The American Straddle Close to Expiry

One of the classic problems of mathematical finance is the pricing of American options and the behavior of the optimal exercise boundary close to expiry. For the uninitiated, financial derivatives are securities whose value is based on the value of some other underlying security, and options are an example of derivatives, carrying the right but not the obligation to enter into a specified transaction in the underlying security. A call option allows the holder to buy the underlying security at a specified strike price E, a put option allows the holder to sell the underlying at the price E, while a straddle, which we consider in the current study, allows the holder the choice of either buying or selling (but not both) the security. If S is the price of the underlying, then the payoff at expiry is max(S−E,0) for a call, max(E− S,0) for a put, andmax(S−E, E− S) for a straddle. From this payoff, it would appear at first glance that the holder of a straddle is holding a call and a put on the same underlying with the same strike and the same expiry, but is only allowed to exercise one. This is true for a European straddle, which can be exercised at expiry, because if the call is in the money, the put must be out of the money and vice versa, and the holder will naturally exercise whichever of the call and the put is in the money at expiry, unless he is unlucky enough that S = E so that they are both exactly at the money, and the payoff is zero.