Rectangularization of the survival probability seems to be an ongoing process. It results from a higher concentration of the ages at death; but it can be reversed by a continuous increase in the limit of life time. In this paper, we assume that these two factors are endogenous and we show that...

This paper studies the properties of bid and ask prices posted by a monopolistic market-maker, without parametric assumptions about the utility function of the market maker or about the probability distribution of the return of the risky asset. We first prove that the two prices can be higher or...

Holding more of the riskless asset and insuring the risky asset are two ways to reduce portfolio risk. These methods can be employed jointly. As a result, the amount of insurance selected to indemnify against possible losses from holding a risky asset depends, in general, on the quantities of...

The price of any asset can be expressed with risk neutral probabilities, which are adjusted to incorporate risk preferences. This paper introduces the concepts of downside (respectively outer) risk neutral probabilities, which are adjusted to incorporate the preferences for downside (resp....