Quant Trading: The Mathematics Behind Systematic Strategies

The mathematics, not the strategy names

Most quant trading material is a tour: here is momentum, here is mean reversion, here is an equity curve. It is enjoyable and it teaches very little, because the thing that separates someone who can do this work is not knowing what a strategy is called.

It is the probability, time series analysis, stochastic processes and optimisation underneath — and above all the statistical judgement to know when a promising result is an artefact. This course is built on that, implemented in Python.

Where retail strategies quietly fail

Three failures account for most of it, and all three are mathematical.

Distributional assumptions. Market returns have heavier tails than the normal distribution, so a model calibrated on normality understates the frequency of large moves — exactly the moves that end accounts. Non-stationarity. A relationship estimated on one period need not hold in the next, and a statistic computed on a non-stationary series can be meaningless rather than merely imprecise. Costs. Transaction costs, spread and market impact are first-order terms. A strategy with a theoretical edge smaller than its execution cost is a losing strategy, and it looks excellent on paper.

The unit nobody wants and everybody needs

An entire unit goes to backtesting and its statistics, and it is the most valuable thing in the course.

A researcher who tests two hundred strategy variants will find several with excellent Sharpe ratios by chance alone. That is arithmetic, not misfortune. Add look-ahead bias — using information unavailable at the time — and survivorship bias, where the dataset silently excludes companies that failed, and you can manufacture an impressive result from pure noise without any intention to deceive.

So we teach train-test discipline, walk-forward analysis, the deflated Sharpe ratio and honest performance measurement as standard practice rather than as advanced refinements. The hardest skill in this subject is recognising that your own result is not real. Most self-taught practitioners never acquire it.

Time series and stochastic processes, properly

Financial data is a time series and most statistical training assumes independent observations, which is the source of a great deal of confusion.

We cover stationarity and why it is the precondition for most of what follows, autocorrelation and partial autocorrelation, the AR, MA, ARMA and ARIMA family, volatility clustering through ARCH and GARCH, and cointegration — which is the actual mathematical basis of pairs trading, as opposed to the folklore version.

From there into stochastic processes: random walks and martingales, Brownian motion and its quadratic variation, geometric Brownian motion, and Ito's lemma. This is the language option pricing is written in, and a student who has it can read the literature instead of taking its conclusions on trust.

Portfolios as a convex problem

Mean-variance optimisation is a constrained convex optimisation problem, and treating it as one explains its notorious behaviour.

The optimiser is extremely sensitive to the estimated covariance matrix, and covariance estimated from limited data carries large error — so the "optimal" portfolio is frequently an artefact of estimation noise rather than a discovery. We cover why this happens, what shrinkage does about it, and how eigenvalue structure and PCA reveal the factor structure underneath. This is a case where understanding the mathematics changes what you would actually do.

Machine learning, used carefully

Machine learning has specific failure modes on financial data and we teach it with them in plain view.

Ordinary k-fold cross-validation leaks future information when applied to a time series, which by itself invalidates a lot of published work. The signal-to-noise ratio is low and the structure is non-stationary, so flexible models overfit readily and convincingly. Used with time-series-aware validation and genuine regularisation, these methods are useful. Used as a black box on price data, they produce confident nonsense.

What this course will not tell you

It will not promise profitability, and you should treat any course that does with suspicion. Markets are competitive, costs are real, and the majority of strategies that look good in a backtest do not survive execution.

What it gives you is the mathematics the field runs on, implemented in code you wrote and understand, plus the statistical discipline to evaluate your own ideas honestly. That is the foundation quantitative research roles assume and the foundation graduate study in financial mathematics builds on. It is not a substitute for either.

Who this suits

Students from Class 11 upward through undergraduate study with calculus, basic linear algebra and basic probability. Python is taught within the course, so prior programming helps but is not assumed.

Classes are live and online in groups of at most eight. Students write code and work problems during the session, and strategy work is critiqued on its evaluation discipline as strictly as on its idea — because in this subject a good idea evaluated badly is indistinguishable from a bad one.