diff --git a/requirements.txt b/requirements.txt
index 39a17f0..bc07e75 100644
--- a/requirements.txt
+++ b/requirements.txt
@@ -2,4 +2,3 @@ numpy
 scipy
 matplotlib
 jupyter
-pip install -r requirements.txt
diff --git a/test_imports.py b/test_imports.py
index cfcac6b..8132d25 100644
--- a/test_imports.py
+++ b/test_imports.py
@@ -1,5 +1,11 @@
 # Test whether all dependencies are installed correctly
-import numpy
+import numpy as np
 import scipy
 import matplotlib.pyplot as plt
-print("✅ All core packages imported successfully.")
+
+
+def test_imports():
+    """Ensure required packages import correctly."""
+    assert np.__version__
+    assert scipy.__version__
+    assert plt is not None
diff --git a/utils/bs_functions.py b/utils/bs_functions.py
index 92baa42..270b6da 100644
--- a/utils/bs_functions.py
+++ b/utils/bs_functions.py
@@ -3,13 +3,75 @@
 
 # d1 and d2 calculations
 def d1(S, K, r, T, sigma):
+    """Compute the d1 term in the Black–Scholes formula.
+
+    Parameters
+    ----------
+    S : float
+        Current stock price.
+    K : float
+        Strike price of the option.
+    r : float
+        Risk‑free interest rate.
+    T : float
+        Time to maturity (in years).
+    sigma : float
+        Volatility of the underlying asset.
+
+    Returns
+    -------
+    float
+        The d1 value used in the Black–Scholes formula.
+    """
     return (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
 
+
+
 def d2(S, K, r, T, sigma):
+    """Compute the d2 term in the Black–Scholes formula.
+
+    d2 is derived from d1 by subtracting sigma * sqrt(T).
+    See Black–Scholes equation for details.
+
+    Parameters
+    ----------
+    S, K, r, T, sigma : float
+        Same as for d1().
+
+    Returns
+    -------
+    float
+        The d2 value used in the Black–Scholes formula.
+    """
     return d1(S, K, r, T, sigma) - sigma * np.sqrt(T)
 
-# Black-Scholes Price
+
+
+# Black‑Scholes Price
+
 def black_scholes_price(S, K, r, T, sigma, option_type='call'):
+    """Calculate the price of a European option using the Black–Scholes formula.
+
+    Parameters
+    ----------
+    S : float
+        Current stock price.
+    K : float
+        Strike price of the option.
+    r : float
+        Risk‑free interest rate.
+    T : float
+        Time to maturity (in years).
+    sigma : float
+        Volatility of the underlying asset.
+    option_type : {'call', 'put'}, optional
+        Type of the option to price. Default is 'call'.
+
+    Returns
+    -------
+    float
+        The price of the option.
+    """
     d_1 = d1(S, K, r, T, sigma)
     d_2 = d2(S, K, r, T, sigma)
     if option_type == 'call':
@@ -19,18 +81,56 @@ def black_scholes_price(S, K, r, T, sigma, option_type='call'):
     else:
         raise ValueError("option_type must be 'call' or 'put'")
 
+
+
 # Greeks
+
+
+
 def delta(S, K, r, T, sigma, option_type='call'):
+    """Calculate the Delta of a European option.
+
+    Delta measures the sensitivity of the option price to changes in the underlying asset price.
+
+    Parameters are the same as for black_scholes_price().
+
+    Returns
+    -------
+    float
+        The Delta of the option.
+    """
     d_1 = d1(S, K, r, T, sigma)
     if option_type == 'call':
         return norm.cdf(d_1)
     else:
         return norm.cdf(d_1) - 1
 
+
+
 def gamma(S, K, r, T, sigma):
+    """Calculate the Gamma of a European option.
+
+    Gamma measures the rate of change of Delta with respect to the underlying asset price.
+
+    Returns
+    -------
+    float
+        The Gamma of the option.
+    """
     d_1 = d1(S, K, r, T, sigma)
     return norm.pdf(d_1) / (S * sigma * np.sqrt(T))
 
+
+
 def vega(S, K, r, T, sigma):
+    """Calculate the Vega of a European option.
+
+    Vega measures the sensitivity of the option price to changes in volatility.
+
+    Returns
+    -------
+    float
+        The Vega of the option.
+    """
     d_1 = d1(S, K, r, T, sigma)
     return S * norm.pdf(d_1) * np.sqrt(T)