When the non-dividend paying stock price is $40, the strike price is $42, the risk-free rate is 3% p.a. (continuously compounded), the volatility is 25% and the time to maturity is 6 months, which of the following is the price of a European call option on the stock?[Fill in the blank]单项选择题
A
a. 41.37N(–0.28) –40N(–0.10)
B
b. 41.37N(–0.10) –40N(–0.28)
C
c. 40N(–0.28) –41.37N(–0.10)
D
d. 40N(–0.10) –41.37N(–0.28)

Question

When the non-dividend paying stock price is $40, the strike price is $42, the risk-free rate is 3% p.a. (continuously compounded), the volatility is 25% and the time to maturity is 6 months, which of the following is the price of a European call option on the stock?[Fill in the blank]单项选择题

A

a. 41.37N(–0.28) –40N(–0.10)

B

b. 41.37N(–0.10) –40N(–0.28)

C

c. 40N(–0.28) –41.37N(–0.10)

D

d. 40N(–0.10) –41.37N(–0.28)

BlackTom AI · Accepted Answer

Black–Scholes formula for a European call is C = S N(d1) − K e^(−rT) N(d2).

1) Relevant concepts:
- Use Black–Scholes for a non‑dividend paying stock.
- d1 = [ln(S/K) + (r + 0.5 σ^2)T] / (σ√T), d2 = d1 − σ√T.
- N(·) is the standard normal cumulative distribution function.
- Discount strike by e^(−rT).

2) Step-by-step reasoning and 3) intermediate calculations:
- Given: S = 40, K = 42, r = 0.03 (continuously compounded), σ = 0.25, T = 0.5.
- ln(S/K) = ln(40/42) = ln(0.95238095) ≈ −0.04879.
- σ^2 = 0.25^2 = 0.0625. r + 0.5σ^2 = 0.03 + 0.03125 = 0.06125. Multiply by T: (r + 0.5σ^2)T = 0.06125 × 0.5 = 0.030625.
- Numerator for d1: ln(S/K) + (r + 0.5σ^2)T = −0.04879 + 0.030625 = −0.018165.
- σ√T = 0.25 × √0.5 ≈ 0.25 × 0.7071068 = 0.1767767.
- d1 = −0.018165 / 0.1767767 ≈ −0.1028 (≈ −0.10).
- d2 = d1 − σ√T = −0.1028 − 0.1767767 ≈ −0.2796 (≈ −0.28).
- Discounted strike: K e^(−rT) = 42 × e^(−0.03×0.5) = 42 × e^(−0.015) ≈ 42 × 0.985112 ≈ 41.3748 (≈ 41.37).
- Plug into Black–Scholes: C = 40 N(−0.10) − 41.37 N(−0.28).

Numeric check (approx): N(−0.103) ≈ 0.459, N(−0.280) ≈ 0.390 → C ≈ 40×0.459 − 41.37×0.390 ≈ 18.36 − 16.14 ≈ 2.22 (sanity check).

4) Why this is correct:
- The algebraic derivation follows the Black–Scholes formula exactly with computed d1, d2 and discounted strike; signs and coefficients match the theoretical expression S N(d1) − K e^(−rT) N(d2).

5) Why other options are incorrect:
- Option a and b swap the order and coefficients (they place the discounted strike first or swap which N(·) multiplies which coefficient), which does not match S N(d1) − K e^(−rT) N(d2).
- Option c uses the same numbers but swaps the N arguments with the coefficients incorrectly (40 paired with −0.28 and 41.37 with −0.10), not matching computed d1 and d2.

6) Conclusion:
The correct expression is 40 N(−0.10) − 41.37 N(−0.28), which corresponds to choice d. This matches the provided answer.

When the non-dividend paying stock price is $40, the strike price is $42, the risk-free rate is 3% p.a. (continuously compounded), the volatility is 25% and the time to maturity is 6 months, which of the following is the price of a European call option on the stock?[Fill in the blank]单项选择题

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