Dupire

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. ...

Why This Matters Most practitioners have seen the Black-Scholes PDE. It is closely related to the backward Kolmogorov PDE: fix a payoff at maturity, and the equation propagates its value back to today. There is also a forward equation, which may be less familiar. The backward equation has current spot and current time as its variables and takes the payoff as a terminal condition. The forward one instead propagates a probability density forward from today, with the terminal value of the underlying and the maturity as its variables. This is Fokker-Planck. ...