Patterns in high-frequency FX data: Discovery of 12 empirical scaling laws

arXiv:0809.1040v2 [q-fin.ST] 22 Jun 2010 Patterns in high-frequency FX data: Discovery of 12 empirical scaling laws J.B. Glattfelder∗†, A. Dupuis†‡ and R.B. Olsen†‡ † Olsen Ltd., Seefeldstrasse 233, 8008 Zurich, Switzerland ‡ Centre for Computational Finance and Economic Agents (CCFEA), University of Essex, UK Abstract We have discovered 12 independent new empirical scaling laws in foreign exchange data-series that hold for close to three orders of magnitude and across 13 currency ex- change rates. Our statistical analysis crucially depends on an event-based approach that measures the relationship between different types of events. The scaling laws give an accurate estimation of the length of the price-curve coastline, which turns out to be surprisingly long. The new laws substantially extend the catalogue of stylised facts and sharply constrain the space of possible theoretical explanations of the mar- ket mechanisms. Keywords: Scaling Laws; High-Frequency Finance; Foreign Exchange; Time-Series Analysis; Gaussian Random Walk Models 1 Introduction The global financial system has recently been rocked by losses that could total four tril- lion USD (International Monetary Fund [2009]). The crisis is seriously undermining the functioning of the financial system, the backbone of the global economy. This suggests an acute deficiency in our understanding of how markets work. Are there “laws of nature” to be discovered in financial systems, giving us new insights? We approach this question by identifying key empirical patterns, namely scaling-law relations. We believe that these universal laws have the potential to significantly enhance our understanding of the markets. Scaling laws establish invariance of scale and play an important role in describing complex systems (e.g. West et al. [1997], Barab´asi and Albert [1999], Newman [2005]). In finance, there is one scaling law that has been widely reported (M¨uller et al. [1990], Mantegna and Stanley [1995], Galluccio et al. [1997], Guillaume et al. [1997], Ballocchi et al. [1999], Dacorogna et al. [2001], Corsi et al. [2001], Di Matteo et al. [2005]): the size of the ∗Corresponding author. Email: jbg@olsen.ch. 1 average absolute price change (return) is scale-invariant to the time interval of its occur- rence. This scaling law has been applied to risk management and volatility modelling (see Ghashghaie et al. [1996], Gabaix et al. [2003], Sornette [2000], Di Matteo [2007]) even though there has been no consensus amongst researchers for why the scaling law exists (e.g., Bouchaud [2001], Barndorff-Nielsen and Prause [2001], Farmer and Lillo [2004], Lux [2006], Joulin et al. [2008]). In the challenge of identifying new scaling laws, we analyse the price data of the foreign exchange (FX) market, a complex network made of interacting agents: corporations, insti- tutional and retail traders, and brokers trading through market makers, who themselves form an intricate web of interdependence. With a daily turnover of more than 3 trillion USD (Bank for International Settlements [2007]) and with price changes nearly every sec- ond, the FX market offers a unique opportunity to analyse the functioning of a highly liquid, over-the-counter market that is not constrained by specific exchange-based rules. In this study we consider five years of tick-by-tick data for 13 exchange rates through November 2007 (see section 3.1 for a description of the data set). It is a common occurrence for an exchange rate to move by 10 to 20% within a year. However, since the seminal work of Mandelbrot [1963] we know about the fractal nature of price curves. The coastline, roughly being the sum of all price moves of a given threshold, at fine levels of resolution may be far longer than one might intuitively think. But how many times longer? The scaling laws described in this paper provide a surprisingly accu- rate estimate and highlight the importance of not only considering tail events (Sornette [2002]), but set these in perspective with the remarkably long coastline of price changes preceding them. It should be noted that our study is not related to the analysis of lead-lag relationships. The remainder of the paper is organised as follows. Our main results are presented in section 2. We start by enumerating the empirical scaling laws, then cross-check our results by establishing quantitative relations amongst them and discuss the coastline. In section 3 the methods and the data are described and we conclude with some final remarks in section 4. Finally, appendix A contains tables with all the estimated scaling-law parameters. 2 The laws and beyond 2.1 The new scaling laws Interest in scaling relations in FX data was sparked in 1990 by a seminal paper relating the mean absolute change of the logarithmic mid-prices, sampled at time intervals ∆t over a sample of size n∆t, to the size of the time interval (M¨uller et al. [1990]) ⟨|∆χ|⟩p =  ∆t Cχ(p) Eχ(p) , (0a) where ∆χi = χi −χi−1 and χi = χ(ti) = (ln bidi + ln

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