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.

BUSFIN 4229 SP2025 (4930) Practice Quiz

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