In this paper we study options on a unit-type closed-end investment fund.
These options are included among the exotic options, because the underlying asset of
the options is the value process of the investment fund and therefore depends on a fund
manager (= an option writer)'s action. We prove that a fair price of such option is represented
as the value function of the associated stochastic exit time control problem. Using
Hajek's mean comparison theorem, we find an explicit form of the fair option premium
in the case of a constant volatility. We also characterize the fair option premium as
a limit of a sequence of classical solutions to the associated Hamilton-Jacobi-Bellman
equations with a classical Dirichlet boundary condition in the case of a diffusion market
model.