Encyclopedia of Quantitative Finance, IV Volume Set

Early Exercise Options: Upper Bounds

Setup and Basic Results

We work, as usual, on a filtered probability space and consider a contingent claim with early exercise rights, that is, the right to accelerate payment on the claim at will. Let the claim in question be characterized by an adapted, nonnegative payout process U(t), payable to the option holder at a stopping time (or exercise policy) τ ≤ T , chosen by the holder. If early exercise can take place at any time in some interval, we say that the derivative security is an American option; if exercise can only take place on a discrete set of dates, we say that it is a Bermudan option.

Let the allowed set of exercise dates larger than or equal to t be denoted (t), and suppose that we are given at time 0 a particular exercise policy τ taking values in (0), as well as a pricing numeraire N inducing a unique martingale measure Q^N. Let C^τ(0) be the time 0 value of a derivative security that pays U(τ). Under technical conditions on U(t), we can write the value of the derivative security as

where E^N (·) denotes expectation in measure Q^N