Question 1
Suppose you are a portfolio manager and you are going to use derivatives to construct certain portfolio. The current stock price for AAPL is $171.01 per share, at 10 am, March 8, 2019. You want to sell 1 unit of European Call on AAPL, with strike of $180 and maturity of 1 year. 

 

Suppose the annualized interest rate is 3%, the annualized drift for AAPL is 5%, no dividend, and the annualized volatility is 10%. In order to hedge the potential risk of your option, you are going to calculate several Greeks based on Black-Merton-Scholes model. 


Instructions and Hints: 

(i) In your calculation, all results should round to four digits, and you should use the convention that 0.00005 will be rounded up to 0.0001.  (It is understandable that calculation normal PDF and CDF can have computational error, so a quite wide range of answers is admitted.)

(ii) You should carefully keep your intermediate calculation results, for example, 
d
1
d 
1
​
 , as they will be helpful for later questions in this assignment.

(iii) You can use online Normal CDF calculators but NO BMS-Calculator (option price or Greeks calculator) is allowed. Since there will be lots of formula-oriented questions, it will be helpful to make your OWN program in Excel, Python, Matlab etc., in order to do calculation.


Part 1: Greeks based on Black-Merton-Scholes Greeks

Calculate the Call option price, at 10 am, March 8, 2019.

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.

Suppose you are a portfolio manager and you are going to use derivatives to construct certain portfolio. The current stock price for AAPL is $171.01 per share, at 10 am, March 8, 2019. You want to sell 1 unit of European Call on AAPL, with strike of $180 and maturity of 1 year. Suppose the annualized interest rate is 3%, the annualized drift for AAPL is 5%, no dividend, and the annualized volatility is 10%. In order to hedge the potential risk of your option, you are going to calculate several Greeks based on Black-Merton-Scholes model.

This page explores how to calculate the price of a European Call option on AAPL using the Black-Scholes-Merton model. The current stock price is $171.01, with a strike price of $180, 1-year maturity, a 3% interest rate, and a 10% volatility. Learn to compute the Greeks and manage the risk of derivatives trading.

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