Optimal Discrete Hedging of American Options Using an Integrated Approach to Options with Complex Embedded Decisions

Abstract

Option pricing literature is usually concerned with financial contracts whose payoffs depend on decisions by only one of the contract's parties. Generalizations to more complex cases with decisions by both parties are impeded by the ad-hoc nature of many contributions. One prominent example still lacking a satisfying treatment is hedging of American options.This paper follows an integrated approach where 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 due to hedging, 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 will significantly deviate from the classical Black-Scholes exercise boundary.