Price Dynamics from First Principles: Random Walks, Rough Volatility, and Neural SDEs — Volume I

Most books on price dynamics assume you already know the mathematics. This one teaches it.

This book begins with a coin flip and builds toward a calibrated volatility surface. Then it takes the framework beyond the Black–Scholes frontier, into regime switching, long memory, rough volatility, agent-based markets, self-exciting processes, machine learning, generative models, and neural stochastic differential equations. Nothing essential is asserted, quoted, or waved through. Where a result is needed, it is derived in front of you, line by line.

WHAT YOU NEED TO START

High-school algebra, calculus, probability, logarithms, and vectors. That is the whole prerequisite. Stochastic calculus, Itô's lemma, martingales, characteristic functions, Fourier transforms, and the mathematics of time series are built from scratch in the appendices, at the point where the argument first needs them.

THE MODELS COVERED IN THIS BOOK

• The random walk — ruin, the arcsine law, and why square-root scaling is not a metaphor

• Arithmetic Brownian motion — Bachelier's forgotten thesis, and the negative prices of April 2020

• Geometric Brownian motion — the log-normal world and exactly where it lies to you

• Mean-reverting models — Ornstein–Uhlenbeck, and the trap of estimating a half-life from short samples

• Jump diffusion and Lévy processes — what a crash is, mathematically

• Stochastic, local, and stochastic-local volatility — Heston, Dupire, and the smile

• GARCH — volatility clustering, and why yesterday still matters

EVERY CHAPTER FOLLOWS THE SAME DISCIPLINED PROGRESSION

Historical motivation; intuition explained without equations; mathematics built one assumption at a time; geometric, physical, and economic interpretations of every parameter; worked examples; model diagnostics and failure modes; comparisons with earlier models; complete Python implementations; and exercises.

THE INTERPRETATION SECTIONS ARE THE POINT OF THE BOOK.

Plenty of readers can fit a model. Far fewer can say why the persistence estimate crept toward one, what a negative estimated jump intensity means, when a calibrated local volatility surface should be disbelieved, or why adding one parameter can flip the sign of another. That gap is what this book was written to close.

VERIFIED, NOT ASSUMED.

Every figure is generated by a seeded Python script whose seed is printed in the caption. Every numerical claim was computed, not recalled. Where the computation contradicted the plan, the computation won and the disagreement became part of the text.

ACROSS THE COMPLETE TWO-VOLUME WORK

19 chapters, 8 mathematical appendices, 156 figures, and 688 exercises spanning easy, medium, hard, research-level, programming, conceptual, interview, and challenge tiers — with full worked solutions to nearly all of them, and one hundred conceptual questions per chapter.

FOR

Self-taught quants, graduate students entering mathematical finance, practitioners who calibrate models they were never shown the inside of, and anyone who would rather understand than remember.

Volume I develops price dynamics from random walks through diffusion, jumps, mean reversion, stochastic and local volatility, and GARCH. Volume II continues with regime switching, fractional Brownian motion, rough volatility, agent-based markets, Hawkes processes, machine learning, deep generative models, neural SDEs, and modelling hygiene.