- Email: [email protected]
Theoretical analysis of potential forces in markets
ARTICLE IN PRESS
Physica A 383 (2007) 115–119 www.elsevier.com/locate/physa
Theoretical analysis of potential forces in markets Misako Takayasua,, Takayuki Mizunoa, Hideki Takayasub a
Department of Computational Intelligence & Systems Science, Interdisciplinary Graduate School of Science & Engineering, Tokyo Institute of Technology, 4259-G3-52 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan b Sony Computer Science Laboratories, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan Available online 3 May 2007
Abstract We show that random walks in a moving potential function, with its center at the moving average of market prices, are represented in the form of the self-modulation model. From this point of view we confirm the existence of non-trivial autocorrelation in real market price changes. By generalizing the formulation of potential function we prove that the ARCH model belongs to the special case of random walk in an asymmetric potential with randomly changing coefficient. r 2007 Elsevier B.V. All rights reserved. PACS: 89.65.Gh; 05.40.a; 05.45.Fb Keywords: Market price; Random walk; Self-modulation; Potential force
1. Introduction Moving averages is popular technique among practitioners in financial markets. However, there has been little scientific study on the meaning and validity of these methods. Recently, the authors developed a new market price model, which is based on the moving averages [1,2]. The model is named potentials of unbalanced complex kinetics (PUCK), and in this model the market price is assumed to follow a random walk in a velocity potential field that shifts according to the change of the moving average value of market prices. The curvature of potential field also changes slowly, so the total behavior is much more complicated than a simple random walk. In this paper we firstly show that the time evolution of PUCK is equivalent to the self-modulation process [3], and discuss the validity of the data analysis method comparing with random time sequences. Then we generalize the formulation of PUCK and show that the ARCH model can also be viewed as a special case of a random walk in a random potential field.
Corresponding author.
E-mail address: [email protected] (M. Takayasu). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.04.094
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
116
2. PUCK model and the self-modulation process The time evolution of market price of PUCK model is given as q Pðt þ DtÞ PðtÞ ¼ FðPÞ þ f ðtÞ, qP P¼PðtÞ FðPÞ ¼
bðtÞ ðP PM ðtÞÞ2 , 2ðM 1Þ
(1)
(2)
where P(t) is the optimal moving average of market price [4], Dt is the time interval, b(t) is the time-dependent curvature of the potential function, and f(t) is a white noise, PM(t) is the simple moving average of latest M terms represented as PM ðtÞ ¼
X 1 M1 Pðt kDtÞ. M k¼0
(3)
As known from this we assume a random walk in a velocity potential field F(P) given by the quadratic function with its center at the moving average PM(t). This quadratic potential function is transformed as ( ) M1 X fP Pðt kDtÞg2 X bðtÞ bðtÞ 1 M1 2 2 PM ðtÞ 2 FðPÞ ¼ þ Pðt kDtÞ . (4) 2ðM 1Þ 2 ðM 1ÞM 2 k¼1 M k¼1 Here, as the second term in the right-hand side vanishes by differentiation in terms of P, so the potential field F(P) can be recognized as the summation of quadratic potentials centered at each trace {P(tkDt)} for k ¼ 1; 2; . . . ; M. The difference of P(t)PM(t) can be represented in terms of price difference as PðtÞ PM ðtÞ ¼
M2 X k¼0
Mk1 DPðt kDtÞ M
M1 ¼ hDPðtÞiM , 2
ð5Þ
where DPðtÞ PðtÞ Pðt DtÞ, and hDPðtÞiM is the moving average of price difference defined as hDPðtÞiM
M2 X k¼0
2ðM k 1Þ DPðt kDtÞ. MðM 1Þ
(6)
Then, the PUCK model is represented simply by the following stochastic equation: DPðt þ 1Þ ¼
bðtÞ hDPðtÞiM þ f ðtÞ. 2
(7)
This type of stochastic equation is known by the name of self-modulation process and peculiar properties are reported such as power law distributions and long range correlation like the 1/f power spectrum [3]. In view of time series analysis, this is a kind of AR model with stochastic multiplicative coefficients and the value of b(t) is an unbiased estimator independent of M. Eq. (7) can be directly applied for analyzing foreign exchange markets. Analyzing so-called the tick data of Yen–Dollar market we use 2000 latest data for estimation of the value of b(t). The estimated values are almost identical for different values of M in the range of M ¼ 11–20 as expected. In Fig. 1 we plot the distribution of the value of b(t) comparing two cases of random time sequences: one is a pure random walk and the other is randomly shuffled original time sequence of Yen–Dollar rates. Both cases follow nearly Gaussian distributions with zero mean and the standard deviations about 0.3, so the probability of finding a value of which absolute value is larger than 1 is less than 0.1%. As demonstrated in Fig. 1 the distribution of b(t) for the real Yen–Dollar rates is much wider than these random cases extending to the ranges, bðtÞo 1 and bðtÞ41.
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
117
Fig. 1. The distributions of b(t) estimated for the raw data, randomly shuffled data, and a pure random walk. The numbers of data points are about 13 million in each case. The smooth lines show the normal distributions with the mean value 0 and the standard deviation 0.30 and 0.29, respectively.
Fig. 2. Semi-log plot of the autocorrelations of b(t) for the raw data (~), randomly shuffled data (’), and a pure random walk (n).
The autocorrelation of b(t) is characterized by a long tail as shown in Fig. 2. The autocorrelation of b-value does not vanish up to about 3 months contrary to the very short correlation decay in the random cases. It is generally very difficult to confirm statistical robustness of a technical analysis, however, the estimation of our b-value is statistically meaningful especially in the case where the absolute value is larger than 1. The results are almost the same for other market data such as stock market prices. 3. Generalization of PUCK model We can generalize the PUCK model in two ways; the generalization of the moving average and the generalization of the potential function. The moving average in Eq. (3) is generalized in the following form introducing arbitrary weight function mk(t): PM 0 ðtÞ ¼
M1 X k¼0
mk ðtÞPðt kDtÞ.
(8)
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
118
Then Eq. (5) becomes PðtÞ PM 0 ðtÞ ¼ OðtÞ
M2 X
Ok ðtÞDPðt kDtÞ
k¼0
OðtÞhDPðtÞiM 0 ,
ð9Þ
where OðtÞ ¼ M 1 ðM 1Þ ( Ok ðtÞ ¼
1
k X
)
M1 X
mk ðtÞ þ
k¼0
M1 X
kmk ðtÞ
k¼0
mj ðtÞ =OðtÞ.
ð10Þ
j¼0
The PUCK formulation Eq. (1) is valid with the following modified version of potential function: FðPÞ ¼
bðtÞ ðP PM 0 ðtÞÞ2 . 4OðtÞ
(11)
Correspondingly, Eq. (7) is now represented as bðtÞ hDPðtÞiM 0 þ f ðtÞ. (12) 2 Thus, the self-modulation formulation does not change much by this generalization of the moving average. So far we have assumed only quadratic term in the potential function. In the general situation nonlinear potential field can be introduced by the following expansion: DPðt þ 1Þ ¼
FðPÞ ¼
1 X
bn ðtÞðP PM 0 ðtÞÞn .
(13)
n¼1
The corresponding self-modulation form is given as DPðt þ 1Þ ¼
1 X
nbn ðtÞOðtÞn fhDPðtÞiM 0 gn1 þ f ðtÞ.
(14)
n¼1
This type of generalized formulation is needed when the market fluctuation is asymmetric. 4. PUCK representation of the ARCH model The ARCH model in financial technology is now widely applied to describe the volatility clustering of market fluctuations [5]. Here, we show that the ARCH model can be described in the PUCK formulation. We introduce the following potential function: P P2 ðtÞ FðPÞ ¼ aðtÞG , (15) 2s0 where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi GðxÞ 12fx 1 þ x2 þ logðx þ 1 þ x2 Þg,
(16)
and a(t) is a white Gaussian noise with zero mean and unit standard deviation. Then, by introducing this potential form into Eq. (1) we have the following ARCH model: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DPðt þ 1Þ ¼ aðtÞ s20 þ DPðtÞ2 . (17) As known from this result the ARCH model corresponds to an asymmetric potential case with purely random coefficient. Owing to the effects of this potential field and the random coefficient, the price fluctuation is magnified so that the volatility clustering is realized.
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
119
5. Discussions The study of random walks in a potential was started in 1930 by Uhlenbeck and Ornstein [6], however such a fixed potential does not fit with the market prices because its large scale behaviors are apparently predictable. If there exists such a predictable behavior in a market, speculators will utilize the chance immediately and the market status will become no longer predictable. This economists’ logic sounds correct in rough sense, however, there must be a time lag for this relaxation phenomenon. Friedrich et al. [7] analyzed the foreign exchange market data intensively and found that attractive potential forces can be observed at some occasions in certain time scales. The PUCK model provides a much more sophisticated data analysis method and we can observe both types of the market potentials: attractive potentials and repulsive potentials [1]. These potentials shift with the moving average of market prices, namely, the potential field itself is randomly walking in a large scale, so the potential is quite different from Uhlenbeck and Ornstein potential in view of predictability. In this paper we showed that the effect of this potential field defined by the moving average of market prices is equivalent to the effect of the self-modulation of price changes. We confirmed the existence of this potential in the Yen–Dollar market data comparing with randomly shuffled time sequences. This self-modulation effect implies that the dealers’ responses take some time and the dealers’ action as a mass is described by a certain moving average. By generalizing the potential formulation we proved that the ARCH model is equivalent to assuming an asymmetric potential with a purely random coefficient. If the coefficient of this asymmetric potential is a constant, then the market price tends to move to one direction enhancing the fluctuation indefinitely. In the case of ARCH model no such directional motion occurs, instead we have volatility clustering as the coefficient is randomly changing with zero mean. The generalized PUCK formulation is not only directly applicable to any market data, it is applicable to any time sequential data which looks roughly random in large time scale but having complicated behaviors in short time scale. Acknowledgements This work is partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research #16540346 (M. T.) and Research Fellowships for Young Scientists (T. M.). The authors appreciate H. Moriya of Oxford Financial Education Co. Ltd. for providing the tick data. References [1] M. Takayasu, T. Mizuno, T. Ohnishi, H. Takayasu, in: H. Takayasu (Ed.), Practical Fruits of Econophysics, Springer, Tokyo, 2005, p. 29. [2] M. Takayasu, T. Mizuno, H. Takayasu, Physica A 370 (2006) 91. [3] M. Takayasu, H. Takayasu, Physica A 324 (2003) 101. [4] T. Ohnishi, T. Mizuno, K. Aihara, M. Takayasu, H. Takayasu, Physica A 344 (2004) 207. [5] R.F. Engle, Econometrica 4 (1982) 987. [6] G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36 (1930) 823. [7] R. Friedrich, J. Peinke, Ch. Renner, Phys. Rev. Lett. 84 (2000) 5224.
Physica A 383 (2007) 115–119 www.elsevier.com/locate/physa
Theoretical analysis of potential forces in markets Misako Takayasua,, Takayuki Mizunoa, Hideki Takayasub a
Department of Computational Intelligence & Systems Science, Interdisciplinary Graduate School of Science & Engineering, Tokyo Institute of Technology, 4259-G3-52 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan b Sony Computer Science Laboratories, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan Available online 3 May 2007
Abstract We show that random walks in a moving potential function, with its center at the moving average of market prices, are represented in the form of the self-modulation model. From this point of view we confirm the existence of non-trivial autocorrelation in real market price changes. By generalizing the formulation of potential function we prove that the ARCH model belongs to the special case of random walk in an asymmetric potential with randomly changing coefficient. r 2007 Elsevier B.V. All rights reserved. PACS: 89.65.Gh; 05.40.a; 05.45.Fb Keywords: Market price; Random walk; Self-modulation; Potential force
1. Introduction Moving averages is popular technique among practitioners in financial markets. However, there has been little scientific study on the meaning and validity of these methods. Recently, the authors developed a new market price model, which is based on the moving averages [1,2]. The model is named potentials of unbalanced complex kinetics (PUCK), and in this model the market price is assumed to follow a random walk in a velocity potential field that shifts according to the change of the moving average value of market prices. The curvature of potential field also changes slowly, so the total behavior is much more complicated than a simple random walk. In this paper we firstly show that the time evolution of PUCK is equivalent to the self-modulation process [3], and discuss the validity of the data analysis method comparing with random time sequences. Then we generalize the formulation of PUCK and show that the ARCH model can also be viewed as a special case of a random walk in a random potential field.
Corresponding author.
E-mail address: [email protected] (M. Takayasu). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.04.094
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
116
2. PUCK model and the self-modulation process The time evolution of market price of PUCK model is given as q Pðt þ DtÞ PðtÞ ¼ FðPÞ þ f ðtÞ, qP P¼PðtÞ FðPÞ ¼
bðtÞ ðP PM ðtÞÞ2 , 2ðM 1Þ
(1)
(2)
where P(t) is the optimal moving average of market price [4], Dt is the time interval, b(t) is the time-dependent curvature of the potential function, and f(t) is a white noise, PM(t) is the simple moving average of latest M terms represented as PM ðtÞ ¼
X 1 M1 Pðt kDtÞ. M k¼0
(3)
As known from this we assume a random walk in a velocity potential field F(P) given by the quadratic function with its center at the moving average PM(t). This quadratic potential function is transformed as ( ) M1 X fP Pðt kDtÞg2 X bðtÞ bðtÞ 1 M1 2 2 PM ðtÞ 2 FðPÞ ¼ þ Pðt kDtÞ . (4) 2ðM 1Þ 2 ðM 1ÞM 2 k¼1 M k¼1 Here, as the second term in the right-hand side vanishes by differentiation in terms of P, so the potential field F(P) can be recognized as the summation of quadratic potentials centered at each trace {P(tkDt)} for k ¼ 1; 2; . . . ; M. The difference of P(t)PM(t) can be represented in terms of price difference as PðtÞ PM ðtÞ ¼
M2 X k¼0
Mk1 DPðt kDtÞ M
M1 ¼ hDPðtÞiM , 2
ð5Þ
where DPðtÞ PðtÞ Pðt DtÞ, and hDPðtÞiM is the moving average of price difference defined as hDPðtÞiM
M2 X k¼0
2ðM k 1Þ DPðt kDtÞ. MðM 1Þ
(6)
Then, the PUCK model is represented simply by the following stochastic equation: DPðt þ 1Þ ¼
bðtÞ hDPðtÞiM þ f ðtÞ. 2
(7)
This type of stochastic equation is known by the name of self-modulation process and peculiar properties are reported such as power law distributions and long range correlation like the 1/f power spectrum [3]. In view of time series analysis, this is a kind of AR model with stochastic multiplicative coefficients and the value of b(t) is an unbiased estimator independent of M. Eq. (7) can be directly applied for analyzing foreign exchange markets. Analyzing so-called the tick data of Yen–Dollar market we use 2000 latest data for estimation of the value of b(t). The estimated values are almost identical for different values of M in the range of M ¼ 11–20 as expected. In Fig. 1 we plot the distribution of the value of b(t) comparing two cases of random time sequences: one is a pure random walk and the other is randomly shuffled original time sequence of Yen–Dollar rates. Both cases follow nearly Gaussian distributions with zero mean and the standard deviations about 0.3, so the probability of finding a value of which absolute value is larger than 1 is less than 0.1%. As demonstrated in Fig. 1 the distribution of b(t) for the real Yen–Dollar rates is much wider than these random cases extending to the ranges, bðtÞo 1 and bðtÞ41.
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
117
Fig. 1. The distributions of b(t) estimated for the raw data, randomly shuffled data, and a pure random walk. The numbers of data points are about 13 million in each case. The smooth lines show the normal distributions with the mean value 0 and the standard deviation 0.30 and 0.29, respectively.
Fig. 2. Semi-log plot of the autocorrelations of b(t) for the raw data (~), randomly shuffled data (’), and a pure random walk (n).
The autocorrelation of b(t) is characterized by a long tail as shown in Fig. 2. The autocorrelation of b-value does not vanish up to about 3 months contrary to the very short correlation decay in the random cases. It is generally very difficult to confirm statistical robustness of a technical analysis, however, the estimation of our b-value is statistically meaningful especially in the case where the absolute value is larger than 1. The results are almost the same for other market data such as stock market prices. 3. Generalization of PUCK model We can generalize the PUCK model in two ways; the generalization of the moving average and the generalization of the potential function. The moving average in Eq. (3) is generalized in the following form introducing arbitrary weight function mk(t): PM 0 ðtÞ ¼
M1 X k¼0
mk ðtÞPðt kDtÞ.
(8)
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
118
Then Eq. (5) becomes PðtÞ PM 0 ðtÞ ¼ OðtÞ
M2 X
Ok ðtÞDPðt kDtÞ
k¼0
OðtÞhDPðtÞiM 0 ,
ð9Þ
where OðtÞ ¼ M 1 ðM 1Þ ( Ok ðtÞ ¼
1
k X
)
M1 X
mk ðtÞ þ
k¼0
M1 X
kmk ðtÞ
k¼0
mj ðtÞ =OðtÞ.
ð10Þ
j¼0
The PUCK formulation Eq. (1) is valid with the following modified version of potential function: FðPÞ ¼
bðtÞ ðP PM 0 ðtÞÞ2 . 4OðtÞ
(11)
Correspondingly, Eq. (7) is now represented as bðtÞ hDPðtÞiM 0 þ f ðtÞ. (12) 2 Thus, the self-modulation formulation does not change much by this generalization of the moving average. So far we have assumed only quadratic term in the potential function. In the general situation nonlinear potential field can be introduced by the following expansion: DPðt þ 1Þ ¼
FðPÞ ¼
1 X
bn ðtÞðP PM 0 ðtÞÞn .
(13)
n¼1
The corresponding self-modulation form is given as DPðt þ 1Þ ¼
1 X
nbn ðtÞOðtÞn fhDPðtÞiM 0 gn1 þ f ðtÞ.
(14)
n¼1
This type of generalized formulation is needed when the market fluctuation is asymmetric. 4. PUCK representation of the ARCH model The ARCH model in financial technology is now widely applied to describe the volatility clustering of market fluctuations [5]. Here, we show that the ARCH model can be described in the PUCK formulation. We introduce the following potential function: P P2 ðtÞ FðPÞ ¼ aðtÞG , (15) 2s0 where pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi GðxÞ 12fx 1 þ x2 þ logðx þ 1 þ x2 Þg,
(16)
and a(t) is a white Gaussian noise with zero mean and unit standard deviation. Then, by introducing this potential form into Eq. (1) we have the following ARCH model: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DPðt þ 1Þ ¼ aðtÞ s20 þ DPðtÞ2 . (17) As known from this result the ARCH model corresponds to an asymmetric potential case with purely random coefficient. Owing to the effects of this potential field and the random coefficient, the price fluctuation is magnified so that the volatility clustering is realized.
ARTICLE IN PRESS M. Takayasu et al. / Physica A 383 (2007) 115–119
119
5. Discussions The study of random walks in a potential was started in 1930 by Uhlenbeck and Ornstein [6], however such a fixed potential does not fit with the market prices because its large scale behaviors are apparently predictable. If there exists such a predictable behavior in a market, speculators will utilize the chance immediately and the market status will become no longer predictable. This economists’ logic sounds correct in rough sense, however, there must be a time lag for this relaxation phenomenon. Friedrich et al. [7] analyzed the foreign exchange market data intensively and found that attractive potential forces can be observed at some occasions in certain time scales. The PUCK model provides a much more sophisticated data analysis method and we can observe both types of the market potentials: attractive potentials and repulsive potentials [1]. These potentials shift with the moving average of market prices, namely, the potential field itself is randomly walking in a large scale, so the potential is quite different from Uhlenbeck and Ornstein potential in view of predictability. In this paper we showed that the effect of this potential field defined by the moving average of market prices is equivalent to the effect of the self-modulation of price changes. We confirmed the existence of this potential in the Yen–Dollar market data comparing with randomly shuffled time sequences. This self-modulation effect implies that the dealers’ responses take some time and the dealers’ action as a mass is described by a certain moving average. By generalizing the potential formulation we proved that the ARCH model is equivalent to assuming an asymmetric potential with a purely random coefficient. If the coefficient of this asymmetric potential is a constant, then the market price tends to move to one direction enhancing the fluctuation indefinitely. In the case of ARCH model no such directional motion occurs, instead we have volatility clustering as the coefficient is randomly changing with zero mean. The generalized PUCK formulation is not only directly applicable to any market data, it is applicable to any time sequential data which looks roughly random in large time scale but having complicated behaviors in short time scale. Acknowledgements This work is partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research #16540346 (M. T.) and Research Fellowships for Young Scientists (T. M.). The authors appreciate H. Moriya of Oxford Financial Education Co. Ltd. for providing the tick data. References [1] M. Takayasu, T. Mizuno, T. Ohnishi, H. Takayasu, in: H. Takayasu (Ed.), Practical Fruits of Econophysics, Springer, Tokyo, 2005, p. 29. [2] M. Takayasu, T. Mizuno, H. Takayasu, Physica A 370 (2006) 91. [3] M. Takayasu, H. Takayasu, Physica A 324 (2003) 101. [4] T. Ohnishi, T. Mizuno, K. Aihara, M. Takayasu, H. Takayasu, Physica A 344 (2004) 207. [5] R.F. Engle, Econometrica 4 (1982) 987. [6] G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36 (1930) 823. [7] R. Friedrich, J. Peinke, Ch. Renner, Phys. Rev. Lett. 84 (2000) 5224.