## Step-by-Step Guide: Implementing the Black-Scholes Model in Python

In the world of finance, the Black-Scholes-Merton model stands out as a pivotal tool for pricing options. Developed through rigorous mathematical derivations, this model calculates the theoretical value of an option based on five essential parameters:

**Underlying Price (S)**: The current market price of the asset.**Strike Price (K)**: The predetermined price at which the option can be exercised.**Time to Expiration (T)**: The time left (in years) until the option’s expiration date.**Risk Free Rate (r)**: The constant rate of return on a risk-free asset, such as a government bond.**Volatility (σ)**: A measure of how much the price of the underlying asset fluctuates.

With these components in mind, let’s dive into the implementation in Python:

**Step 1: Import Necessary Libraries**

Before we begin our calculations, it’s crucial to have the right tools at our disposal. Thus, we’ll import the required Python libraries:

` ````
```import math
from scipy.stats import norm

**Step 2: Define the Variables**

Here, we’ll establish the variables based on the parameters mentioned earlier:

` ````
```S = 45 # Underlying Price
K = 40 # Strike Price
T = 2 # Time to Expiration
r = 0.1 # Risk-Free Rate
vol = 0.1 # Volatility (σ)

**Step 3: Calculate d1**

With our variables set, we first compute d1, a crucial intermediary value:

` ````
```d1 = (math.log(S/K) + (r + 0.5 * vol**2)*T ) / (vol * math.sqrt(T))

**Step 4: Calculate d2**

Subsequently, we derive d2:

Here’s the code:

` ````
```d2 = d1 - (vol * math.sqrt(T))

**Step 5: Calculate Call Option Price**

To determine the theoretical price of a call option:

` ````
```C = S * norm.cdf(d1) - K * math.exp(-r * T) * norm.cdf(d2)

**Step 6: Calculate Put Option Price**

For the put option:

` ````
```P = K * math.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)

**Step 7: Print the Results**

Lastly, let’s display our results:

` ````
```print('The value of d1 is: ', round(d1, 4))
print('The value of d2 is: ', round(d2, 4))
print('The price of the call option is: $', round(C, 2))
print('The price of the put option is: $', round(P, 2))

In conclusion, the Black-Scholes-Merton model, while built on complex mathematics, can be easily implemented in Python. With just a few lines of code, you can determine the theoretical price of both call and put options.