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Short-term predictions in forex trading
ARTICLE IN PRESS
Physica A 344 (2004) 190–193 www.elsevier.com/locate/physa
Short-term predictions in forex trading A. Muriel Data Transport Systems Rue de la Vallee, 2 Geneva 1204, Switzerland Received 15 December 2003 Available online 22 July 2004
Abstract Using a kinetic equation that is used to model turbulence (Physica A, 1985–1988, Physica D, 2001–2003), we redefine variables to model the time evolution of the foreign exchange rates of three major currencies. We display live and predicted data for one period of trading in October, 2003. r 2004 Elsevier B.V. All rights reserved. PACS: 05.40.a; 89.90.þm; 05.90.þm
1. Introduction There have been numerous recent applications of statistical mechanics to financial markets. Most of these applications use equilibrium concepts [1]. In this paper, we apply a recent model in kinetic theory that is used to model turbulence [2]. As far as we can tell, this is the first application of this kinetic approach to a financial system. Consider the following kinetic equation: g21 g12 J f2 q f2 ¼ ; ð1Þ qt f 1 g21 J g12 f1 where f 2 and f 1 are momentum distribution functions for the two states 2 and 1. Here the momentum will be reinterpreted later as we please. We will specify the jump Tel.: +41-22-310-1229; fax: +41-22-310-1229.
E-mail address: [email protected], [email protected] (A. Muriel). 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.06.114
ARTICLE IN PRESS A. Muriel / Physica A 344 (2004) 190–193
191
operator J later. For the origin, solution and application of this equation we refer the reader to the literature [2–4] from which it could be shown that the resulting time evolution equations for f 2 is given by " 2m m 1 X X t=2 m! Gt 2j f2 ¼ e 4j g2ðmjÞ g2j o J f 2 ð0Þ j!ðj mÞ! ð 2m Þ! m¼0 j¼0 2nþ1 n 1 X t=2 X n! 2j g 4j g2ðnjÞ g2j o J f 2 ð0Þ ð Þ! j!ðj nÞ! 2n þ 1 n¼0 j¼0 # 1 n 2nþ1 X X ðt=2Þ n! j 2ðnjÞ 2j 2j þ g12 go J f 1 ð0Þ ; ð2Þ 4g ð2n þ 1Þ! j¼0 j!ðj nÞ! n¼0 where G ¼ g21 þ g12 ; g ¼ g21 g12 ; go2 ¼ g21 g12 . The corresponding solution for f 1 is obtained by interchanging the subscripts 2 and 1. This will hold as well for other subscripted variables.
2. The financial model 2
2
Assume that f 2 ð0Þ ¼ c2 ebð pp2 Þ , f 1 ð0Þ ¼ c1 ebð pp1 Þ and define J k f ðpÞ ¼ f ½1 Dk p ;
ð3Þ
where D is some fractional loss (that violates the traditional conservation of momentum in physics). Then we could simplify the equation and calculate the average of p to get, after resummation of the resulting series, 2 6 t pffi 2 4g2o Gt 6 p2 ¼ e 6p2 ð0Þ cosh ½g þ 4 2 ð1 DÞ4
t pffi 2 4g2o ½g þ g sinh 2 ð1 DÞ4 p2 ð0Þ pffi 2 4g2o ½g þ ð1 DÞ4 3 t pffi 2 4g2o ½g þ g12 sinh 7 2 ð1 D Þ4 7 þ p1 ð0Þ 7: 2 5 p ffi 4g o ð1 DÞ2 ½g2 þ 4 ð1 D Þ
ð4Þ
As remarked earlier, the analogous expression for p1 is obtained by interchanging the subscripts 1 and 2.
ARTICLE IN PRESS 192
A. Muriel / Physica A 344 (2004) 190–193
In principle, the variable p could be negative, but in our application, for short intervals of time, and for positive averages in the initial condition, we do not have to worry about this as justified by the demonstrated success of the algorithm below. We now take a leap of interpretation and assume that hp2 i and hp1 i are the average exchange rates of currency 2 and currency 1 relative to a stable currency 0 which does not change too much over short periods. In this world there will be only three major currencies, not too far from the truth sometimes. g21 is interpreted as the transition rate from currency 2 to 1 and g12 is the transition rate from currency 1 to 2; they could be positive or negative. We may then convert Eq. (3) and its paired equation into difference equations and use the following strategy: (1) Take two preceding times, time step 1, and time step 2, with known currency exchange rates for each step. (2) Invert the two time evolution equations to get the values of the transition rates from time step 1 to time step 2. (3) Use these transition rates to predict the currency exchange rates for time step 3. Repeat the process for the next time step, and so on, always recalculating the new transition rates. In this way, one could always be ahead by one time step, predicting the exchange rates at step 3, presumably allowing a trading strategy to yield profit. One must of course choose the value of the fraction D and fine tune the predictive scheme. The method is not quite Markoffian, but not completely non-Markoffian in the traditional sense. Without discussing the details, we illustrate one application of the above algorithm to produce Fig. 1.
Fig. 1. Illustrative live United States dollar data from a period in October, 2003. The longer line is the prediction line, while the short line is the live data. Time step 18 is the prediction for the next time step. In each graph, the middle horizontal line is the mean of all the data, while the top horizontal line and the bottom horizontal line delimit the spread used by a trading platform for the period during which the data was collected. The above example is always accompanied by a similar graph for the euro, in USD, Euro and Swiss franc trio. The two graphs may be accompanied by a recommendation to buy dollars using Swiss francs and sell Euros for Swiss francs.
ARTICLE IN PRESS A. Muriel / Physica A 344 (2004) 190–193
193
3. Comments There are more currencies than this model can cope with. In the universe of three currencies, we assume that the other currencies are simply background information that violate any conservation principles that one may invoke about the value of money in a zero-sum game. But any set of three currencies may be used, and it is possible to develop a methodology that uses many sets of three currencies to cover more ground. In addition, there will be redefined models for the short-time evolution of the stock market. That a model arising from turbulence research may be utilized as well for the socalled turbulent market is a typical extension of physical theories to other fields, sometimes, without justification. Such has been the character of several attempts to use the Heisenberg Uncertainty Principle, or even collective behavior. Identifying or redefining variables is the main difficulty. For example, in this model, why did we choose to average the momentum instead of the kinetic energy? As it turns out, the qualitative behavior of the model using any of these two variables are the same, averaging the momentum variable is analytically easier, and the purist’s gain with the use of energy, a measure of value, or money, is not justified by the performance of that model. In the end, only success justifies the model once the leap of imagination is made.
Acknowledgements I acknowledge my colleagues at Citigroup, New York, for introducing me to the vagaries of the foreign currency market. References [1] [2] [3] [4]
J. Voit, The Statistical Mechanics of Financial Markets, Springer, New York, 2003. A. Muriel, Physica D 124 (1998) 225–247. A. Muriel, Physica A 304 (2002) 379. A. Muriel, Physica A 322 (2003) 139.
Physica A 344 (2004) 190–193 www.elsevier.com/locate/physa
Short-term predictions in forex trading A. Muriel Data Transport Systems Rue de la Vallee, 2 Geneva 1204, Switzerland Received 15 December 2003 Available online 22 July 2004
Abstract Using a kinetic equation that is used to model turbulence (Physica A, 1985–1988, Physica D, 2001–2003), we redefine variables to model the time evolution of the foreign exchange rates of three major currencies. We display live and predicted data for one period of trading in October, 2003. r 2004 Elsevier B.V. All rights reserved. PACS: 05.40.a; 89.90.þm; 05.90.þm
1. Introduction There have been numerous recent applications of statistical mechanics to financial markets. Most of these applications use equilibrium concepts [1]. In this paper, we apply a recent model in kinetic theory that is used to model turbulence [2]. As far as we can tell, this is the first application of this kinetic approach to a financial system. Consider the following kinetic equation: g21 g12 J f2 q f2 ¼ ; ð1Þ qt f 1 g21 J g12 f1 where f 2 and f 1 are momentum distribution functions for the two states 2 and 1. Here the momentum will be reinterpreted later as we please. We will specify the jump Tel.: +41-22-310-1229; fax: +41-22-310-1229.
E-mail address: [email protected], [email protected] (A. Muriel). 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.06.114
ARTICLE IN PRESS A. Muriel / Physica A 344 (2004) 190–193
191
operator J later. For the origin, solution and application of this equation we refer the reader to the literature [2–4] from which it could be shown that the resulting time evolution equations for f 2 is given by " 2m m 1 X X t=2 m! Gt 2j f2 ¼ e 4j g2ðmjÞ g2j o J f 2 ð0Þ j!ðj mÞ! ð 2m Þ! m¼0 j¼0 2nþ1 n 1 X t=2 X n! 2j g 4j g2ðnjÞ g2j o J f 2 ð0Þ ð Þ! j!ðj nÞ! 2n þ 1 n¼0 j¼0 # 1 n 2nþ1 X X ðt=2Þ n! j 2ðnjÞ 2j 2j þ g12 go J f 1 ð0Þ ; ð2Þ 4g ð2n þ 1Þ! j¼0 j!ðj nÞ! n¼0 where G ¼ g21 þ g12 ; g ¼ g21 g12 ; go2 ¼ g21 g12 . The corresponding solution for f 1 is obtained by interchanging the subscripts 2 and 1. This will hold as well for other subscripted variables.
2. The financial model 2
2
Assume that f 2 ð0Þ ¼ c2 ebð pp2 Þ , f 1 ð0Þ ¼ c1 ebð pp1 Þ and define J k f ðpÞ ¼ f ½1 Dk p ;
ð3Þ
where D is some fractional loss (that violates the traditional conservation of momentum in physics). Then we could simplify the equation and calculate the average of p to get, after resummation of the resulting series, 2 6 t pffi 2 4g2o Gt 6 p2 ¼ e 6p2 ð0Þ cosh ½g þ 4 2 ð1 DÞ4
t pffi 2 4g2o ½g þ g sinh 2 ð1 DÞ4 p2 ð0Þ pffi 2 4g2o ½g þ ð1 DÞ4 3 t pffi 2 4g2o ½g þ g12 sinh 7 2 ð1 D Þ4 7 þ p1 ð0Þ 7: 2 5 p ffi 4g o ð1 DÞ2 ½g2 þ 4 ð1 D Þ
ð4Þ
As remarked earlier, the analogous expression for p1 is obtained by interchanging the subscripts 1 and 2.
ARTICLE IN PRESS 192
A. Muriel / Physica A 344 (2004) 190–193
In principle, the variable p could be negative, but in our application, for short intervals of time, and for positive averages in the initial condition, we do not have to worry about this as justified by the demonstrated success of the algorithm below. We now take a leap of interpretation and assume that hp2 i and hp1 i are the average exchange rates of currency 2 and currency 1 relative to a stable currency 0 which does not change too much over short periods. In this world there will be only three major currencies, not too far from the truth sometimes. g21 is interpreted as the transition rate from currency 2 to 1 and g12 is the transition rate from currency 1 to 2; they could be positive or negative. We may then convert Eq. (3) and its paired equation into difference equations and use the following strategy: (1) Take two preceding times, time step 1, and time step 2, with known currency exchange rates for each step. (2) Invert the two time evolution equations to get the values of the transition rates from time step 1 to time step 2. (3) Use these transition rates to predict the currency exchange rates for time step 3. Repeat the process for the next time step, and so on, always recalculating the new transition rates. In this way, one could always be ahead by one time step, predicting the exchange rates at step 3, presumably allowing a trading strategy to yield profit. One must of course choose the value of the fraction D and fine tune the predictive scheme. The method is not quite Markoffian, but not completely non-Markoffian in the traditional sense. Without discussing the details, we illustrate one application of the above algorithm to produce Fig. 1.
Fig. 1. Illustrative live United States dollar data from a period in October, 2003. The longer line is the prediction line, while the short line is the live data. Time step 18 is the prediction for the next time step. In each graph, the middle horizontal line is the mean of all the data, while the top horizontal line and the bottom horizontal line delimit the spread used by a trading platform for the period during which the data was collected. The above example is always accompanied by a similar graph for the euro, in USD, Euro and Swiss franc trio. The two graphs may be accompanied by a recommendation to buy dollars using Swiss francs and sell Euros for Swiss francs.
ARTICLE IN PRESS A. Muriel / Physica A 344 (2004) 190–193
193
3. Comments There are more currencies than this model can cope with. In the universe of three currencies, we assume that the other currencies are simply background information that violate any conservation principles that one may invoke about the value of money in a zero-sum game. But any set of three currencies may be used, and it is possible to develop a methodology that uses many sets of three currencies to cover more ground. In addition, there will be redefined models for the short-time evolution of the stock market. That a model arising from turbulence research may be utilized as well for the socalled turbulent market is a typical extension of physical theories to other fields, sometimes, without justification. Such has been the character of several attempts to use the Heisenberg Uncertainty Principle, or even collective behavior. Identifying or redefining variables is the main difficulty. For example, in this model, why did we choose to average the momentum instead of the kinetic energy? As it turns out, the qualitative behavior of the model using any of these two variables are the same, averaging the momentum variable is analytically easier, and the purist’s gain with the use of energy, a measure of value, or money, is not justified by the performance of that model. In the end, only success justifies the model once the leap of imagination is made.
Acknowledgements I acknowledge my colleagues at Citigroup, New York, for introducing me to the vagaries of the foreign currency market. References [1] [2] [3] [4]
J. Voit, The Statistical Mechanics of Financial Markets, Springer, New York, 2003. A. Muriel, Physica D 124 (1998) 225–247. A. Muriel, Physica A 304 (2002) 379. A. Muriel, Physica A 322 (2003) 139.