Naoya Sazuka

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

Possible distributions are discussed for intertrade durations and first-passage processes in financial markets. The view-point of renewal theory is assumed. In order to represent market data with relatively long durations, two types of distributions are used, namely, a distribution derived from the so-called Mittag-Leffler survival function and the Weibull distribution. For Mittag-Leffler type distribution, the average waiting time (residual life time) is strongly dependent on the choice of a cut-off parameter t_ max, whereas the results based on the Weibull distribution do not depend on such a cut-off. Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type one if one wishes to evaluate relevant statistics such as average waiting time in financial markets with long durations. On the other side, we find that the Gini index is rather independent of the cut-off parameter. Based on the above considerations, we propose a good candidate for describing the distribution of first-passage time in a market: The Weibull distribution with a power-law tail. This distribution compensates the gap between theoretical and empirical results much more efficiently than a simple Weibull distribution. We also give a useful formula to determine an optimal crossover point minimizing the difference between the empirical average waiting time and the one predicted from renewal theory. Moreover, we discuss the limitation of our distributions by applying our distribution to the analysis of the BTP future and calculating the average waiting time. We find that our distribution is applicable as long as durations follow a Weibull-law for short times and do not have too heavy a tail.

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

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 the 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 transition between a Weibull-law and a power-law in the long time asymptotic regime.

Analysis of binarized high frequency financial data

Abstract.A non-trivial probability structure is evident in the binary data extracted from the up/down price movements of very high frequency data such as tick-by-tick data for USD/JPY. In this paper, we analyze the Sony bank USD/JPY rates, ignoring the small deviations from the market price. We then show there is a similar non-trivial probability structure in the Sony bank rate, in spite of the Sony bank rates having less frequent and larger deviations than tick-by-tick data. However, this probability structure is not found in the data which has been sampled from tick-by-tick data at the same rate as the Sony bank rate. Therefore, the method of generating the Sony bank rate from the market rate has the potential for practical use since the method retains the probability structure as the sampling frequency decreases.

Fluctuations in time intervals of financial data from the view point of the Gini index

We propose an approach to explain fluctuations in time intervals of financial markets data from the view-point of the Gini index. We show the explicit form of the Gini index for a Weibull distribution: A good candidate to describe the first passage time of foreign exchange rate. The analytical expression of the Gini index compares well with the value obtained from empirical data.

Queueing theoretical analysis of foreign currency exchange rates

We propose a useful approach for investigating the statistical properties of foreign currency exchange rates. Our approach is based on queueing theory, particularly, the so-called renewal-reward theorem. For the first passage processes of the Sony Bank US dollar/Japanese yen (USD/JPY) exchange rate, we evaluate the average waiting time which is defined as the average time that customers have to wait between any instant when they want to observe the rate (e.g. when they log in to their computer systems) and the next rate change. We find that the assumption of exponential distribution for the first-passage process should be rejected and that a Weibull distribution seems more suitable for explaining the stochastic process of the Sony Bank rate. Our approach also enables us to evaluate the expected reward for customers, i.e. one can predict how long customers must wait and how much reward they will obtain by the next price change after they log in to their computer systems. We check the validity of our prediction by comparing it with empirical data analysis.

Crossover between Lévy and Gaussian regimes in first-passage processes

We propose an approach to the problem of the first-passage time. Our method is applicable not only to the Wiener process but also to the non-Gaussian Lévy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first-passage time problems in the truncated Lévy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first-passage time t distribution changes from t(-(alpha+1)/alpha)-law (non-Gaussian Lévy regime) to t(-32)-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy flight but also in the first-passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the Lévy distribution are discussed.

Waiting time analysis of foreign currency exchange rates: Beyond the renewal-reward theorem

We evaluate the average waiting time between observing the price of financial markets and the next price change, especially in an on-line foreign exchange trading service for individual customers via the Internet. Basic technical idea of our present work is dependent on the so-called renewal-reward theorem. Assuming that stochastic processes of the market price changes could be regarded as a renewal process, we use the theorem to calculate the average waiting time of the process. In the conventional derivation of the theorem, it is apparently hard to evaluate the higher order moments of the waiting time. To overcome this type of difficulties, we attempt to derive the waiting time distribution Omega(s) directly for arbitrary time interval distribution (first passage time distribution) of the stochastic process PW(r) and observation time distribution PO (t) of customers. Our analysis enables us to evaluate not only the first moment (the average waiting time) but also any order of the higher moments of the waiting time. Moreover, in our formalism, it is possible to model the observation of the price on the internet by the customers in terms of the observation time distribution PW(t). We apply our analysis to the stochastic process of the on-line foreign exchange rate for individual customers from the Sony bank and compare the moments with the empirical data analysis