Sufficient and necessary conditions for perpetual multi-assets exchange options

This paper considers the general problem of optimal timing of the exchange of the sum of n Ito-diffusions for the sum of m others (e.g., the optimal time to exchange a geometric Brownian motion for a geometric mean reverting process). We first contribute to the literature by providing analytical sufficient conditions and necessary conditions for optimal stopping (i.e. sub- and super- sets of the stopping region) for some sub-cases of the general problem. We then exhibit a connection between the problem of finding sufficient conditions for optimal stopping and linear programming. This connection provides a unified approach which does not only allow to recover previous analytically determinable subsets of the stopping region, but also allows to characterize (more complex) subsets of the stopping region that do not have an analytical expression. In the particular case where all assets are geometric Brownian motions, this connection gives us new insights. In particular, it simplifies the expression of the subset of the stopping region identified by Nishide and Rogers (2011). Our numerical examples finally confirms the good behavior of the candidate investment rule introduced by Gahungu and Smeers (2011) for this particular case, which seems to comfort a conjecture that their rule might be optimal.