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2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 39
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
5. Consequences for rational expectation
bubbles
We have seen in section 3.3 from proposition 1 that, as a
result of the no-arbitrage condition, the spectral radius of the
matrix EP[A] = 1
δ Id is greater than 1. As a consequence, by
application of the converse of proposition 2, this proves that
the tail index κ1 of the distribution of (X) is smaller than 1.
Using the same arguments as in Lux and Sornette (1999),
that we do not recall here, it can be shown that the distribution
of price differences and price returns follows, at least over
an extended range of large returns, a power law distribution
whose exponent remains lower than 1. This result generalizes
to arbitrary d-dimensional processes the result of Lux and
Sornette (1999). As a consequence, d-dimensional rational
expectation bubbles linking several assets suffer from the
same discrepancy compared to empirical data as the onedimensional
bubbles. It therefore appears that accounting for
possible dependences between bubbles is not sufficient to cure
the Blanchard and Watson model: a linear multi-dimensional
bubble dynamics such as (16) is hardly reconcilable with some
of the most fundamental stylized facts of financial data at a very
elementary level.
This result does not rely on the diagonal property of the
matrices E[At] but only on the value of its spectral radius
imposed by the no-arbitrage condition. In other words, the fact
that the introduction of dependences between asset bubbles is
not sufficient to cure the model can be traced to the constraint
introduced by the no-arbitrage condition, which imposes that
the averages of the off-diagonal terms of the matrices At
vanish. As we indicated before, this implies zero correlations
(but not the absence of dependence) between asset bubbles.
It thus seems that the multi-dimensional generalization is
constrained so much by the no-arbitrage condition that the
multi-dimensional bubble model almost reduces to an average
one-dimensional model. With this insight, our present result
generalizing that of Lux and Sornette (1999) is natural.
To our knowledge, there are two possible remedies. The
first one is based on the rational bubble and crash model
of Johansen et al (1999, 2000) which abandons the linear
stochastic dynamics (8) in favour of an essentially arbitrary
and nonlinear dynamics controlled by a crash hazard rate.
A jump process for crashes is added to the process, with a
crash hazard rate evolving with time such that the rational
expectation condition is ensured. This model is squarely
based on the rational expectation framework and shows that
changing the dynamics of the Blanchard and Watson model
allows satisfying results to be obtained, as the corresponding
return distributions can be made to exhibit reasonable fat tails.
The second solution (Sornette 2001) requires the existence
of an average exponential growth of the fundamental price at
some return rate rf > 0 larger than the discount rate. With
the condition that the price fluctuations associated with bubbles
must on average grow with the mean market return rf , it can be
shown that the exponent of the power law tail of the returns is
no more bounded by 1 as soon as rf is larger than the discount
rate r and can take essentially arbitrary values. This second
approach amounts to abandoning the rational pricing theory
(6) with (4) and keeping only the no-arbitrage constraint (7)
on the bubble component. This hypothesis may be true in the
case of a firm, sometimes in the case of an industry (railways in
the 19th century, oil and computer in the 20th for instance), but
is hard to defend in the case of an economy as a whole. The
real long run interest rate in the US is approximately 3.5%,
and the real rate of growth of profits since World War II is
about 2.1%. Thus, for the economy as a whole, the discounted
sum is always finite. It would be interesting to investigate the
interplay between inter-dependence between several bubbles
and this exponential growth model.
As a final positive remark balancing our negative result,
we stress that the stochastic multiplicative multi-asset rational
bubble model presented here provides a natural mechanism for
the existence of a ‘universal’ tail exponent µ across different
markets.
Acknowledgments
We acknowledge helpful discussions and exchanges with
J P Laurent, T Lux, V Pisarenko and M Taqqu. We also
thank T Mikosch for providing access to Le Page (1983) and
V Pisarenko for a critical reading of the manuscript. Any
remaining error is ours.
Appendix A. Proof of proposition 1 on
the no-arbitrage condition
Let t be the value at time t of any self-financing portfolio:
t = W ′
t Xt, (A.1)
where W ′
t = (W1,...,Wd) is the vector whose components
are the weights of the different assets and the prime denotes
the transpose. The no-free-lunch condition reads:
t = δ · EQ[t+1|Ft] ∀{t}t0. (A.2)
Therefore, for all self-financing strategies (Wt), wehave
W ′
t+1 EQ[A] − 1
δ Id
Xt = 0 ∀ Xt ∈ R d , (A.3)
where we have used the fact that (Wt+1) is (Ft)-measurable
and that the sequence of matrices {At} is i.i.d.
The strategy W ′
t = (0,...,0, 1, 0,...,0) (1 in ith
position), for all t, is self-financing and implies
(ai1,ai2,...,aii − 1
δ ,...,aid)
·(X (1)
t ,X (2)
t ,...,X (i)
t ,...,X (d)
t ) ′ = 0, (A.4)
for all Xt ∈ Rd . We have called aij the (i, j)th coefficient of
the matrix EQ[A]. As a consequence,
(ai1,ai2,...,aii − 1
δ ,...,aid) = 0 ∀i, (A.5)
and
EQ[A] = 1
δ Id. (A.6)
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