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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 $\Delta x(\Delta(t))$ distribution functions for different time lags $\Delta(t)$. From the probability distribution functions at various $\Delta(t)$, the Fokker-Planck equation for $p(\Delta x(t), \Delta(t))$ 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., Comment: 8 pages, 5 figures; to appear in the Proceedings of the Proceedings of the Second Nikkei Symposium: Toward control of Economic Change -- Application of Econophysics, Nov 12-14, 2002, Tokyo, Japan

この論文は国立情報学研究所の電子図書館事業により電子化されました。, 解説

In this work the analytical characterization of the probability density of financial returns in the exponential Ornstein-Uhlenbeck model is addressed. Since prices are driven by a Geometric Brownian motion, whose diffusion coefficient is expressed through an exponential function of the mean-reverting process Y , an exact expression for the density is hard to be drawn. However, we derive the closed-form formula for the characteristic function and for the cumulants of the approximating linear model under the low volatility fluctuations regime. Theoretical results are confirmed numerically by use of Monte Carlo simulations. The effectiveness of the analytical predictions is tested on the German DAX30 Index, finding a good agreement between the empirical data and the theoretical description.

We analyze the sequence of time intervals between consecutive stock trades of five large companies representing different sectors of the US economy over a period of four years. We show that independent of the industry sector and average level of activity, the series of intertrade times exhibit common statistical features. Specifically, we find that: (1) the tail of the probability density function of intertrade times may be fit by a stretched exponential form; (ii) the probability densities of the intertrade times for all five companies collapse onto a single curve when appropriately resealed, and (iii) the intertrade times exhibit correlated behaviour over hundreds of trades within a trading day and an even greater degree of correlation over longer time scales.