Exit option for a class of profit functions

ABSTRACT

In this paper we propose a formula to derive the value of a firm which is currently producing a certain product and faces the option to exit the market, whose demand follows a geometric Brownian motion. The problem of optimal exiting is an optimal stopping problem that can be solved using the dynamic programming principle. This is a free-boundary problem. We propose an approximation for the original model and, using the Implicit Function Theorem, we obtain the solution of the original problem. Finally we show, analytically, that the exit threshold is decreasing with the volatility as well as the drift of the geometric Brownian motion.

KEYWORDS:

• Real options
• exit option
• free-boundary problem
• optimal stopping time

2010 AMS Subject Classifications:

• 49L20
• 60G40
• 65L10

Acknowledgments

We would like to thank the three anonymous reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCiD

C. Oliveira http://orcid.org/0000-0002-2156-3609

Additional information

Funding

This work was supported by the Portuguese Scientific Foundation (FCT) under Grant [UTA_CMU/MAT/0006/2009] and [SFRH/BD/102186/2014]; European Union FP7 Marie Curie Actions Grant Multi-ITN STRIKE: Novel Methods in Computational Finance; and Foundation Calouste Gulbenkian.