Suppose we know that the price of a 1-year put on a stock of Company ABC with an exercise price of $70 is $5 and the price of a stock of Company ABC today is $78/share. Knowing that the annual continuously compounded risk-free interest rate is 5%, determine the price of a call option on a stock of Company ABC, with an exercise price of $70 that expires in 1- year.

Question

BlackTom AI · Accepted Answer

We start by identifying the relevant put-call parity for European options on a non-dividend-paying stock: C - P = S0 e^{-qT} - K e^{-rT}, where q is the continuous dividend yield (assumed 0 here), S0 is the current stock price, K is the strike, r is the annual risk-free rate, and T is time to expiration.

Step 1: Plug in the given values. Here, P = 5, S0 = 78, K = 70, r = 0.05, T = 1, and q = 0, so e^{-qT} = 1 and e^{-rT} = e^{-0.05} ≈ 0.951229.

Step 2: Compute the present value term for K: K e^{-rT} = 70 × 0.951229 ≈ 66.586.

Step 3: Compute the stock price term adjusted for carry: S0 e^{-qT} = 78 × 1 = 78 (since q = 0).

Step 4: Apply put-call parity to solve for the call price: C = P + S0 e^{-qT} − K e^{-rT} = 5 + 78 − 66.586 ≈ 16.414.

Step 5: Compare with the answer choices. The computed value rounds to 16.41, which matches the option $16.41.

Why the other options are not correct:
- $16.41 vs $17.24, $18.45, $19.50: Only $16.41 aligns with the parity relationship given the inputs (no dividends, one-year maturity, and the stated r). The discrepancy in the other numbers would imply a different parity input (e.g., a nonzero dividend yield, a different r, or a misapplication of the formula).
- If one misapplies C = P + S0 − K e^{-rT} without discounting the strike appropriately, or omits the e^{-rT} factor, results would drift away from the correct parity-based value, illustrating why careful use of the exponentiated terms is essential.
- Misinterpreting q as positive or assuming immediate exercise features (American vs European) could also lead to incorrect estimates, but the problem context implies European options and zero dividends for parity calculation.

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