Installment Options Close to Expiry

In the last thirty years, there has been a dramatic growth in the trading of options, which are contracts between two parties giving one party the right but not the obligation to partake in a financial transaction with the other party at or before a specified date in the future. The majority of options involve the right to buy or sell an underlying asset at a prescribed price, known as the strike or exercise price. An option carrying the right to buy an asset is a call, while the right to sell is a put. One way to classify options is according to exercise style. It is straightforward to price European options, which can only be exercised at expiry, using the Black-Scholes option pricing formula [6]. American options, which can be exercised at any time at or prior to expiry, are harder to price because the possibility of early exercise leads to a free boundary separating the region where it is optimal to hold from that where exercise is optimal. In theory, exercise should take place only on this free boundary, known as the optimal exercise boundary. This sort of free boundary problem is common in diffusion problems such as melting and solidification problems and is referred to as a Stefan problem, and a large number of studies have focused on the optimal exercise boundary for American options, and on the behavior of this boundary close to expiry, including the studies in [1, 2, 5, 13, 16, 17, 20, 21, 25]. In addition to standard American and European options, known as vanilla options, there exist numerous exotic options, which have either unusual payoffs or some other