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Optimality and Risk - Modern Trends in Mathematical Finance

pp 197-210

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

  • Yuliya MishuraAffiliated withKyiv National Taras Shevchenko University Email author 
  • , Georgiy ShevchenkoAffiliated withKyiv National Taras Shevchenko University

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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.

Keywords

Optimal stopping Geometric Brownian motion Finite horizon Free boundary problem

Mathematics Subject Classification (2000)

60G40 60J65 35R35