Optimal time to invest when the price processes are geometric Brownian motions. A tentative based on smooth fit

This paper considers the problem of the optimal timing of the exchange of the sum of n geometric Brownian motions for the sum of m others. We propose a closed form determinable stopping time based on the heuristic principle of smooth fit. We cannot prove that this stopping time is optimal. However, we show numerically on examples that it is a potentially useful candidate: letting S^O denote the stopping region induced by our stopping time we show that (i) S^- c S^O c S^+ where S^- and S^+ are well-known subset and superset of the optimal stopping region; (ii) stopping at the first entry time of S^O offers a better payoff than stopping at the first entry time of S^- or S^+, especially when assets are correlated.

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