No, for the put (under constant interest rates), the fair value is [$]\max_{t \le u \le T} E[e^{-r (u - t)} (K - S_u)^+ | S_t][$], where [$]u[$] is actually a stopping-time *policy*. The optimal exercise policy is to stop at the first time after [$]t[$] such that [$]S_u \le S_c(u)[$]. For the case ppauper was asking about, [$]S_t < S_c(t)[$], and so the optimal policy rule is to stop immediately, namely at [$]u = t[$].

Alan, It makes sense but idea was not fully realised. Indeed the deterministic function

[$] E[e^{-r (u - t)} (K - S_u)^+ | S_t][$]

if we assume that S ( t ) = x of the variable u represents the value of the option at a future moment u given information about underlying stock at a current moment t. Taking maximum in u represents maximum BS value of the option during lifetime of the option. Nevertheless if buyer purchases option at t for C ( t , S ( t ) ) then the value of the option at a future moment u will be C ( u , S ( u )). Here S ( u ) is the real world not the risk neutral value of stock which is involved in definition of the C ( u , S ).. Hence we should look at the maximum in of the return at unknown u which presents maximum of C ( u , S ( u ))/ C ( t , S ( t )) . C ( t , S ( t ) ) is a constant and we actually need to know max C ( u , S ( u )) and here we have random function. I do not think that maximum u is reached within set of markov moments as far as for a more simpler problem to find max S ( u ) on [ 0 , T ] we should know S for each [$]\omega[$] on all [ o , T ]. Of course if we simplify the problem and decide to use a reduced optimal principal we can follow reduced optimal function given in your message. One can see that using the reduced goal function we lose something in accuracy pricing rule. Lost is the market risk that can be expressed as a difference between two goal functions. In general the difference could be illustrated in similar situation

[$]\max_{ t < u < T }\, S ( u )[$] and [$]\max_{ t < u < T }\, E S ( u )[$]