Local Volatility: From the Implied Vol Surface to Risk-Neutral Dynamics
Why This Matters In an earlier article we constructed the implied volatility surface and used it primarily to price vanilla options. But a great deal of what trades is not vanilla. Products like barriers and autocallables depend on the path the underlying takes, not only where it lands. Suppose I price a barrier by Monte Carlo. At each step the spot sits at some level, and I need a volatility to advance it. What vol do I use? The surface gives me a vol for every strike, but simulation does not ask about strikes. It asks what volatility the spot experiences at this level, at this moment, which the surface cannot answer. ...