Black-Scholes Option Pricing - Implied Volatility Surface Analysis

Overview

This Jupyter notebook implements a comprehensive Black-Scholes-Merton option pricing framework with advanced implied volatility (IV) surface analysis, market data integration, and accuracy validation. The notebook extracts real-time options data, computes implied volatilities using robust numerical methods, and visualizes volatility surfaces in 2D and 3D.

Objectives

Calculate theoretical option prices using Black-Scholes-Merton with dividend yield adjustments
Compute implied volatilities from market prices using three numerical methods
Integrate real market data from Yahoo Finance API for validation
Generate volatility surfaces with 2D/3D visualizations and interpolation
Validate pricing accuracy through comprehensive error metrics

Key Features

Core Functionality

Feature	Description
Black-Scholes Pricing	Call/put pricing with continuous dividend yield support
Implied Volatility Calculation	Three methods: Newton-Raphson, Brent’s, and Hybrid
Market Data Integration	Live options chains from Yahoo Finance (yfinance)
Volatility Surface Generation	2D scatter plots and 3D interpolated surfaces
Greeks Computation	Vega calculation for Newton-Raphson convergence
Data Quality Filtering	Strike range, time-to-expiry, volume, and open interest filters
Advanced Interpolation	Cubic/linear griddata with Gaussian smoothing
Comprehensive Metrics	MAE, RMSE, MAPE for price and IV accuracy

Methodology

Analysis Pipeline

Data Acquisition → 2. Filtering → 3. IV Computation → 4. Pricing → 5. Validation → 6. Visualization

Step 1: Market Data Retrieval

Fetches current spot price and historical data via yfinance
Retrieves options chains for multiple expiration dates
Extracts bid-ask spreads, volume, open interest, and market-quoted IVs

Step 2: Data Quality Filtering

Strike Range: 70-130% of spot price (configurable: 85-115% in example)
Minimum Time to Expiry: 25 days (~0.07 years)
Price Validation: Positive last traded prices only
Optional Thresholds: Volume and open interest filters
Moneyness Calculation: K/S ratio and log-moneyness

Step 3: Implied Volatility Calculation

Three Computational Methods:

Newton-Raphson Method:
- Fast iterative approach using Vega for gradient descent
- Initial guess: Brenner-Subrahmanyam approximation for ATM options
- \[\sigma_{n+1} = \sigma_n - \frac{C_{BS}(\sigma_n) - C_{market}}{Vega(\sigma_n)}\]
- Max 50 iterations, tolerance: 1e-8
- Damping for stability when approaching bounds
Brent’s Method:
- Root-finding algorithm using bracketing
- Robust fallback when Newton-Raphson fails
- Dynamic bound adjustment (0.001 to 10.0)
- Max 150 iterations
Hybrid Method (Default):
- Attempts fast Newton-Raphson first
- Falls back to robust Brent’s method if:
  - Newton fails to converge
  - Result outside reasonable bounds (0.1% - 1000%)
  - Verification fails (re-pricing error too large)
- Success Rate Tracking: Reports % of successful IV calculations

Convergence Parameters:

Tolerance: 1e-8
IV Search Bounds: 0.001 to 10.0 (0.1% to 1000% annualized)
Minimum Volatility: 1e-10 for numerical stability
Initial Guess Clipping: 0.05 to 2.0

Step 4: Theoretical Pricing

Recalculates option prices using computed IVs
Generates price error metrics:
- Absolute error: Theoretical - Market
- Percentage error: (Absolute / Market) × 100
Compares computed IV vs. market-quoted IV (when available)

Step 5: Accuracy Validation

Error Metrics Computed:

Metric	Description	Formula
MAE	Mean Absolute Error	\(\frac{1}{n}\sum\|Theo - Market\|\)
RMSE	Root Mean Squared Error	\(\sqrt{\frac{1}{n}\sum(Theo - Market)^2}\)
MAPE	Mean Absolute % Error	\(\frac{1}{n}\sum\frac{\|Theo - Market\|}{Market} \times 100\)
Median Error	Robust central tendency	Median of absolute errors

Segmented Analysis:

All Options: Complete dataset statistics
ATM Options: Focused analysis within ±12-15% moneyness range
IV Comparison: Computed vs. market-quoted IV errors (when available)

Step 6: Visualization

2D Visualizations:

IV Smile by Strike: Scatter plot showing volatility smile pattern
- Separate markers for calls vs. puts
- Overlay market-quoted IV for comparison
- Spot price reference line
IV by Moneyness: Color-coded by time to expiry
- Shows term structure effects
- ATM reference line at K/S = 1.0

3D Surface Plots:

Computed IV Surface: Interpolated from calculated IVs
- X-axis: Time to expiry
- Y-axis: Moneyness (K/S)
- Z-axis: Implied Volatility
- Grid size: 50×50 with Gaussian smoothing (σ=0.7)
Market IV Surface: Based on market-quoted IVs (when available)
Difference Surface: Computed - Market IV
- Identifies systematic biases
- Highlights potential arbitrage opportunities
- Color scale: Red (overpriced) to Blue (underpriced)

Interpolation Techniques:

Primary: Cubic griddata interpolation
Fallback: Linear interpolation if cubic fails
Smoothing: Gaussian filter with configurable sigma
Outlier Removal: 3-sigma filter before interpolation
Edge Handling: 5% padding around data range

Technical Implementation

Core Functions

Black-Scholes Pricing

black_scholes_price(S, K, r, T, sigma, q=0.0, option_type='call')
black_scholes_vega(S, K, r, T, sigma, q=0.0)

Includes dividend yield adjustment
Handles edge cases (at expiration, zero volatility)
Returns intrinsic value on numerical errors

Implied Volatility Calculation

calculate_iv_newton()     # Newton-Raphson method
calculate_iv_brent()      # Brent's method
calculate_iv()            # Hybrid method (default)

Data Pipeline

get_stock_data()          # Fetch spot price and history
get_options_data()        # Retrieve options chains
filter_options_data()     # Apply quality filters
compute_iv_for_options()  # Batch IV calculation

Analysis & Metrics

calculate_theoretical_prices()  # Reprice with computed IVs
calculate_accuracy_metrics()    # MAE, RMSE, MAPE

Visualization

plot_iv_comparison_2d()   # 2D scatter plots
plot_iv_surfaces_3d()     # 3D surface plots
create_iv_surface_grid()  # Interpolation helper