On perpetual American options in a multidimensional Black–Scholes model

Abstract

We consider the problem of pricing perpetual American options written on dividend-paying assets whose price dynamics follow a multidimensional Black and Scholes model. For convex Lipschitz continuous reward functions, we give a probabilistic characterization of the fair price in terms of a reflected BSDE, and an analytical one in terms of an obstacle problem. We also provide the early exercise premium formula.

Keywords:

• Perpetual American option
• backward stochastic differential equation
• obstacle problem

2010 Mathematics Subject Classifications:

• Primary 91G20
• Secondary 60H10
• 60H30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Polish National Science Centre under Grant 2016/23/B/ST1/01543).