Rev. Acad. Canar. Cienc., XII (Núms. 1-2), 135-146 (2000)- (Publicado enjulio de 2001)
THE EFFECT OF EXTERNAL NOISE ON THE DYNAMICS OF
SPECULATIVE MARKETS
JOSÉ M. PACHECO, JOSÉ M. LÓPEZ AND CÉSAR RODRÍGUEZ
ABSTRACT. A general model for asset price dynamics in speculative markets
is considered. The choice of a chartist demand function allows to check out
the agreement between analytical predictions and numerical simulations of this
model. A discussion is presented on the effect of market noise on speculative
behaviour. First, by solving a particular Fokker-Planck equation we show that
the white noise has only a disorganizing effect around the deterministic equilibrium
state. Second, a separation condition is used to deduce the existence
of limit cycles in the slow-fast dynamics for small values of the average time
needed for the chartist formation of the price trend. Numerical results show
that the effect of noise can double the wavelength of alternate slow and fast
transitions.
l. INTRODUCTION
Severa! categories of models have been assessed with the aim of analyzing cyclelike
behaviour in economic theory. In particular, prediction of asset market prices
is an area where these models are broadly in use and there is active research where
various mathematical techniques are employed. Among these dynamical systems
are known to be essential tools in economic analysis, both in the continuous and in
the discrete cases.
In this paper emphasis is put on the investigation of sudden changes in the
behaviour of market asset prices. These changes are idealized as the onset of a
slow-fast cyclic dynamics for sorne parameter value. In turn, cyclic behaviour itself
appears as a result of a Hopf bifurcation. But in order to mimic real market
behaviour, this is still too rigid a frame and dynamical systems are extended to
other categories of equations.
The importance of random walks in many classical economic assumptions is
the starting point for the use of the theory of stochastic differential equations in
the modelling of economic proQlems, though alternative approaches, as reactiondiffusion
equations have also been applied (see [l]). As a rule, an adequate deterministic
dynamical system is built and analyzed in phase space, afterwards it is
perturbed with noise or diffusion and studied ( or solved) in this new setting. Usually
the stochastic models are treated numerically, because for higher dimensional
systems the theoretical Fokker-Planck equation approach is cumbersome and applicable
only in a few cases with specific side conditions. This is the way followed
in this study.
2. A MODEL FOR AN SPECULATIVE MARKET
Severa! drawbacks in the application of random walk theory and the paradigm
of efficients markets led Beja and Goldmann [2] and Chiarella [3] to build more
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JOSÉ M. PACHECO, JOSÉ M. LÓPEZ AND CÉSAR RODRÍGUEZ
realistic models of the price evolution of an asset in a speculative market. The
excess demand function is defined as the time derivative P' ( t), where P stands for
the logarithm of price. The basic hypothesis is that the excess demand function
can be split into two components: a fundamental one D(t) anda speculative one
d(t) which correspond to the two basic types of agents acting in the market:
P'(t) = D(t).+ d(t)
The two types are called respectively "fundamentalists" and "chartists". The
first ones make their decisions on a theoretical basis (rational expectatives), while
chartists ( also known as speculators) estimate prices on the basis of past price trends
(adaptive expectatives). Chiarella [3] defines the fundamentalist excess demand to
be proportional to the difference between the equilibrium (walrasian) price W(t) in
an ideal market and the actual price P(t):
D(t) = a[W(t) - P(t)]
where a > O is the slope of the fundamentalist demand. On the other hand, the
chartist excess demand is represented by a nonlinear function of the difference
between an average estimation 'lj;(t) of the actual trend P'(t) of P(t) and the yield
g(t) of sorne other less risky alternative reference, ( e.g. bonds):
d(t) = h('lj;(t) - g(t))
where h is sorne bounded (both above and below) increasing function with a single
inflection and such that h(O) =O (see [3]). The model is completed by specifying
how 'lj;(t) is built. Here adaptive expectatives are used, with the following equation
defining 1f; ( t):
1/J'(t) = c[P'(t) - 'lj;(t)]
where e > O is a measure of how quickly chartists adjust their offers. Its inverse
~ can be considered as the time lag r needed for building expectatives and will
play an important role in the seque!. Therefore the model reads, dropping the t
dependence:
P' = a[W-P]+h('lj;-g)
r'lj;' = a[W - P] -1/J + h('lj; - g)
This differential system has a single equilibrium point (Pe, 1/Je) = (W - h(;?', O),
and under the hypothesis that W and g be constant a change of origin to this point
yields the new system
p' - ap + k('lj;)
r'lj;' = -ap -1/J + k('lj;)
where p = P - (W - h(;:-ul) and k(1f;) = h('lj; - g) - h(-g).
A linear stability analysis can be carried on in order to show how the qualitative
behaviour of the equilibrium point depends on the parameter r . The jacobian for
this system is
(
-a
.=Q:
T
h'('lj; - g) )
h1 (1/J-g)-l
T
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THE EFFECT OF EXTERNAL NOISE ON THE DYNAMICS OF SPECULATIVE MARKETS 3
and at the equilibrium point it becomes
(
-a A= .=!!
T
h'(-g) ) ( -~aa h'(-g)-1
T ~~1 )
where b = h'(-g) = k'(O) > O, this last inequality being true because of the
hypotheses made on the function h. The quadratic equation for the eigenvalues is
2 2 b - 1 ab a(b - 1)
>. - Tr(A)>. + det(A) = >. +(a - --)>. + - - =O
r r r
Therefore, whenever a- (b:;:l) < O, i.e. if r > r* = b:l, one has a stable equilibrium.
If the reverse inequality holds, an unstable equilibrium appears. Under application
of the Hopf bifurcation theorem, Chiarella [3] showed that a limit cycle exists when
r crosses the critica! value r*.
The features of the cycle can be determined with a little effort. In the last system
p can be eliminated in the following way. First, substitute the first equation in the
second one to obtain
n// = p' - '!/;
Now, taking the time derivative in this equation:
r'I/;" = p" - '!/;'
but p" = -ap' + k' ( '!/; )'!/;' from the first equation, so one has
r'I/;" = - ap' + k'('l/;)'I/;' - '!/;' = -a[-ap + k('I/;)] + k'('l/;)'I/;' - 'l/;1
and on taking -ap = r'I/;' - k( '!/;) + '!/; from the second equation it follows that
r'I/;" = -a[r'I/;' - k('I/;) + '!/; + k('I/;)] + k'('l/;)'I/;' - 'l/;1 = -ar'I/;' - a'I/; + k'('l/;)'I/;' - '!/;'
and from this last expression a more familiar looking ODE of Liénard type is found:
r'I/;" +[ar+ 1 - k'('l/;)]'I/;' + a'I/; =O
Now, using the trick ar = ar - ar* +ar*, one has ar+ 1 = ar - ar* +ar* + 1,
and remembering the definition r* = b:l ( or b =ar*+ 1), one can write ar+ 1 =
ar - ar*+ ar*+ 1 = a(r - r*) +ar*+ 1 = ac + b to yield
r'I/;" + [ac + b - k'('l/;)]'I/;' + a'I/; = O
A straightforward application of the Olech and Levinson-Smith theorems ( see e.g.
[4, p. 52 ]) shows that for e > O the equilibrium point is globally asymptotically
stable, while for e < O there exists a unique stable limit cycle. In other words, this
model exhibits an asymptotic behaviour with either a stable fixed point or a stable
limit cycle. Transition from one to another is achieved through a neutral Hopf
bifurcation arising when the time lag involved in building expectatives is reduced.
In the limit case, as this time lag becomes extremely small, i.e. when chartists revise
their estimate of the price trend infinitely rapidly, slow-fast cycles limit appear.
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4 JOSÉ M. PACHECO, JOSÉ M. LÓPEZ AND CÉSAR RODRÍGUEZ
3. NUMERICAL SIMULATIONS OF NONLINEAR MODEL
In this section our goal is to select a specific chartist demand function in order
to simulate the dynamics of the model.
The qualitative form chosen for the chartist demand function has led to an special
choice for the function h, defined in the following way:
b(e2(s+g) -1) b(e2Y - 1)
h(s) = e2(s+g) + 1 - --'-e-2Y_+_l-'-
which for the value s = 'ljJ - g yields the following expressions:
h('l/J - g)
k('l/J)
b(e2..P - 1) b(e29 - 1)
e2..P + 1
b(e2..P - 1)
e2..P + 1
e2Y + 1
It is clear (see Fig 1) that the graph of this function is obtained by modifying the
well-known function Th(s) = :::¡:::::::: in order to fulfill the properties requested in
the model. Now the second order ODE becomes
11 [ 4be2.P J ,
T'l/J + a€+ b - (e2.P + l)2 'ljJ + a'ljJ =O
and the equivalent system for phase-plane analysis is:
1/J' <P
<P'
The numerical simulations carried out (see Fig 2a, 2b, 2c) coincide in detail with
the predictions (Hopf bifurcation) of the qualitative analysis of the model.
As it was pointed out, the appearance of a small parameter in the second equation
suggests a slow-fast cyclic behaviour for this model. This can be easily checked by
applying a separation condition for the existence of limit cycles in slow-fast systems
(see (5]). This separation principle, based on singular perturbation arguments, is
purely geometric: If the nullcline p = kt;!) of the slow variable separates both stable
branches of the nullcline p = k(.Pl-.P of the fast variable, as sketched in Fig.2, a
stable cycle exists in the limit case. Numerical simulations (see Fig. 3a, 3b, 3c,
4a, 4b, 4c) show clearly that this occurs and relaxation oscillations can be easily
observed.
The interpretation is the following: Whenever T-+ O, chartists estímate prices on
the basis of only recent past prices, thus generating sudden and violent fluctuations
corresponding to the fast part of the cycle. From an economic viewpoint, persistent
information gathering will trigger quick reactions from chartist agents.
4. THE EFFECT OF NOISE ON SPECULATIVE BEHAVIOUR: SOME
CONCLUSIONS
Markets are always under the influence of externa! fluctuations of a random
nature that perturb their dynamics. The observation (see (6], (2]) that the evolution
of walrasian prices can be represented vía a random walk approach by introducing a
Wiener process Wt = .,/2{if,t, where f,t is a gaussian 8-correlated white noise signal
with zero mean and noise intensity u, suggests the introduction of random terms
in the model, thus becoming a system of stochastic differential equations.
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THE EFFECT OF EXTERNAL NOISE ON THE DYNAMICS OF SPECULATIVE MARKETS 5
To take into account the effect of this random infiuence, we replace the parameter
W in the deterministic system by a stationary random process Wt = W + v'2CTet,
where w corresponds to the average state of the environment, and et describes
fl.uctuations of strenght a around it. Thus, by collecting all noise terms on the
right hand side, a formal calculation analogous to the one in the deterministic
case leads to the following nonlinear stochastic differential equations, where a new
parameter, viz. the noise intensity a, is added to the Hopf bifurcation parameter
T:
1/J'
4be2"1
-a1/J - [aé + b - ( e2.P + 1 )2 ]4> + V2aa2et
Systems of this type have been dealt with by the authors in sorne other works
(see for example [7], [8]) on geomorphological processes, where they showed to
be extremely useful and accurate. The airo of the study is to analyze how the
qualitative structure is modified under changes in a.
A first approach can be obtained by developping the expresion (e1~e::)2 as a
McLaurin series and retaining only the first term b. This simplifies the system to
1/J' 4>
T4>1 -a1/J - aé4> + V2aa2et
where the second equation can be written conveniently in the form
T4>' = -(V'(1/J) + aé4>) + V2aa2et
with V(1/J) = 4, a single well potential. Under these assumptions the stationary
solution to the Fokker-Planck equation of the system can be obtained in closed form
(see [8, p. 156]) whenever é > O:
Ps (·.',·, , .,,¡,, ) -- N exp[ - é(a 1/J22 a+a T4>2) J
an elliptic bell shaped density function where N stands for the normalization constant.
Sketching several graphs of this stationary probability density (see Fig. 5a,
5b, 5c), we note that the externa! noise obviously has a disorganizing infiuence.
lndeed, since for é > O in the deterministic case the equilibrium point is globally
asymptotically stable, the stationary " probability density " mass will be entirely
concentrated on it, e. g., it consists of a delta peak centered on the equilibrium
point. Then the effect of noise will fl.atten and spread this sharp peak, depending
on its strength. Nevertheless, as was pointed out in the introduction, the general
nonlinear case can not be dealt with so easily in this way, so a numerical attack
was chosen.
Numerical simulations show that noise distorts the limit cycle giving rise to a
crater-like probability density function and a main feature is that a modulation can
be observed in the oscillation frequency of the variables (see Fig. 6a, 6b, 6c, 7a,
7b, 7c). Noise seems to amplify by a factor of two the wavelength of the sample
paths: This opens the way to the application of averaging techniques in order to
determine when the chartist agents change abruptly their price estimations.
It is to be noted that the effect of noise when T is small ( e.g. T = 0.02) is reduced
to a minimum, showing that intrinsically irregular behaviours are robust to noise
infl.uence.
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6 JOSÉ M. PACHECO, JOSÉ M. LÓPEZ AND CÉSAR RODRÍGUEZ
REFERENCES
[1] J. M. Pacheco, Reaction diffusion equations and economic cycles, Jour. o/ Math. & Comp.
Sci. {Math. Ser.) 9(2) 137-144 (1996).
[2] A. Beja and M. B. Goldman, On the dynamic behaviour of prices in disequilibrium, Journal
o/ Finance, XXXV(2) 235-248 (1980)
[3] C. Chiarella, The dynamics of speculative behaviour, Annals o/ Operations Research, 1992.
[4] H.-W. Lorenz, Nonlinear dynamical economics and chaotic motion, Springer-Verlag, Berlin
(1983)
[5] S. Muratori and S . Rinaldi, A separation condition for the existence of limit cycles in slow-fast
systems, Appl. Math. Modelling 15, 312-318 {1991)
[6} P. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management
Review 6 41-49 (1965)
[7] J. M. Pacheco, A simple modelling of braid-like structure {rill marks) appearing on sandy
beaches, Revista de Geofísica 48, 159-164 (1992)
[8} J . M . López, J. M. Pacheco and C. Rodríguez, Un modelo matemático para ciertas estructuras
sedimentarias primarias, Rev. de la Academia Canaria de Ciencias, VIIl{l), 9-17 {1996)
[Spanish]
[9} C . Gardiner, Handbook o/ stochastic methods, Springer-Verlag, Berlin {1983)
DEPARTAMENTO DE MATEMÁTICAS, UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA, CAMPUS
DE TAFIRA, 35017 LAS PALMAS, SPAIN
E-mail address: pacheco!Ddma.ulpgc .es ; jlopezlDdma.ulpgc.es ; cesarlDdma.ulpgc.es
140
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FIGURES
Fig 1 : A typical graph of a chartist demand function
Fig. 3 : Typical isoclines of the dynamics
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X
y
X
y
X
Fig. 2 : The Hopf bifurcation for a = 0.5 ; b = 2
2a} e= 1 (spiral sink} ; 2b} e= O (centre} ; 2c} e= -0.5 (spiral source)
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11
3.000
1.500
i o.o
-1.500
-3.000
-z.ooo -1 .000
Z.000
1.000
o.o
-1.000
·Z.000 o.o Z.000
o.o
fl
4.000
TI me
1.000 Z.000
6.000 8.000
Figs. 4a, 4b : Slow-fast limit cycle for b = 1.05, a = 0.5 , i:: = -0.02
and relaxation oscillations.
Fig. 5 : A typical stationary probability density for i:: > O.
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i
2.000
1.000
o.o
-1.000
-2.000 o.o
2.000
1.000
o.o < -1.000
-2.000
-2.000 -1.000 o.o
"
z.ooo 4.000
Time
~
1.000 Z.000
. 6.000
Figs. 6a, 6b : effect of noise for u = 0.2
144
8.000
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fj
2.000
1.000
o.o
-1.000
-2.000 o.o
2.000
1.000
i o.o
-1.000
-2.000
-2.000 -1.000
2.000
psi vs fl
o.o
fl
4.000
Time
1.000 2.000
6.000
Figs. 7a, 7b : effect of noise for a= 0.4
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2.000
1.000
o.o
·1.000
-2.000 o.o
2.000
1.000
i o.o
-1.000
-2.000
-2.000
2.000
-1.000
psi vs ti
4.000
Time
o.o
fl
1.000
6.000
Figs. 8a, 8b : effect of noise for a = 1
146
2.000