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.

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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