Hedging and Portfolio Optimization Under Transaction Costs: A Martingale Approach¹²

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous‐time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the “wealth process” is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model, we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state‐price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state‐price density for pricing contingent claims in this market. the mathematical tools are those of continuous‐time martingales, convex analysis, functional analysis, and duality theory.