Articles

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 you receive the payoff at expiry, or when you choose to exercise for American options. That framing was so ingrained that I assumed it was the general case. When I started working on commodity derivatives, I encountered a different world. Many options are traded under futures-style margining. No premium changes hands at inception; instead, the option is margined daily like a futures contract. The convention tends to split along venue lines rather than by underlying. US exchanges are predominantly equity-style: options on WTI crude futures at the CME and options on corn and wheat futures at the CBOT all require an upfront premium. European venues lean the other way. Options on ICE Brent futures and options on EUA carbon futures at ICE Endex and EEX are both margined futures-style. ...

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 became 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 plan to write about calibrating a full volatility surface, and that work rests on 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 quote, we look for the volatility at which the model price matches the market price. A common algorithm for this is Newton’s method. I learned it in class, saw it in textbooks, and implemented it in the traders’ tools at work. Most of the time it just works. I knew convergence wasn’t guaranteed, but that stayed abstract until I hit an edge case. ...

Why This Matters One of the first results I learned in my derivatives pricing course is that an American call on a non-dividend-paying stock is worth the same as its European counterpart, while an American put can be worth more. The result is easy to remember, but what actually produces the asymmetry did not register with me at the time, and that is what I wanted to pin down in this article. ...

Why This Matters Brownian motion, the mathematical model underlying everything from stock prices to heat diffusion, has one of its most elegant properties: the variance of its position at time $t$ grows linearly with time. Not $t^2$, not $\sqrt{t}$, but exactly $t$. This seemingly abstract fact has a concrete consequence in financial markets: under the idealised conditions of an at-the-money option with zero rates, it is precisely why option prices scale with $\sqrt{T}$ rather than $T$, a direct fingerprint of Brownian motion inside Black-Scholes. Understanding why requires looking at both physical observations and the mathematical construction of Brownian motion. ...