  In a discrete time setting, we study the central problem of giving a fair
price to some financial product. For several decades, the no-arbitrage
conditions and the martingale measures have played a major role for solving
this problem. We propose a new approach for estimating the super-replication
cost based on convex duality instead of martingale measures duality: The prices
are expressed using Fenchel conjugate and bi-conjugate without using any
no-arbitrage condition.The super-hedging problem resolution leads endogenously
to a weak no-arbitrage condition called Absence of Instantaneous Profit (AIP)
under which prices are finite. We study this condition in details, propose
several characterizations and compare it to the no-arbitrage condition.
