Quant Fundamentals

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. ...

Why This Matters While practitioners price American puts correctly in production systems, the deeper question of why early exercise is sometimes optimal, and the precise conditions under which it occurs, is less often articulated rigorously. This article works through the argument, starting with why early exercise is never optimal for calls without dividend, and then showing, using the Black–Scholes PDE, when and why it becomes mandatory for puts. For those working with options pricing, hedging, or products with embedded American optionality, a rigorous understanding of the early exercise boundary can offer useful intuition beyond what standard pricing tools provide. ...

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. ...