  A make-your-mind-up option is an American derivative with delivery lags. We
show that its put option can be decomposed as a European put and a new type of
American-style derivative. The latter is an option for which the investor
receives the Greek Theta of the corresponding European option as the running
payoff, and decides an optimal stopping time to terminate the contract. Based
on this decomposition and using free boundary techniques, we show that the
associated optimal exercise boundary exists and is a strictly increasing and
smooth curve, and analyze the asymptotic behavior of the value function and the
optimal exercise boundary for both large maturity and small time lag.
