Pricing and Hedging of American Knock-In Options

C omplex derivatives have become accepted instruments to tailor risk coverage for risk managers and investors. Barrier-type options have become important instruments, particularly for the valuation of structured products (see Banks [1994]). They are also widely used in currency markets. The holder of a barrier option acquires option coverage on only a subset of the risky outcomes for which a plain vaniUa option pays off; this reduces the cost of the resulting coverage so that the holder of the contract does not have to pay for contingencies the holder thinks are unlikely to occur. Because of this flexibility, barrier options were traded over the counter long before the opening of the Chicago Board Options Exchange, and have become some of the most commonly traded derivative contracts. American barrier options offer the added flexibility of early exercise but have to be priced using numerical algorithms. as they do not have closed-form solutions, unlike their European-style counterparts (see Merton [1973] and Rubinstein and Reiner [1991]). Naive application of the Cox-RossRubinstein binomial tree method for barrier options has been shown by Boyle and Lau [1994] to yield inaccurate values, even with many steps. To address this problem, which stems from the position of the barrier relative to the grid, a number of variants of the tree method have been advanced. Ritchken [1995] implements a trinomial tree method. Cheuk and Vorst [1996| develop a time-dependent shift for the trinomial tree, and Figlewski and Gao [1999[ introduce an adaptive mesh model that grafts high-resolution lattices around points that cause the inaccuracies in the binomial model. In an alternative to tree methods, Gao, Huang, and Subrahmanyam [2000] and AitSahlia, Imhof, and Lai [2003] extend the price decomposition approach originally developed for standard American options by Kim [1990], Jacka [1991], and Carr, Jarrow, and Myneni [1992] to derive analytic approximations for American knock-out prices and hedge parameters. The sum of American knock-in and knock-out prices does not equal the standard American option price, as is the case for European barrier options. Moreover, the knockin option value process is non-Markovian, so the classic binomial or trinomial tree methods or numerical partial differential equations for