Algorithmic Trading Using Phase Synchronization

Transcription

1 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST Algorithmic Trading Using Phase Synchronization Alireza Ahrabian, Clive Cheong Took, Member,IEEE, and Danilo P. Mandic Abstract A novel trading algorithm which performs trading decisions by making use of phase synchronization between oscillatory components of asset pairs is proposed. This way the phase information ascertains a leading asset, which is then used to predict the lagging asset. The oscillatory components of asset pairs are identified using the Synchrosqueezed Transform (SST), which facilitates stable and online implementation. The performance of the proposed approach is compared with existing algorithms used extensivelybytraders,suchasthemoving average cross-over, and an extrapolation algorithm based upon the multichannel-least mean square (MLMS). Index Terms Algorithmic trading, lead-lag relationship, phase synchronization, synchrosqueezed transform. I. INTRODUCTION F ORECASTING price trends accurately in the stock market is a formidable task, yet it is an economic reality. Despite the many caveats associated with financial forecasting, estimating values of financial indices is an important component of business decision-making [1]. There exist a wide variety of forecasting approaches, ranging from simple data analysis (e.g., moving average-based methods), to more complex techniques such as genetic algorithms [2]. These techniques can be categorized into three main classes [1, p. 88]: Subjective forecasting, which is performed based on experience, intuition and guesswork. It is usually inferred from both macroeconomic (e.g., inflation rate) and microeconomic factors (e.g., yield rate of a particular stock). Extrapolation techniques, whose aim is to project past trends into the future. Common extrapolation techniques include regression analysis andmethodsbasedonerror criteria, such as the mean absolute deviation (MAD) and the mean squared error (MSE), to name a few. Causal modeling, where the goal is to predict a lagging variable based on a leading variable. The relationship between these two variables can be modeled as a cause and effect, and is typically inferred from Bayesian methods. In this work, we use phase synchrony to model this relationship. An algorithmic gold standard currently used in trading is the so-called Technical Analysis (TA), where based on the history of trading (e.g., price changes, volume of transactions) in a certain stock or in the averages, probable future trend is Manuscript received June 10, 2011; revised October 14, 2011; accepted October 17, Date of publication October 27, 2011; date of current version July 13, The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ali Akansu. The authors are with the Department of Electrical and Electronic Engineering, Imperial College London SW7 2AZ, U.K. ( alireza. ahrabian06@imperial.ac.uk; c.cheong-took@imperial.ac.uk; d.mandic@ imperial.ac.uk). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JSTSP deduced [3]. Technical analysis indicators include relative strength index, stochastics, moving average-based methods, convergence/divergence, momentum oscillator, and commodity channel index. A commonly used technical indicator in trading is the moving average cross-over, calculated from two moving average processes of different time lengths (usually the 5- and 20-day moving average pairs). Whenever the slower 20-day moving average trace crosses below the faster 5-day moving average, this is considered a buy indicator (signal); conversely, whenever the 20-day moving average trace crosses above the 5-day moving average this is a sell indicator. More advanced computational techniques advocated by the academic community include machine learning methods which aim to minimize an error criterion and the Black Scholes approach which models the stock price as a geometric Brownian motion [4]. In 2001, a special issue on financial engineering in the IEEE TRANSACTIONS ON NEURAL NETWORKS [2] provided an exposé on how machine-learning techniques can be used to perform forecasting of financial time series, including volatility, interest rates and model calibration, portfolio analysis, stock selection, and credit assessment. These techniques, however, were mostly black-box neural network-based approaches which utilize heuristics at multiple scales and may not be optimal for real-world processing, a crucial issue for algorithmic trading. It is not uncommon to see stocks in the same industry following similar patterns, e.g., the credit contagion spread across several stocks during a credit crunch. In such a scenario, the volatility index, which measures the variation of price of a financial instrument, is likely to be high and implies a high degree of nonstationarity. In these cases, technical analysis may not be of great help to traders, whereas advanced computational techniques which generally do not operate in real-time are too complextobe used. For instance, the class of Black Scholes models rely on time-consuming Monte Carlo simulations [4]; hence, they are not appropriate for real-time forecasting and are used mainly for pricing of assets. There is empirical evidence of a lead-lag relationship in financial data [5, pp ], and in this context, trading strategies have been developed to exploit this relationship which can be regarded as a special case of pair trading [6]. More recently, a study on the lead-lag relationship in the property market was investigated in [7]. For a forecasting technique to be useful, it must add information to what is already known: the main issue is how to take advantage of this lead-lag relationship to perform prediction. In this spirit, we propose a real-time trading algorithm that exploits the lead-lag relationship between a pair of stock prices to forecast the price of the lagging asset; this is achieved by using the recently proposed Synchrosqueezed Transform to quantify the lead-lag relationship [8]. For rigor, we statistically verify the existence of the synchronization between the assets using a method based on surrogate data analysis. In this framework, since phase synchrony between two stocks can be employed for forecasting purposes, whether the data is stationary or not [9], the forecasting task thus /$ IEEE

2 400 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012 amounts to using the leading asset to trade on the lagging asset. The analysis is supported by illustrative simulations. II. BACKGROUND SIGNAL PROCESSING TECHNIQUES A. Synchrosqueezed Transform The Synchrosqueezed Transform (SST) [8] is a novel signal decomposition technique developed as an extension of the continuous wavelet transform (CWT), given by (1) The CWT convolves the original signal with a wavelet (a finite energy oscillation) for different scales and time shifts, thus effectively projecting the original signal from the time domain to the time-scale domain. The CWT spreads the energy contained in a narrow range of frequencies in a signal around a particular scale, by ; this also introduces redundancy in the wavelet coefficients. To address this issue, the Synchrosqueezed Transform reallocates the wavelet coefficients from the timescale domain to the time-frequency domain through a procedure known as synchrosqueezing. The SST performs CWT of the data, ensuring that a wavelet in the positive spectrum is selected, from which the instantaneous frequency of the resulting CWT coefficients (1) is given by for each time-scale. The next step is to perform frequency binning of the wavelet coefficients by synchrosqueezing. In other words, all the wavelet coefficients are combined from the same time frequency bin to reduce the redundancy 1 in the CWT. Mathematically, synchrosqueezing (in the discrete case) can be formulated as B. Phase Synchronization Phase Synchrony (PS) ascertains whether the phase difference between two oscillatory systems is constant in time. The advantage of this approach is that while the amplitudes between the two systems may not be linearly correlated, synchronization usually still exist [10]. Phase synchrony between two oscillatory systems can be calculated from the difference between the instantaneous phase of two oscillatory systems. It is measured from the argument of an analytic signal, for instance, by performing the Hilbert Transform on a real-valued signal and combining with the original to make them complex-valued. The instantaneous phase for atime can then be calculated from the complex-valued SST filter as 1 The implementation of the Synchrosqueezed Transform is detailed in [9]. (2) (3) (4) Fig. 1. The top panel shows the trend (dashed), superimposed on the original asset price. The bottom panel shows the oscillations which exist in the asset after removing the very low frequency trend. Both graphs begin on , and end on All data used in this investigation has been acquired from If and represent the instantaneous phases of the two coupled systems and, then the instantaneous phase difference is given by,where and represent the phase locking values (assumed to be unity in many studies) [11]. Once the phase difference has been obtained, the next step is to calculate the phase synchrony score, which measures whether the phase difference is constant. In this work, an information theoretic criterion for measuring phase synchrony is used, based on the entropy of the distribution of. In this way, a uniform distribution yields a zero synchrony score, whereas for a delta function probability distribution its corresponding phase synchrony is unity (since the phase difference focuses on one value). A comprehensive explanation can be found in [10, p. 2], where the multivariate empirical mode decomposition (MEMD) [12] was used to obtain the oscillatory components. Notice however, that MEMD does not operate in real time, making it unsuitable for online trading. We next exploit the online processing ability of the SST to perform phase synchrony-based trading. III. PHASE SYNCHRONY AND TRADING ALGORITHM IMPLEMENTATION Financial data usually contain a low-frequency trend component upon which a variety of different frequencies are superimposed. In this work, the frequencies of interest are the lowest narrowband frequency components (not including the trend information and the high frequency noise components). The preprocessing stage, prior to performing any phase synchronization, consists of de-trending the financial data and removing the high-frequency noise components. Fig. 1 illustrates the de-trending question, where a low-pass moving average filter is used to determine the trend to be subtracted from the original signal. The filtering stage uses the SST as a bandpass filter (SST is a block algorithm, so the data was first windowed), applied to only the low frequency oscillations of the de-trended signal. An important criterion for the band-pass filtering is to ensure that the resulting signal is as narrowband as possible (shown in Fig. 2). As a rule of thumb, the number

3 AHRABIAN et al.: ALGORITHMIC TRADING USING PHASE SYNCHRONIZATION 401 A. Trading Based Upon the Phase Synchrony of Assets In order to perform a trading strategy based on the lead-lag relation, we use the leading asset to predict the lagging asset. Our main assumption is that the two systems exhibit synchrony in some sense; the main trading strategy is then to wait for aconfirmed maximum or minimum of the leading asset, and then trade on the lagging asset based upon the phase difference. An outline of the proposed trading algorithm is given in Algorithm 2. Algorithm 2 Trading based upon phase synchrony 1) Check if the phase synchrony score,where is a user defined parameter. 2) Identify the leading asset,using Fig. 2. After the SST filtering, the high-frequency components were successfully removed from the signal in Fig. 1 (bottom panel). of extreme points of a narrowband signal should be almost equal to the number of its zero crossings (as in [12] and [13]). Algorithm 1 provides a summary of the de-trending and SST filtering procedures for online implementation using a sliding window. Algorithm 1 De-trending and SST bandpass filtering 1) De-trend the data where W denotes the length the of moving average filter (window length). 2) SST the de-trended signal, selecting a narrowband signal 3) Calculate the phase difference and the phase synchronization score. 4) Convert the phase difference from radians to days (5) (6) where is the frequency that is common between both filtered assets. if if 3) Use the leading asset to identify a extreme point, and then trade on the lagging asset, as shown in (8) at the bottom of the page, where is the current price and is a time constraint such that. The SST filtered asset prices are narrowband (time-varying sinusoidal signals). In the context of our algorithm, the maximum can be defined as the point where is greater than the previous and the next samples. Similarly for a minimum, we have. IV. STATISTICAL ANALYSIS OF PHASE SYNCHRONIZATION To demonstrate phase synchronization between two financial assets, we provide a rigorous statistical analysis on the datasets using a variant of the surrogate statistical method proposed in [14]. Surrogate data are statistically similar to the original data, with equivalent mean, amplitude spectrum, and variance; however, the phase information pertained to the original data is completely randomized (by drawing phase information from a uniform distribution between 0 and ). The process of phase randomization destroys any phase information which may have existed; thus, a statistical comparison can be made between the surrogate phase synchrony scores and the assets synchrony score (via hypothesis testing). The null hypothesis used was that no phase synchronization exists between the assets and the surrogates were employed to calculate the threshold synchronization score which gave a 95% confidence interval. This way if a asset data (phase synchronization score) fell outside the confidence interval then the null hypothesis was rejected (thus, phase synchrony must exist). We generated 300 surrogate time series, and carried out offline (7) Trade action Buy if and Sell if and No action else (8)

4 402 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012 TABLE I STATISTICAL VERIFICATION OF THE PHASE SYNCHRONIZATION BETWEEN ASSET PAIRS USING THE SURROGATE DATA GENERATION METHOD. THE BEMD METHOD WAS USED FOR CALCULATING PHASE SYNCHRONIZATION Fig. 3. Outcome of the phase synchrony algorithm, with the indications for buying and selling for a window length of 800 trading days. (Top) Price of the leading asset Sainsburys. (Middle) Lagging asset Tesco with the buy (solid vertical line) and sell (dashed vertical line) indicators. (Bottom) Volatility of Tesco. Graphs begin on and end on phase synchronization estimation based upon the bivariate empirical mode decomposition (BEMD) [15]. The BEMD-based phase synchrony is nonparametric, and has been shown to be very accurate, thus giving more robust results then when using parametric methods. The results are summarized in Table I and show that phase synchronization between asset prices is statistically significant. It thus supports our online trading method based the synchronization. V. SIMULATIONS The proposed algorithm was assessed against the moving average crossover (5-day and 20-day moving average was used), a conventional tool used by traders, and the multichannel least mean square (MLMS) algorithm, a standard multivariate signal processing technique [16]. MLMS was chosen since 1) it is a stochastic algorithm that can operate in a highly nonstationary environment in real-time (high volatility); 2) its multivariate nature accounts for the inter-channel coupling. The MLMS algorithm was employed in a prediction configurationinorderto extrapolate the SST filtered signals. The main metric to compare the performance of the algorithms considered was the compound rate of return (RoR). The algorithms were tested on three pairs of assets from the same sector, and all the data were approximately between the years The three company pairs were Sainsburys & Tesco, BAE Systems & Rolls Royce, and Anglo American Plc & BHP Billiton. Since phase synchrony based methods use one asset for prediction and the other for trading, only one of the pairs was traded. An example of the outcome of the algorithm is shown in Fig. 3; the top panel shows the leading asset, the middle Fig. 4. RoR of the phase synchrony algorithm (thick solid line), LMS extrapolation (dashed), and MA crossover (thin solid line). Graphs begin on the 1000th trading day and end on 1500th trading day of the Tesco stock. panel the asset which is lagging and thus traded upon (where the dashed vertical and solid vertical lines indicate the sell and buy signals, respectively); the bottom panel shows the volatility of the lagging asset. Phase synchrony has been designed to exploit the coupling between oscillations in stock pairs; we thus expected it to perform at its best when the volatility 2 was at its highest. In order to compare the performances (between the different methods), the Annualized Rate of Return was next used during a volatile period between the trading days (2 trading years) and the results are shown in Fig. 3 (for all three asset pairs the volatility was at its highest during the same period). The annualized RoR was calculated as in [17], by,where is the number of trading years (in this example ), the initial investment (where is the initial capital at risk), and the Profit and Loss accumulated through trading. From Figs. 4 6, the compound RoR for the proposed algorithm during the volatile period performed well for all the three stocks. The simulations demonstrate that using phase synchrony for forecasting during volatile periods is quite promising, given its real-time operation; however, during low volatility periods, other methods would have similar or superior results. One of the main advantages of the proposed approach is that little training is required; the only requirement is an adequate length of the sliding window and that the SST bandpass filtering is carried out correctly. From the RoR graphs in Figs. 4 6, the phase synchronization algorithm outperformed the other two methods, with a consistently higher rate of return. Under high volatility, the proposed method therefore has the ability to detect a lead-lag situation, 2 The volatility has been measured using exponential moving average with [4, p. 471].

5 AHRABIAN et al.: ALGORITHMIC TRADING USING PHASE SYNCHRONIZATION 403 between stocks. The technique has been shown to yield robust estimates of oscillatory components in financial data and promising forecasting results in highly volatile and nonstationary environments. The synchrony between two stocks has been verified statistically, with a confidence of 95%. The simplicity of our proposed method and its real-time operation capability have been illustrated on comparative analyses against the moving average crossover and the least mean square algorithms. REFERENCES Fig. 5. RoR of the proposed phase synchrony algorithm (thick solid line), LMS extrapolation (dashed), and MA crossover (thin solid line). Graphs begins on the 1000th trading day and end on 1500th trading day of the Rolls Royce stock. Fig. 6. RoR of the phase synchrony algorithm (thick solid line), LMS extrapolation (dashed), and MA crossover (thin solid line). Graphs begins on the 1000th trading day and end on 1500th trading day of the Anglo-American stock. TABLE II PERCENTAGE RETURN ON DAY 500 OF THE ROR GRAPHS IN FIGS. 4 6 provided that the synchronization score is high enough to exploit this effect. Table II shows the results on the final day of trading (during the volatile period); observe that the phase synchrony algorithm generated a profit from the initial investment over a consistent time period, whereas the other methods considered in general generated losses. VI. CONCLUSION We have introduced a synchrosqueezed wavelet transform-based method for trading, based on phase synchrony [1] R. Stuteley, Numbers guide: The essentials of business numeracy, The Economist, 2003, 5th ed. [2] Y. S. Abu-Mostafa, A. Atiya, Magdon-Ismail, and M. White, Introduction to the special issue on neural networks in financial engineering, IEEE Trans. Neural Networks, vol. 12, no. 4, pp , Jul [3] R. D. Edwards, J. Magee, and W. Bassetti, Technical Analysis of Stock Trends, 9th ed. Boca Raton, FL: CRC, [4] J. Hull, Options, Futures and Other Derivatives, 7th ed. Upper Saddle River, NJ: Prentice-Hall, [5] F. R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields & Stock Prices in the United States Since Philadelphia, PA: Ayer, [6] C. Brooks, A. G. Rew, and S. Ritson, A trading strategy based on the lead-lag relationship between the spot index and futures contract for the FTSE 100, Int. J. Forecast., vol. 17, no. 1, pp , [7] C. Y. Yiu, C. M. Hui, and S. K. Wong, Lead-lag relationship between the real estate spot and forward contracts markets, J. Real Estate Portfolio Manage., vol. 11, no. 3, pp , [8] I. Daubechies, J. Lu, and H. T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool, Appl. Comput. Harmon. Anal., vol. 30, no. 2, pp , [9] E. Brevdo, N. Fuckar, G. Thakur, and H. T. Wu, The synchrosqueezing algorithm: A robust analysis tool for signals with time-varying spectrum, 2011, (Arxiv preprint arxiv: ), submitted for publication. [10] A. Y. Mutlu and S. Aviyente, Multivariate empirical mode decomposition for quantifying multivariate phase synchronization, EURASIP J. Adv. Signal Process., vol. 2011, p , [11] R. Q. Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, Performance of different synchronization measures in real data: A case study on electroencephalographic signals, Phys. Rev. E, vol. 65, no. 4, p , [12] N. Rehman and D. P. Mandic, Multivariate empirical mode decomposition, in Proc. R. Soc. A, 2010, vol. 466, no. 2117, pp [13] N. Huang, Z. Shen, S. Long, M. Wu, H. Shih, Q. Zheng, N. Yen, C. Tung, and H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, in Proc.R.Soc.A, 1998, vol. 454, pp [14] C. M. Sweeny-Reed and S. J. Nasuto, A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition, J. Comput. Neurosci., vol. 23, no. 1, pp , [15] D. Looney, C. Park, P. Kidmose, M. Ungstrup, and D. P. Mandic, Measuring phase synchrony using complex extensions of EMD, in Proc. 15th IEEE/SP Workshop Statist. Signal Process. (SSP 09), 2009, pp [16] Y. Huang and J. Benesty, Audio Signal Processing for Next Generation Multimedia Communication Systems. Norwell, MA: Kluwer, [17] K. E. Hild and V. D. Calhoun, The fourth annual MLSP competition on stock market prediction, in Proc. IEEE Workshop Mach. Learn. Signal Process., 2008, pp Alireza Ahrabian received the M.Eng degree in electrical and electronic engineering from Imperial College London, London, U.K, graduating in He is currently a Ph.D student in Signal Processing Group at Imperial College London. His main research interests include time-frequency methods, brain computer interface systems, and statistical signal processing.biography to come.

6 404 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012 Clive Cheong Took (S 06 M 07) received the B.S. degree in telecommunication engineering from Kings College London University, London, U.K., where he was the top departmental graduate in 2004, and the Ph.D. degree in blind signal processing from Cardiff University, U.K. in His research interests include adaptive, blind, and multidimensional signal processing, with applications in biomedicine, renewable energy, telecommunication and finance. Danilo P. Mandic is a Professor of Signal Processing at Imperial College London, and is working in the area of adaptive signal processing and multiscale modeling. His publication record includes two research monographs entitled Recurrent Neural Networks for Prediction (1st ed., August 2001) and Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models (1st ed., Wiley, April 2009), an edited book entitled Signal Processing for Information Fusion (Springer, 2008) and more than 200 publications on signal processing. He has been a Guest Professor at K.U. Leuven Belgium and a Frontier Researcher in RIKEN Japan. He is a Member of the IEEE Technical Committee on Theory and Methods for Signal Processing, Associate Editor for the IEEE TRANSACTIONS ON NEURAL NETWORKS and has been on editorial boards for IEEE TRANSACTIONS ON SIGNAL PROCESSING and IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II. He has produced award winning papers and products resulting from his collaboration with Industry.