# Option Pricing and Volatility Modeling with Python: Implementing Black-Scholes and GARCH Models

Option Pricing is a common problem in Finance that requires pricing financial options. Volatility Modeling is another important concept that focuses on predicting the magnitude of price changes in the stock market. Python is a powerful tool that can be used to model these concepts.

In this tutorial, we will explore how to implement two popular models for Option Pricing and Volatility Modeling in Python: the Black-Scholes Model and the GARCH Model. You will learn how to apply the Black-Scholes model to value European call and put options. The GARCH Model will be used to estimate volatility for stock prices.

To follow along with this tutorial, you should have a basic understanding of Python and finance concepts.

## Understanding the Black-Scholes Model

The Black-Scholes Model is a popular method for pricing European-style options. It is used to determine a fair price for a call or put option based on several factors, including the strike price, time to expiration, underlying asset price, risk-free rate, and expected volatility. The Black-Scholes formula is a partial differential equation where the option price is the unknown function. This formula can be solved using numerical techniques to obtain the option value.

To implement the Black-Scholes Model in Python, we need to import the necessary libraries. We will be using the `numpy`

library for mathematical operations and the `matplotlib`

library for creating plots.

`import numpy as np`

import matplotlib.pyplot as plt

Next, we need to define the Black-Scholes formula in Python.

`def black_scholes_call(S, K, T, r, sigma):`

d1 = (np.log(S/K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))

d2 = d1 - sigma * np.sqrt(T)

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

return C

def black_scholes_put(S, K, T, r, sigma):

d1 = (np.log(S / K) + (r + sigma ** 2 / 2) * T) / (sigma * np.sqrt(T))

d2 = d1 - (sigma * np.sqrt(T))

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

return P