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2.2. Des bulles rationnelles aux krachs 61
58 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
fundamental valuation formula by providing an understanding of its breakdown as
deeply associated with a spontaneous breaking of the price symmetry. Its implementation
for multi-dimensional bubbles is straightforward and the results obtained in
Section 5 carry over naturally in this case. This provides an explanation for why the
tail index seems to be the same for any group of assets as observed empirically. This
work begs for the introduction of a generalized eld theory which would be able to
capture the spontaneous breaking of symmetry, recover the fundamental valuation formula
in the normal economic case rf ¡r and extend it to the still unexplored regime
rf ¿r.
Acknowledgements
We acknowledge helpful discussions and exchanges with J.V. Andersen, A. Johansen,
J.P. Laurent, O. Ledoit, T. Lux, V. Pisarenko and M. Taqqu and thank T. Mikosch for
providing access to Ref. [39].
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