To calculate $d_1$ for both distributions $s_1$ and $s_2$, we will use the Black-Scholes formula components, which are given by:

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

Where:

• $S$ is the stock price at the time ($s_1 = 101$, $s_2 = 103$)
• $K$ is the strike price (100)
• $r$ is the risk-free rate (0.02)
• $\sigma$ is the volatility (5% for $s_1$ and 4.5% for $s_2$)
• $T$ is the time to maturity (5 years)

Step 1: Calculate $d_1$ for $s_1$

Given:

• $S = 101$
• $K = 100$
• $r = 0.02$
• $\sigma = 0.05$
• $T = 5$

$d_1(s_1) = \frac{\ln\left(\frac{101}{100}\right) + \left(0.02 + \frac{(0.05)^2}{2}\right) \times 5}{0.05 \sqrt{5}}$

Calculating each component:

1. $\ln\left(\frac{101}{100}\right) \approx 0.00995033$
2. $\left(0.02 + \frac{0.0025}{2}\right) \times 5 = (0.02 + 0.00125) \times 5 = 0.02125 \times 5 = 0.10625$
3. $0.05 \sqrt{5} \approx 0.1118$

Now substitute into the formula:

$d_1(s_1) = \frac{0.00995033 + 0.10625}{0.1118} \approx \frac{0.11620033}{0.1118} \approx 1.039$

Step 2: Calculate $d_1$ for $s_2$

Given:

• $S = 103$
• $K = 100$
• $r = 0.02$
• $\sigma = 0.045$

$d_1(s_2) = \frac{\ln\left(\frac{103}{100}\right) + \left(0.02 + \frac{(0.045)^2}{2}\right) \times 5}{0.045 \sqrt{5}}$

Calculating each component:

1. $\ln\left(\frac{103}{100}\right) \approx 0.0295588$
2. $\left(0.02 + \frac{0.002025}{2}\right) \times 5 = (0.02 + 0.0010125) \times 5 = 0.0210125 \times 5 = 0.1050625$
3. $0.045 \sqrt{5} \approx 0.1006$

Now substitute into the formula:

$d_1(s_2) = \frac{0.0295588 + 0.1050625}{0.1006} \approx \frac{0.1346213}{0.1006} \approx 1.339$

Step 3: Difference Between $d_1$

$d_1(s_2) - d_1(s_1) \approx 1.339 - 1.039 = 0.300$

Impact on Delta of a Put Option

The delta of a put option is given by:

$\Delta = N(d_1) - 1$

Where $N(d_1)$ is the cumulative distribution function of the standard normal distribution.

Calculate Delta for Both Options

• For $d_1(s_1) = 1.039$:

  • $N(1.039)$ (approximately 0.8508)
  • $\Delta(s_1) = 0.8508 - 1 \approx -0.1492$

• For $d_1(s_2) = 1.339$:

  • $N(1.339)$ (approximately 0.9115)
  • $\Delta(s_2) = 0.9115 - 1 \approx -0.0885$

Summary of Results

• $d_1(s_1) \approx 1.039$
• $d_1(s_2) \approx 1.339$
• Difference in $d_1$: $0.300$
• Delta of put option $s_1 \approx -0.1492$
• Delta of put option $s_2 \approx -0.0885$

This means that the put option for $s_2$ is less sensitive to changes in the stock price than the put option for $s_1$, due to its higher $d_1$ value.

Would you like to explore further details or have any questions? Here are some related topics to consider:

1. How do changes in volatility impact option pricing?
2. What is the significance of delta in options trading?
3. How does the time to maturity affect options pricing?
4. Can you explain the differences between call and put options in more detail?
5. What are the implications of using different distributions for modeling stock prices?

Tip: Always consider using simulations or historical data to validate the assumptions made in pricing models.