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Handbook of Exchange Rates by Lucio Sarno, Ian Marsh, Jessica James

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20.3 Scaling Laws

Scaling laws establish invariance of scale and play an important role in describing complex systems (Barabasi and Albert, 1999; Newman, 2005; West et al., 1997. In finance, there is one scaling law that has been widely reported (Ballocchi et al., 1999; Corsi et al., 2001; Dacorogna et al., 2001; Di Matteo et al., 2005; Galluccio et al., 1997; Mantegna and Stanley, 1995; Müller et al., 1990; Guillaume et al., 1997): the size of the average absolute price change (return) is scale-invariant to the time interval of its occurrence. This scaling law has been applied to risk management and volatility modeling (Di Matteo, 2007; Gabaix et al., 2003; Ghashghaie et al., 1996; Sornette, 2000), even though there has been no consensus among researchers for why the scaling law exists (Barndorff-Nielsen and Prause, 2001; Bouchaud, 2001; Farmer and Lillo, 2004; Joulin et al., 2008; Lux, 2006).

Searching for new scaling laws, we analyze the price data of the FX market, which is a complex network of interacting agents: corporations, institutional and retail traders, and brokers trading through market makers, who themselves form an intricate web of interdependence. We consider five years of tick-by-tick data for 13 exchange rates through November 2007 (see Glattfelder et al. (2010) for a description of the data set).

An exchange rate often moves by 10–20% within a year. However, since the seminal work of Mandelbrot (1963), we know about the fractal nature of price curves. The coastline, ...

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