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Answer: USD 5.78
## Explanation This question uses the **put-call parity** formula adjusted for dividends: $$c = S_0 + p - PV(K) - PV(D)$$ Where: - $c$ = call price (to be determined) - $S_0$ = current stock price = USD 32.00 - $p$ = put price = USD 4.00 - $PV(K)$ = present value of strike price - $PV(D)$ = present value of dividend **Step 1: Calculate PV(K)** $$PV(K) = K \cdot e^{-rT} = 30.00 \cdot e^{-0.035 \cdot 0.5} = 29.4796$$ **Step 2: Calculate PV(D)** $$PV(D) = D \cdot e^{-rt} = 0.75 \cdot e^{-0.035 \cdot 0.25} = 0.7435$$ **Step 3: Calculate call price** $$c = 32.00 + 4.00 - 29.4796 - 0.7435 = 5.7769 \approx \text{USD } 5.78$$ **Key Points:** - Put-call parity must be adjusted for dividends - Present values are calculated using continuous compounding - The dividend is discounted for 3 months (0.25 years) - The strike price is discounted for 6 months (0.5 years) - The result matches option C (USD 5.78)
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An options trader wants to price a European-style call option on a stock with a strike price of USD 30.00 and a time to maturity of 6 months. The trader observes that the current price of a 6-month, USD 30.00 strike price, European-style put option on the same underlying stock is USD 4.00. The current stock price is USD 32.00. A special one-time dividend of USD 0.75 per share is expected in 3 months. The continuously compounded risk-free rate for all maturities is 3.5% per year. Which of the following is closest to the no-arbitrage value of the call option?
A
USD 2.22
B
USD 5.26
C
USD 5.78
D
USD 6.52
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