okay so assuming the stock price could follow two distributions, s1 is N(-1%, 5%) and s2 is (3%, 4.5%). Now at one point in time s1= 101, whilst s2=103, k=100 for both, Everything else is the same, whats the difference between the two d1 and ultimately whats the impact on delta of a put option. R=0.02 T=5

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.

Explore the difference in d1 values for two stock price distributions and the impact on delta for put options. Given parameters include s1 at 101, s2 at 103, K at 100, R at 0.02, and T at 5. Learn key concepts of the Black-Scholes model.

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