Dynamic Hedging of Options, Realized vs. Implied Volatility

A quantitative framework that treats a delta-hedged option as a bet on realized versus implied volatility, where a perfectly hedged long option’s P&L reduces to the gamma-weighted variance identity ½ Σ Γ S² (realized − implied). The project pairs a Black-Scholes-Merton pricing and Greeks engine (with a Newton-Raphson implied-vol solver) with a Monte-Carlo hedging lab that simulates thousands of GBM paths, validates the variance decomposition path-by-path, recovers the √Δt discrete-hedging error scaling law, and optimizes rebalancing against transaction costs. A real SPY call option trade is analyzed minute-by-minute, and an interactive Streamlit dashboard ties the modules together.