Black-Scholes Equation

The Black-Scholes equation is a partial differential equation that describes the price evolution of a European option over time: \[ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0 \] where \( V \) is the option price, \( t \) is time, \( S \) is the underlying asset price, \( \sigma \) is volatility, and \( r \) is the risk-free interest rate.