How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market

Abstract

It is shown that prize changes of the US dollar -German Mark exchange rates upon different delay times can be regarded as a stochastic Marcovian process. Furthermore we show that from the empirical data the Kramers-Moyal coefficients can be estimated. Finally, we present an explicite Fokker-Planck equation which models very precisely the empirical probabilitiy distributions. PACS: 02.50-r;05.10GSince high-frequency intra-day data are available and easy to access, research on the dynamics of financial markets is enjoying a broad interest. [1,2,3,4,5,6,7]. Well-founded quantitative investigations now seem to be feasible. The identification of the underlying process leading to heavy tailed probability density function (pdf) of price changes and the volatility clustering (see fig.1) are of special interest. The shape of the pdf expresses an unexpected high probability of large price changes on short time scales which is of utmost importance for risk analysis. In a recent work [8], an analogy between the short-time dynamics of foreign exchange (FX) market and hydrodynamic turbulence has been proposed. This analogy postulates the existence of hierarchical features like a cascade process from large to small time scales in the dynamics of prices, similar to the energy cascade in turbulence c.f. [9]. This postulate has been supported by some work [7] and questioned by others [6].1 Friedrich, Peinke, Renner 2 One main claim of the hypothesis of cascade processes is that the statistics of the time series of the financial market can be determined. The aim of the present paper is to discuss a new kind of analysis capable to derive the underlying mathematical model directly from the given data. This method yields an estimation of an effective stochastic equation in the form of a Fokker-Planck equation (also known as Kolmogorov equation). The solutions of this equation yields the probability distributions with sufficient accuracy (see fig. 1). This means that our method is not based on the conventional phenomenological comparison between models and several stochastic aspects of financial data. Our approach demonstrates how multiplicative noise and deterministic forces interact leading to the heavy tailed statistics. [10] Recently it has been demonstrated that from experimental data of turbulent flow the hierarchical features induced by the energy cascade can be extracted in form of a Fokker-Planck equation for the length scale dependence of velocity fluctuations [12]. In the following it will be shown the dynamics of foreign exchange rates can also be described by a Fokker-Planck equation. Here we use a data set consisting of 1,472,241 quotes for US dollar-German mark exchange rates from the years 1992 and 1993 as used in reference [8].We investigate the statistical dependence of prize changes ∆x i := x(t+ ∆t i ) − x(t) upon the delay time ∆t i . Here x(t) denotes the exchange rate at time t. We will show that the prize changes ∆x 1 , ∆x 2 for two delay times ∆t 1 , ∆t 2 are statistically dependent, provided the differ...

Fokker–Planck equation of distributions of financial returns and power laws

“…In a recent letter [22], Friedrich et al have proposed an alternative approach in terms of a Fokker-Planck equation for the distribution of asset returns at different time-scales, relying on their previous similar Fokker-Planck approach for the description of the distribution of velocity increments in turbulent cascades [23] and their proposed analogy between turbulent cascades in hydrodynamics and information cascades in stock markets.…”

“…Friedrich et al [22] then arrive at the following Langevin equation for the price increment ∆x(τ ) as a function of logarithmic time-scale τ…”

“…over the time interval ∆t (which also defines the time-scale), the purpose of Friedrich et al [22] is to write down an equation for the statistical process underlying the price changes ∆x over a series of nested time delays ∆t i (or time-scales) of decreasing durations. The controlling parameter is thus ∆t while t is dummy variable which, when varied, gives different realizations of the same process at the given time-scale ∆t.…”

“…dW is the increment of the random walk W (τ ), with white noise spectrum and variance equal to 2. This Langevin equation (3) is equivalent to the Fokker-Planck equation describing the distribution P (∆x, τ ) of price variations ∆x at a given logarithmic time-scale τ [22]:…”

A (reactive) lattice-gas approach to economic cycles

“…This leads to Langevin-like, and/or Fokker-Planck equations for price evolutions [8,9,10,11], even e.g. Black-Scholes equation for options [5,12] or simply partial distribution functions description [4,9].…”