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Physica A 344 (2004) 190193
Using a kinetic equation that is used to model turbulence
(Physica A, 19851988, Physica D,
There have been numerous recent applications of statistical
mechanics to nancial
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
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www.elsevier.com/locate/physa
0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights
reserved.
doi:10.1016/j.physa.2004.06.114
Tel.: +41-22-310-1229; fax: +41-22-310-1229.
E-mail address: [email protected], [email protected] (A.
Muriel).markets. Most of these applications use equilibrium
concepts [1]. In this paper, weapply a recent model in kinetic
theory that is used to model turbulence [2]. As far aswe can tell,
this is the rst application of this kinetic approach to a nancial
system.Consider the following kinetic equation:
qqt
f 2
f 1
g21 g12J
g21J g12
f 2
f 1
; 120012003), we redene 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.m; 05.90.m
1. IntroductionShort-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
-
momentum in physics). Then we could simplify the equation and
calculate theaverage of p to get, after resummation of the
resulting series,
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A. Muriel / Physica A 344 (2004) 190193 191p2 eGt p20 cosht
2
pg2 4g
2o
1 D 4 26664
p20g sinh
t
2
pg2 4g
2o
1 D 4
p g2 4g2o1 D 4
p10g12 sinh
t
2
pg2 4g
2o
1 D 4
1 D 2 p g2 4g2o1 D 4
37775 : 4
As remarked earlier, the analogous expression for p1 is obtained
by interchangingoperator J later. For the origin, solution and
application of this equation we referthe reader to the literature
[24] from which it could be shown that the resulting timeevolution
equations for f 2 is given by
f 2 eGtX1m0
t=2 2m2m !
Xmj0
m!
j!j m!
4jg2mjg2jo J
2j f 20"
gX1n0
t=2 2n12n 1 !
Xnj0
n!
j!j n!
4jg2njg2jo J
2j f 20
g12X1n0
t=22n12n 1!
Xnj0
n!
j!j n!
4jg2njg2jo J
2j f 10#; 2
where G g21 g12; g g21 g12; go2 g21g12. The corresponding
solution for f 1 isobtained by interchanging the subscripts 2 and
1. This will hold as well for othersubscripted variables.
2. The nancial model
Assume that f 20 c2eb pp2 2
, f 10 c1eb pp1 2
and dene
Jkf p f 1 D kp ; 3where D is some fractional loss (that violates
the traditional conservation ofthe subscripts 1 and 2.
-
In principle, the variable p could be negative, but in our
application, for shortintervals of time, and for positive averages
in the initial condition, we do not have toworry about this as
justied by the demonstrated success of the algorithm below.We now
take a leap of interpretation and assume that hp2i and hp1i are the
average
exchange rates of currency 2 and currency 1 relative to a stable
currency 0 whichdoes not change too much over short periods. In
this world there will be only threemajor currencies, not too far
from the truth sometimes. g21 is interpreted as thetransition 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 evolutionequations 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 newtransition rates. In this way, one
could always be ahead by one time step, predictingthe exchange
rates at step 3, presumably allowing a trading strategy to yield
prot.One must of course choose the value of the fraction D and ne
tune the predictivescheme. The method is not quite Markofan, but
not completely non-Markofan inthe traditional sense.Without
discussing the details, we illustrate one application of the
above
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A. Muriel / Physica A 344 (2004) 190193192algorithm 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 usingSwiss francs and sell Euros for
Swiss francs.
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3. Comments
There are more currencies than this model can cope with. In the
universe of threecurrencies, we assume that the other currencies
are simply background informationthat violate any conservation
principles that one may invoke about the value ofmoney in a
zero-sum game. But any set of three currencies may be used, and it
ispossible to develop a methodology that uses many sets of three
currencies to covermore ground. In addition, there will be redened
models for the short-time evolutionof the stock market.That a model
arising from turbulence research may be utilized as well for the
so-
use of energy, a measure of value, or money, is not justied by
the performance of
Acknowledgements
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A. Muriel / Physica A 344 (2004) 190193 193I acknowledge my
colleagues at Citigroup, New York, for introducing me to
thevagaries of the foreign currency market.
References
[1] J. Voit, The Statistical Mechanics of Financial Markets,
Springer, New York, 2003.
[2] A. Muriel, Physica D 124 (1998) 225247.
[3] A. Muriel, Physica A 304 (2002) 379.
[4] A. Muriel, Physica A 322 (2003) 139.that model. In the end,
only success justies the model once the leap of imaginationis
made.called turbulent market is a typical extension of physical
theories to other elds,sometimes, without justication. Such has
been the character of several attempts touse the Heisenberg
Uncertainty Principle, or even collective behavior. Identifying
orredening variables is the main difculty. For example, in this
model, why did wechoose to average the momentum instead of the
kinetic energy? As it turns out, thequalitative behavior of the
model using any of these two variables are the same,averaging the
momentum variable is analytically easier, and the purists gain with
the
Short-term predictions in forex tradingIntroductionThe financial
modelCommentsAcknowledgementsReferences
