Explanation:
To price the call option using the one-step binomial tree model (Cox-Ross-Rubinstein approach), we calculate the up and down factors, the risk-neutral probability, and then discount the expected payoff.
Given: Current stock price (S) = 88 Strike price (K) = 85 Volatility (σ) = 21% = 0.21 Risk-free rate (r) = 3% = 0.03 Time to maturity (T) = 1 year
Calculate the up (u) and down (d) factors: u = e^(σ√T) = e^(0.21 × 1) ≈ 1.233678 d = e^(-σ√T) = 1 / u ≈ 0.810584
Calculate the up and down stock prices: S_u = S × u = 88 × 1.233678 ≈ 108.5637 S_d = S × d = 88 × 0.810584 ≈ 71.3314
Calculate the payoff of the call option in both states: C_u = Max(S_u - K, 0) = Max(108.5637 - 85, 0) = 23.5637 C_d = Max(S_d - K, 0) = Max(71.3314 - 85, 0) = 0
Calculate the risk-neutral probability of an up move (p): p = (e^(rT) - d) / (u - d) p = (e^(0.03 × 1) - 0.810584) / (1.233678 - 0.810584) p = (1.030455 - 0.810584) / 0.423094 p = 0.219871 / 0.423094 ≈ 0.519674
Calculate the option price today by discounting the expected payoff: C = e^(-rT) × [p × C_u + (1 - p) × C_d] C = e^(-0.03) × [0.519674 × 23.5637 + (1 - 0.519674) × 0] C = 0.970446 × 12.2454 ≈ 11.88
The price of the one-year European call option is approximately USD 11.88.
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Q.56 A stock is currently trading at USD 88 per contract. The volatility of the stock is 21%. What is the price of a one-year European call option with a USD 85 strike price on this stock using the one-step binomial tree model if the risk-free rate is 3% per year?
A
USD 10.24
B
USD 11.88
C
USD 12.21
D
USD 23.56
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