Options Analytics Engine

Black-Scholes Pricer & Implied Volatility Solver

A Python-based derivatives analytics library that implements the Black-Scholes-Merton model for European options pricing. It features a custom Newton-Raphson solver for calculating Implied Volatility and includes a visualization engine for Option Greeks (Delta, Gamma, Vega).

Project Architecture

options-analytics/
|-- black_scholes.py      # Core Pricing Engine (Price + Greeks).
|-- implied_volatility.py # Numerical Solver (Newton-Raphson) for IV.
|-- plot_greeks.py        # Visualization script for sensitivity analysis.
|-- requirements.txt      # Dependencies.
|-- README.md             # Documentation.

Features

1. Pricing Engine: Calculates fair value for Call and Put options.
2. Risk Metrics (The Greeks):

• Delta ($\Delta$): Sensitivity to underlying price.
• Gamma ($\Gamma$): Sensitivity to Delta (Convexity).
• Vega ($\nu$): Sensitivity to Volatility.

3. Implied Volatility Solver: Reverse-engineers market fear ($\sigma$) from option prices using numerical optimization.

The Math

The engine implements the closed-form Black-Scholes solution:

$$C = S N(d_1) - K e^{-rT} N(d_2)$$ $$P = K e^{-rT} N(-d_2) - S N(-d_1)$$ Where: $$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$

Implied Volatility (Newton-Raphson) Since $\sigma$ cannot be isolated algebraically, we solve for the root of the difference between Market Price and Model Price using the Newton-Raphson iteration, using Vega as the derivative:

$$\sigma_{new} = \sigma_{old} - \frac{Price_{BS}(\sigma) - Price_{Market}}{Vega(\sigma)}$$

Visualizations: The Physics of Options

The engine plots the Greeks to visualize risk exposure across different strike prices.

• Red Line: Current Strike Price ($100).
• Green Curve (Gamma): Shows that risk/acceleration is highest At-The-Money.

How to Run1.

1. Install Dependencies

   pip install -r requirements.txt
   

2. Run the Visualization

   python plot_greeks.py
   

3. Calculate Implied Volatility Check implied_volatility.py to see the solver in action:

   # Example Usage
   iv = implied_volatility(S=100, K=100, T=1, r=0.05, price_market=10.45)
   print(f"Implied Volatility: {iv}")