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Natural Time Analysis in Financial Markets Algorithmic Finance 5 (2016) 37–46 37 DOI:10.3233/AF-160057 IOS Press Natural time analysis in financial markets A. Mintzelas and K. Kiriakopoulos∗ Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens, Greece Abstract. In this paper we introduce natural time analysis in financial markets. Due to the remarkable results of this analysis on earthquake prediction and the similarities of earthquake data to financial time series, its application in price prediction and algorithmic trading seems to be a natural choice. This is tested through a trading strategy with very encouraging results. Keywords: Natural time, complex systems, financial time series, trend’s prediction, market’s energy, trading strategy 1. Introduction et al., 2010; Sornette, 2002; Bhattacharyya et al., 2007; Chakrabarti et al., 2008). As a representative It’s a widespread belief that economic systems, example, it is argued that exchange-traded stock such as financial markets, constitute one of the most prices follow the dynamics of built-up and release vivid and rich example of complex systems (Man- stress, similar to earthquakes (Andersen et al., 2011). tegna and Stanley, 2000; Sornette, 2002, 2003, 2009; Moreover, the distribution of market volatility before Yan et al., 2012; Preis et al., 2012; Munnix¨ et al., and after a stock market crash is well described 2012; Gvozdanovic et al., 2012; Borland, 2005). by the Gutenberg-Richter law, which reflects the Complex systems are governed by dynamic char- scale-invariance and self-similarity of the underlying acteristics which are founded on common and well dynamics by a robust power-law relation (Sarlis established principles used to describe a great variety et al., 2010; Varotsos et al., 2004). Finally, the of scientific and technological approaches of different cascade of volatility “aftershocks” triggered by the types of natural, artificial and social systems (Tsallis, “main financial shock” is quantitatively similar to 2009; Bar-Yam, 1997; Picoli Jr et al., 2007; Sor- earthquakes, which have been described by three nette and Helmstetter, 2002; Abe and Suzuki, 2004; empirical laws, the Omori law, the productivity law Fukuda et al., 2003; Peters et al., 2001). and the Bath law (Petersen et al., 2010). In order to comprehend the behaviour of finan- The aforementioned suggests that methods based cial markets and acquire a deeper knowledge of the on knowledge of earthquakes’ studies seems to be rules which govern them, experience is drawn from very promising in analysing the dynamics of financial the study of complex systems in diverse scientific markets and in predicting their evolution. fields. One of these fields is the field of geophysics In the present paper, a new method for analysing and earthquake prediction. financial markets with the help of Natural time anal- A lot of articles have contributed to the attempt of ysis is proposed. The prime pillar of this theory showing the similarities of the dynamic character- is Natural time χ (Varotsos et al., 2001), a new istics of earthquakes to financial markets (Petersen frame for understanding the timing of the events, which can help us to analyse financial time series and reveal hidden dynamic characteristics. Moreover, ∗Corresponding author: K. Kiriakopoulos, Lecturer in Finance. Department of Mathematics, Panepistimiopolis, the applicability of this theory to financial markets is Zografos 157 84, Athens, Greece. Tel.: +30 210 6516460 E-mail: demonstrated via the development and the implemen- k [email protected]. tation of a trading strategy. 2158-5571/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved 38 A. Mintzelas and K. Kiriakopoulos / Natural time analysis in financial markets Natural time analysis is a well established theory which is really difficult to describe with one single which has been applied successfully in prediction of comprehensive definition because of its many forms. earthquakes (in terms of time and magnitude) for Depending on the application, “energy” can take one more than ten years including the strongest earth- of the usual forms encountered in physics or can be quake that occurred in Greece on 14 February 2008 a measure suitable to the modelling purposes. For (Varotsos et al., 2007; Sarlis et al., 2008). Natural example in earthquake prediction, Qk stands for the time enables the determination of the occurrence time duration of the k-th seismic electric signal (SES)1 of an impending major earthquake (Varotsos et al., (Varotsos et al., 2008). In detection of syndrom of 2011b) since it can identify when a complex system Sudden Cardiac Death (SCD) we use the RR2 interval approaches a critical point (Varotsos et al., 2011c). as Qk. In other Complex systems, such as Olami- With Natural time analysis we are also able to fore- Feder-Christensen (OFC) model(Olami et al., 1992), cast the trajectory of dynamical models and systems the size of avalanche is used as a quantity proportional (Ising model, OFC model) (Sarlis et al., 2011) as well to the “energy” (Qk) (Varotsos et al., 2011c). as their transitions to a critical regime. In the standard rebound theory of earthquakes, The rest of this article is constructed as follows. In elastic deformation energy3 is gradually stored in the Section 2 the concept of Natural time and its basic crust of earth. As rocks on opposite sides of a fault are features is presented. In Section 3 it is shown how subjected to force and shift, they accumulate energy Natural time can be applied in financial time series. and slowly deform until their internal strength thresh- In Section 4 a trading strategy based on this is applied old is exceeded. This is the critical point at which on major financial markets (major FX pairs, DJIA energy is suddenly released in an earthquake. In other stocks). Also a portfolio implementation of this strat- words, earthquake is a recurrent phenomenon which egy on the stocks of S&P500 index shows that this is a result of a continuous built-up stress and release theory can give very promising results. Finally, in process of the tectonic plates of the earth. Section 5 conclusions follow. On the other hand, Andersen et al. (2011) show that world⣙s stock exchanges follow the dynam- ics of build-up and release of stress and they have 2. Natural time a non-linear threshold response to events, similar to earthquakes. The non-linear response based on Conventional time is modelled as the continuum R change blindness, a phenomenon in which humans of real numbers. This continuity does not stem from (traders in this case) react disproportionally to big any fundamental principle. It is rather used for rea- changes, results in only large changes to be taken sons of convenience and mathematical tractability. into account whereas small changes go unnoticed. On the other hand, Natural time, a new time domain, A key principle in finance states that as new infor- is not continuous and its values as well as those of mation is revealed, it immediately becomes reflected energy form a countable sets (Varotsos et al., 2001, in the price of an asset and thereby loses its relevance 2011c). (Famma, 1969). As a consequence, focusing initially on stock indices, stress (elastic deformation energy) 2.1. Definition of Natural time enters the system because of price movements of the indices. The idea is that a large (eventually cumu- Specifically, in a time series comprising of N lative) price movement of a given stock index can events, the Natural time is defined as: induce stresses on other stock indices world-wide to follow its price movement. Similar to the BK k model of earthquakes (Burridge and Knopoff, 1967), χ = k N (1) 1SES: Low frequency ≤1Hz electric signals that precede earth- and serves as the index for the occurrence of the k-th quakes. event. It is smaller than or equal to unity (χk (0, 1]). 2RR is (beat-to-beat) interval time serie of an electrocardio- In Natural time analysis we are concerned about gram. 3Elastic energy is the potential mechanical energy stored in the the evolution of the pair of two quantities (χk,Qk), configuration of a material or physical system as work is performed χk where is the Natural time as aforementioned, and to distort its volume or shape. Elastic energy occurs when objects Qk is a quantity proportional to the “energy” of the are compressed and stretched, or generally deformed in any manner k-th event. “Energy” constitutes a scientific definition (Landau and Lifshitz, 1986). A. Mintzelas and K. Kiriakopoulos / Natural time analysis in financial markets 39 we assume a “stick-slip” motion of the indices so 2.2. The normalized power spectrum Π(ω) and that only a large (eventually cumulative) movement the variance κ1 of Natural time of a given index has a direct impact in the pricing of the remaining indices world-wide. In this line of Considering the evolution of the pair (χk,Qk), we thinking, “price-quakes” can happen in the financial define the function: system as cascades of big price movements (Ander- sen et al., 2011). N iω k Inspired by the aforementioned, we make the F(ω) = Qke N (4) assumption that price can be a natural choice for a k=1 quantity proportional to energy. Various functions of where ω = 2πφ and φ stands for the frequency in price, price trend, volume or a suitable combination of F ω F them could be possible canditates as well. The selec- Natural time. We normalize ( ) dividing by (0) tion depends on the application and the objectives and we end up to the following equation: pursued. N Qk Equivalently with , we consider the quantities: iω k (ω) = pke N (5) k=1 Qk p = p = Qk k N (2) where k N .
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