Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

LÚvy term structure models : no-arbitrage and completeness


Code(s) de Classification MSC:

RÚsumÚ: We study the term structure models which are driven by a LÚvy process, from the point of view of arbitrage and completeness. Exactly as for the Heath--Jarrow--Morton model, which fits into our class of models, we observe that the conditions on the coefficients for having no arbitrage opportunity are rather stringent. For the completeness problem, the results are quite surprising: namely the model is complete if the driving LÚvy process is $1$--dimensional, provided the coefficient are non--random, or they satisfy a very mild non--degeneracy assumption if they are random. On the other hand, an example suggests that the model is no longer complete when the LÚvy process is genuinely multidimensional. This is in deep contrast with the completeness of stock prices models, where typically we have completeness if the number of stock prices is bigger than or equal to the dimension of the driving Brownian motion in the continuous case, while completeness usually fails when the driving process is discontinuous.

Mots ClÚs: LÚvy processes ; representation of martingales ; term structure model ; Heath-Jarrow-Morton ; arbitrage

Date: 2003-10-16

Prépublication numéro: PMA-854