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*Learning objectives: Identify the six factors that affect an option’s price and describe how these six factors affect the price for both European and American options. Identify and compute upper and lower bounds for option prices on non-dividend and dividend paying stocks.*

**Questions:**

725.1. Consider a European call option on a non-dividend-paying stock that has a current price, c = $6.37, if we make the following assumptions:

- S(0) = K = $100.00 and this option has a delta, N(d1) = 0.570
- Volatility, σ = 20.0% and this option has vega = 27.8
- Riskfree rate, Rf = 3.0%
- Time to expiration, T = 0.5 years or six months

a. Double the riskfree rate to 6.0%

b. Increase the stock price by $5.00 to $105.00

c. Increase volatility by 10.0% to 30.0%

d. Double the time to expiration to T = 1.0 year

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(-q*T)-1]; in this case, D = $50.00*[exp(-0.0160*0.5)-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

725.3. The risk-free rate is 3.0% while the price of a stock is $30.00. Consider two European at-the-money (ATM) options, a call and a put, such that both have a strike price of $30.00. Let c = value of the call option and p = value of the put option. If nothing else changes except the options' terms double from six months to one year, then

**value of the call option increases by +$1.440**. Put simply, c(S=30, K=30, σ=?, Rf=0.030, T=1.0) - c(..., T=0.5) = $1.440.

What is

**nearest**to the increase in the value of the corresponding put option on the same stock, if its time to expiration similarly doubles from six months to one year; i.e., what is p(S=30, K=30, σ=?, Rf=0.030, T=1.0) - p(..., T=0.5)?

a. +$0.35

b. +$1.00

c. +$1.27

d. +$1.440

**Answers here:**

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