the second-most important equation

The Second-Most Important Equation

Posted At : February 4, 2007 8:10 PM | Posted By : Emanuel Derman
Related Categories: Models

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*sigma^2 S^2(realized vol^2 - implied vol^2).

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.