An option has strike price of $8 and 2 months to expiry.The current price of the underlying share is $26 and its volatility (sigma) is 22%.The riskfree rate of interest is 6% per annum. Calculate d1 for this option. [your answer should have at least 2 decimal places]

Question

BlackTom AI · Accepted Answer

Black‑Scholes d1 formula is d1 = [ln(S/K) + (r + σ^2/2)·T] / (σ·√T). This is the relevant concept: S = spot price, K = strike, r = continuously compounded risk‑free rate, σ = volatility (as decimal), T = time to expiry in years.

Step‑by‑step calculation and intermediate steps:
1. Convert inputs to consistent units:
   - S = 26, K = 8
   - σ = 22% = 0.22
   - r = 6% = 0.06 (per annum)
   - T = 2 months = 2/12 = 1/6 ≈ 0.1666667 years

2. Compute ln(S/K):
   - S/K = 26 / 8 = 3.25
   - ln(3.25) ≈ 1.178655

3. Compute σ^2/2 and (r + σ^2/2)·T:
   - σ^2 = 0.22^2 = 0.0484
   - σ^2/2 = 0.0484 / 2 = 0.0242
   - r + σ^2/2 = 0.06 + 0.0242 = 0.0842
   - (r + σ^2/2)·T = 0.0842 × (1/6) ≈ 0.01403333

4. Numerator of d1:
   - ln(S/K) + (r + σ^2/2)·T = 1.178655 + 0.01403333 = 1.19268833

5. Denominator of d1:
   - √T = √(1/6) ≈ 0.40824829
   - σ·√T = 0.22 × 0.40824829 ≈ 0.08981462

6. Compute d1:
   - d1 = 1.19268833 / 0.08981462 ≈ 13.275 → rounded to two decimals = 13.28

Why this answer is correct: each term in the Black‑Scholes d1 formula was computed using standard conversions (percent → decimal, months → years) and evaluated precisely; the large ln(S/K) because S is much larger than K yields a very large positive d1.

Common mistakes (why alternatives would be wrong):
- Using time as 0.02 (i.e., 2 months = 0.02 years) or forgetting to convert σ to decimal would give incorrect results.
- Using r as simple instead of continuously compounded in the Black‑Scholes inputs would be inconsistent.

Conclusion: The computed d1 = 13.28, which matches the provided answer.

An option has strike price of $8 and 2 months to expiry.The current price of the underlying share is $26 and its volatility (sigma) is 22%.The riskfree rate of interest is 6% per annum. Calculate d1 for this option. [your answer should have at least 2 decimal places]数值题

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