The Optimal Time to Exchange one Asset for Another on Finite Interval

Abstract

Let $S^{1}_{t}$, $S_{t}^{2}$ be correlated geometric Brownian motions. We consider the following problem: find the stopping time τ ^*≤T such that

$$\sup_{\tau\in[0,T]}\mathsf{E}[S_\tau^1-S_\tau^2]=\mathsf{E}[S_{\tau^*}^1-S_{\tau^*}^2]$$

where the supremum is taken over all stopping times from [0,T]. A similar problem, but on infinite interval, was studied by MacDonald and Siegel (Int. Econ. Rev. 26:331–349, 1985), and by Hu and Oksendal (Finance Stoch. 2(3):295–310, 1998), who also considered multiple assets. For a finite time horizon, the problem gets considerably more complicated and cannot be solved explicitly. In this paper we study generic properties of the optimal stopping set and its boundary curve, and derive an integral equation for the latter.