Project Solution: Derivative & Option Pricing Python Script | Black-Scholes & Monte Carlo

Unlock the world of quantitative finance with this all-in-one Python script!

Stop struggling with complex formulas. This ready-to-use script is your personal toolkit for pricing European, American, and even exotic Barrier options. It includes both the classic Black-Scholes model and advanced Monte Carlo simulations. Perfect for students, aspiring quants, and professionals looking to sharpen their skills.

Download now and turn theory into practice!

Bonus:

Additional features and visualizations: 

1. Greeks sensitivity analysis and visualization 
2. Implied volatility calculations 
3. Portfolio risk metrics (VaR, CVaR) 
4. Dynamic delta hedging simulation 
5. Additional exotic options (Asian, Lookback) 
6. Volatility smile visualization 
7. Early exercise boundary for American options 
8. Enhanced performance metrics and backtesting

For theoretical Foundation of the concepts visit SimplifiedZone

Introduction & Motivation

This project builds upon the foundational concepts and challenges you to apply them to price and hedge various financial options. In the world of finance, accurately pricing derivatives and managing their associated risks are critical skills for portfolio managers, traders, and financial analysts. This project will provide you with hands-on experience in implementing and comparing two of the most fundamental option pricing models: the Black-Scholes model and the Monte Carlo simulation.

You will step into the role of a quantitative analyst tasked with valuing European and American options. You will explore how factors like volatility and "moneyness" affect option prices and their sensitivities (the "Greeks"). Furthermore, you will investigate hedging strategies to neutralize the risk of an options portfolio. This project is designed to not only test your programming skills but also to deepen your understanding of the theoretical underpinnings of derivative pricing in a practical, real-world context.

Learning Objectives

By successfully completing this project, you will be able to:

• Implement the Black-Scholes closed-form solution to price European call and put options.
• Develop a Monte Carlo simulation model to price European and American options under a Geometric Brownian Motion (GBM) framework.
• Calculate and interpret key option sensitivities ("Greeks"), including Delta, Gamma, Vega, Theta, and Rho.
• Analyze the impact of changing market parameters, such as volatility and moneyness, on option prices.
• Verify and discuss the Put-Call parity relationship for European options.
• Construct and evaluate a delta-hedged portfolio.
• Price exotic barrier options (Up-and-Out and Up-and-In) using Monte Carlo methods..

Step-by-Step Project Plan

Milestone 1: Data Understanding and Setup

1. Familiarize Yourself with the Parameters: Review the provided market data: initial stock price (), risk-free interest rate (r), volatility (σ), and time to maturity (T).
2. Environment Setup: Ensure your Python environment is ready with all the required libraries installed. Set up your Jupyter Notebook or Python script file.

Milestone 2: European Option Pricing

1. Black-Scholes Implementation: Write a Python function to price a European call and put option using the Black-Scholes formula. Calculate the option's Greeks.
2. Monte Carlo Simulation: Develop a function to price a European call and put option using a Monte Carlo simulation with daily time steps.
3. Analysis and Comparison:
   • Price an at-the-money (ATM) European call and put option using both methods.
   • Compare the prices and Greeks obtained from both models. Do they converge? Discuss why or why not.
   • Verify that the Put-Call Parity holds for the prices you've calculated.

Milestone 3: American Option Pricing

1. Monte Carlo for American Options: Adapt your Monte Carlo simulation to price an American call and put option. Remember to account for the possibility of early exercise.
2. Moneyness Analysis: Using your Monte Carlo simulation, price American call and put options for five different strike prices to represent deep out-of-the-money (OTM), OTM, ATM, in-the-money (ITM), and deep ITM scenarios.
3. Visualization: Create plots showing the relationship between the option price and moneyness for both call and put options and comment on the observed trends.

Milestone 4: Hedging and Barrier Options

1. Delta Hedging:
   • Price a European call with 110% moneyness and a European put with 95% moneyness.
   • Construct a portfolio by (1) buying both the call and the put, and (2) buying the call and selling the put.
   • Calculate the delta of each portfolio and determine the position in the underlying asset required to create a delta-hedged (delta-neutral) portfolio.
2. Barrier Option Pricing:
   • Implement a Monte Carlo simulation to price an Up-and-Out (UAO) barrier call option.
   • Calculate the price of an Up-and-In (UAI) barrier call option with the same parameters.
   • Price a vanilla call option with the same characteristics. Discuss the relationship between the prices of the UAO, UAI, and vanilla options.

Milestone 5: Final Analysis and Reporting

1. Synthesize Findings: Consolidate all your results, code, and visualizations.

Python Script Output Screenshots: