Preface

Algorithmic trading sits at the intersection of several mathematical disciplines — probability theory, stochastic calculus, statistical inference, optimisation, and machine learning — applied to one of the most complex and dynamic systems human societies have built: financial markets. Each of these disciplines has an extensive literature of its own, and so does the practical craft of building trading systems. What is harder to find is a treatment that integrates them — that derives the Avellaneda–Stoikov market-making equations from first principles in the same breath as it explains how a Kalman filter tracks a fair mid-price, or that connects Bayesian decision theory to the design of a client quoting strategy.

This book is my attempt at that integration. It grew out of lecture notes developed across several teaching contexts: internal courses for my team at BBVA, where the focus was always on problems we faced in practice; a graduate course on algorithmic trading at the Universidad de Alcalá; and courses at the Instituto de Empresa. Each audience shaped the material differently — practitioners pushed for concrete models and implementable intuitions, while students pushed for accessible introductions to the core subjects — and the tension between those two demands is, I hope, visible in a positive way throughout these pages. The starting point for all of it was my own work at BBVA on algorithmic trading problems, from optimal execution in equity markets to market making in fixed income and FX. The result is a treatment that tries to be mathematically rigorous without sacrificing the practical intuition that makes models useful.

Although the book’s origins are in algorithmic trading, its scope is broader. The analytical models it develops — for market microstructure, order book dynamics, fair price estimation, and liquidity measurement — are useful to anyone working quantitatively on financial markets, whether or not their primary interest is in building trading algorithms. Readers focused on quantitative research, risk modelling, or financial data science will find the first three parts of the book a self-contained introduction to advanced analytics for financial markets, and can engage with the algorithm-focused parts selectively according to their interests.

The book is also designed to serve as course material for advanced degrees and professional programmes. Its six parts map naturally onto distinct course modules that can be combined or taught independently. A master’s programme in quantitative finance or financial data science could draw on the full book as a spine: Parts I and II as a foundations course covering markets, probability, stochastic calculus, and machine learning; Part III as a course on advanced financial analytics covering microstructure, pricing models, and liquidity; and Parts IV through VI as applied courses on execution, market making, and systematic investment. Instructors can also assemble more focused offerings — a single-semester course on algorithmic trading from Parts IV–VI, or a course on quantitative financial modelling from Parts II–III — without loss of coherence, since each part opens with the necessary context and cross-references prerequisites explicitly.

The book is, and will remain for some time, a work in progress. I release it in regular updates and welcome feedback from readers through the issue tracker on GitHub.

Prior Knowledge¶

The mathematical prerequisites are those of a strong undergraduate degree in mathematics, physics, engineering, or a quantitatively oriented economics or finance programme. Specifically, I assume familiarity with:

• Calculus and linear algebra: partial derivatives, the chain rule, integration, matrix operations, eigendecompositions, and the singular value decomposition. These are used throughout without review.

• Probability and statistics: probability spaces, random variables, expectations, conditional distributions, the Gaussian family, basic hypothesis testing, and the method of maximum likelihood. The Bayesian Modelling chapter introduces the Bayesian perspective from first principles, but a frequentist background is assumed.

• Python: the numerical examples and all figures are produced in Jupyter notebooks using standard scientific computing libraries (NumPy, SciPy, Matplotlib, pandas, scikit-learn). Readers comfortable with Python at an intermediate level should be able to run and modify the notebooks without difficulty.

Prior knowledge of finance or economics is helpful but not strictly required. The Financial Markets and Mechanics of Financial Instruments chapters introduce the institutional landscape, market structure, and the main asset classes from first principles. Readers with a finance background may skim these chapters and use them as a reference for notation.

How to Read This Book¶

The book is divided into six parts, which can be read sequentially or, for readers with relevant background, selectively.

Part I — Financial Markets Fundamentals (Financial Markets, Mechanics of Financial Instruments, Algorithmic Trading) establishes the institutional context in which algorithmic trading operates. It covers market participants, the main asset classes, and the taxonomy of algorithmic trading strategies. Readers who already work in financial markets can treat this part as a notation reference.

Part II — Mathematical and Computational Foundations (Bayesian Modelling, Causal inference, Stochastic Calculus, Stochastic Optimal Control, Data-Driven Methods, Generative Artificial Intelligence) develops the mathematical toolkit used throughout the rest of the book. It covers Bayesian probability and graphical models, causal inference, Itô calculus and stochastic differential equations, stochastic optimal control via the Hamilton–Jacobi–Bellman equation, the core of supervised and unsupervised machine learning, and the foundations of large language models and AI agents. These chapters can be read independently of each other, though Stochastic Calculus is a prerequisite for Stochastic Optimal Control, and Data-Driven Methods provides useful context for Generative Artificial Intelligence.

Part III — Advanced Analytics for Financial Markets (Market microstructure, Modelling the Limit Order Book, Modelling RfQs in Dealer to Client Markets, Fair value estimation, Liquidity Modelling) develops the quantitative models of financial markets that underpin the algorithmic chapters and stand on their own as a treatment of advanced financial analytics. It covers market microstructure and price formation theory, models of limit order book dynamics and order flow, request-for-quote market models, methods for fair price estimation (Kalman filters, Black–Scholes–Merton, utility indifference, stochastic discount factors), and the measurement and modelling of market liquidity. Readers interested in quantitative financial analysis — pricing, liquidity research, or risk modelling — can read Parts I through III as a self-contained introduction, independently of the algorithm-focused parts that follow.

Part IV — Execution Algorithms (Execution fundamentals, Optimal Execution Theory, Execution Tactics) covers the problem of executing a pre-determined order in the market at minimum cost and risk. It develops the taxonomy of execution benchmarks and cost models, derives the Almgren–Chriss optimal execution trajectory, and covers practical execution tactics including adaptive slicing and dark-pool routing.

Part V — Market Making Algorithms (Market Making Fundamentals, Optimal Market Making, Optimal Hedging) addresses the dealer’s problem: continuously quoting bid and ask prices to clients while managing inventory and adverse selection risk. The chapters progress from the conceptual decomposition of the bid–ask spread to the mathematical theory of optimal market making derived from stochastic optimal control, and the hedging of residual inventory risk.

Part VI — Investment Algorithms (Quantitative Investment Fundamentals, Optimal Investment Theory) covers systematic investment strategies: the construction and evaluation of signals, the principles of backtesting and avoiding overfitting, and the mathematical foundations of the main strategy families — mean reversion, momentum, statistical arbitrage, factor models, and portfolio optimisation including Markowitz mean-variance optimisation, risk parity, and the Black–Litterman framework.

The natural reading order is Parts I through VI sequentially. Readers primarily interested in execution can proceed from Parts I–III directly to Part IV, returning to Part II as individual chapters require it. Readers focused on market making will find Bayesian Modelling, Stochastic Calculus, and Stochastic Optimal Control most directly relevant from Part II. Investment-focused readers will draw most heavily on Data-Driven Methods and the portfolio theory sections of Bayesian Modelling. Readers interested in financial markets analytics without a specific focus on algorithmic trading strategies can read Parts I through III as a self-contained treatment.

Each chapter closes with a set of exercises. These range from short derivations intended to consolidate the mathematical content to longer computational problems whose solutions require the accompanying Jupyter notebooks. The notebooks are collected in the final section of the book and reproduce all figures in the main text.