Optimal exercise of executive stock options

In the absence of frictions if a portfolio strategy replicates the payoff of one unit of a claim, an appropriately scaled strategy replicates any amount of the claim. If assets are priced by arbitrage, the value per-unit is invariant to the amount of the asset considered. In particular, in the case of American claims, the optimal exercise time is independent of the amount of the claim that is considered. In this paper we show that this result does not necessarily hold in the presence of portfolio constraints or other frictions. We produce an example in which the absence of short sales leads the holder of an American option on a (possibly non-dividend paying) stock to exercise parts of his option over time.