Pricing American options

An American warrant (call option) is a security issued by a company (individual) that gives its owner the right to purchase a share of stock at a given exercise price on or before a given date. In the case of an American put option, the owner has the right to sell a share of stock in the same conditions as the call option. Unlike with European options, at any exercise time the holder of an American option compares the optimal payout from immediate exercise with the expected payout from continuation, and then exercises if the immediate payout is higher. Thus, the optimal exercise strategy is fundamentally determined by the conditional expectation of the payout from continuing to keep the option alive. Samuelson (1965b, 1972) proved that these two contracts may not have the same value. Merton (1973) considered American option pricing, focusing on put options when the underlying stock pays dividends. The notion of dominant and dominated securities was used to show that an American put option was not uniquely determined by the price of a call option and the stock.