A variational inequality arising from optimal exercise perpetual executive stock options

We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qinet al.(Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN:http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Laiet al.(2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qinet al.(2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.