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Recent reports suggest that the stochastic process underlying financial time series is nonstationary with nonstationary increments. Therefore, time averaging techniques through sliding intervals are inappropriate and ensemble methods have been proposed. Using daily ensemble averages we analyze two different measures of intraday volatility, trading frequency and the mean square fluctuation of increments for the three most active FX markets; we find that both measures indicate that the underlying stochastic dynamics exhibits nonstationary increments. We show that the two volatility measures are equivalent. In each market we find three time intervals during the day where the mean square fluctuation of increments can be fit by power law scaling in time. The scaling indices in the intervals are different, but independent of the FX market under study. We also find that the fluctuations in return in these intervals lie on exponential distributions.

Analysis of financial time series has provided new insights about the underlying stochastic processes (Bouchaud and Potters, 2000, Dacorogna et al., 2001, Mantegna and Stanley, 2007 and McCauley, 2009). Techniques from statistical physics have been adapted to analyze and model financial time series, to access risk, and price options. Early work by Osborne, 1959, Osborne, 1977 and Samuelson, 1965 laid the foundation for the Black and Scholes option pricing model (Black and Scholes, 1973 and Merton, 1973), which assumed that the stochastic dynamics of the underlying asset was a geometric Brownian motion. However, the hypothesis of Gaussian fluctuations disagrees with fluctuations seen in commodity markets as reported by Mandelbrot, 1963 and Mandelbrot, 1966. Empirical studies conducted over the last two decades found that distributions of intraday fluctuations are non-Gaussian and contain fat tails (Cont, 2001, Dacorogna et al., 1993, Gopikrishnan et al., 1999, Müller et al., 1990, Olsen et al., 1997, Schmitt et al., 1999 and Xu and Gencay, 2003). For example, these distributions were found to follow a power law outside the Lévy stable domain (Gopikrishnan et al., 2000, Gopikrishnan et al., 1999 and Plerou and Stanley, 2007). Furthermore, empirical analysis suggests that the distributions scale with the length of the time interval analyzed (Galluccio et al., 1997, Gopikrishnan et al., 2000 and Vandewalle and Ausloos, 1998). Many of these analyses employed sliding interval methods, which implicitly assume that the underlying stochastic process Xt has stationary increments, i.e., the increments ΔXτ = ΔXt, τ = Xt + τ − Xt are independent of time t and are functions of τ only. However, other reports have suggested that the increments are nonstationary, i.e., the increment ΔXt, τ is an explicit function of time. First, it was shown that the trading frequency is not uniform within a day. In fact, it was shown that the frequency varies by a factor of ~ 20 ( Dacorogna et al., 2001, Müller et al., 1990 and Zhou, 1996). Many authors proposed that financial market fluctuations are best analyzed in transaction time ( Ane and Geman, 2000, Baviera et al., 2001, Clark, 1973, Griffin and Oomen, 2008, Mandelbrot and Taylor, 1967, Oomen, 2006 and Silva and Yakovenko, 2007). A second approach inferred that the volatility of the Euro–Dollar exchange rate (in real time) was not uniform and varied by a factor of around 3 within a day ( Bassler et al., 2007, Dacorogna et al., 2001, Müller et al., 1990 and Zhou, 1996). Both approaches suggest that intraday increments are generally time dependent and one conclusion of the present work is that they are equivalent. Bassler et al. (2007) demonstrated that there were several time intervals during which the Euro–Dollar exchange rate can be fit by power laws in time. Moreover, the scaling indices within these intervals were different. The second result of our work is that the scaling intervals and scaling indices are common for the three major currency exchange rates, EUR/USD, USD/JPY, and GBP/JPY. In fact, the volatility in these markets exhibits similar characteristics even outside the scaling intervals. We also ask whether price variations outside of the scaling intervals lie on exponential distributions as reported in Silva et al., 2004 and Bassler et al., 2007. We address this issue using low order absolute moments of the distributions. Our studies are conducted on FX rates, which have the most active and liquid markets. The daily turnover in traditional FX market transactions in 2009 was approximately 3 trillion Dollars (BIS, 2007 and IFSL, 2009). The market is open 24 hours on weekdays, i.e., Sunday 20:15 Greenwich Mean Time (GMT) till Friday 22:00 GMT. The global turnover can be accounted in three main geographical regions: Asia, Europe and North America (BIS, 2007 and Galati and Heath, 2007). The UK accounts for 35.8% of exchange trading, while the US and Japan account for 13.9% and 6.7% respectively (IFSL, 2009). The three FX rates considered here were the most traded between 2001 and 2009 (BIS, 2007 and IFSL, 2009). We restrict our analyses to trading days on which each recorded trade is reported with the bid and ask quote and approximate the spot price p as the average of the bid and ask price: View the MathML sourcep=12(pbid+pask). Following Osborne (1959), we analyze market fluctuations using the return View the MathML sourcex(t;τ)=logp(t+τ)p(t), where p(t) represents the price of the commodity at time t. If the increments were stationary, the distribution of x(t ; τ) would be independent of the starting time t, and would only depend on the time-lag τ. As we already mentioned, intra-day variations in trading frequency ( Ane and Geman, 2000, Clark, 1973 and Mandelbrot and Taylor, 1967) and volatility ( Müller et al., 1990, Dacorogna et al., 2001, Zhou, 1996 and Bassler et al., 2007) were used to argue that increments in FX rates were nonstationary. We define volatility of returns as the root mean square fluctuation, see Eq. (2). If successive transactions are uncorrelated and the returns for each transaction are from the same unknown underlying distribution with finite variance σ0 constant over time, then the standard deviation after M transactions is proportional to View the MathML sourceMσ0. Assuming that M transactions have been reported in a (short) time interval [t, t + τ], the standard deviation can be expressed as equation(1) View the MathML sourceσ(t;τ)∝τντ(t)σ0, Turn MathJax on where View the MathML sourceντ=Mτ is the trading frequency. Thus, we suspect the volatility at a time t to be a function of the trading frequency. Here we define trading frequency as the number of recorded trades within a fixed time interval. Alternatively, we can define trading frequency by only considering trades within the time interval that change the price (tick time sampling) ( Griffin & Oomen, 2008). We find however, that the choice of transaction time is the most appropriate for our analysis. Fig. 1 illustrates the daily behavior of tick frequency and volatility according to Eq. (2). Both measures vary over the course of a day and exhibit similar complicated daily behavior. This means that the underlying stochastic process is not independent of the time of day. Full-size image (27 K) Fig. 1. A. The average number of ticks ντ of the EUR/USD exchange rate is plotted against time of day, with time lag τ = 10 min. B. Volatility σ(t ; τ) of the EUR/USD exchange rate is plotted against time of day, also with time lag τ = 10 min to ensure that autocorrelations have decayed. The plots indicate that the underlying stochastic process is nonstationary and exhibits nonstationary increments, depending on starting time t. If the increments were stationary, σ would be flat. Times of high volatility (and high trading frequency) coincide with opening times of banks and financial markets in major financial centers. The peaks in plot B can be assigned to characteristic times during the trading day. Both measures exhibit similar daily behavior raising the question if they are related. Figure options Bassler et al. (2007) demostrated that the intra-day volatility for the EUR/USD exchange rate contained several intervals during which the fluctuations exhibited scaling; the scaling indices in these intervals were different. Here we wish to determine if the volatility in other FX markets are similarly time dependent, if there are scaling regions, and how the scaling intervals and indices of different markets are related. These studies are conducted using the mean-square-fluctuation of increments during the time interval [t, t + τ] over different trading days. Specifically, equation(2) View the MathML sourceσ2t;τ:=x2t;τ=1N∑k=1Nxk2t;τ Turn MathJax on where N is the number of trading days and τ is chosen to be 10 min to eliminate correlations ( Bassler et al., 2007). xk(t ; τ) represents the return in the interval [t, t + τ] on the kth trading day. Note that applying an ensemble average is necessary because of the nonstationarity of the stochastic process. Methods based on sliding time averages of increments are not appropriate because the underlying dynamics exhibits nonstationary increments ( McCauley, 2008). On the other hand, the use of ensemble averages is justified due to the approximate daily repetition of σ(t) ( Bassler et al., 2007). Next, consider the distribution W(x, τ ; 0) of fluctuations over a time lag τ starting from t = 0. In the scaling region, the scaling ansatz given by Bassler et al. (2007) asserts that equation(3) W(x,τ;0)=τ−HF(u),Wx,τ;0=τ−HFu, Turn MathJax on where FF is the scaling function of the scaling variable View the MathML sourceu=xτH with the scaling index H. Note that t is set to zero at the beginning of each scaling interval. It was shown that the scaling function FF of the EUR/USD rate within the scaling region between 9:00 AM and 12:00 AM New York time was close to bi-exponential. Here we compute the scaling functions for other scaling intervals and other FX markets. Also note that we only have ~ 2000 ensembles in our study. This is insufficient to obtain accurate distribution functions. The method outlined in Bassler et al. (2007) first determines the scaling index H and subsequently uses the scaling ansatz, Eq. (3), and increments from multiple time intervals to compute FF. The first step was to note that within the scaling interval, the moments of x(0 ; τ) satisfy equation(4) View the MathML sourcex0;τβ1β∝τH−12. Turn MathJax on Computations for several moments β can be used to estimate H. Next, we use Eq. (3) at a range of intervals τ, in order to compute FF. It is found that FF is a bi-exponential distribution ( Bassler et al., 2007). The next step is to determine if the fluctuations in return outside the scaling regions lie on the same distribution. Without scaling, we do not have sufficient data to compute the distributions of x(t ; τ). Instead, we look at non-dimensional low-order moments equation(5) View the MathML sourcemβt;τ=∫dxxt;τβWx,τ;t1β∫dxx(t;τ)Wx,τ;t. Turn MathJax on The equality of two distributions will imply that the corresponding moments are identical. Thus, we compare the moments for the EUR/USD within a single scaling interval with those in a second interval that does not lie within a single scaling interval. The moments mβ(t ; τ) are calculated over the ensemble of daily returns x(t ; τ), whereas the returns are calculated over the whole time interval, i.e., τ equals the interval length. We compare the non-dimensional moments for an interval within a scaling region and for one not contained in a scaling region.

FX markets can be regarded as large complex systems exhibiting stochastic behavior resulting from interactions between market participants at different levels. Properties of the resulting stochastic process can be inferred using statistical features. We analyzed two different measures of intraday volatility, trading frequency and the mean square fluctuation of increments, for the three most active FX markets and found that both measures indicate that the underlying stochastic dynamics is nonstationary. Consequently, the use of sliding interval techniques will not give an accurate characterization of the underlying stochastic process. The primary peak in Fig. 1B at 8:30 AM EST coincides with the time of announcement of important economic indicators, e.g., jobless claims, international trade. We also note that opening hours of financial institutions in the three main financial centers Asia, Europe, and North America are associated with peaks in volatility. Typically, the volatility decreases systematically following the opening. Recent studies proposed two approaches to demonstrate that the stochastic process underlying FX markets is non-stationary with nonstationary increments. Each approach assumes that the process is repeated every trading day (Dacorogna et al., 2001, Dacorogna et al., 1993, Galluccio et al., 1997, Müller et al., 1990 and Zhou, 1996). The assumption has been justified using the behavior of markets during a week (Bassler et al., 2007). The first technique relies on the variation in trading frequency during the day, and suggests the use of tick or transaction time to analyze the stochastic process. The second approach analyzes the time dependence of volatility (Dacorogna et al., 2001 and Dacorogna et al., 1993). We showed that the two approaches are equivalent during times at which financial institutions in at least one of the major trading centers (Japan, Britain, and USA) are open. Although previous studies reported the proportionality of the variance and the frequency of trades (Bouchaud et al., 2008, Plerou and Stanley, 2007 and Silva and Yakovenko, 2007), they used sliding interval techniques which do not apply for nonstationary processes. All three FX markets exhibit time intervals in the course of the day during which the volatility decays from a peak. We analyzed each scaling region for the different markets and found that the dynamical scaling indices differ from the often reported value of 0.5. The scaling indices differ between scaling intervals, but are consistent between all three markets. Earlier work by Bassler et al., 2007 and Bassler et al., 2008 found that the scaling index, or Hurst exponent, of H ≈ 0.5 can arise artificially from the use of sliding interval techniques. An additional misconception regarding H lies in the conclusion of long time correlations for View the MathML sourceH>12 like fractional Brownian motion (fBm). It has been shown that View the MathML sourceH>12 does not necessarily imply long term correlations ( Bassler et al., 2006 and Preis et al., 2007). The empirical scaling functions for each FX market and scaling interval are identical suggesting that the dynamics during the scaling intervals in all three FX markets are governed by the same underlying process. Our calculations were based on an underlying variable diffusion process, which was shown to exhibit volatility clustering (Gunaratne, Nicol, Seemann, & Torok, 2009), i.e., a slow decay of the autocorrelation function of the absolute or squared values of the time series, which is another characteristic feature of financial markets (Heyde and Leonenko, 2005 and Heyde and Yang, 1997). The scaling functions do not exhibit fat tails and are exponential, in agreement with previous work (Bassler et al., 2007, Bassler et al., 2008 and Silva et al., 2004). We further supported the finding of the bi-exponential behavior using low order moments. Our result suggests that reported fat tails might be caused by inappropriate use of sliding interval techniques since moving average procedures can not only give rise to artificial Hurst exponents but also to artificial fat tails (Bassler et al., 2007).