Option Pricing Project

A quantitative finance project implementing option pricing models, sensitivity analysis (Greeks), Monte Carlo simulation, and 3D visualizations.

This project builds an end-to-end understanding of option pricing, from analytical solutions (Black-Scholes) to numerical methods and multi-dimensional visual analysis.

Full Report

For a detailed explanation of the methodology, mathematical derivations, and experimental observations, please refer to the full report:

Black-Scholes Option Pricing and Greeks Surface Analysis

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Project Overview

This project explores:

• Analytical pricing using the Black-Scholes model
• Sensitivity analysis using option Greeks
• Time decay and volatility effects
• Monte Carlo simulation for numerical pricing
• 2D and 3D visualizations of option behavior

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Sample Visualizations

Delta Surface

Call Price Surface

Monte Carlo Convergence

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How to Run

Install dependencies:

pip install -r requirements.txt

Run any module:

python main/main_log1.py

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Key Components

1. Black-Scholes Pricing

• Implemented closed-form solutions for:
  • European Call Options
  • European Put Options

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2. Sensitivity Analysis (Greeks)

• Delta

  • Measures sensitivity of option price to stock price
  • Interpreted as approximate probability of finishing in-the-money
  • Used in delta hedging and risk management

• Gamma

  • Measures rate of change of Delta
  • Highlights curvature and sensitivity near the strike price

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3. Time Decay (Theta Intuition)

• Demonstrates how option prices converge to payoff as expiration approaches
• Shows that time value diminishes as T → 0

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4. Volatility Analysis

• Explores how option prices increase with volatility
• Demonstrates that both calls and puts benefit from higher uncertainty

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5. Monte Carlo Simulation

• Simulates stock price paths using Geometric Brownian Motion
• Estimates option price via expected payoff
• Validates convergence to Black-Scholes analytical solution

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6. 3D Visualizations

• Call Price Surface
  C = f(S, σ)

• Delta Surface
  Δ = f(S, T)

These visualizations provide a geometric understanding of option pricing and sensitivities.

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Key Insights (Financial Interpretation)

• Option prices are nonlinear functions of their inputs
• Volatility increases option value due to asymmetric payoff structure
• Delta represents sensitivity and approximates probability of finishing in-the-money
• Gamma captures curvature and is highest near the strike
• Monte Carlo simulations converge to analytical solutions as sample size increases

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Project Structure

option-pricing-project/
│
├── src/
│   ├── black_scholes.py
│   ├── greeks.py
│   └── utils.py
│
├── main/
│   ├── main_log1.py
│   ├── main_log2.py
│   ├── main_log3.py
│   ├── main_log4.py
│   ├── main_log5.py
│   ├── main_log6.py
│   ├── main_log7.py
│   ├── main_log8.py
│   ├── main_log9.py
│   └── main_log10_delta_surface.py
│
├── figures/
│   ├── *.png
│
├── README.md
└── requirements.txt

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Tech Stack

• Python
• NumPy
• Matplotlib

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Future Work

This project serves as a foundation for a larger quantitative system, which will include:

• Time series modeling (ARIMA / GARCH)
• Technical indicators (RSI, ADX, Bollinger Bands)
• Signal generation engine
• Integration of a GenAI-based explanation layer

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Disclaimer

This project is for educational purposes only and does not constitute financial advice.

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Summary

This project builds a complete understanding of option pricing from first principles, combining analytical models, simulation techniques, and visual exploration.

It demonstrates how mathematical finance concepts can be translated into practical, interpretable implementations.