Scale free effects in world currency exchange network

A large collection of daily time series for 60 world currencies’ exchange rates is considered. The correlation matrices are calculated and the corresponding Minimal Spanning Tree (MST) graphs are constructed for each of those currencies used as reference for the remaining ones. It is shown that multiplicity of the MST graphs’ nodes to a good approximation develops a power like, scale free distribution with the scaling exponent similar as for several other complex systems studied so far. Furthermore, quantitative arguments in favor of the hierarchical organization of the world currency exchange network are provided by relating the structure of the above MST graphs and their scaling exponents to those that are derived from an exactly solvable hierarchical network model. A special status of the USD during the period considered can be attributed to some departures of the MST features, when this currency (or some other tied to it) is used as reference, from characteristics typical to such a hierarchical clustering of nodes towards those that correspond to the random graphs. Even though in general the basic structure of the MST is robust with respect to changing the reference currency some trace of a systematic transition from somewhat dispersed – like the USD case – towards more compact MST topology can be observed when correlations increase. PACS. 89.65.Gh Economics; econophysics, financial markets, business and management – 89.75.Fb Structures and organization in complex systems – 05.45.Tp Time series analysis The world currency exchange market (foreign exchange – FOREX, FX) is the world largest financial market and it constitutes an extremely complex network. The FX daily takeover volume is of the order of 10 USD. Any other financial market can hardly approach such volume. Also, this market has direct influence on all other markets because any price is expressed in terms of a currency. The large volume makes it virtually impossible to control from outside and there is no friction (transactions are basically commission free). Due to time differences FX transactions are performed 24 hours a day, 5.5 day a week with maximum volume between 1 and 4 p.m. GMT, when both American and European markets are open. Hence, the FX time series relations represent an exceptionally complex network indeed, and they are therefore especially worth of detailed analysis. The FX market can be viewed as a complex network of mutually interacting nodes, each node being an exchange rate of two currencies. In principle, all nodes are interconnected with complex nonlinear interactions. Any currency can be expressed in terms of particular one that is called the base currency. In spite of its importance, much less attention has been paid in literature to the FX cross-correlation analysis, than to such analysis of stock markets[1,2,3]. Therefore in the following the currency network will be analyzed. Our motivation to investigate correlations of FX time series is twofold: theoretical and practical. More detailed correlation analysis can give insight into the structure of links between various currencies and, in particular, it potentially may provide quantitative arguments in favor of an often postulated hierarchical organization of world currency exchange market. Knowledge of correlations is also essential for the portfolio management. Usually, for a financial time series of an ith asset (i = 1, . . . , n) at time t, xi(t) = xi, one defines its return over time period τ as Gi(t; τ) = lnxi(t + τ) − lnxi(t). For FX series instead of a value xi(t) one has x B A(t), an exchange rate, i.e. a value of currency A expressed in terms of a base currency B. Hence, the returns can be denoted as GBA(t) and they are clearly antisymmetric G B A(t; τ) = −GB(t; τ). FX returns, due to the lack of commission and high liquidity satisfy the triangle rule: GBA(t; τ) + GCB(t; τ) +G A C(t; τ) = 0 , already for relatively small values of τ [4]. As a result, for a set of n currencies we have N = n − 1 independent values and the same number of nodes with a given base currency. In the following we analyze time series of daily data for 60 currencies, including gold, silver and platinum[5]. The data taken covers the time period Dec 1998–May 2005. In order to automatically get rid of possible misprints in the original data the daily jumps greater than 5σ (less than 0.3% of data points) were removed. Also, the gaps related to non-trading days were synchronized. For each exchange rate we thus obtain a time series of 1657 data points. The 2 A. Z. Górski et al.: Scale free effects in world currency exchange network currencies are denoted according to ISO 4217 standard, and they can be formally divided into four groups, according to their liquidity. The major currencies, that we call the A group, include USD, EUR, JPY, GBP, CHF, CAD, AUD, NZD, SEK, NOK, DKK (11 currencies). All other liquid currencies belong to the group A (CYP, CZK, HKD, HUF, IDR, ILS, ISK, KRW, MXN, MYR, PHP, PLN, SGD, SKK, THB, TRY, TWD, XAG, XAU, XPT, ZAR, 21 currencies). Less liquid currencies (group B) include: ARS, BGN, BRL, CLP, KWD, RON, RUB, SAR, TTD (9 currencies). Finally, the non-tradable currencies (group C) taken into account are: AED, COP, DZD, EGP, FJD, GHC, HNL, INR, JMD, JOD, LBP, LKR, MAD, PEN, PKR, SDD, TND, VEB, ZMK (19 currencies). In the latter group the exchange rates are usually fixed daily by national central banks. Dividing currencies into such four groups of different liquidity is common among finance practitioners. This therefore opens an additional interesting issue to be verified if the different dynamics that stays behind such a division according to the liquidity (implying the fixing method) is also reflected in correlations of daily exchange rates. For a given choice of the base currencyX the (symmetric) correlation matrix (CM) can be computed in terms of the normalized returns, g A (t). To this end one takes N time series {gX A (t0), g A (t0 + τ), . . . , g A (t0 + (T − 1)τ)} of length T . These series can form an N × T rectangular matrix M , and the CM can be written in the matrix notation as