Curriculum map

Deep Quantitative Mathematics

The complete path from college foundations to advanced quant finance — and exactly where each course fits. Follow it in order, or jump to what you need.

10 of 10 modules with content liveAvailable nowCore coveredPlanned

Stage 1 · Foundations

Months 1–4

Linear algebra, probability basics, pre-calculus. Establish proof fluency.

Module 1

College Mathematics Foundations

Core covered

The bedrock every advanced topic rests on — algebra, intro linear algebra, and intro probability.

Pre-calculus & proof logicVectors, matrices, linear systemsProbability axioms & BayesDistributions, expectation, varianceOLS regression

Full syllabus · 3 sub-modules · 21 topics▾

1.1 Pre-Calculus & Mathematical Logic

·Sets, functions, relations
·Real number system: sup, inf, density
·Inequalities & absolute values
·Polynomial, rational, exp, log functions
·Sequences & series
·Induction & proof by contradiction
·Binomial theorem

1.2 Introductory Linear Algebra

·Vectors in Rⁿ, dot product, norms
·Matrices, transpose, inverse
·Gaussian elimination
·Determinants
·Eigenvalues & eigenvectors
·Gram–Schmidt orthogonality
·Basis, dimension, rank–nullity

1.3 Introductory Probability & Statistics

·Kolmogorov axioms, conditional, Bayes
·Discrete & continuous distributions
·CDF / PDF / PMF
·Expectation, variance, covariance
·LLN & Central Limit Theorem
·Hypothesis testing, confidence intervals
·OLS regression

Reference texts

Strang — Introduction to Linear Algebra · DeGroot & Schervish — Probability and Statistics

Probability for Quants Vectors & Matrices

Stage 2 · Calculus & Analysis

Months 5–8

Calculus through multivariable; real analysis up to Lebesgue integration.

Module 2

Calculus (Single & Multivariable)

Available now

The language of continuous change — derivatives, integrals, gradients, and multiple integration.

Limits & continuityDifferentiation & Taylor seriesIntegration techniquesGradient, Hessian, Lagrange multipliersDouble/triple integrals, Jacobians

Full syllabus · 3 sub-modules · 14 topics▾

2.1 Single-Variable Calculus

·Limits (ε–δ), continuity, IVT
·Derivative & differentiation rules
·Taylor & Maclaurin series
·MVT, L'Hôpital, optimization
·Riemann integral & the FTC
·Techniques & improper integrals

2.2 Multivariable Calculus

·Partial derivatives & gradient
·Tangent planes, chain rule
·Hessian & second-derivative test
·Lagrange multipliers
·Double/triple integrals, Jacobian

2.3 Vector Calculus

·Vector fields, curl, divergence
·Line & surface integrals
·Green's, Stokes', Divergence theorems

Reference texts

Stewart — Multivariable Calculus · Apostol — Calculus Vol. 2

Calculus for Quants

Module 3

Real Analysis

Available now

Rigorous foundations under calculus and probability — the backbone for measure theory and Itô.

Completeness, sup/infSequences, series, CauchyTopology of the real lineRiemann & Lebesgue integrationL^p spaces, Banach & Hilbert

Full syllabus · 4 sub-modules · 16 topics▾

3.1 Reals, Sequences & Series

·Completeness axiom, sup/inf
·Cauchy sequences, Bolzano–Weierstrass
·Convergence tests
·Power series, radius of convergence

3.2 Topology & Continuity

·Open, closed, compact sets
·Heine–Borel theorem
·Uniform continuity
·MVT, Taylor with remainder

3.3 Riemann & Lebesgue Integration

·Darboux sums, integrability
·Lebesgue measure & measurable functions
·Monotone & Dominated Convergence, Fatou
·L^p spaces, Hölder & Minkowski

3.4 Metric & Functional Analysis

·Metric spaces, completeness
·Banach & Hilbert spaces
·Contraction mapping theorem
·Bounded operators, Riesz representation

Reference texts

Rudin — Principles of Mathematical Analysis · Kreyszig — Introductory Functional Analysis

Real Analysis for Quants

Stage 3 · Advanced LA & Probability

Months 9–12

Advanced linear algebra and measure-theoretic probability — run in parallel.

Module 4

Advanced Linear Algebra & Matrix Analysis

Available now

Spectral theory, decompositions, and optimization geometry behind portfolio theory and PCA.

LU/QR/Cholesky/SVDSpectral theorem, positive definitenessQuadratic forms & optimizationPCA & factor modelsOLS as projection, regularization

Full syllabus · 4 sub-modules · 17 topics▾

4.1 Matrix Decompositions

·LU with pivoting
·QR (Gram–Schmidt, Householder)
·Singular Value Decomposition (SVD)
·Cholesky (correlated normals)
·Spectral decomposition

4.2 Spectral Theory

·Eigenvalue sensitivity, condition number
·Symmetric matrices, orthogonal eigenvectors
·Positive definiteness
·Jordan form, Cayley–Hamilton

4.3 Quadratic Forms & Optimization Geometry

·Definite / semidefinite classification
·Second-order optimality (Hessian)
·Rayleigh quotient, Courant–Fischer
·Markowitz as a quadratic program

4.4 Applied Linear Algebra in Finance

·PCA: covariance eigendecomposition
·Factor models, beta
·OLS as projection onto column space
·Ridge regression (Tikhonov)

Reference texts

Horn & Johnson — Matrix Analysis · Trefethen & Bau — Numerical Linear Algebra

Linear Algebra & Optimization for Quants

Module 5

Advanced Probability Theory

Available now

Measure-theoretic probability — the precise language of modern mathematical finance.

Sigma-algebras, measures, filtrationsExpectation as Lebesgue integralConditional expectation (L² projection)Modes of convergence, CLTMultivariate normal, copulas, EVT

Full syllabus · 5 sub-modules · 20 topics▾

5.1 Measure Theory Foundations

·Sigma-algebras, Borel sets, filtrations
·Probability measures
·Lebesgue–Stieltjes & the CDF
·Random variables as measurable maps
·Pushforward / distribution

5.2 Integration & Expectation

·Expectation as a Lebesgue integral
·Convergence theorems
·MGF & characteristic functions

5.3 Conditional Expectation

·E[X|G] as an L² projection
·Tower property
·Regular conditional probability
·Core of martingale theory

5.4 Limit Theorems

·Modes of convergence
·Strong & weak LLN
·CLT with Lindeberg condition
·Slutsky, delta method

5.5 Multivariate & Extreme Value

·Multivariate normal
·Copulas (Gaussian, t), tail dependence
·Fisher–Tippett–Gnedenko, GEV
·Peaks-over-threshold (GPD), VaR/ES

Reference texts

Durrett — Probability: Theory and Examples · McNeil, Frey & Embrechts — Quantitative Risk Management

Measure-Theoretic Probability

Stage 4 · Differential Equations & Numerics

Months 13–18

ODEs and PDEs alongside numerical methods — implement an FDM solver.

Module 6

Ordinary & Partial Differential Equations

Core covered

ODEs and PDEs — the Black–Scholes PDE reduces option pricing to a boundary-value problem.

First & second-order ODEsLaplace transform methodsHeat equation & separation of variablesWave & Laplace equationsFourier methods, BS↔heat equation

Full syllabus · 2 sub-modules · 12 topics▾

6.1 Ordinary Differential Equations

·First-order: separable, integrating factor
·Existence/uniqueness (Picard–Lindelöf)
·Second-order linear, characteristic equation
·Systems via matrix exponential
·Laplace transform methods
·Lyapunov stability

6.2 Partial Differential Equations

·Elliptic / parabolic / hyperbolic classification
·Heat equation, separation, heat kernel
·Wave equation (D'Alembert)
·Laplace/Poisson, Green's functions
·Fourier series & transform
·Black–Scholes PDE; American as free-boundary

Reference texts

Evans — Partial Differential Equations · Wilmott, Howison & Dewynne — Mathematics of Financial Derivatives

Differential Equations for Finance

Module 7

Numerical Methods

Core covered

How quant finance is actually practiced — every real pricing engine is a numerical algorithm.

Root finding (Newton for implied vol)Interpolation & splinesQuadrature & Monte CarloNumerical ODEs (Runge–Kutta)Finite differences (Crank–Nicolson)

Full syllabus · 4 sub-modules · 15 topics▾

7.1 Numerical Linear Algebra

·LU factorization
·Jacobi, Gauss–Seidel, SOR
·Conjugate Gradient, GMRES
·Conditioning & stability

7.2 Root Finding & Optimization

·Bisection, Newton–Raphson, secant
·Brent's method
·Implied volatility solve

7.3 Interpolation & Quadrature

·Lagrange / Newton interpolation, splines
·Vol-surface & yield-curve interpolation
·Gaussian quadrature
·Monte Carlo & quasi-MC (Sobol, Halton)

7.4 Numerical ODEs & PDEs

·Euler, Runge–Kutta (RK4), stiffness
·Finite differences: explicit, implicit, Crank–Nicolson
·Theta method, Thomas algorithm
·FEM overview; Monte Carlo & variance reduction

Reference texts

Glasserman — Monte Carlo Methods in Financial Engineering · Duffy — Finite Difference Methods in Financial Engineering

Numerical Methods C++ & HPC for Quants

Stage 5 · Stochastic Calculus

Months 19–24

The hardest stage: Itô's lemma, Girsanov, Feynman–Kac. This is the core.

Module 8

Stochastic Processes & Stochastic Calculus

Available now

The mathematical core — Brownian motion, the Itô integral, Girsanov, Feynman–Kac.

Martingales & filtrationsBrownian motion, quadratic variationItô integral & Itô's lemmaSDEs: GBM, OU, CIRGirsanov, Feynman–Kac, Lévy & jumps

Full syllabus · 4 sub-modules · 17 topics▾

8.1 Discrete-Time Processes

·Filtrations, adapted/predictable processes
·Martingales, optional stopping
·Random walks, reflection principle
·Markov chains, Doob decomposition

8.2 Brownian Motion

·Independent increments, continuous paths
·Quadratic variation [B,B]_t = t
·Nowhere differentiable
·Brownian bridge, reflection principle

8.3 Itô Calculus

·Itô integral, Itô isometry
·Itô's lemma, (dB)²=dt
·SDEs: GBM, Ornstein–Uhlenbeck, CIR
·Girsanov & change of measure
·Feynman–Kac (PDE ↔ expectation)

8.4 Jumps & Lévy Processes

·Poisson & compound Poisson
·Jump-diffusion (Merton)
·Lévy–Khintchine representation
·Variance Gamma, NIG, subordination

Reference texts

Shreve — Stochastic Calculus for Finance II · Øksendal — Stochastic Differential Equations

Stochastic Calculus Lévy Models in Finance Hawkes & Microstructure

Stage 6 · Mathematical Finance

Months 25–30

Black–Scholes, Greeks, exotics, stochastic vol, interest rates. Implement everything.

Module 9

Mathematical Finance & Derivatives Pricing

Available now

No-arbitrage pricing, Black–Scholes, the Greeks, exotics, interest rates, and volatility models.

FTAP & risk-neutral pricingBlack–Scholes & the GreeksImplied vol, smile & surfaceExotic & American optionsHeston, SABR, local vol, jumps

Full syllabus · 5 sub-modules · 20 topics▾

9.1 No-Arbitrage Pricing

·Arbitrage, replication, completeness
·Fundamental Theorem of Asset Pricing
·Risk-neutral pricing V₀ = E^Q[e^{-rT}V_T]
·Change of numeraire, forward measure

9.2 Black–Scholes & the Greeks

·BS PDE & closed-form solution
·Put–call parity
·Delta, gamma, vega, theta, rho
·Implied vol, smile, skew, surface

9.3 Exotic & American Options

·Binary, barrier, Asian, lookback
·Margrabe, quanto, cliquet
·Optimal stopping / free boundary
·Binomial tree, Longstaff–Schwartz

9.4 Rates & Volatility Models

·Vasicek, CIR, Hull–White, HJM, LMM
·Caps, floors, swaptions
·Heston, SABR, local vol (Dupire)
·Jump models: Merton, Kou, CGMY; Fourier pricing

9.5 Risk & Portfolio

·VaR & Expected Shortfall (coherent)
·Markowitz mean–variance, CAPM
·Fama–French / factor models
·Stress testing

Reference texts

Hull — Options, Futures, and Other Derivatives · Gatheral — The Volatility Surface · Brigo & Mercurio — Interest Rate Models

The Black–Scholes Model Stochastic Volatility & Measure Change

Stage 7 · Advanced Frontiers

Month 31+

HJB, Malliavin, rough vol, BSDEs/XVA, ML in finance — as your research demands.

Module 10

Advanced Topics & Frontiers

Available now

Optimal control, Malliavin calculus, rough vol, ML, XVA/credit, BSDEs, and convex optimization.

HJB & stochastic controlMalliavin calculusRough volatility (fBm)BSDEs & nonlinear PDEs for XVAML in finance; convex optimization

Full syllabus · 4 sub-modules · 17 topics▾

10.1 Optimal Control & Malliavin

·Hamilton–Jacobi–Bellman equation
·Merton portfolio problem
·Malliavin derivative, Greeks via Malliavin
·Clark–Ocone formula

10.2 Rough Volatility & ML

·Fractional Brownian motion, Hurst H
·Rough Heston (El Euch–Rosenbaum)
·Deep hedging, neural SDEs
·Gaussian-process & PCA yield-curve models

10.3 BSDEs, XVA & Credit Risk

·Backward SDEs & nonlinear Feynman–Kac
·Semilinear PDEs for valuation adjustments
·CVA / DVA / FVA / KVA
·Exposure simulation, wrong-way risk
·Deep BSDE solvers (high-dimensional PDEs)

10.4 Convex Optimization

·Convex sets & functions, KKT conditions
·LP, QP, SOCP, SDP
·Interior-point methods
·Robust & risk-parity portfolios

Reference texts

Boyd & Vandenberghe — Convex Optimization · Nualart — The Malliavin Calculus · Crépey — Counterparty Risk and Funding

BSDE, XVA & Credit Risk Optimization for Finance

Start where the path begins

The Foundations track covers Stage 1 today — probability, linear algebra, and more.

Start the Foundations →

Curriculum adapted from the “Deep Quantitative Mathematics” syllabus · 10 modules.