Pricing exotic type options such as American and lookback options often translates to optimal stopping time problems. The key to solving these problems is to apply the ``principle of smooth fit'' to find the ``free boundary'' between the continuation region and the stopping region.
We will review some recent free boundary problems in option pricing in both classical Black-Scholes and regime switching models. For the Black-Scholes model, we will talk about pricing perpetual American type lookback options where the smooth fit principle is necessary but not sufficient in solving this problem. For the regime switching model, we will investigate some examples where explicit solutions can be obtained via extending the smooth fit technique to allow jump discontinuities.
Part of the talk is from a joint work with Larry Shepp.