Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets

Abstract

Fat-tailed distributions have been reported in fluctuations of financial markets for more than a decade. Sliding interval techniques used in these studies implicitly assume that the underlying stochastic process has stationary increments. Through an analysis of intraday increments, we explicitly show that this assumption is invalid for the Euro-Dollar exchange rate. We find several time intervals during the day where the standard deviation of increments exhibits power law behavior in time. Stochastic dynamics during these intervals is shown to be given by diffusion processes with a diffusion coefficient that depends on time and the exchange rate. We introduce methods to evaluate the dynamical scaling index and the scaling function empirically. In general, the scaling index is significantly smaller than previously reported values close to 0.5. We show how the latter as well as apparent fat-tailed distributions can occur only as artifacts of the sliding interval analysis.fat tails ͉ Fokker-Planck equation ͉ Langevin simulations A rguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies conducted over the last decade have reported that they are non-Gaussian, scale in time, and have power law (or fat) tails (1-6). However, by combining increments at multiple times in their statistical analyses (sliding interval techniques), these studies implicitly assume that the stochastic process has stationary increments. For financial markets, it is not clear whether this assumption is valid. For example, it is possible that trading activity at the beginning of a trading day may differ from that at the end of the day. How is it possible to test whether intraday fluctuations are timeindependent? If they are time-dependent, how can statistical analyses be conducted? Will results from previous studies be invalidated?Our analysis is conducted on intraday Euro-Dollar exchange rates (traded 24 h per day) during 1999-2004 recorded in 1-min intervals. It is based on the assumption that intraday variations in the market follow the same underlying stochastic process every day. Then, a statistical analysis for fluctuations at a given time of the day can be conducted by using data from multiple trading days within the sample.We find from this analysis the following. (i) The stochastic process is time-dependent and there are several intervals during the day where the standard deviation of increments exhibits power law behavior. Stochastic dynamics during these intervals is given by variable diffusion processes (2). (ii) Dynamical scaling indices and empirical scaling functions within these scaling intervals are different from previously reported results. We show how the latter can result from the application of sliding interval methods to a time-dependent stochastic process. (iii) Autocorrelation functions for var...

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

“…Standard correlation measures like the Pearson correlation coefficient and the cross-correlation function require stationary data in order to provide reliable results, which is a requirement that is hard to fulfill in many real-world situations (the financial and physiological data are the negative examples here [1][2][3][4][5][6][7][8][9][10]). (By stationarity we mean stability of the probability distribution functions of the data over time; from this perspective nonstationarity can be produced both by the long-range autocorrelations and by the pdf's heavy tails that make any signal length effectively insufficient.)…”

Martingale option pricing

“…The quantity p 2 d(p,t) is the price diffusion coefficient. The function d(p,t) is constant in the Black-Scholes model (lognormal pricing, or Gaussian returns) but real market data [1] forces us to consider models where d(p,t)≠constant. A merely t-dependent function d(t), independent of p, is trivially lognormal as well, as one can easily see by transforming to logarithmic returns.…”

“…The empirical market density (the 1-point density) is then given by f(x,t)=g(x,t;0,0), a quantity that one hopes to extract as histograms from financial time series [1].…”

“…To describe variable diffusion models we begin with the sde for the stock price p(t) at present time t, ! dp = µpdt + p 2 d(p, t)dB(t), (1) where B(t) is the Wiener process and µ is the (hard to estimate) stock 'interest rate'. The quantity p 2 d(p,t) is the price diffusion coefficient.…”

“…time intervals [1]: (i) Finance markets are nonstationary with no observable approach to statistical equilibrium, ruling out asymptotically stationary processes like those considered in [2], e.g. (ii) To the extent that reliable histograms can be constructed, the market distribution f(x,t) is exponential in x where x is the log return.…”