Quant Fundamentals

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 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 In my earlier article on Brownian motion, I worked through the forward view: a process starting at a known value, diffusing into an uncertain future. Sometimes we know more than just the starting point. We also know where the process ended up, and we want to characterise the path in between. The object that answers this is the Brownian bridge: a Brownian motion conditioned on its terminal value. ...

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 In the article on Girsanov’s Theorem, we studied how the real-world measure $\mathbb{P}$ and the risk-neutral measure $\mathbb{Q}$ relate, and showed that switching between them amounts to reweighting paths via the Girsanov exponential. Throughout, the risk-free bond was the numéraire: the asset against which all prices were expressed. But this is a convenient choice, not a fundamental one. Any strictly positive self-financing wealth process can serve as a numéraire, and each choice gives a different probability measure under which asset prices, expressed in units of that numéraire, become martingales. The price of a derivative is invariant; what changes is how the problem is represented. So instead of viewing pricing as a fixed-measure expectation problem, it is often more natural to think of it as choosing the numéraire that best matches the structure of the payoff. ...

Why This Matters We want to price a derivative. Under the real world measure $\mathbb{P}$, we face two problems. First, we do not know the true drift $\mu$ of the underlying, and historical estimates are notoriously unreliable. Second, even if we knew $\mu$, taking the expected payoff under $\mathbb{P}$ would still not give the market price. Risky cash flows must be discounted more heavily than guaranteed ones because investors are risk averse. Pricing under $\mathbb{P}$ requires both the true probabilities of outcomes and a model for how the market prices risk. Both are fundamentally unobservable. So what can we do? ...

Why This Matters The first time I encountered the Feynman-Kac theorem, I found it fascinating but unintuitive. The theorem claims that a deterministic PDE and the expectation of a stochastic process are two representations of the same object. A PDE is smooth and deterministic. A stochastic expectation involves randomness, probability measures, and averaging over infinitely many paths. How could these be the same thing? I understood the steps of the proof, but I still didn’t have a clear intuition for why this equivalence should exist. ...

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. ...

Why This Matters Most of my early intuition about options came from the Black-Scholes model, which is clean and widely used. But once I started working with real option data, it becomes clear that the Black-Scholes assumption of a lognormal distribution is too restrictive. For a given maturity, the implied volatility is not constant across strikes, and its shape suggests asymmetry and heavy tails in the risk-neutral distribution. That leads to a more basic question. Instead of imposing a parametric model and calibrating its parameters, is there a way to extract volatility, skewness, and kurtosis directly from option prices in a model-free way? This is where the Bakshi, Kapadia, and Madan (2003) framework becomes useful. Their key idea is that smooth payoff functions can be represented as a continuum of vanilla options across strikes. In this view, volatility, skewness, and kurtosis are not model assumptions or calibration outputs. They are quantities that can be recovered from market prices through static option replication. ...

Why This Matters I started writing about calibrating a full volatility surface and realised it first requires a clear understanding of a simpler problem: solving for implied vol from a single option price. At its core, this is a root-finding problem: given a market price, we need to find the volatility that makes the model match that price. Once framed this way, the question becomes how to solve this nonlinear problem efficiently and reliably. ...