When the non-dividend paying stock price is $30, the strike price is $29, the risk-free rate is 4% with continuous compounding, the volatility is 20% and the time to maturity is 6 months which of the following is the price of a European put option on the stock?

Question

BlackTom AI · Accepted Answer

The question asks for the price of a European put option on the stock, given S0 = 30, K = 29, r = 0.04 (continuous compounding), T = 0.5 years, and sigma = 0.20. We use the Black-Scholes formula for a put: P = K e^{-rT} N(-d2) − S0 N(-d1), where d1 = [ln(S0/K) + (r + 0.5 sigma^2) T] / (sigma sqrt(T)) and d2 = d1 − sigma sqrt(T).

First, calculate the necessary components: ln(S0/K) = ln(30/29) ≈ 0.03385. The term (r + 0.5 sigma^2) T = (0.04 + 0.5*(0.20)^2) * 0.5 = (0.04 + 0.02) * 0.5 = 0.06 * 0.5 = 0.03. So the numerator for d1 is 0.03385 + 0.03 ≈ 0.06385.

The denominator is sigma sqrt(T) = 0.20 * sqrt(0.5) ≈ 0.20 * 0.7071 ≈ 0.1414. Therefore d1 ≈ 0.06385 / 0.1414 ≈ 0.451. Then d2 = d1 − sigma sqrt(T) ≈ 0.451 − 0.1414 ≈ 0.3096.

Thus N(-d1) = N(-0.451) and N(-d2) = N(-0.310). Since N(-0.451) ≈ 0.326 and N(-0.310) ≈ 0.379 (using standard normal CDF values), we can assemble the put price:
P ≈ 29 * e^{−0.04*0.5} * N(-0.310) − 30 * N(-0.451).
Compute e^{−0.02} ≈ 0.9802, so 29 * 0.9802 ≈ 28.426. Then 28.426 * 0.379 ≈ 10.79. The second term is 30 * 0.326 ≈ 9.78. The difference is about 1.01, consistent with the Black-Scholes framework for these inputs.

Now evaluate the given answer options against the put formula structure: P = K e^{-rT} N(-d2) − S0 N(-d1). The correct expression must place 28.43 (which is 29 * e^{−0.02}) multiplied by N(-d2) with the smaller magnitude of the two d-values (approximately N(-0.31)), minus 30 multiplied by N(-d1) (approximately N(-0.45)). The option that matches this arrangement is 28.43N(-0.31) − 30N(-0.45).

Option-by-option review:
- Option 1: 30N(-0.45) − 28.43N(-0.31). This swaps the coefficients and the order of the terms, and uses the wrong pairing of N(d) values with S0 and K. It does not reflect P = K e^{-rT} N(-d2) − S0 N(-d1).
- Option 2: 28.43N(-0.45) − 30N(-0.31). Here the d-values are swapped relative to the required d1 and d2, and the coefficients are attached to the wrong N(-d). This misrepresents how N(-d1) and N(-d2) should appear and inflates the mismatch with the standard formula.
- Option 3: 30N(-0.31) − 28.43N(-0.45). This again misplaces both coefficients and the N(-d) terms; it effectively inverts the intended structure and sign pattern of the put price formula.
- Option 4: 28.43N(-0.31) − 30N(-0.45). This aligns with P = K e^{-rT} N(-d2) − S0 N(-d1), using 28.43 for K e^{-rT} and N(-d2) with −0.31, and 30 for S0 times N(-d1) with −0.45. This is the correct configuration.

In summary, the essence of the correct choice lies in applying P = K e^{−rT} N(−d2) − S0 N(−d1) and plugging d1 ≈ 0.451 and d2 ≈ 0.310, which yields the form shown in the final option.

2025FallDYN-T-FIN530-86763-86762 Quiz 6 - "Black-Scholes-Merton Model and Dynamic Hedging"

When the non-dividend paying stock price is $30, the strike price is $29, the risk-free rate is 4% with continuous compounding, the volatility is 20% and the time to maturity is 6 months which of the following is the price of a European put option on the stock?

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