Asymptotic behavior of optimal exercise strategy for a small number of executive stock options

We investigate the asymptotic behavior, as the number of options approaches zero, of the optimal exercise strategy of continuous-exercise perpetual executive stock options. Here the optimal strategy is described in terms of the free boundary of a degenerate variational inequality. To our knowledge, our method based on the convexity of the variational inequality at the initial time is new. Also, the unique formal asymptotic expansions of the free boundary and the solution of the variational inequality are derived.