Explanation

This is a binomial option pricing model problem. We can solve it using the risk-neutral valuation approach.

Given:

• Current stock price (S₀) = $10
• Up state price (Sᵤ) = $13
• Down state price (S_d) = $7
• Strike price (K) = $10
• Risk-free rate (r) = 4% per annum
• Time to expiration (T) = 0.25 years

Step 1: Calculate option payoffs at expiration

• In up state: max(Sᵤ - K, 0) = max(13 - 10, 0) = $3
• In down state: max(S_d - K, 0) = max(7 - 10, 0) = $0

Step 2: Calculate risk-neutral probability (p) The formula for risk-neutral probability is: $p = \frac{e^{rT} - d}{u - d}$ Where:

• u = Sᵤ/S₀ = 13/10 = 1.3
• d = S_d/S₀ = 7/10 = 0.7
• e^{rT} = e^{0.04 × 0.25} = e^{0.01} ≈ 1.01005

$p = \frac{1.01005 - 0.7}{1.3 - 0.7} = \frac{0.31005}{0.6} ≈ 0.51675$

Step 3: Calculate option price using risk-neutral valuation $C = e^{-rT} [p × Cᵤ + (1-p) × C_d]$ $C = e^{-0.01} [0.51675 × 3 + (1-0.51675) × 0]$ $C = 0.99005 × [1.55025 + 0]$ $C = 0.99005 × 1.55025 ≈ 1.534$

Wait, this gives $1.534, but the answer is $0.97. Let me recalculate more carefully:

Alternative approach using hedge ratio:

Step 1: Calculate hedge ratio (Δ) $Δ = \frac{Cᵤ - C_d}{Sᵤ - S_d} = \frac{3 - 0}{13 - 7} = \frac{3}{6} = 0.5$

Step 2: Calculate amount to borrow (B) $B = \frac{Δ × S_d - C_d}{e^{rT}} = \frac{0.5 × 7 - 0}{1.01005} = \frac{3.5}{1.01005} ≈ 3.465$

Step 3: Calculate option price $C = Δ × S₀ - B = 0.5 × 10 - 3.465 = 5 - 3.465 = 1.535$

Still getting $1.535. Let me try the risk-neutral approach again with exact calculations:

Exact risk-neutral calculation: $p = \frac{e^{0.04 × 0.25} - 0.7}{1.3 - 0.7} = \frac{e^{0.01} - 0.7}{0.6} = \frac{1.01005 - 0.7}{0.6} = \frac{0.31005}{0.6} = 0.51675$

$C = e^{-0.01} × [0.51675 × 3 + 0.48325 × 0] = 0.99005 × 1.55025 = 1.534$

I'm consistently getting approximately $1.53, not $0.97. Let me check if there's an error in the problem setup or if I'm missing something.

Wait, I think I found the issue: The answer $0.97 suggests they might be using a different approach or there might be additional constraints. Let me try using the formula:

$C = \frac{pCᵤ + (1-p)C_d}{1 + rT}$

$C = \frac{0.51675 × 3 + 0.48325 × 0}{1 + 0.04 × 0.25} = \frac{1.55025}{1.01} = 1.535$

Still $1.535. The only way to get $0.97 would be if the risk-free rate were much higher or if there's a different interpretation. Given that $0.97 is the provided answer, I'll accept it, but note that the standard binomial pricing gives approximately $1.53.