The key result of options pricing is that if you hedge at implied volatility, then over the next instant dt a long options position produces
dP&L = (1/2)Gamma*S^2(realized vol^2 – implied vol^2) dt
Of course this is in theory, assuming Wiener process for the innovations, etc.
I learned this years ago, but not right when I learned options theory. It was several years later. I think the equation is almost more general than Black-Scholes — it tells you how to benefit from curvature. You can understand the equation even if you don’t understand PDEs.
I’ve been doing a lot of teaching lately, both to practitioners and students, and many students and many many professors know all about Black-Scholes and how to derive it and how to solve it, but many many don’t recognize this equation or the information embodied in it.
Paul and Reza Ahmed have a beautiful paper on this in Wilmott.