Optimal partial hedging of an American option: shifting the focus to the expiration date

As a main contribution we present a new approach for studying the problem of optimal partial hedging of an American contingent claim in a finite and complete discrete-time market. We assume that at an early exercise time the investor can borrow the amount she has to pay for the option holder by entering a short position in the numéraire asset and that this loan in turn will mature at the expiration date. We model and solve a partial hedging problem, where the investor’s purpose is to find a minimal amount at which she can hedge the above-mentioned loan with a given probability, while the potential shortfall is bounded above by a certain number of numéraire assets. A knapsack problem approach and greedy algorithm are used in solving the problem. To get a wider view of the subject and to make interesting comparisons, we treat also a closely related European case as well as an American case where a barrier condition is applied.