Black-Scholes

Why This Matters For vanilla options, the simple models are usually sufficient. A plain call or put can be priced off Black-Scholes directly; you often do not need to reach for local volatility or a stochastic volatility model. Those heavier models earn their place with exotics, where the payoff depends on how the smile behaves rather than just its level today. For a vanilla, you take the market’s implied volatility at the relevant strike and maturity and feed it into Black-Scholes. But that assumes a volatility surface already exists: before Black-Scholes can price anything, the surface it reads from has to be built, and building it is less straightforward than it appears. ...

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. ...

Why This Matters A derivative price can be computed two equivalent ways: as a risk-neutral expectation, or as the solution of a PDE. This is the Feynman-Kac result, which I explored in the earlier article. Monte Carlo is the natural way to handle the expectation, and the previous article worked through techniques for making it more efficient. Here I want to look at the other side, where the price is the solution of a PDE and we solve it on a grid. ...

Why This Matters Many of the world’s most actively traded commodities are priced in USD, yet end investors and corporates often operate in other currencies. A Canadian oil producer hedging output, a European airline managing jet fuel costs, or an Asian sovereign wealth fund allocating to commodity exposure all face the same underlying issue: commodity risk does not exist in isolation from FX risk. The standard approach is to hedge the commodity leg with USD-denominated futures or swaps and manage FX separately through forwards or options. This works, but it treats the two risks as independent. Quanto and compo options take a different approach by packaging both risks into a single instrument, but the way each handles FX risk creates some pricing and hedging subtleties that I find are easy to miss. ...