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Optimal hitting time and perpetual option in a non-Lévy model: application to real options

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

P. Barrieu  and
N. Bellamy
P. Barrieu*
Affiliation:
London School of Economics
N. Bellamy*
Affiliation:
Université d'Evry Val d'Essonne
*
∗ Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: p.m.barrieu@lse.ac.uk
∗∗ Postal address: Equipe d'Analyse et Probabilités, Université d'Evry Val d'Essonne, Rue du Père Jarlan, 91025 Evry Cedex, France. Email address: nadine.bellamy@univ-evry.fr
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Abstract

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We study the perpetual American option characteristics in the case where the underlying dynamics involve a Brownian motion and a point process with a stochastic intensity. No assumption on the distribution of the jump size is made and we work with an arbitrary positive or negative jump. After proving the existence of an optimal stopping time for the problem and characterizing it as the hitting time of an optimal boundary, we provide closed-form formulae for the option value, as well as for the Laplace transform of the optimal stopping time. These results are then applied to the analysis of a real option problem when considering the impact of a fundamental and brutal change in the investment project environment. The consequences of this impact, that can seriously modify, positively or negatively, the project's future cash flows and therefore the investment decision, are illustrated by numerical examples.

Keywords

Optimal stopping timeperpetual American optionpoint processreal option

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G55: Point processes
Secondary: 60K99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 510 - 530
Copyright
Copyright © Applied Probability Trust 2007 

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