Early Exercise Options: Upper Bounds

In this article, we discuss how to generate upper bounds for American or Bermudan securities by Monte Carlo methods. These techniques provide a useful supplement to strategies that provide lower bound estimates (e.g., eqf13-006 and eqf13-025), allowing one to both generate valid confidence intervals for the true option price and to test the accuracy to any proposed approximation to the optimal exercise strategy. 1. Setup and Basic Results As usual, we work on a filtered probability space and consider a contingent claim with early exercise rights, i.e., the right to accelerate payment on the claim at will. Let the claim in question be characterized by an adapted, non-negative 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 D(t), and suppose that we are given at time 0 a particular exercise policy τ taking values in D(0), as well as a pricing numeraire N inducing a unique martingale measure Q . Let C (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 (1) C (0) = E ( U(τ) N(τ) )