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Answer: (III.) European put option on 1.60% dividend stock and risk-free rate of 3.0%
For lower bounds: - **European call on non-dividend stock**: \(\max(0, S - Ke^{-rT})\) - **European call on dividend-paying stock**: \(\max(0, S - D - Ke^{-rT})\) - **European put on dividend-paying stock**: \(\max(0, Ke^{-rT} - S + D)\) Compute each: 1. **I**: \(50 - 50e^{-0.03\cdot0.5} \approx 0.7445\) 2. **II**: \(50 - 0.40 - 50e^{-0.03\cdot0.5} \approx 0.3445\) 3. **III**: \(50e^{-0.03\cdot0.5} - 50 + 0.40 \approx -0.3445\), so lower bound is **0** 4. **IV**: \(50 - 50 + 0.40 = 0.40\) The **lowest minimum value** is therefore **III**, which has a lower bound of **0**.
Author: Manit Arora
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Q-725.2. Consider an at-the-money (ATM) stock option with a strike price of $50.00 and six months time to expiration; i.e., 50.00`T=0.5$ 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?
A
(I.) European call option on non-dividend stock and risk-free rate of 3.0%
B
(II.) European call option on 1.60% dividend stock and risk-free rate of 3.0%
C
(III.) European put option on 1.60% dividend stock and risk-free rate of 3.0%
D
(IV.) European put option on 1.60% dividend stock and risk-free rate of zero
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