What I find fascinating about this equation is that is in the form of the diffusion (heat) equation. The fundamental solution to the diffusion equation is the G

What I find fascinating about this equation is that is in the form of the diffusion (heat) equation. The fundamental solution to the diffusion equation is the Gaussian. So you can think of solving the equation as applying a Gaussian blur to the boundary conditions (given by the terminal price of the option, which turns into an initial condition after a change of variables). The implication is that the further out the option, the blurrier the picture gets, because we're applying more Gaussian blur. One thing that Veritasium glosses over is that BS assumes that the stock price is lognormal distributed, so it follows a geometric brownian process instead of just a regular one. That's why the drift term on a stock price is (r - sigma^2/2) instead of just r in terms of geometric returns. Volatility lowers compounding returns. This is called volatility drag [0]. [0] https://www.kitces.com/blog/volatility-drag-variance-drain-mean-arithmetic-vs-geometric-average-investment-returns/ https://www.kitces.com/blog/volatility-drag-variance-drain-m...