Turbulence hierarchy in foreign exchange markets

We present a multiscale stochastic analysis of foreign exchange rates using the H-theory formalism, which provides a hierarchical intermittency model for the information cascade in the currency market. We examine the distributions of returns and volatilities for the three most traded currency pairs: euro–U.S. dollar, U.S. dollar–Japanese yen, and British pound–U.S. dollar. We find that these markets have a hierarchy of timescales, with larger markets exhibiting more hierarchy levels. We provide a theoretical framework for understanding why the number of levels in the information cascade increases with market size, in analogy with similar behavior for the energy cascade in turbulence as a function of Reynolds number. We briefly argue that using turbulence-like models for financial markets can also provide valuable insights for developing efficient algorithmic trading strategies.

Figure 1

One-hour quotes for the exchange rates of the three most traded currency pairs: EUR-USD (blue), JPY-USD (orange), and GBP-USD (green). Here we have normalized the initial point of all three series to unity for convenience of visualization.

Figure 2

(a) Plots of the distribution $P_N\left(x\right)$ given in Eq. (12) for $v_0=1, \beta =1$, and $N=1,2,3,4,5$. (b) Same as in (a) in log-log scale for the positive side ($x>0$) of the distributions.

Figure 3

Errors of the joint fit as a function of the number $N$ of hierarchy levels for the three most traded currency pairs: EUR-USD (blue circles), USD-JPY (orange triangles), and GBP-USD (green squares).

Figure 4

(a) Distribution of exchange rates returns (black circles) for EUR-USD and the H-theory prediction (lines) for $N=1$ and $\beta =0.53$ (green dashed), $N=2$ and $\beta =0.93$ (orange dotted), $N=3$ and $\beta =1.36$ (purple dashed), $N=4$ and $\beta =1.85$ (blue dashed), and $N=5$ and $\beta =2.27$ (red). Here the best joint fit was attained for $N=5$; see Fig. 3. (b) Histogram (black circles) of the volatility series $v\left(t\right)$ and model prediction (lines) with the same parameters and colors convention as in (a).

Figure 5

(a) Distribution of exchange rates returns (black circles) for USD-JPY and the H-theory prediction (lines) for $N=1$ and $\beta =0.38$ (green dashed) and $N=2$ and $\beta =0.85$ (red). Here the best joint fit was attained for $N=2$; see Fig. 3. (b) Histogram (black circles) of the volatility series $v\left(t\right)$ and model prediction (lines) with the same parameters and colors convention as in (a).

Figure 6

(a) Distribution of exchange rates returns (black circles) for GBP-USD and the H-theory prediction (red line) for $N=1$ and $\beta =0.35$. Here the best joint fit was attained for $N=1$; see Fig. 3. (b) Histogram (black circles) of the volatility series $v\left(t\right)$ and model prediction (line) with the same parameters and colors convention as in (a).