To calculate the price of a one-year, at-the-money European call option using the Black-Scholes formula, we use the following formula:

$C = S_0 N(d_1) - X e^{-rT} N(d_2)$

Variables:

• $S_0 = 40$ (current stock price)
• $X = 40$ (strike price, as it's at-the-money)
• $\sigma = 0.20$ (volatility)
• $r = 0.07$ (continuously compounded risk-free rate)
• $T = 1$ year (time to maturity)

Steps:

1. Calculate $d_1$ and $d_2$:

$d_1 = \frac{\ln(S_0 / X) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$ $d_2 = d_1 - \sigma \sqrt{T}$

Substituting values:

$d_1 = \frac{\ln(40 / 40) + (0.07 + 0.20^2 / 2) \cdot 1}{0.20 \cdot \sqrt{1}}$ $d_1 = \frac{0 + (0.07 + 0.02)}{0.20}$ $d_1 = \frac{0.09}{0.20} = 0.45$ $d_2 = 0.45 - 0.20 \cdot \sqrt{1} = 0.45 - 0.20 = 0.25$

2. Find $N(d_1)$ and $N(d_2)$:

$N(d_1)$ and $N(d_2)$ are the cumulative probabilities from the standard normal distribution.

Using a standard normal distribution table or calculator: $N(0.45) \approx 0.6736$ $N(0.25) \approx 0.5987$

3. Plug into the Black-Scholes formula:

$C = 40 \cdot N(0.45) - 40 \cdot e^{-0.07 \cdot 1} \cdot N(0.25)$ $C = 40 \cdot 0.6736 - 40 \cdot e^{-0.07} \cdot 0.5987$ First, calculate $e^{-0.07}$: $e^{-0.07} \approx 0.9324$

Now calculate the terms: $C = 40 \cdot 0.6736 - 40 \cdot 0.9324 \cdot 0.5987$ $C = 26.944 - 40 \cdot 0.558$ $C = 26.944 - 22.32 = 4.624$

Final Answer:

The price of the one-year, at-the-money European call option is approximately $4.62.

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Related Questions:

1. What are the differences between European and American options in terms of pricing?
2. How does the Black-Scholes model change for dividend-paying stocks?
3. What happens to the option price if volatility increases?
4. How would the price of a put option differ for the same parameters?
5. What are the limitations of the Black-Scholes model in real-world applications?

Tip:

When using the Black-Scholes formula, ensure all inputs are consistent in terms of units (e.g., time to maturity should be in years).