Options

Why This Matters In the article on the Feynman-Kac theorem, we saw that the price of a derivative can be expressed equivalently as the solution to a deterministic PDE or as the expectation of a discounted payoff under the risk-neutral measure. This gives us two complementary numerical approaches to pricing. For low-dimensional problems with smooth payoffs, finite difference methods on the PDE side are efficient and accurate. For high-dimensional problems, path-dependent payoffs, or models where the PDE is hard to derive, Monte Carlo (MC) on the expectation side becomes the natural choice. ...

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. ...

Why This Matters When I first studied options, most textbook examples were equity-style: you pay a premium upfront, and at expiry (or whenever you choose to exercise, if the option is American), you receive the payoff. That framing stuck with me for a long time. When I started working on commodity derivatives, I encountered a different world. Many options in commodity markets are traded under futures-style margining. No premium changes hands at inception, and instead the option is margined daily like a futures contract. This is common across a wide range of exchange-traded products: options on WTI crude oil futures at the CME, options on Henry Hub natural gas futures, options on corn and wheat futures, and options on carbon emissions futures, to name a few. ...