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Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions

In order to solve the problem of optimal discrete hedging of American options, this paper utilizes an integrated approach in which the writer’s decisions (including hedging decisions) and the holder’s decisions are treated on equal footing. From basic principles expressed in the language of acceptance sets we derive a general pricing and hedging formula and apply it to American options. The result combines the important aspects of the problem into one price. It finds the optimal compromise between risk reduction and transaction costs, i.e. optimally placed rebalancing times. Moreover, it accounts for the interplay between the early exercise and hedging decisions. We then perform a numerical calculation to compare the price of an agent who has exponential preferences and uses our method of optimal hedging against a delta hedger. The results show that the optimal hedging strategy is influenced by the early exercise boundary and that the worst case holder behavior for a sub-optimal hedger significantly deviates from the classical Black–Scholes exercise boundary.