# what is exercise frontier in option pricing

What's exercise frontier in option pricing? It kept popping up but I was never fully introduced to the concept.

Follow up question: And is the optimal exercise time the first time the option is in-the-money or is it the time where (value_exercise_now - value_continue) is at its largest?

Someone please shed some light thanks!

Exercise frontier represents the decision boundary when you would exercise an option.

This is what a exercise frontier would look like in American option. A more common name is "exercise region". This is the region where it's optimal your option.

L is the optimal exercise price. It's a convex function of maturity. Far away from maturity, the optimal price is significantly lower than K because we'd expect a deep in-the-money intrinsic value to compensate giving up the option rights early. The price approaches to the strike price as shown in the plot because you would have to be less selective for exercising (i.e: you don't have much time to wait).

The concept of exercise region is closely related to optimal stopping time, I recommend Sheve's Stochastic Calculus book if you're interested to learn more. The early exercise boundary (or frontier) for American puts is the level $S^*(t)$ where it is optimal to exercise the put if $S(t)<S^*(t)$. There is no known analytical formula for it, but it can be approximated in various ways.

• So for one simulated path, there will be only one frontier? It is when you simulate a large a mount of paths then you can get an actual graph of frontier? – butterbetter Nov 18 '15 at 20:40
• I thought that the efficient time to exercise an american option was the first time at which the option leaves the "out-of-money area"... Am I wrong? – Louis. B Nov 18 '15 at 22:13