The put-call parity equation establishes a relationship between the prices of European call and put options that have the same strike price and expiration date. The formula is:

$C + PV(X) = P + S$

Where:

• $C$ = Call option price
• $PV(X)$ = Present value of the strike price (exercise price), discounted at the risk-free rate
• $P$ = Put option price
• $S$ = Stock price

We are given:

• Call option price $C = 12$
• Stock price $S = 80$
• Exercise price $X = 75$
• Risk-free rate $r = 3\%$ per year
• Time to expiration $T = 6$ months = $0.5$ years

First, we calculate the present value of the strike price $PV(X)$:

$PV(X) = X \times e^{-rT} = 75 \times e^{-0.03 \times 0.5}$

Using the approximation $e^{-0.015} \approx 0.9851$:

$PV(X) \approx 75 \times 0.9851 = 73.88$

Now, substitute the known values into the put-call parity formula:

$12 + 73.88 = P + 80$

Solve for $P$ (put option price):

$P = 12 + 73.88 - 80 = 5.88$

Thus, the put option price is approximately $5.88.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How does the put-call parity change if the stock pays dividends?
2. How would a change in the risk-free rate affect the put option price?
3. What is the impact of a longer expiration date on put-call parity?
4. Can put-call parity be used for American options?
5. What happens to the parity relationship if the stock price drops significantly?

Tip: Put-call parity is crucial for spotting arbitrage opportunities in options markets!