On the gap between an empirical distribution and an exponential distribution of waiting times for price changes in a financial market

Abstract

We analyze waiting times for price changes in a foreign currency exchange rate. Recent empirical studies of high frequency financial data support that trades in financial markets do not follow a Poisson process and the waiting times between trades are not exponentially distributed. Here we show that our data is well approximated by a Weibull distribution rather than an exponential distribution in a non-asymptotic regime. Moreover, we quantitatively evaluate how much an empirical data is far from an exponential distribution using a Weibull fit. Finally, we discuss a phase transition between a Weibull-law and a power-law in the asymptotic long waiting time regime.

Queueing theoretical analysis of foreign currency exchange rates

“…III, we introduce a general formula to evaluate average waiting time using the renewal-reward theorem and calculate it with regard to Sony Bank customers. Recently, one of the authors [10] provided evidence implying that the first-passage time (FPT) distribution of the Sony Bank rate obeys the Weibull distribution [28]. This conjecture is regarded as a counter part of studies that suggest that the FPT should follow an exponential distribution (see example in [29]).…”

“…Obviously, the rate (or price) change of the trading behaves according to some unknown stochastic processes, and numerous studies have been conducted to reveal the statistical properties of its nonlinear dynamics [1,2,3]. In fact, several authors have analysed tick-by-tick data of price changes including the currency exchange rate in financial markets [4,5,6,7,8,9,10]. Some of these studies are restricted to the stochastic variables of price changes (returns) and most of them are specified by terms such as the fat or heavy tails of distributions [1].…”

The distribution of first-passage times and durations in FOREX and future markets

“…In previous studies, we found that a Weibull distribution is a good candidate to describe the Sony Bank USD/JPY exchange rate time statistic [13]. The average waiting time was also evaluated to investigate to what extent the Sony Bank rate is well-explained by the Weibull distribution [9,14].…”

“…The next problem is how to choose the parameters γ, t × , m and a. Fortunately, we know these parameters from empirical data analysis [13,14]. Substituting those parameters γ = 4.67, m = 0.585 and a = 49.63 into our formula (15), we evaluate the average waiting time w as a function of the crossover point t × .…”

“…For these data, the relevant random variable is an intertrade duration. Both data sets have already been studied and extensively described in previous papers ( [9,11,12,13,14,15]). In both cases, we assume that the empirical random variables are a realization of a renewal process.…”