Pricing finite horizon fund protection with early withdrawal option

Abstract

We present an efficient method to compute the prices and early exercise boundaries of American dynamic fund protection options with finite maturity. A crucial step involves a change of numeraire which reduces the number of states in the valuation problem. This change of numeraire simplifies our valuation procedure by allowing a space-time transformation to be performed to reduce the valuation to a single canonical optimal stopping problem for standard Brownian motion, indexed by two parameters. Further, we prove that, after the space-time transformation, the boundaries are piecewise linear functions for fixed canonical time. This in turn allows us to completely specify the boundaries by considering their behaviour at z*=0, where z* is the transformed maximum asset price. Extensive computations using the Bernoulli walk show that the boundary values at z*=0 are well approximated by piecewise linear functions with a few pieces, which in turn provides us with a fast approximate method to obtain the American dynamic fund protection option values using a decomposition formula.