Why This Matters The local vol model we discussed in the previous article assumes the vol is a deterministic function of spot and time, so the vol has no randomness of its own. But the market does not treat vol that way. The VIX is the market’s implied vol for SPX, and options on the VIX are themselves quoted with an implied vol, a vol of vol. The market is pricing vol itself as a random quantity. ...
Why This Matters The previous article on finite difference methods solved the heat equation by brute force: lay down a grid, step it through time, let the solution emerge step by step. It works, and for many pricing problems it is the practical choice. But can we instead solve the equation analytically? For a class of PDEs, the heat equation among them, we can. The idea is to stop viewing a function as a shape over space and instead see it as a combination of frequencies. That change of view is the Fourier transform. In this article, we explore it on the heat equation, where the diffusion structure shows through without the variable coefficients of Black-Scholes to clutter it, and then turn to its application in option pricing, where it works even when the distribution of prices has no closed form. ...