| Derivatives have become increasingly important in recent years. Options are one of the major groups. American options which permit exercise any time up to expiry are free boundary problems and, as such, are difficult, especially in higher dimensions.; The aim of this work is to explore American options with higher dimensional state variables. We start by establishing an existence and uniqueness theorem on a general group of American options by the penalized method. Two specific American put options, on the geometric mean and the arithmetic mean of the underling assets, are studied in the following two chapters. We obtain some properties of the free boundaries. An exact asymptotic expression for the free boundary of the American put option on the arithmetic mean of the underlying assets is derived. Finally, a transformed binomial tree method is developed and some numerical results are given, including those of Monte Carlo simulations. |
