The efficient hedging problem for American options

In this paper, we prove the existence of efficient partial hedging strategies for a trader unable to commit the initial minimal amount of money needed to implement a hedging strategy for an American option. The attitude towards the shortfall is modeled in terms of a decreasing and convex risk functional satisfying a lower semicontinuity property with respect to the Fatou convergence of stochastic processes. Some relevant examples of risk functionals are analyzed. Numerical computations in a discrete-time market model are provided. In a Lévy market, an approximating solution is given assuming discrete-time trading.