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2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 35
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
for instance Hommes (2001) and Challet et al (2001) and
references therein). Following many others before us, we
conclude that the hypotheses of rational expectations and of noarbitrage
condition are useful approximation or starting points
for model constructions.
Within this framework, the price Pt of an asset at time t
should obey the equation
Pt = δ · EQ[Pt+1|Ft]+dt ∀{Pt}t0, (4)
where dt is an exogeneous ‘dividend’, δ is the discount factor
and EQ[Pt+1|Ft] is the expectation of Pt+1 conditioned upon the
knowledge of the filtration up to time t under the risk-neutral
probability Q.
It is important to stress that the rationality of both
expectations and behaviour does not necessarily imply that
the price of an asset be equal to its fundamental value. In
other words, there can be rational deviations of the price from
this value, called rational bubbles. A rational bubble can arise
when the actual market price depends positively on its own
expected rate of change. This is thought to sometimes occur in
asset markets and constitutes the very mechanism underlying
the models of Blanchard (1979) and Blanchard and Watson
(1982). Indeed, the ‘forward’ solution of (4) is well-known to
be
+∞
Ft =
i=0
δ i · EQ[dt+i|Ft]. (5)
It is straightforward to check by replacement that the general
solution of (4) is the sum of the forward solution (5) and of an
arbitrary component Xt
where Xt has to obey the single condition:
Pt = Ft + Xt, (6)
Xt = δ · EQ[Xt+1|Ft]. (7)
With only two fundamental and quite reasonable assumptions,
it is thus possible to derive an equation (6) (with (7)) justifying
the existence of price fluctuations and deviations from the
fundamental value Ft, for equilibrium markets with rational
agents. However, the dynamics of the bubbles remains
unknown and is a priori completely arbitrary apart from the
no-arbitrage constraint (7).
2.2. Bubble dynamics
One of the main contributions of the model of Blanchard
and Watson (1982), and its possible generalizations (8), is
to propose a bubble dynamics which is both compatible with
most of the empirical stylized facts of price time series and
sufficiently simple to allow for a tractable analytical treatment.
Indeed, the stochastic autoregressive process
Xt+1 = atXt + bt, (8)
where {at} and {bt} are i.i.d. random variables, is well-known
to lead to fat-tailed distributions and volatility clustering.
Kesten (1973) (see Goldie (1991) for a modern extension)
has shown that, if E[ln |a|] < 0, the stochastic process {Xt}
admits a stationary solution (i.e. with a stationary distribution
function), whose distribution density P(X) is a power law
P(X) ∼ X −1−µ with exponent µ satisfying
E[|a| µ ] = 1, (9)
provided that E[|b| µ ] < ∞. It is easy to show that
without other constraints every exponent µ > 0 can be
reached. Consider, for example, a sequence of variables {at}
independently and identically distributed according to a lognormal
law with localization parameter a0 < 1 and scale
ln a0
parameter σ . Equation (9) simply yields µ =−2 σ 2 , which
shows that, varying σ , µ can range over the entire positive real
line.
The second interesting point is that the process (8) allows
volatility clustering. It is easy to see that a large value of Xt will
be followed by a large value of Xt+1 with a large probability.
The change of variables Xt = Y 2
t with at = vZ2 t and bt = uZ2 t
maps exactly the process (8) onto an ARCH(1) process
u + vYt 2 , (10)
Yt+1 = Zt
where Zt is a Gaussian random variable. The ARCH processes
are known to account for volatility clustering. Therefore, the
process (8) also exhibits volatility clustering.
This class (8) of stochastic processes thus provides
several interesting features of real price series (Roman et al
2001). There is however an objection related to the fact that,
without additional assumptions, the bubble price can become
arbitrarily negative and can then lead to a negative price:
Pt < 0. In fact, a negative price is not as meaningless as
often taken for granted, as shown in Sornette (2000). But,
even without allowing for negative prices, it is reasonable to
argue that near Pt = 0, other mechanisms come into play and
modify equation (8) in the neighbourhood of a vanishing price.
For instance, when a market undergoes a too strong and abrupt
loss, quotations are interrupted.
2.3. The independent bubbles assumption
The Blanchard and Watson model assumes that there is only
one asset and one bubble. In other words, the evolution of
each asset does not depend on the dynamics of the others.
But, in reality, there is no such thing as an isolated asset.
Stock markets exhibit a variety of inter-dependences, based
in part on the mutual influences between the USA, European
and Japanese markets. In addition, individual stocks may be
sensitive to the behaviour of the specific industry as a whole
to which they belong and to a few other indicators, such as
the main indices, interest rates and so on. Mantegna (1999)
and Bonanno et al (2001) have indeed shown the existence
of a hierarchical organization of stock interdependences.
Furthermore, bubbles often appear to be not isolated features
of a set of markets. For instance, Flood et al (1984) tested
whether a bubble simultaneously existed across the nations,
such as Germany, Poland, and Hungary, that experienced
hyperinflation in the early 1920s. Coordinated corrections
to what may be considered to be correlated bubbles can
sometimes be detected. One of the most prominent examples
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