myjournal manuscript No. (will be inserted by the editor)
Which is better for describing first-passage processes of financial markets: a Weibull distribution or a Mittag-Leffler survival function? Naoya Sazuka1 , Jun-ichi Inoue2 , Enrico Scalas3 1 2
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Sony Corporation, 1-7-1 Konan Minato-ku, Tokyo 108-0075, Japan, e-mail:
[email protected] Complex Systems Engineering, Graduate School of Information Science and Technology, Hokkaido University, N14-W9, Kita-ku, Sapporo 060-0814, Japan, e-mail: j
[email protected] Dipartimento di Scienze e Technologie, Universita del Piemonte Orientale, Via Bellini, 25 g I-15100 Alessandria, Italy, e-mail:
[email protected]
September 10, 2007
Abstract A possible choice for distributions of duration to describe firstpassage processes of financial markets is discussed. To represent market data which possess a relatively long duration, we use two types of distributions, namely, a distribution derived from the so-called Mittag-Leffler survival function and a Weibull distribution. For the survival function of Mittag-Leffler type, we find that the average waiting time (residual life time), which is a relevant statistics to specify the markets, is strongly dependent on the choice of the cut-off parameter tmax , whereas the Weibull distribution is free from such parameters. This fact means that a Weibull distribution is more convenient than a Mittag-Leffler one in order to evaluate the relevant statistics in financial markets with long durations. Based on the above considerations, we propose a good candidate for describing the first passage time distribution in the market, namely, a Weibull distribution with a power-law tail compensating the gap between theoretical and empirical results much more efficiently than a pure Weibull distribution.
1 Introduction The distribution of the time interval between price changes gives us important pieces of information about the market [1]. Recently, various on-line trading services on the internet were established by several major banks.
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For instance, the Sony Bank uses a trading system in which their own foreign currency exchange rate changes according to the so-called first-passage process. Namely, their USD/JPY exchange rate is renewed only if the market rate changes more than 0.1 yen. As the result, for the case of the Sony Bank rate, the average duration becomes longer, passing from 7 seconds to 20 minutes, in comparison with many other market rates such as BTP futures (BTP is the middle and long term Italian Government bonds with fixed interest rates) once traded at LIFFE (LIFFE stands for London International Financial Futures and Options Exchange) and with liquid stocks traded at major stock exchanges. From the view’point of complex systems engineering, a relevant quantity used to specify the stochastic process of the market rate is the average waiting time (residual life time) rather than the average duration. In a series of recent studies by the present authors, the average waiting time of the Sony Bank USD/JPY exchange rate was evaluated extensively under the assumption that the first-passage time (FPT) distribution might obey a Weibull-law. On the other side, the so-called Mittag-Leffler survival function has been used to represent the distribution of durations in several markets. For example, Raberto et al. showed that BTP’future intertrade durations are well-described by a survival function of Mittag-Leffler type. However, the Mittag-Leffler survival function has never yet been applied to evaluation of the average waiting time. In this paper, we compare a Weibull distribution with a Mittag-Leffler survival function in order to evaluate the average waiting time for financial markets. We give an analytical formula for the average waiting time under the assumption that the FPT distribution might be described by a Mittag-Leffler survival function. We find that the average waiting time diverges linearly with respect to the cut-off parameter tmax . This fact tells us that it is hard to handle the Mittag-Leffler survival function to evaluate relevant statistics such as the average waiting time. We also provide a good candidate for the description of the first passage process of the market rate a Weibull distribution in which the behaviour of the distribution changes from a Weibull-law to a power-law at some crossover point t× . We find that the average waiting time becomes much closer to empirical value for the Sony Bank USD/JPY exchange rate than for a pure Weibull distribution. This paper is organized as follows. In the next section, we introduce both the Mittag-Leffler survival function and the Weibull distribution and discuss their properties. In section 3, we evaluate the average waiting time for the Mittag-Leffler function. We find that the average waiting time diverges linearly on the cut-off paramter tmax . In section 4, we introduce a Weibull distribution with a power-law tail to compensate a small gap between the results of theoretical and empirical data analyses for the average waiting time. In the last section, we summarize our results.
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2 Mittag-Leffler survival function and Weibull distribution For market data of BTP futures, the successive time intervals are welldescribed in terms of the Mittag-Leffler survival function [2]: Eβ (−(t/t0 )β ) =
∞
(−1)n
n=0
(t/t0 )βn Γ (βn + 1)
(0 < β < 1)
(1)
where Γ (z) denotes a Gamma function and we set the upper bound of the sum to a large value nmax in practical use. The above Mittag-Leffler survival function has asymptotic forms: Eβ (−(t/t0 )β ) exp[−(t/t0 )β /Γ (1 + β)] (t/t0 → 0) and Eβ (−(t/t0 )β ) (t/t0 )−β /Γ (1 − β) (t/t0 → ∞). Then, the density function of the duration t is given by PML (t : β) ≡ −
∞ 1 (t/t0 )βn+β−1 ∂Eβ (−(t/t0 )β ) = . (−1)n ∂t t0 n=0 Γ (βn + β)
(2)
In the limiting case β = 1, the Mittag-Leffler distribution coincides with the exponential distribution. On the other hand, the so-called Weibull distribution is given by m t tm−1 PW (t : m, a) = m exp − , (3) a a and is a good approximation of the Sony Bank USD/JPY exchange rate in a non-asymptotic regime t ∞. It can be directly verified that the Weibull distribution (3) is reduced to an exponential distribution for m = 1 and a Rayleigh distribution for m = 2. For these two candidates, we check the validity of the assumption for the distribution of the duration through a relevant statistics, namely, the average waiting time, which is defined as the average time of the interval before the next price change after a Sony-bank customer starts observing a price via internet. It is a more informative quantity than the duration between two consecutive price changes in the context of queueing theory. 3 Divergence of the average waiting time for the Mittag-Leffler survival function For the density (2) derived by the Mittag-Leffler function, the average waiting time, which is obtained by the ratio of the first two moments ∞ of the FPT distribution as w = E(t2 )/2E(t) with a definition: E(tn ) = 0 tn PML (t)dt, is given by n (t/t0 )βn+β+2 n t0 n=0 (−1) (βn+β+2)Γ (βn+β) (4) w(t0 , tmax : β) = n (t/t0 )βn+β+1 n 2 n=0 (−1) (βn+β+1)Γ (βn+β) In Figure 1, we plot the w for several values of tmax with t−1 0 = 1/12. We
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4000
tmax=1000 tmax=5000 tmax=10000
3000
w 2000
1000
0
0
0.2
0.4
β
0.6
0.8
1
Fig. 1 The average waiting time w by (4) with t−1 0 = 1/12.
should keep in mind that the above w diverges as tmax → ∞. As we saw, the asymptotic form of the above density function is ∼ t−1−β when t/t0 → ∞. The divergence of the w might come from only this power-law regime. Actually, we see this fact by evaluating the first two t moments of the density function. These two moments behave as t2 0 max r−β−1 r2 dr = t2−β max and tmax −β−1 1−β t 0 r rdr = tmax for (4) as tmax → ∞. Thus, the average wait1−β ing time diverges linearly as a function of tmax as w t2−β max /tmax = tmax . Thus, we should take into account the finite upper bound of the integral with respect to t, namely, the maximum value of the duration or the cut-off parameter tmax . Then, we have a problem. Namely, how do we determine tmax to obtain a reasonable w that is consistent with the result by the empirical data analysis? Of course, it also depends on the crossover point t× at which the density function changes its shape from a stretched exponentiallaw to the power-law. If t× is close to the tmax , the w is not sensitive to the value of tmax , however, if t× is far from tmax , w depends on the value of tmax because the integral for the power-law tail becomes dominant. This might be an evidence to conclude that the Mittag-Leffler function is hard to use in order to evaluate the average waiting time for the market rates with a relatively long duration such as the Sony Bank USD/JPY exchange rate.
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4 A Weibull distribution with a power-law tail In the previous studies, we found that a Weibull distribution is a good candidate to describe the Sony Bank USD/JPY exchange rate time statistics [4]. The average waiting time was also evaluated to investigate to what extent the Sony Bank rate is well-explained by the Weibull distribution [5, 6]. Moreover, it was shown that the analytical prediction of the Gini index calculated for a Weibull distribution is in good agreement with the value obtained from the empirical data of the Sony Bank rate [7]. However, there exists a manifest small gap between the theoretical prediction (∼ 42 [min]) and the empirical result for w (∼ 49 [min]). In this section, we consider to what extent the average waiting time can be modified by taking into account a power-law behavior for the tail in the FPT distribution. In our previous paper, we assumed that the FPT of the Sony Bank rate might obey a pure Weibull distribution (3). However, several empirical data analyses have shown that the shape of the FPT distribution changes from a pure Weibull-law to a power-law at some crossover point t× . Therefore, here it is natural to assume that the FPT distribution should be modified in terms of the following form: m−1 m mt exp − ra (t < t× ) a (5) PW (t : m, a, γ, t× ) = λ t−γ (t > t× ) As at the critical point t× , the distribution must be continuous, the condim−1 /a) exp(−tm tion t−γ × λ = (mt× × /a) is required. This condition determines the parameter λ as m t mtm+γ−1 × exp − × . λ= (6) a a Thus, the modified FPT distribution is given by ⎧ m−1 m ⎨ mt (t < t× ) exp − ta a
m m+γ−1 PW (t : m, a, γ, t× ) = mt× t× −γ ⎩ exp − a t (t > t× ) a
(7)
From the FPT distribution, we have the average waiting time w from the renewal-reward theorem as follows. mtm+1
m 1 1 tm t × a1/m × + a(γ−2) exp − a× m Γ m B m + 1, a w(t× : m, a, γ) = mtm+2
tm (8) 2 2 tm 2a2/m × × + B Γ + 1, exp − a× m m m a a(γ−3) where B(a, x) denotes the following incomplete Gamma function: x 1 ta−1 e−t dt. B(a, x) = Γ (a) 0
(9)
The next problem we encounter is how to choose the parameters γ, t× , m and a. Fortunately, we know these parameters from empirical data analysis
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m=0.58 m=0.585 m=0.59
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50
48
w 46
44
42
40
38
2
4
tx
6
8
10 (x 10000)
Fig. 2 The average waiting time w as a function of t× . We set γ = 4.66, a = 50.855. For two cases of the choice for m, namely, for m = 0.58, 0.585 and 0.59, the w is plotted.
[4]. Substituting γ = 4.66, m = 0.59, a = 50.855, we plot the average waiting time w as a function of the critical time point t× in Figure 2. In this figure, we present the average waiting time for two values of m, namely, m = 0.58, 0.585 and 0.59. From the empirical data analysis, we have t× = 18, 000. Therefore, we conclude that for m = 0.59, the waiting time is estimated as w = 43.982 [min]. This value is closer to the sampling value w = 49 [min] than the value obtained under the assumption of a pure Weibull distribution (w = 42 [min]). Therefore, we conclude that a correction by taking into account the tail behavior of the Weibull distribution works in the right direction for estimating the average waiting time.
5 Concluding remarks In this paper, we have compared a Weibull distribution and a Mittag-Leffler distribution and the average waiting time has been studied in both cases. Our theoretical analysis revealed that the average waiting time diverges linearly as a function of the cut-off parameter tmax for the Mittag-Leffler distribution. This fact implies a more difficult treatment to check the validity of modeling the market renewal process by means of the Mittag-Leffler distribution. On the other side, we also find that a Weibull distribution with a power-law tail is an efficient way to describe renewal processes in markets with a long duration such as the Sony Bank USD/JPY exchange rate seen
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as a first passage process. We conclude that the Weibull distribution with power-law tail is more suitable to evaluate the relevant statistics for financial markets with a long duration. Acknowledgements E.S. is grateful to JSPS for a short-term fellowship in Japan during which this paper has been discussed. The authors wish to thank Prof. T. Kaizoji for useful discussion.
References 1. E. Scalas, Mixtures of compound Poisson processes as models of tick-by-tick financial data, Chaos, Soliton & Fractals, 34 33-40 (2007). 2. M. Raberto, E. Scalas, R. Gorenflo and F. Mainardi, Quantitative Finance, 4 695-702 (2004). 3. http://moneykit.net 4. N. Sazuka, Physica A 376 500-506 (2007). 5. J. Inoue and N. Sazuka, Submitted to Quantitative Finance, physics/0606040. 6. N. Sazuka and J. Inoue, in the proceedings of the first IEEE Symposium of Foundations of Computational Intelligence 2007, physics/0702003. 7. N. Sazuka and J. Inoue, 383, 49-53 (2007). 8. E. Scalas, Physica A 362 225-239 (2006).
