Journal of Financial Economics, 78, pp [2] Dourra, H. and Siy, P. (22). Investment using technical analysis and fuzzy logic. Fuzzy Sets and Systems, Vol. 27 (22), pp [3] Ellul, A., C. Holden, P. Jain, and R. Jennings, (23), Determinants of order choice on the New York Stock Exchange. Working Paper. [4] Fama, E. F. (965). The behaviour of stock market prices. The Journal of Business, Vol. 38, pp [5] Griffin, J. (27). Stock market trading and market conditions, forthcoming, Review of Financial Studies. [6] Jang, R. (993). ANFIS: Adaptive network-based fuzzy inference system, IEEE Transactions on Systems, Man and Cybernetics, Vol. 23 (3) pp [7] Mamdani, E. and Assilian, S. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies 7 (), pp. -3. [8] Song, Q. and Chissom, B.S. (993). Forecasting enrollments with fuzzy time-series. Part I. Fuzzy Sets and Systems 54, pp. -. [9] Song, Q. and Chissom, B. S. (994). Forecasting enrollments with fuzzy time-series. Part II. Fuzzy Sets and Systems 62, pp. -8. [] T. Takagi and M. Sugeno. Fuzzy identification of systems and its application to modeling and control, IEEE Transactions on Systems, Man and Cybernetics, Vol. 5, pp 6-32, 985. [] Wong, F.S. and Wang, P. Z. (99). A stock selection strategy using fuzzy neural networks. Neurocomputing 2 (99/9), pp Figure : Optimised FULMAS outperformance to SVS system for selling VOD ISBN: ISSN: (Print); ISSN: (Online) WCE 2

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