5. Two ways of understanding the market: mathematical models and statistical models

Level 1: Foundations of Algorithmic Trading

When it comes to building a financial model there are, fundamentally, two different schools of thought about how the market behaves, and both are worth knowing because most real strategies are a blend of the two.

The first is mathematical modeling. It starts from the idea that the market is fundamentally random, and that the future movement of price cannot be predicted with certainty from past data. Instead of trying to predict, this school builds models that describe how price evolves as if it were a random process — the best-known example being the Black-Scholes model for pricing options. Here the goal is not to guess where the price is going, but to manage risk and put a fair price on derivative products, assuming the market already reflects all available information.

A widely used example within this school is Geometric Brownian Motion, a model that describes the evolution of an asset's price as the sum of two forces: an expected trend, called drift, and a purely random component proportional to the asset's volatility. In practice, it is used to simulate thousands of possible futures for a single asset — not to know which one will happen, but to get a sense of the range of reasonable outcomes and of tail risk, that is, how badly things could realistically go. You do not need to master the exact mathematical formula to benefit from the idea: what matters is understanding that price is modeled as a mixture of trend plus random noise, and that the more volatile an asset is, the wider the fan of possible futures becomes.

The second school is statistical, or data-driven, modeling. Its premise is different: even if the market is broadly efficient, it is not perfectly so, and exploitable patterns may exist in historical data, however small and short-lived. This school gives rise to regression models, time series analysis and, increasingly, machine learning — techniques capable of detecting non-linear relationships between variables that a classical mathematical model does not capture.

Neither school is 'the correct one'. Mathematical modeling is stronger at managing risk and valuing derivatives; statistical modeling is stronger at hunting for exploitable signals in the data. What they do share is something worth never forgetting: every model rests on assumptions, and those assumptions — that volatility is constant, that errors are normally distributed, that the relationships between variables do not change over time — almost never hold entirely in reality. A good quant is not the one with the most sophisticated model, it is the one who knows exactly which assumptions their model is breaking and how much that could cost.