Deterministic and stochastic influences on Japan and US stock and foreign exchange markets. A Fokker-Planck approach

TL;DR

The paper addresses distinguishing deterministic and stochastic forces in major stock indices and currency rates by deriving a Fokker-Planck equation directly from data. It estimates Kramers-Moyal coefficients to obtain the drift $D^{(1)}$ and diffusion $D^{(2)}$ and tests the Markov property via Chapman-Kolmogorov consistency. The leading drift term is largely universal across series, while the NASDAQ diffusion term is about twice as large as the others, indicating stronger stochasticity and possibly electronic-trading effects; the Chapman-Kolmogorov test confirms a Markovian pdf evolution. The approach is model-independent and scalable to both short and long time horizons, providing a framework that could enable Langevin-type forecasting and deeper econophysics insight.

Abstract

The evolution of the probability distributions of Japan and US major market indices, NIKKEI 225 and NASDAQ composite index, and $JPY/DEM$ and $DEM/USD$ currency exchange rates is described by means of the Fokker-Planck equation (FPE). In order to distinguish and quantify the deterministic and random influences on these financial time series we perform a statistical analysis of their increments $Δx(Δ(t))$ distribution functions for different time lags $Δ(t)$. From the probability distribution functions at various $Δ(t)$, the Fokker-Planck equation for $p(Δx(t), Δ(t))$ is explicitly derived. It is written in terms of a drift and a diffusion coefficient. The Kramers-Moyal coefficients, are estimated and found to have a simple analytical form, thus leading to a simple physical interpretation for both drift $D^{(1)}$ and diffusion $D^{(2)}$ coefficients. The Markov nature of the indices and exchange rates is shown and an apparent difference in the NASDAQ $D^{(2)}$ is pointed out.

Figures (5)

• Figure 1: Daily closing price of (a) NIKKEI 225, (b) NASDAQ, (c) $JPY$/$DEM$ and (d) $DEM$/$USD$ exchange rates for the period from Jan. 01, 1985 till May 31, 2002
• Figure 2: Probability distribution function $p(\Delta x,\Delta t)$ of (a) NIKKEI 225, (b) NASDAQ, (c) $JPY$/$DEM$ and (d) $DEM$/$USD$ from Jan. 01, 1985 till May 31, 2002 for different delay times. Each pdf is displaced vertically to enhance the tail behavior; symbols and the time lags $\Delta t$ are in the insets. The discretisation step of the histogram is (a) 200, (b) 27, (c) 0.1 and (d) 0.008 respectively
• Figure 3: Typical contour plots of the joint probability density function $p(\Delta x_2,\Delta t_2; \Delta x_1,\Delta t_1)$ of (a) NIKKEI 225, (b) NASDAQ closing price signal and (c) $JPY$/$DEM$ and (d) $DEM$/$USD$ exchange rates for $\Delta t_2=1\quad day$ and $\Delta t_1=3\quad days$. Contour levels correspond to $log_{10}p=-1.5,-2.0,-2.5,-3.0,-3.5$ from center to border
• Figure 4: Functional dependence of the drift and diffusion coefficients $D^{(1)}$ and $D^{(2)}$ for the pdf evolution equation (3); $\Delta x$ is normalized with respect to the value of the standard deviation $\sigma$ of the pdf increments at delay time 32 days: (a,b) NIKKEI 225 and (c,d) NASDAQ closing price signal, (e,f) $JPY$/$DEM$ and (g,h) $DEM$/$USD$ exchange rates
• Figure 5: Equal probability contour plots of the conditional pdf $p(\Delta x_2,\Delta t_2|\Delta x_1,\Delta t_1)$ for two values of $\Delta t$, $\Delta t_1=8 \quad$ days, $\Delta t_2=1 \quad$ day for NASDAQ. Contour levels correspond to $log_{10}p$=-0.5,-1.0,-1.5,-2.0,-2.5 from center to border; data (solid line) and solution of the Chapman Kolmogorov equation integration (dotted line); (b) and (c) data (circles) and solution of the Chapman Kolmogorov equation integration (plusses) for the corresponding pdf at $\Delta x_2$ = -50 and +50