Before revision:  Here, we propose a new approach  based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and bi-conjugate.
After revision:  Here, we propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and bi-conjugate.