Implementations of Black-Scholes, binomial trees, Monte Carlo simulations, and risk models for option pricing. Includes Greeks analysis and implied volatility surfaces.
This repository contains notebooks that provide implementation of options pricing and risk analysis:
| # | Notebook | Details | Implementation |
|---|---|---|---|
| 1 | Black-Scholes Option Pricing and Monte Carlo | Implements the classic Black-Scholes formula and Monte Carlo simulations for European options. | Documentation , Kaggle Notebook , Github Notebook |
| 2 | Black-Scholes Option Pricing with Comprehensive Greeks Analysis | Calculates and visualizes Delta, Gamma, Theta, Vega, and Rho. | Documentation , Kaggle Notebook , Github Notebook |
| 3 | Black-Scholes Option Pricing - Implied Volatility Surface Analysis | Builds and analyzes IV surfaces using interpolation. | Documentation , Kaggle Notebook , Github Notebook |
| 4 | Option Pricing Binomial Tree | American and European option pricing via Cox-Ross-Rubinstein binomial model. | Documentation , Github Notebook, Kaggle Notebook |
| 5 | Delta Hedging | Delta hedging strategies for European options, with detailed profit & loss (P&L) attribution and hedging error analysis | Documentation , Kaggle Notebook , Github Notebook |
Each notebook combines rigorous mathematical implementation with practical applications, featuring real market data integration and professional-grade visualizations.
File: 01_black_scholes_monte_carlo.ipynb
Purpose: Implement and validate Black-Scholes-Merton analytical pricing formulas using Monte Carlo simulation methods.
Key Features:
- Analytical Black-Scholes pricing for European call/put options
- Monte Carlo simulation using Geometric Brownian Motion (GBM)
- Convergence analysis (100 to 300,000 simulations)
- Real market data comparison (Boeing stock example)
- Terminal price distribution validation
- Sample path visualization (20+ paths)
Highlights:
- Accuracy: <0.5% error with 100,000 Monte Carlo simulations
- Convergence: Demonstrates O(1/√N) theoretical behavior
- Validation: Compares theoretical vs market prices from Yahoo Finance
File: 02_implied_volatility_surface.ipynb
Purpose: Calculate implied volatilities from market prices and generate 2D/3D volatility surfaces with advanced numerical methods.
Key Features:
- Three IV calculation methods: Newton-Raphson, Brent's, Hybrid
- Real-time market data: Yahoo Finance API integration
- Volatility surface generation: 2D scatter plots and 3D interpolated surfaces
- Data quality filtering: Strike range, time-to-expiry, volume, open interest
- Accuracy metrics: MAE, RMSE, MAPE for price and IV errors
- Greeks calculation: Vega for Newton-Raphson convergence
Technical Capabilities:
| Feature | Implementation |
|---|---|
| Pricing Model | Black-Scholes-Merton with continuous dividend yield |
| IV Algorithms | Hybrid Newton-Raphson/Brent's with adaptive bounds |
| Interpolation | Cubic/linear griddata with Gaussian smoothing |
| Success Rate | Typically >90% IV computation success |
| Convergence | Tolerance 1e-8, max 150 iterations |
Volatility Surface Visualizations:
- Computed IV Surface: From calculated implied volatilities
- Market IV Surface: From market-quoted IVs (when available)
- Difference Surface: Identifies pricing discrepancies and arbitrage opportunities
File: 03_comprehensive_greeks_analysis.ipynb
Purpose: Calculate, visualize, and analyze all five primary Option Greeks with multi-dimensional sensitivity analysis and portfolio risk management applications.
Key Features:
- All five Greeks: Delta, Gamma, Vega, Theta, Rho
- Multi-dimensional visualization: 1D line plots, 2D heatmaps, 3D surfaces
- Sensitivity analysis: Greeks vs spot price, volatility, time to maturity
- Monte Carlo validation: Verify analytical Greeks through simulation
- Real market data: Boeing (BA) options analysis
- Portfolio Greeks aggregation: Multi-position risk management
Greeks Suite:
| Greek | Measures | Formula | Practical Use |
|---|---|---|---|
| Delta (Δ) | Price sensitivity to spot | ∂V/∂S | Hedge ratio, directional exposure |
| Gamma (Γ) | Delta's rate of change | ∂²V/∂S² | Convexity risk, hedge rebalancing frequency |
| Vega (ν) | Volatility sensitivity | ∂V/∂σ | Volatility trading, implied vol exposure |
| Theta (Θ) | Time decay | ∂V/∂t | Daily P&L erosion, calendar strategies |
| Rho (ρ) | Interest rate sensitivity | ∂V/∂r | Rate risk, long-dated options |
Visualization Types:
- 5-Panel Line Charts: Greeks vs spot price (200 points)
- 4-Panel Heatmaps: Greeks vs spot & volatility (50×50 grid)
- 3D Surface Plots: Greeks vs spot & time (30×30 grid)
- Greeks Dashboard: Tabular analysis across multiple strikes
Portfolio Risk Management:
Example Portfolio:
├── +10 CALL K=$200, T=182 days
├── -5 CALL K=$220, T=182 days
└── +5 PUT K=$180, T=182 days
Aggregate Greeks:
├── Delta: +4.23 (net long)
├── Gamma: +0.012 (positive convexity)
├── Vega: +2.35 (long volatility)
├── Theta: -0.46 (time decay cost)
└── Rho: +1.23 (benefits from rate increases)- Python 3.8+
- Jupyter Lab or Notebook
- Libraries: See
requirements.txt