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.