Project Proposal ME/ECE/CS 539 Stock Trading via Fuzzy Feedback Control

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1 Project Proposal ME/ECE/CS 539 Stock Trading via Fuzzy Feedback Control Saman Cyrus May 9, 216 Abstract In this project we would try to design a fuzzy feedback control system for stock trading systems. The goal of this project is to tune parameters of the control system with fuzzy logic. This is a famous application of fuzzy logic into control engineering (for example see [5] for examples of tuning PID controllers with fuzzy logic). The output of the controller (control law, control input) is the amount of investment and the input of the controller is the instantaneous price of security we are trading. The goal is to have positive profit from investments at each time instance. Recently, Simultaneous Long-Short (SLS) controller has been introduced in the literature and it has been shown theoretically and by simulation that having SLS controller would lead to positive output of the system (i.e. positive profit of the trading in the stock market). This project would be an effort to extend the concepts of fuzzy logic into SLS controller and build a fuzzy tuned-sls controller. 1 Introduction There is plenty of literature about implementing control engineering ideas into financial markets. From 196 s researchers in the field of control engineering tried to see if the advantages of a control system can be extended to financial markets. In these efforts, system has been modeled first and then using stochastic calculus and stochastic control systems theory the control sequence is defined. Recently a new line of research has been introduced in which instead of trying to find a model for the financial market, a model-free controller has been designed [1, 3, 2]. This leads to design of a robust controller in which the only information needed for the controller is security price p(t). Therefore the advantage of this method is its independence from having a model for the financial markets. The main disadvantage, on the other hand, is the general disadvantage of feedback control systems which is their 1

2 need for having an error in order to correct it. In other words, in feedback control systems controller needs the system to commit a mistake and produce an error to be able to correct it. Fuzzy controllers are famous of being model-free, hence using fuzzy logic to tune the controller might result in a better performance of the crisp robust model-free feedback control system. Data Source & Implementation Details: Back-testing is use of historical data to test the proposed algorithm. The implementation platform in this project is Matlab. A code is written which implements crisp SLS controller without tuning of the parameters and fuzzy-sls controller with fuzzy tuning of the parameters of the controller simultaneously using the same data. At the end of the back-testing the profit of both algorithms can be compared. The historical data can be taken from Yahoo finance 1 website or Wharton Research Data Services (WRDS) 2. In this method, the training can be done by historical data of a security (like GOOGL which is trademark of google in the trading market) in a specific time interval, e.g., 2-21, and then the testing would be done by historical data of the market in interval. 2 The Crisp SLS (Simultaneous Long -Short) Controller First we start with the crisp controller. The idea is that we want to design a robust controller such that no matter what is the noise of the system (changes in the stock price p(t)), the output of the system (cumulative profit up to the time t or g(t)) is always positive. In [2, 1] it has been shown that with using two simultaneous controllers and superposing their generated control law, this goal would be achieved. The formulas for the control system in crisp case are [2, 1] ρ(t) = 1 dp p(t) (1) dg = 1 dp I(t) p (2) I L (t) = I + g L (t) (3) I S (t) = I g S (t) (4) I(t) = I L (t) + I S (t) (5) Here, p(t) is the price, g S is cumulative profit of the short controller until time t, g L is cumulative profit of the long controller until time t, dp is stock price increment, I is investment, I L and I S are investment of the long and short controllers respectively. By

3 substituting equations (3) and (4) into equation (2), we would get The Arbitrage theorem [2] claims that g(t) = I dg L = ρ(t)(i + g L ) (6) dg S = ρ(t)(i + g S ) (7) [ ( ) p(t) + p() where g(t) > for all nonzero price variations. ( ) p(t) 2] p() Proof. If solve the differential equations (6),(7), we would have [ ( g L (t) = I ) p(t) 1] p() g S (t) = I adding these two equations, one gets [ ( ) p(t) 1] p() g(t) = 1 (p (t) + p (t) 2) If we write this function as a function of p, we would get a strictly convex function. Since the function g(p) above is a strictly convex function and has its critical point at p = 1, and since for p = 1 function s value and the derivative equal zero and the second derivative is positive, the function is non-negative. This signal would be fed-back to the controller. Controller uses this signal as well as the stock price p(t) to generate control input I(t) (please see Figure 1 ). 3 Fuzzy SLS-Controller Now we want to make a fuzzy-simultaneous Long-Short controller. By fuzzy controller we mean a controller whose parameters are tuned via fuzzy logic. 3

4 Figure 1: Feed-back control system for trading. Figure is taken from [1] 3.1 Components of a Fuzzy Controller: To design a fuzzy controller we need to design four main components of a fuzzy controller: 1) rule-base 2) fuzzy inference system 3)Fuzzification interface 4) Defuzzification interface.[4] Rule-Base: Rule-base includes the rules for the controller to work. It tells us how should the controller work to have the best results. Inference Mechanism: Briefly, interference mechanism decides which rule should be used regarding the current state of the system and hence the control input is chosen by the interference mechanism. Fuzzification Interface: Fuzzification is the process of changing the input in a form that is understandable and comparable with the rules in the rule-base. Defuzzification Interface: Defuzzification is the process which is needed to interpret the output of the interference mechanism to an understandable input for the plant. 3.2 Designing a Fuzzy Controller: To design a fuzzy controller we should design each of the mentioned components in section 3.1 and also decide what are the inputs and outputs of the fuzzy controller. In this example, the target is to keep system s output g(t) always positive. In other words, from classical control theory point of view this can be interpreted as a pseudoregulator problem. The difference is that in regulators the target is to make the output of the system zero while here the goal is to keep it positive. 4

5 In this project we are trying to tune the SLS controller such that it has a better performance. The parameter which should be tuned is (Feedback gain of the static feedback controller). To find out what should the rules be, pay attention to the fact that the output is g(t) = 1 ( p (t) + p (t) 2 ) Hence, if take partial derivative with respect to we would get dg() d = 1 ( p ln p p ln p ) 1 2 ( p + p 2 ) dg d = 1 ( p 2 ln p p ln p p p + 2 ) If put it equal to zero p ln p p ln p p p + 2 = By solving this equation we would get that for all values of the price p, the answer is the trivial answer =. Hence at = the profit would not change (see Fig. 2). If draw g(t) as function of we would get Figure dg()/d Figure 2: Evolution of the slope of the profit dg/d with changes of As it is obvious from figure 2, by increasing the profit would be more and the slope would also increase.now let s see the behavior of the function itself with. (see Figure 3) By paying attention to figure 3 it is obvious that we should be interested to increase as much as possible, but increasing has other effects. 5

6 g() Figure 3: Evolution of the profit g with changes of Now let s see the rules of behaving in a market with SLS controller. If the price is increasing, we should go long (buy shares) and investment I should be positive. On the other hand, if price is going down a wise strategy would be to go short (sell stocks and receive money. If we don t have stocks going short means to borrow share from the broker and sell it and receive money) Rule-Base: Rules are: 1. IF d d2 p(t) is neglarge and p(t) is neglarge THEN is neglarge 2. IF d d2 p(t) is neglarge and p(t) is negsmall THEN is neglarge 3. IF d d2 p(t) is poslarge and p(t) is poslarge THEN is poslarge 4. IF d d2 p(t) is poslarge and p(t) is possmall THEN is poslarge 5. IF d d2 p(t) is neglarge and p(t) is poslarge THEN is negsmall 6. IF d d2 p(t) is neglarge and p(t) is possmall THEN is negmedium 6

7 7. IF d d2 p(t) is poslarge and p(t) is neglarge THEN is possmall 8. IF d d2 p(t) is poslarge and p(t) is negsmall THEN is posmedium 9. IF d d2 p(t) is possmall and p(t) is neglarge THEN is possmall 1. IF d d2 p(t) is possmall and p(t) is negsmall THEN is posmedium 11. IF d d2 p(t) is negsmall and p(t) is poslarge THEN is negsmall 12. IF d d2 p(t) is negsmall and p(t) is possmall THEN is negmedium 13. IF d d2 p(t) is negsmall and p(t) is negsmall THEN is negmedium 14. IF d d2 p(t) is negsmall and p(t) is neglarge THEN is neglarge 15. IF d d2 p(t) is possmall and p(t) is possmall THEN is posmedium 16. IF d d2 p(t) is possmall and p(t) is poslarge THEN is poslarge Also we can see the results in Table 1 Table 1: Rule table for the Fuzzy Controller d 2 p NL NS PS PL NL NL NL NM NS NS NL NM NM NS dp PS PS PM PM PL PL PS PM PL PL Fuzzy Quantification of nowledge: In this part, using membership functions, these linguistic values which we have get in the previous part would be quantified. We use triangular and trapezoidal membership functions. For membership functions which are describing the beginning and the ending of the interval, trapezoidal MF (membership function) has been taken into account and for membership functions in the middle, triangular MFs has been considered. 7

8 The interval for dp has four divisions. Depending on the application, these intervals are determined. For FOREX market, normally the difference between two consecutive prices are around 1-2 pips (1 pip =.1 of the price). Hence, less than 2 pips would be considered as small value for dp, values between 2 and 4 pips are considered as medium and larger than 4 pips are large values. (see figure 4) 1 Membership Function of dp NL NS PS PL Degree of membership dp Figure 4: Membership Function of input 2: dp For d2 p we can again define four divisions: NL (Negative Large), NS (Negative Small), PS (Positive Small), PL (Positive Large).(see Figure 5) 1 Membership Function of d2 p NL NS PS PL Degree of membership d 2 p Figure 5: Membership Function of input 1: d2 p For feedback gain () six membership function has been defined: NL (Negative Large), NM (Negative Medium), NS (Negative Small), PS (Positive Small), PM (Positive Medium), PL (Positive Large). See figure 6. Fuzzy inference system is shown in Figure 7 It is a Mamdani fuzzy inference system. For AND operation of the rules, minimum is considered. Defuzzification is also centriod. 8

9 Membership Function of the Feedback Gain 1 NL NM NS PS PM PL Degree of membership Figure 6: Membership Function of the Feedback gain dp/ (4) project (mamdani) 16 rules (6) d 2 / p (4) System project: 2 inputs, 1 outputs, 16 rules Figure 7: Fuzzy Inference System for this problem. There are two inputs, Input1: dp, Input2: and the output is d 2 p 9

10 Control surface of the controller can be seen in Figure 8. Fuzzy Inference System output surface d 2 p dp Figure 8: Control surface of the fuzzy controller 4 Numerical Example: Now let s see a numerical example. SLS controller has been implemented on the Euro to US Dollar price changes at year 2. Minute data is used (Minute data means the changes of the price is saved once per minute. Other available data are 5-Min, 15-Min, 1 Hour, 1 Day, 1 Week). Also it is useful to mention that market is closed in the weekends. FOREX data (Foreign Exchange Market data) is available easily through internet. One good website is Yahoo Finance. 4.1 Case I: SLS without tuning If we use SLS without fuzzy tuning, the gain would eventually be negative after the end of the year (see Figure 9). In this case is considered to be 5 and is not changing during the trade. 4.2 Case II: SLS with Fuzzy tuning: In this case, the feedback gain changes all the time. We tune the feedback gain based on dp and d2 p and the result for year 2 can be seen at Figure 1. As it is obvious by using Fuzzy tuning for the same situation, the profit would be much higher, therefore tuning is working properly. 1

11 Cumulative Profit Function g(t) without Fuzzy tunning 2 1 Cumulative Profit(g(t)) Time (Minute) 1 4 Figure 9: Cumulative profit g(t) for SLS controller without gain tuning. Data: EUR/USD minute data for year 2 Cumulative Profit Function g(t) using Fuzzy logic Cumulative Profit(g(t)) Time (Minute) 1 4 Figure 1: Cumulative profit g(t) for SLS controller with feedback gain tuned with fuzzy logic. Data: EUR/USD minute data for year 2 11

12 References [1] B. R. Barmish and J. A. Primbs. On a new paradigm for stock trading via a modelfree feedback controller. IEEE Transactions on Automatic Control, 61(3): , March 216. [2] B Ross Barmish. On performance limits of feedback control-based stock trading strategies. In American Control Conference (ACC), 211, pages IEEE, 211. [3] Shirzad Malekpour, James A Primbs, and B Ross Barmish. On stock trading using a pi controller in an idealized market: the robust positive expectation property. In Decision and Control (CDC), 213 IEEE 52nd Annual Conference on, pages IEEE, 213. [4] evin M Passino and Stephen Yurkovich. Fuzzy control, volume 42. Citeseer, [5] E Yeşil, M Güzelkaya, and I Eksin. Self tuning fuzzy pid type load and frequency controller. Energy Conversion and Management, 45(3):377 39,