Black-Scholes Option Pricing with Greeks Analysis

Overview

This Jupyter notebook provides an advanced implementation of Black-Scholes-Merton option pricing with comprehensive Option Greeks analysis, featuring multi-dimensional sensitivity analysis, interactive visualizations, Monte Carlo validation, and real market data integration. The notebook demonstrates both theoretical foundations and practical applications for options trading and risk management.

Objectives

Implement analytical Black-Scholes-Merton pricing formulas for European options with dividend yield support
Calculate and visualize all five primary Option Greeks (Delta, Gamma, Vega, Theta, Rho)
Perform multi-dimensional sensitivity analysis across spot price, volatility, and time parameters
Compare Monte Carlo simulations with analytical solutions for validation
Analyze real market data from Yahoo Finance (Boeing stock example)
Demonstrate portfolio-level Greeks aggregation for risk management

Components

1. Black-Scholes-Merton Pricing Implementation

Core Functions:

black_scholes_call() - European call option pricing
black_scholes_put() - European put option pricing
Includes dividend yield (q) parameter for comprehensive modeling
Edge case handling (at-maturity scenarios)

Formulas Implemented:

Call Option: $C = Se^{-q\tau}N(d_1) - Ke^{-r\tau}N(d_2)$

Put Option: $P = Ke^{-r\tau}N(-d_2) - Se^{-q\tau}N(-d_1)$

Where: $d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)\tau}{\sigma\sqrt{\tau}}$ $d_2 = d_1 - \sigma\sqrt{\tau}$

2. Option Greeks - Complete Suite

Greeks Calculation Function

black_scholes_greeks() computes all five primary Greeks:

Greek	Description	Formula Representation	Units
Delta (Δ)	Sensitivity to spot price	$\frac{\partial V}{\partial S}$	Change per $1 move
Gamma (Γ)	Rate of change of Delta	$\frac{\partial^2 V}{\partial S^2}$	Delta change per $1
Vega (ν)	Sensitivity to volatility	$\frac{\partial V}{\partial \sigma}$	Change per 1% vol
Theta (Θ)	Time decay	$\frac{\partial V}{\partial t}$	Change per day
Rho (ρ)	Interest rate sensitivity	$\frac{\partial V}{\partial r}$	Change per 1% rate

Key Formulas:

Delta (Call): $\Delta_c = e^{-q\tau}N(d_1)$
Delta (Put): $\Delta_p = e^{-q\tau}(N(d_1) - 1)$
Gamma: $\Gamma = \frac{e^{-q\tau}n(d_1)}{S\sigma\sqrt{\tau}}$
Vega: $\nu = Se^{-q\tau}\sqrt{\tau}n(d_1) / 100$
Theta (Call): $\Theta_c = -\frac{1}{365}\left[\frac{S\sigma e^{-q\tau}n(d_1)}{2\sqrt{\tau}} + rKe^{-r\tau}N(d_2) - qSe^{-q\tau}N(d_1)\right]$

3. Example Calculation - Boeing (BA) Stock

Test Parameters:

Spot Price (S): $214 (Boeing as of Dec 19, 2025)
Strike Price (K): $225
Volatility (σ): 28%
Risk-Free Rate (r): 3.6%
Dividend Yield (q): 0%
Time to Maturity: 33 days (0.0904 years)

Visualization Suite

1. Greeks vs Spot Price (5-Panel Line Charts)

Comprehensive 1D sensitivity analysis:

Panel 1: Delta for calls and puts
Panel 2: Gamma (maximum at ATM)
Panel 3: Vega (sensitivity to volatility changes)
Panel 4: Theta (time decay patterns)
Panel 5: Rho (interest rate sensitivity)

Features:

Spot price range: $50 to $350
Strike reference line at K=$225
Clear visualization of moneyness effects (ITM/ATM/OTM)
Delta: 0 to 1 for calls, -1 to 0 for puts
Gamma: peaked at ATM, demonstrates curvature risk
Vega: maximum for ATM options

2. Greeks Heatmaps (2D Contour Plots)

4-panel heatmap showing sensitivity to TWO variables simultaneously:

Axes:

X-axis: Spot Price ($80 to $380)
Y-axis: Volatility (10% to 50%)
Color intensity: Greek value

Panels:

Delta (Call): Shows progression from 0 (deep OTM) to 1 (deep ITM)
Gamma: Concentrated band around strike price, highest for ATM
Vega: Maximum in ATM region, demonstrates vol risk concentration
Theta: Time decay patterns across strike and volatility space

Color Schemes:

Delta: Red-Yellow-Green gradient
Gamma: Viridis (blue to yellow)
Vega: Plasma (purple to yellow)
Theta: Coolwarm (red to blue)

3. 3D Surface Plots

3D surfaces showing Greeks evolution:

Two primary surfaces generated:

Delta Surface: Shows smooth transition from 0 to 1
- X-axis: Spot Price ($90-$360)
- Y-axis: Time to Maturity (1-365 days)
- Z-axis: Delta value
Gamma Surface: Shows concentration around strike
- Peaks at ATM for shorter maturities
- Flattens as time to expiry increases

Viewing Parameters:

Elevation: 25°
Azimuth: 45°
Color mapping: Viridis
Grid resolution: 30×30

Advanced Analysis Features

4. Greeks Dashboard - Multiple Strikes

Tabular analysis across strike prices:

Strikes Analyzed: [170, 195, 205, 215, 225, 255, 280, 295]

Columns Provided:

Strike price
Moneyness (S/K ratio)
Status (ITM/ATM/OTM)
Call & Put prices
All Greeks for both calls and puts

5. Monte Carlo Validation

Purpose: Verify analytical formulas through simulation

Configuration:

Simulation Method: Geometric Brownian Motion (GBM)
Number of Simulations: 100,000 paths
Random Seed: 3407 (reproducibility)

GBM Formula: $S_T = S_0 \exp\left[(r - q - \frac{\sigma^2}{2})T + \sigma\sqrt{T}Z\right]$

Where Z ~ N(0,1)

6. Real Market Data Integration

Data Source: Yahoo Finance (yfinance library)

Ticker Example: Boeing (BA)

Workflow:

Fetch Current Price: Real-time spot price retrieval
Get Options Chain: All available expiration dates
Extract Market Data: Strikes, last prices, implied volatilities
Calculate Greeks: Using market-implied volatilities
Compare Theoretical vs Market Prices

Risk-Free Rate Curve (as of Dec 2025):

Uses U.S. Treasury yield curve
Interpolates based on time to maturity
Range: 30 days (3.62%) to 30 years (4.83%)

Observations:

Implied volatility smile visible (higher vol for OTM options)
Pricing discrepancies due to bid-ask spreads and liquidity
Greeks provide risk metrics for each strike

Portfolio Risk Management

7. Portfolio Greeks Aggregation

Function: calculate_portfolio_greeks()

Purpose: Calculate net Greeks exposure for multi-position portfolios

Example Portfolio:

Position 1:  +10 x CALL K=$200, σ=28%, T=182 days
Position 2:   -5 x CALL K=$220, σ=28%, T=182 days  # Short call spread
Position 3:   +5 x PUT  K=$180, σ=28%, T=182 days

Aggregate Greeks Output:

``` Greek Value Interpretation ————— ————— ————————————————– Delta 3.7090 Portfolio moves $3.71 for $1 move in underlying Gamma 0.061512 Delta changes by 0.061512 for $1 move Vega 3.9330 Portfolio moves $3.93 for 1% vol change Theta -0.3609 Portfolio loses $0.36 per day (time decay) Rho 2.9519 Portfolio moves $2.95 for 1% rate change ``

Greek	Description	Formula Representation	Units
Delta (Δ)	Sensitivity to spot price	\(\frac{\partial V}{\partial S}\)	Change per $1 move
Gamma (Γ)	Rate of change of Delta	\(\frac{\partial^2 V}{\partial S^2}\)	Delta change per $1
Vega (ν)	Sensitivity to volatility	\(\frac{\partial V}{\partial \sigma}\)	Change per 1% vol
Theta (Θ)	Time decay	\(\frac{\partial V}{\partial t}\)	Change per day
Rho (ρ)	Interest rate sensitivity	\(\frac{\partial V}{\partial r}\)	Change per 1% rate