Roadmap
The theory line: from calculus and analysis through measure-theoretic probability to stochastic volatility, jumps, and credit — everything derived.
The calculus a quant uses every day — limits, derivatives, Taylor series, integration, multivariable gradients and Lagrange multipliers, and vector calculus — built with full derivations and worked examples.
The other half of quant math — vectors, matrices, eigenvalues, SVD, PCA, least-squares regression, and convex optimization — built to where you can extract PCA factors and solve a Markowitz portfolio by hand.
The math a working quant actually uses — probability, differential equations, stochastic calculus, the Black–Scholes model, and numerical methods — built from the ground up with full derivations, worked examples, and graded problems.
The rigorous foundation under calculus and probability — completeness, sequences and series, topology and continuity, Riemann and Lebesgue integration, and an introduction to functional analysis (Banach/Hilbert/L^p).
Probability made rigorous — sigma-algebras and measures, random variables as measurable maps, expectation as a Lebesgue integral, conditional expectation as an L^2 projection, and the limit theorems (LLN, CLT, extreme value) that underpin risk.
The Heston model and the Girsanov change of measure behind modern pricing.
Why Brownian motion underprices tail risk, and how Lévy processes fix it.
The frontier of derivatives valuation — backward SDEs and nonlinear Feynman–Kac, counterparty credit risk, the full XVA stack (CVA/DVA/FVA/KVA), and the Monte-Carlo and deep-BSDE methods that price them.
Regression through deep learning and transformers, applied to alpha discovery, credit risk, and backtest overfitting.
Build portfolios from expected returns, covariance estimates, constraints, transaction costs, and audit-ready risk evidence.
