*Learning objectives: Identify the six factors that affect an option’s price. Identify and compute upper and lower bounds for option prices on non-dividend and dividend paying stocks. Explain put-call parity and apply it to the valuation of European and American stock options, with dividends and without dividends, and express it in terms of forward prices. Explain and assess potential rationales for using the early exercise features of American call and put options.*

**Questions:**

22.25.1. A stock currently trades at $60.00 and its volatility is 20.0% while the riskfree rate is 3.0%. The underlying stock pays a 4.0% dividend yield. The per annum interest rate and yield are expressed with continuous compounding.

Sally is considering each of the following six-month options (i.e., for each, the maturity, T = 0.5 years) on the stock:

- European call option with K = $50.00 strike price
- American call option with K = $50.00 strike price
- European put option with K = $70.00 strike price
- American put option with K = $70.00 strike price

**TRUE**?

a. Neither of the American options should ever be exercised early

b. The lower bound (aka, minimum value) of both European options is at least $10.00

c. A doubling of the riskfree rate (from 3.0 to 6.0%) will increase the value of all four options

d. A doubling of either the options' terms (to one year) or stock's volatility (to 40.0%) will increase the value of all four options

22.25.2. A stock that pays a 1.0% dividend yield has a current price of $100.00 and volatility of 22.0% while the riskfree rate is 4.0%. Consider two European options with identical strike prices and one-year maturities:

- A European at-the-money call option with a strike price of $100.00 that has a price of $10.09
- A European at-the-money put option with a strike price of $100.00 that has a price of $7.16

a. $7.16 (unchanged)

b. $9.04 (+22% of $$8.40)

c. $10.31 (+ 44% of $7.16)

d. $15.56 (+$8.40)

22.25.3. Barry owns a portfolio of options. He is stress-testing his options under the following two interest rate scenarios:

I. Sudden spike up: The risk-free interest rate quickly jumps up by +150 basis points

II. Shift from positive to negative rate regime: The risk-free interest rate suddenly goes into negative territory

Each of the following is true

**EXCEPT**which is false?

a. Neither regime change breaks (aka, invalidates) put-call parity; ie, put-call parity survives interest rate shocks

b. After the sudden jump up, any non-zero lower bound of an in-the-money call (put) will increase (decrease)

c. In the negative rate regime, early exercise of deeply in-the-money American puts becomes more attractive

d. In the negative rate regime, early exercise of deeply in-the-money American calls on dividend-paying stocks becomes more attractive

**Answers here:**