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This question demonstrates the binomial option pricing model using the risk-neutral valuation approach. **Given:** - Current stock price = $10 - Up state price = $13 - Down state price = $7 - Call option payoff in up state = $3 ($13 - $10 strike) - Call option payoff in down state = $0 - Risk-free rate = 1% **Delta calculation:** A riskless portfolio is created by: - Long Δ shares - Short 1 call option Portfolio values: - Up state: 13Δ - 3 - Down state: 7Δ Set equal for riskless portfolio: 13Δ - 3 = 7Δ 6Δ = 3 Δ = 0.5 **Option price calculation:** Current portfolio value = 10 × 0.5 - f = 5 - f This equals discounted riskless payoff: 5 - f = 3.5 × e^{-0.01} f = 5 - 3.5 × 0.99005 = 5 - 3.46517 = $1.53483 **Alternative risk-neutral approach:** u = 13/10 = 1.3, d = 7/10 = 0.7 p = [e^{rΔt} - d] / (u - d) = [e^{0.01} - 0.7] / (1.3 - 0.7) = [1.01005 - 0.7] / 0.6 = 0.51675 f = e^{-r} × [p × 3 + (1-p) × 0] = 0.99005 × 1.55025 = $1.53483
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Following Hull, a riskless portfolio consists of long delta (d) shares + short one option.
If the stock moves up, value of the riskless portfolio = $13 \times \text{delta} - ` loss on the written call option; and
If the stock moves down, value of the riskless portfolio = $7 \times \text{delta}$. Setting them equal (i.e., riskless payoff):
$13 \times d - `3` = \`7\times d$, and, so .
If delta (d) = 0.5, then value of portfolio today is:
\`10 \times 0.5 - f = 5 - f = \, such that
f = 5 - \`3.5 \times e^{-1\%} = \
Notationally,
;
f = e^{-rT} \times (0.51675 \times \`3 + 0) = \
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