Q-725.2. Consider an at-the-money (ATM) stock option with a strike price of $50.00 and six months time to expiration; i.e., $S(0)=K=`$50.00`$and$T=0.5$ years. Now imagine the following four variations (I., II., III. and IV.) on this option:

I. It is a European CALL option on a non-dividend-paying stock while the risk-free rate is 3.0%
II. It is a European CALL option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is 3.0%
III. It is a European PUT option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is 3.0%
IV. It is a European PUT option on a stock that pays 1.60% dividend yield (D = $0.40) while the risk-free rate is ZERO!

For the three variations where the stock pays a continuous 1.60% dividend, the equivalent present value (over the life of the option) is given by the lump sum, D = $0.40. For those interested, although it is beyond the scope of this question, this translation is given by the following: the PV of dividend, D = -S(0)[exp(-qT)-1]; in this case, D = $50.00[exp(-0.01600.5)-1] = $0.3980.

Each of the above options has a different minimum value (aka, lower bound). However, among the four, which has the LOWEST minimum value?