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This question calculates the value of a call option using a three-step binomial tree: Given: - Initial stock price = $75 - Strike price = $90 - Volatility (σ) = 18.25% - Risk-free rate = 5% - Time period = 3 years - Up probability = 60% Calculations: - u = e^(σ√Δt) = e^(18.25%×√1) = 1.2 - d = e^(-σ√Δt) = 0.83 Ending stock prices: - S_uuu = $75 × 1.2 × 1.2 × 1.2 = $129.60 - S_uud = S_duu = S_udu = $75 × 1.2 × 1.2 × 0.83 = $89.64 - S_udd = S_dud = S_ddu = $75 × 1.2 × 0.83 × 0.83 = $62.00 - S_ddd = $75 × 0.83 × 0.83 × 0.83 = $42.89 Only S_uuu is in-the-money (129.60 > 90) Probability of 3 up moves = 60%^3 = 21.6% Option value = (129.60 - 90) × 21.6% × e^(-5%×3) = 7.36 The correct answer is D based on the solution provided.
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u = e^(σ√Δt) = e^(18.25%×√1) = 1.2 d = e^(-σ√Δt) = 0.83
Next, we project the various paths the stock's price can follow over the 3 year period. The stock has 4 potential ending values:
S_uuu = $75 × 1.2 × 1.2 × 1.2 = $129.60
S_uud = S_duu = S_udu = $75 × 1.2 × 1.2 × 0.83 = $89.64
S_udd = S_dud = S_ddu = $75 × 1.2 × 0.83 × 0.83 = $62.00
S_ddd = $75 × 0.83 × 0.83 × 0.83 = $42.89
The only point at which the option finishes in the money is after 3 upward moves, with a probability of 60%^3 = 21.6%.
The value of the option today is therefore (129.60 – 90) × 21.6% × e^(-5%×3) = 7.36.
A
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B
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C
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D
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E
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F
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