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168 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
Fr,m(p)–p
10 0
10 –1
10 –2
10 –3
10 –4
10 –5
10 –6
10 –7
m = 30
m = 60
m = 100
10 –3 10 –2
slope m(m = 60)=3
p–1/2
10 –1
Figure 6. The logarithm of Fm(p) − p versus the logarithm of
p − 1/2 for three different values of m = 30, 60 and 100, with
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85. Note the transition from a
slope 1 to a large effective slope before the reinjection due to the
contrarian mechanism.
κ>0) followed by a super-exponential growth accelerating
so much as to give the impression of reaching a singularity in
finite-time.
To understand this phenomenon, we plot the logarithm of
Fm(p)−p versus the logarithm of p −1/2 in figure 6 for three
different values of m = 30, 60 and 100. Two regimes can be
observed.
(1) For small p − 1/2, the slope of log 10 (Fm(p) − p) versus
log 10 (p − 1/2) is 1, i.e.
10 0
p ′ − p ≡ Fm(p) − p α(m)(p − 1
). (13)
2
This expression (13) explains the exponential growth
observed at early times in figure 5.
(2) For larger p − 1/2, the slope of log 10 (Fm(p) − p) versus
log 10 (p − 1/2) increases above 1 and stabilizes to a
value µ(m) before decreasing again due to the reinjection
produced by the contrarian mechanism. The interval in
p − 1/2 in which the slope is approximately stabilized at
the value µ(m) enables us to write
Fm(p)−p β(m)(p − 1
2 )µ(m) , with µ>1. (14)
These two regimes can be summarized in the following
phenomenological expression for Fm(p):
Fm(p) = 1
+ (1 − 2gm(1/2)
2
− g ′ 1
1
m (1/2))(p − ) + β(m)(p − 2 2 )µ(m) , (15)
= 1
2
+ (p − 1
2
) + α(m)(p − 1
2 )
+ β(m)(p − 1
2 )µ(m) with µ>1, (16)
and
α(m) =−2gm(1/2) − g ′ m (1/2). (17)
Figure 7. Approximation of the function Fm(p) − 1
2
function f(p)= [1 + α](p + 1
2
by the
) + β(p + 1
2 )µ over different
p-intervals, for m = 60 interacting agents and parameters
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
Table 1. Optimized parameters α, β and µ for several optimization
intervals with m = 60 interacting agents and ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85.
Optimization domain α β µ
0 p − 1
0.05 2 0.011 11.67 3.27
0 p − 1
0.10 2 0.013 43.66 3.77
0 p − 1
0.15 2
0 p −
0.014 60.32 3.91
1
0.20 2 0.004 30.64 3.54
This expression can be obtained as an approximation of the
exact expansion derived in the appendix.
In order to check the hypothesis (16), we numerically solve
the following problem
min
{α,β,µ} Fm(p) − 1
2
1
1
− [1 + α](p − ) − β(p − 2 2 )µ 2 , (18)
which amounts to constructing the best approximation of
the exact map Fm(p) in terms of an effective power-law
acceleration (see (20) below). The results obtained for m =
60 interacting agents and ρhb = ρbh = 0.72 and ρhh =
ρbb = 0.85 are given in table 1 and shown in figure 7.
The numerical values of α are in good agreement with the
theoretical prediction: α(m) = F ′ m (1/2) − 1 which yields
α(m) 0.011 in the present case (m = 60, ρhb = ρbh = 0.72
and ρhh = ρbb = 0.85). As a first approximation, we
can consider that the exponent µ is fixed over the interval
of interest, which is reasonable according to the very good
quality of the fits shown in figure 7. We can conclude from
this numerical investigation that µ(m) ∈ [3, 4]. A finer
analysis shows, however, that the exponent µ is in fact not
perfectly constant but shifts slowly from about 3 to 4 as p
increases. This should be expected as the function Fm(p)
contains many higher-order terms. We can also note that the
parameter pc = (β/α) −1/µ , which defines the typical scale
of the crossover remains constant and equal to pc 0.70 for
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