Early Exercise Regions for Exotic Options

This article examines early exercise regions for American options on multiple assets and average-price options. One feature of the early exercise region for a standard American put option on a single asset is that, at any given time, there exists a critical stock price below which it is optimal to exercise the option. In other words, if it is optimal to exercise the option at a particular stock price at a particular time, it is also optimal to exercise in any state where the option has higher intrinsic value. We show that this property does not hold in general for non- standard options. In the case of options on multiple assets, there can be many states where it is better to hold an option whose intrinsic value is higher than in states where it is better to exercise. The number of such states depends on factors such as the nature of the option, the time remaining until maturity, the correlation between the assets, and their volatilities. In the case of average-price put options, an interesting feature is that the option is not exercised optimally at points with the highest intrinsic value at any time prior to maturity unless the interest rate is very high.