Ashkan Nikeghbali (Uni Zürich): From last passage times in the Black-Scholes formula to some paradox under the usual assumptions

In this talk we shall discuss a recent formula of Madan,
Roynette and Yor which relates the price of a European put option to
some last passage times. It is suggested by the authors that this
formula is a special case of a more general formula which also appears
in the study of penalization of the Brownian paths on the Wiener
space. The problem of finding situations in which this general formula
holds is then posed by them. We will give a solution to this problem,
thus unifying these two problems and putting them in a more general
framework. On our way, we show the existence of a remarkable sigma-
finite measure and we reveal a paradox which has so far not received
much attention in the literature: given a reference probability
measure, the celebrated usual conditions are not compatible with the
existence of a locally absolutely continuous but singular probability
measure with respect to the reference measure. We shall propose a new
kind of augmentation of filtrations which fixes this problem and which
has a natural interpretation in financial modeling.