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Stock exchange: A statistical model
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Stock Exchange: A Statistical Model C. TSALLIS, A.M.C. de SOUZA" and E.M.F. CURADO Centro Brasileiro de Peaquisaz F[sicaz R u a Dr. X a v i e r Sigaud, 150 - 22290-180
Rio de Janeiro - ILl, Brazil
A b s t r a c t - In the spwit of microfoundations of macroeconomic theory, we in~ roduce a coupled map lattice model to describe time evolution of relevant quantities associated with stock exchange. In particular, computer simulations exhibit the stabilizing effect, on stock exchange, of dispersion of the external parameters which control the coupled map. The methods used in Statistical Physics have proved their efficiency in "extraphysical" areas such as Biogenesis ([1] and references therein), Immunology ([2] and references therein), Evolutionary Genetics ([3] and references therein), Neural Networks ([4] and references therein) and Economics ([5,6] and references therein). In wh~t concerns the area of Economics, all the attempts of ~aicrofoundation of macroeeonomic theory can be, in some sense, put into this category. This is, in par~'icular, the case of L. Summers proposal for treating fluctuations in the stock market in terms of nolinear dynamics. To do this, he introduced the concept of "noisy traders" and "sophisticated traders" and achieved some success. Nevertheless his model has not been considered fully convincing (page 248 of [5]). Within this type of philosophy, we propose here a simple theoretical model, based on coupled map lattice[?], for understanding some interesting phenomena currently occuring in stock exchanges, such as the influence of the massive use of computers by the brokers. Let us assume N stock brokers (traders) buying (X =
1) or not buying (X = 0) stocks of an
unique and big company. All the brokers are assumed to operate in competitive regime, and having free access to market information• We associate with each broker the following distribution law
Pi(t)(.X) = (1 -p!t))6(X)+p!t)6(X
- 1)
(i = 1,2,...,N)
(1)
where 6 denotes Kroenecker's delta function (6(X) equals 1 if X = 0, and vanishes otherwise), and
p~t) • [0, 1] represents the probability that he wishes, at time t, to buy stocks (we refer to the "potential demand", not to be confused with the probability that he indeed buys the stocks). The {p~t)} are coupled as follows:
Z,d,
,pl" Ed, 561
(i= 1,2,...,lv)
(2)
C. TSALUSet al.
562
e [0,(a + fl)/7]; consequently, a realistic situation corresponds to a + fl equal 7, or slightly below 7); (ii) if a +/3 e (3.57,4) the system can be chaotic (positive Lyapunov exponent). Let us now go back to the more general case in which the values for {ai, fli,Ti} are in principle different for each broker. We shall now assume that the set {ai} is drawn from the distribution law
D~(a,), one and the same for all brokers; analogously, {fl/} and {7/} are respectively drawn from D~(fl~) and D~(7/) These distributions are basically characterized by their mean value ~, ~ and V, as well by their widths W~, W~ and W~. Human brokers typically correspond to large values for (W~, W~, W~), whereas co:'~puter brokers (i.e., brokers that blindly follow computer recomendations) correspond to small value for (Wo, W0, W~) since the softwares they use are (most frequently) very similar. One of the important effects we want to focus with the present model is that the risk of stock exchange collapse will be shown to monotonously decrease with increasing Wo, Wa and W~. Two basically different models can be discussed. The first of them (referred to as q~enched) consists in fixing once for ever {a,,~i, Ti} (drawn respectively from (D~, D~, DT) at t = 0), i.e., each brokers will evolve, at all times, under the same values for (a,/~, 7), but these values might differ from broker to broker. The second model (referred to as annealed) consists in randomly choosing, at eve~ new time
step, {a~,/3~, 7~} (still drawn respectively from (D~, Da, D~)). Intuition suggests that the annealed model yield ~ more randomized results with respect to those obtalne, i from the quenched model. Our numerical results will confirm this intuition, and the quenched model with widths (W~, W~, W,) behaves ver-¢ similarly to the annealed model with somewhat smaller widths. It suffices consequently to discuss the details of say the annealed model.
1.0
0.3
O.S
0.2, 0.6 A Q. V 0.4
o,t: 0.2
, ....
0.0 0
20
40
60
80
I00
o.o
i 0
.......
....
•. . . .
50
IO0
t
Fig. 1. N = 300 annealed model without noise for & = ~ = 9/2 = 1.85 and Wa = W~ = W~/2 --" W: (a) < p > ( t ) for W = 0.2; (b) ~(t) for W = 0.2 (full circles) and W -- 0 (empty circles).
Stock exchange: a statistical model
563
where {ai, all, 7i} represent a set of external parameters, and ~ 1 P l 0 ~ [0,N]. measures the inertial persistency of thb wish of the i - th broker (if a~ = with time if 0 < ai
<
1, remains constant if ai =
The ai-parameter
7i =
0, p}0 decreases
1, and increases with time if c~i >
1). The
fli-parameter measures the influence, on the individual wish of a given broker, of the average wish of all the other brokers. (if c~i = 7i = 0 and ai = a, the average value < p >(t) --- (E~I P ! O ) / N (which somehow reflects the stock prize) decreases with time if 0 < and increases with time if a
a
< 1, remains constant if a
=
1,
> 1). Finally, 7 / i s associated with the tendency of a given broker not
to buy stocks whenever most of the other brokers wi~h to buy (in the absence of such tendency, his individual profit will eventually be quite modest or impossible); by the way, let us recall a famous anecdote which states that Joseph Kennedy (President Kennedy's father) providentially escaped from the 1929 USA-wide stock exchange collapse because he sold his stock as soon as he realized that his humble shoe-shine-boy was buying stocks! It is clear that if {c~i, ali, 7/} are arbitrarily chosen, Eq.
(2) does not automatically guarantee
0 < p~0 < 1 for all values of i and t, thus violating the definition of probability. This difficulty can, in principle, be overcome through various procedures (e.g., conveniently introducing the hyperbolic tangent function which transforms the interval (0, oe) into (0, 1)). For simplicity, we adopt here the following prescription: whenever p!0 is above one (below zero) it is replaced by p!0 = 1 (p~0 = 0). Let us also remark that p}0 = 0 (for all values of i) is a fixed point of Eq. (2). Another fixed point is given, in the N ~ co limit, by Pi = fliPod(1 - c~i +. 7iP.~) (Vi), where the average P,v =-< P > (co) satisfies limN--oo ~ E ~ I n i l ( 1 -- ai + 7iPav) = 1 (in particular, ( a i , a i , Ti) -- ( a , a , 7 ) implies pi = pov = (,~ + ~ -
1)/7, vi).
An interesting property of Eq. (2) exist in the particular case ai -- c~, ai -- a and 7/-= 7. If we apply ~
~,
on both members of Eq. (2) we obtain
[
(t+l)]=(c~+a)[
(t)]-7
N_~ -
1 N [
(t)]2+TN(N-1)~[PI0]'
(3)
iml
hence in the N --* co limit,
[< p > (t + 1)1 = (~ + a ) [ < p > (t)] - 7[< p > (t)l ~.
(4)
By introducing now Z ( t ) = (7[< P > (t)])/(a + a), Eq. (4) becomes z ( t + i) = (a + a ) Z ( t ) ( : - Z(t))
(5)
which is the well known logistic equation (see [8] and references therein). Let us recall two remarkable properties associated with Eq. (5): (i) if ~ + fl _< 4, Z(t) automatically remains within the interval [0,1] if this was true at t = 0, hence, the same occurs with [< p >(t)] if a + fl _< 7 (hence, [< p >(t)]
564
C. TSALLIS et al.
Another mechanism which increases randomness (note that Eq. (2) for the quenched model is strictly determinist) is the presence of a (centered) white-noise additive term z}}0 in the second member of Eq. (2). No drastic influence is observed: if z}!0 is not too large, the same phenomena occur, but in a smoother manner. In fact, a more general formulation of the present model demands the inclusion of terms of this type. It can be seen tL.at a large variety of these coupled m a p s yields a value < p > ( t ) which satisfies a logistic-like equation, thus exhibiting a sort of universal behavior. Let us now discuss our numerical results for the mean value < p > ( t ) and for the mean dev'ation
o(t~ - i 1 ~
N °~C[pl
< p > (t)]=.
(6)
i=l
These quantities depend of course on the initial conditions {pl°)}. We have used a great variety of them, drawn from both large and narr(,w random distributions in the interval [0,1]. In all the (generic) cases, after a short transient, the systera "forgets" the initial distribution of {p!0)}. So, for convenience, we have chosen to work sistematically with {p~0)} drawn from a white-noise distribution in the interval [0,1].
Clearly, < p > ( t ) and a ( t ) also depend, in principle, on N. We have varied N in the range
[30,3000] with no significative influence. We have consistently chosen to work with N=300 (which, by the way, coincides with a reasonable c t d e r of magnitude for the number of brokers in a typica~ stock exchange). We have also verified that the particular shape of distributions {Da(cq), D~(/3i), D.,(7,)} is not very important, their influence being very well characterized through their mean value (Eq 3, 7) and their widths. Consequently, for coi~venience, we have chosen to work with (~i,/3i, 3'/) independently drawn from white-noise distributions in the intervals N - - ~ _< cx, < N + - ~ , ~ - ~
_
- - ~ -< 7~ -< ~ + !~z, respectively. To illustrate the most interesting situation (i.e.,chaos) we have mainly worked with ~ = fl = 7 / 2 ~ [1.78,2] and
wa
= wa = w , / 2 ~ [0,0.2]. Finally, since the effects
we want to emphasize are more clear-cut for the annealed model without noise, we have primarily addressed this case. In Fig. l ( a ) we show a typical time-dependence of < p > ( t ) (corresponding to = ~ = 7 / 2 = 1.85); although Wa = I'V~ = W ~ / 2 = 0.2 was in fact used, the situation is practically the same for, say, Wo = W# = W ~ / 2 = 0. In Fig.
l(b)
we present, for the same two situations, the
time evolution of a(t); we see now clearly the influence of the widths (W~, W~, IVy). It is now convenient to introduce the stock collapse index ~(t) as follows:
~(t) -
~
~
[o, 1]
(7)
Stock exchange: a statistical model
565
O0
06
04 @
0
0.2 •
00 2
'o
''
•
'
40
60
oe
.'o
I00
Fig. 2. N = 300 annealed model without noise for ~ = ~ = "~/2 = 1.9 and Wo = WO = W~/2 = 0.1: time evolution of the stock collapse index ~(t).
where N~ t) (N~ 0) is the number of brokers whose p~0 is above (below) 1/2, i.e., who basically want to buy (not to buy) stocks. In practice, N~ t) + N~ ') = N (Vt). ~(t) vanishes if the brokers are equally distributed above and be:ow pit) = 1 / ~ ~(t) equals unity if all brokers have simcltaneously the same wish, either to buy or not to buy. We show in Fig. 2 a typical time evolution of ~(t) (corresponds to = ~ = 7 / 2 = 1.9 and I ~ = W~ = W~/2 = 0.1). The time average ~ of the stock collapse index is given by 1 r -
(8) ~=1
We have chosen T = 100 to be a typical value (which, in fact, correctly represents the T ~ oo limit). We present, in Fig. 3, ~ as a function of Wo = W~ = W~/2 =- W for a typical case ( 5 = ~ = ~/2 = 1.85) and for all four cases, annealed or quenched, with or without noise. We clearly see that the risk of collapse of the stock marl.et is ~maller for human brokers(large W) than for computer brokers(small W). To summarize the present work let us recall that the nolinear dynamical model proposed through Eq. (2) (either in its quenched or its annealed version, with or without additive white-noise randomness) (i) presents the typical chaotic-like behaviour associated with the time evolution of stock exchanges, (ii) exhibits that computer brokers (narrow distributions of the external parameters) practically share, alter a short transient, the same wishes (i.e., p~t) ~ < P > (/),Vi), and (iii) exhibits that computer brokers can more easily destabilize the stock market as compared to human brokers (large distributions of the external parameters).
C. TSALLISet aL
566 1.0.
0.9-
O.S IX O.T
06.
~rlf
QO
0.1
0.2
W Fig. 3. The time average stock collapse index • as a function of the w,dth Wa -- Wp = W,/2 =_ W for N = 300 and ~ = / ) = 9/2 = 1.85: • annealed without noise, O annealed with noise
(It/~t)l _< 0.1), • quenched without noise, A quenched with noise ([r/~t) I _< 0.1).
Let us conclude by mentionning the present simplified model could be quite naturally extended to more realistic ones along the following lines:
(i) A term ~ike /~iN-~ i ~j#~ N PJ could be generalized into ~ -i~ ~x'-N ,. (analogously for the -),-term); j # i a. ~',~r~ (ii) Stocks of an a r b i t r a r y number of companies (and not only one) could be focused, and those companies could be not very big ones; (iii) Distribution (1) could be generalized into P~') = q!O6(X + 1) + (1 - p~t) _ q~t))6(X ) + p~t)6(X - 1), being possible, for each broker, to buy stocks ( X = 1), to sell stocks ( X = - 1 ) , or not buy nor sell ( X = 0). (iv) The broker population could be formed by, say, two constituents, for instance, both "computer" and "human" brokers.
In this cases, W would become a r a n d o m variable itself determined by a
distribution law such as R ( W )
= (1 - r ) 6 ( W - W I ) + r 6 ( W - B 2 ) , with, say, 0 < W1 ,~ W2 and
O_
Stock exchange: a statistical model
567
States, where everyone tends to hold the s a m e o p i n i o n at the s a m e time, t h a n in E n g l a n d where differences of o p i n i o n are more usual".
Acknowledgement- We acknowledge very fruitful remarks from M. H. Simonsen, R. Guenzburger, P. Turner and D. Ellis. REFERENCES " On leave for absence from Departarnento de Fisica, Universidade Federal de Sergipe,49000-000 - Aracaju-SE,Brazil. [1] P. W. Anderson, Suggested Model for Prebiotic Evolution: the Use of Chaos, Proc. Nat. Acad. Sci.
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