Martingale Methods in Financial Modelling

5. American Options (p. 171-172)

In contrast to the holder of a European option, the holder of an American op- tion is allowed to exercise his right to buy (or sell) the underlying asset at any time before or at the expiry date. This special feature of American-style op- tions - and more generally of American claims - makes the arbitrage pricing of American options much more involved than the valuation of standard Eu- ropean claims. We know already that arbitrage valuation of American claims is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, un- der risk-neutral probability). Thus, the uniqueness of the arbitrage price of an American claim holds. One of the earliest works to examine the rela- tionship between the early exercise feature of American options and optimal stopping problems was the paper by McKean (1965).

As the arbitrage valu- ation of derivative securities was not yet discovered at this time, the optimal stopping problem associated with the optimal exercise of American put was studied by McKean (1965) under an actual probability IP, rather than under the martingale measure IP*, as is done nowadays. For further properties of the optimal stopping boundary, we refer the reader toVan Moerbeke (1976). Ba- sic features of American options, within the framework of arbitrage valuation theory, were already examined in Merton (1973). However, mathematically rigorous valuation results for American claims were first established by means of arbitrage arguments in Bensoussan (1984) and Karatzas (1988, 1989). An exhaustive survey of results and techniques related to the arbitrage pricing of American options was given by Myneni (1992). For an innovative approach to American options and related issues, see Bank and Follmer (2003).

The purpose of this chapter is to provide the most fundamental results concerning the arbitrage valuation of American claims within the continuous- time framework of the Black-Scholes financial model. Firstly, we discuss the concept of the arbitrage price of American contingent claims and its ba- sic properties. As a consequence, we present the well-known result that an American call option with a constant strike price, written on a non-dividend- paying stock, is equivalent to the corresponding European call option. Sub- sequently, we focus on the features of the optimal exercise policy associ- ated with the American put option. Next, the analytical approach to the pricing of American options is presented. The free boundary problem as- sociated with the optimal exercise of American put options was studied by, among others, McKean (1965) and Van Moerbeke (1976). More recently, Jaillet et al. (1990) applied the general theory of variational inequalities to study the optimal stopping problem associated with American claims.
Finally, the most widely used numerical procedures related to the ap- proximate valuation of American contingent claims are reviewed. An an- alytic approximation of the American put price on a non-dividend-paying stock was examined by Brennan and Schwartz (1977a), Johnson (1983) and MacMillan (1986). We close this chapter with an analysis of an American call written on a dividend-paying stock (this was examined in Roll (1977)).