To calculate the price of the European Call option based on the Black-Scholes-Merton (BSM) model, we need to use the following formula for a call option price:

$C = S_0 \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)$

where:

• $C$ = call option price
• $S_0$ = current stock price (AAPL stock price) = 171.01 USD
• $K$ = strike price = 180 USD
• $r$ = annualized risk-free interest rate = 3% = 0.03
• $T$ = time to maturity in years = 1 year
• $N(x)$ = cumulative distribution function (CDF) of the standard normal distribution
• $\sigma$ = annualized volatility = 10% = 0.10
• $d_1$ and $d_2$ are calculated as follows:

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

Step 1: Calculate $d_1$

Substitute the known values into the $d_1$ formula:

$d_1 = \frac{\ln\left(\frac{171.01}{180}\right) + \left(0.03 + \frac{0.10^2}{2}\right) \cdot 1}{0.10 \cdot \sqrt{1}}$

Step 2: Calculate $d_2$

Once $d_1$ is calculated, $d_2$ is just:

$d_2 = d_1 - 0.10 \cdot \sqrt{1}$

Step 3: Calculate the Option Price

With $d_1$ and $d_2$ computed, you can now calculate the price of the call option using the BSM formula.

I'll calculate these values step by step.The calculations yield the following results:

• $d_1 = -0.1623$
• $d_2 = -0.2623$

Using these values in the Black-Scholes-Merton formula, the price of the European Call option on AAPL is:

$C = 5.2122 \text{ USD}$

Would you like more details on this calculation, or help with the next part?

Here are 5 related questions for practice:

1. How does the option price change if volatility increases?
2. What happens to the call option price if the strike price is reduced?
3. How would the option price change if the maturity is 6 months instead of 1 year?
4. What is the effect of a higher interest rate on the call option price?
5. How would the Greeks be affected if the volatility increased to 15%?

Tip: Understanding the impact of each input (stock price, volatility, interest rate, time to maturity) on the option price can help you better manage risk in derivative trading.