%0 Book Section %1 statphys23_0050 %A Inoue, J. %A Sazuka, N. %A Scalas, E. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K compound continuous-time econophysics first normal passage poisson process queueing random statphys23 theory topic-11 walk %T First passage process in financial markets described by normal compound Poisson processes: Queueing theoretical analysis of the average waiting time %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=50 %X Fluctuation between price changes in financial markets gives us important information for building on-line systems to support customers in decision making1. The time interval between two consecutive trades is one of the relevant quantities, however, the average waiting time, which is defined as the average time of the time interval before the next price change after a customer starts checking a price via internet, is a more informative quantity for on-line trading systems2. A similar on-line foreign exchange trading service, Sony bank (http://moneykit.net), is available for individual customers in the currency exchange market. The Sony back rate can be regarded as a first-passage process (FPT), and if the variation of USD/JPY rate is greater or equal to 0.1 yen, the Sony bank USD/JPY exchange rate is updated to the market rate, otherwise it stays constant. When we estimate the FPT distribution and use the so-called renewal-reward theorem, the average waiting time w is given by $w = E(t^2)/2E(t)$, where $E(\cdots)$ denotes the average over the FPT distribution $P(t)$2. It is instructive to see what happens in different markets. Raberto et.al.3 applied NCCPs (Normal Compound Poisson Processes), which are CTRW (Continuous-Time Random Walks)3,4 characterized by i.i.d. exponentially distributed waiting times with activity parameter $\mu$ and i.i.d. normally distributed jumps with mean $þeta$ and standard deviation $\sigma$, to analyze the BTP futures (BTP is the middle and long term Italian Government bonds with fixed interest rates) at LIFFE (London International Financial Futures and options Exchange) in 1997. However, the first passage process and its statistical properties of the BTP futures have never yet investigated. In this paper, we consider the first passage process of the financial markets and derive the FPT distribution analytically for the NCCPs. For the FPT distribution, we evaluate the average waiting time via the renewal-reward theorem and compare the theoretical expression with the empirical data analysis of the BTP futures. For market data of the BTP futures, the successive time intervals is well described by the Mittag-Leffler function3. Unfortunately, for the Sony bank rate, the underlying market data are not available. To specify the market data, we carry out computer simulations based on the GARCH model in which time intervals of the price change are described by the Mittag-Leffler function and consider the first passage process with rate window with width 0.1 yen. In order to investigate the effect of the rate window, we fit the FPT of the output data from the window by a Weibull distribution5. \\ 1) E. Scalas, Econophysics: tick-by-tick fluctuations in finacial markets and modelling. \\ 2) J. Inoue and N. Sazuka, Submitted to Quantitative Finance, physics/0606040. \\ 3) M. Raberto, E. Scalas, R. Gorenflo and F. Mainardi, Quantitative Finance, condmat/ 0012497. \\ 4) E. Scalas, Physica A 362 225-239 (2006). \\ 5) N. Sazuka, Eur. Phys. J. B 50, 129 (2006).

@incollection{statphys23_0050, abstract = {Fluctuation between price changes in financial markets gives us important information for building on-line systems to support customers in decision making[1]. The time interval between two consecutive trades is one of the relevant quantities, however, the average waiting time, which is defined as the average time of the time interval before the next price change after a customer starts checking a price via internet, is a more informative quantity for on-line trading systems[2]. A similar on-line foreign exchange trading service, Sony bank (http://moneykit.net), is available for individual customers in the currency exchange market. The Sony back rate can be regarded as a first-passage process (FPT), and if the variation of USD/JPY rate is greater or equal to 0.1 yen, the Sony bank USD/JPY exchange rate is updated to the market rate, otherwise it stays constant. When we estimate the FPT distribution and use the so-called renewal-reward theorem, the average waiting time w is given by $w = E(t^2)/2E(t)$, where $E(\cdots)$ denotes the average over the FPT distribution $P(t)$[2]. It is instructive to see what happens in different markets. Raberto et.al.[3] applied NCCPs (Normal Compound Poisson Processes), which are CTRW (Continuous-Time Random Walks)[3,4] characterized by i.i.d. exponentially distributed waiting times with activity parameter $\mu$ and i.i.d. normally distributed jumps with mean $\theta$ and standard deviation $\sigma$, to analyze the BTP futures (BTP is the middle and long term Italian Government bonds with fixed interest rates) at LIFFE (London International Financial Futures and options Exchange) in 1997. However, the first passage process and its statistical properties of the BTP futures have never yet investigated. In this paper, we consider the first passage process of the financial markets and derive the FPT distribution analytically for the NCCPs. For the FPT distribution, we evaluate the average waiting time via the renewal-reward theorem and compare the theoretical expression with the empirical data analysis of the BTP futures. For market data of the BTP futures, the successive time intervals is well described by the Mittag-Leffler function[3]. Unfortunately, for the Sony bank rate, the underlying market data are not available. To specify the market data, we carry out computer simulations based on the GARCH model in which time intervals of the price change are described by the Mittag-Leffler function and consider the first passage process with rate window with width 0.1 yen. In order to investigate the effect of the rate window, we fit the FPT of the output data from the window by a Weibull distribution[5]. \\ 1) E. Scalas, Econophysics: tick-by-tick fluctuations in finacial markets and modelling. \\ 2) J. Inoue and N. Sazuka, Submitted to Quantitative Finance, physics/0606040. \\ 3) M. Raberto, E. Scalas, R. Gorenflo and F. Mainardi, Quantitative Finance, condmat/ 0012497. \\ 4) E. Scalas, Physica A 362 225-239 (2006). \\ 5) N. Sazuka, Eur. Phys. J. B 50, 129 (2006).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Inoue, J. and Sazuka, N. and Scalas, E.}, biburl = {https://www.bibsonomy.org/bibtex/217fab455e4d228c867eb28158efb5288/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {a333a03360b6f02cc35edbf46aa6aa92}, intrahash = {17fab455e4d228c867eb28158efb5288}, keywords = {compound continuous-time econophysics first normal passage poisson process queueing random statphys23 theory topic-11 walk}, month = {9-13 July}, timestamp = {2007-06-20T10:16:10.000+0200}, title = {First passage process in financial markets described by normal compound Poisson processes: Queueing theoretical analysis of the average waiting time}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=50}, year = 2007 }