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Courses Mathematical Foundations for Quant Finance

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Mathematical Foundations for Quant Finance

Algebra, calculus, linear algebra, probability and differential equations, each taught with the finance it is used for. A Proof Gym runs alongside for learners who want to read and write proofs.

  • 7 levels
  • 49 modules
  • 256 lessons
  • About 164 hours
  • Before you start: School algebra and basic Python

Your progress

0 of 256 lessons

Start here

0.2.1 Symbols, sets and function notation

Concept · About 15 minutes

Start the course

How each module works

  1. 1Concept lessonsOne idea at a time, with runnable code
  2. 2Practice setQuick questions with instant feedback
  3. 3Case studiesThe ideas applied to real finance problems
0

Level 0 · About 6 hours

Orientation and fluency

Rebuild algebra fluency and learn to read the symbols in a formula; the placement check lets you skip what is solid.

0 of 9 lessons

0.1Placement checkSkip what you know
You will learn:
One or two questions for each module of Levels 0 to 6, Each module marked skip, review or learn.
Applied to:
Your map of Foundations, so you can skip what you know and test out of a level for your track
Take the placement check
0.2Reading maths4 lessons
You will learn:
Σ and Π, indices, ∈ ∀ ∃ and function notation, 'let' and 'such that', Translating a formula into code.
Applied to:
Portfolio return Σ wᵢrᵢ and net present value written as sums
In Python:
Write NPV three ways (loop, comprehension, NumPy) and check they agree
  1. 0.2.1 Symbols, sets and function notationConcept15 min
  2. 0.2.2 Sums and products: Σ and ΠConcept15 min
  3. 0.2.3 Practice: reading formulasPractice set15 min
  4. 0.2.4 Case study: net present value from formula to codeCase study20 min
0.3Algebra fluency5 lessons
You will learn:
Fractions and percentages, Order of operations, Rearranging equations, Inequalities, Absolute value.
Applied to:
Rearranging simple-interest and break-even formulas
In Python:
SymPy as your answer-checker
  1. 0.3.1 Fractions, percentages and order of operationsConcept15 min
  2. 0.3.2 Rearranging equationsConcept15 min
  3. 0.3.3 Inequalities and absolute valueConcept15 min
  4. 0.3.4 Practice set: algebraPractice set15 min
  5. 0.3.5 Case study: simple interest and break-evenCase study20 min
1

Level 1 · About 18 hours

Precalculus

Functions, roots, exponents, logarithms and series. Each module teaches the rules first, then applies them to bonds, returns and payoffs.

0 of 34 lessons

1.1Functions and graphs5 lessons
You will learn:
Domain and range, Transformations, Piecewise functions, Composition, Inverses.
Applied to:
Option payoff diagrams; price and yield as inverse functions
In Python:
Plot call, put and straddle payoffs from one function
  1. 1.1.1 What a function is: domain, range and notationConcept25 min
  2. 1.1.2 Transformations and piecewise functionsConcept30 min
  3. 1.1.3 Composition and inverse functionsConcept25 min
  4. 1.1.4 Practice set: functionsPractice set20 min
  5. 1.1.5 Case study: option payoff diagramsCase study30 min
1.2Lines and quadratics4 lessons
You will learn:
Slope and intercept, Completing the square, Vertex of a parabola.
Applied to:
Security market line; portfolio variance as a parabola in the weight
In Python:
Find the minimum-variance weight of a two-asset portfolio
  1. 1.2.1 Lines: slope and interceptConcept25 min
  2. 1.2.2 Quadratics and completing the squareConcept25 min
  3. 1.2.3 Practice set: lines and quadraticsPractice set20 min
  4. 1.2.4 Case study: the minimum-variance two-asset portfolioCase study30 min
1.3Equations and roots5 lessons
You will learn:
Polynomials and their roots, Bisection, Newton's method (preview).
Applied to:
Internal rate of return and yield to maturity as roots
In Python:
Write bisection from scratch, then compare with scipy.optimize.brentq
  1. 1.3.1 Polynomials and their rootsConcept25 min
  2. 1.3.2 Bisection: finding a root by halvingConcept25 min
  3. 1.3.3 Newton's method, a first lookConcept25 min
  4. 1.3.4 Practice set: rootsPractice set20 min
  5. 1.3.5 Case study: IRR and yield to maturityCase study35 min
1.4Exponents and logarithms7 lessons
You will learn:
Exponent rules, Exponential functions and ee, Logarithms as the inverse of exponentials, Log rules, Solving for an unknown exponent.
Applied to:
Compounding conventions, continuous rates, simple and log returns
In Python:
Check each rule numerically; convert between compounding conventions; compute and add up returns
  1. 1.4.1 Exponents and their rulesConcept25 min
  2. 1.4.2 Exponential functions and the number eConcept25 min
  3. 1.4.3 Logarithms and their rulesConcept25 min
  4. 1.4.4 Solving exponential and logarithmic equationsConcept25 min
  5. 1.4.5 Practice set: exponents and logarithmsPractice set25 min
  6. 1.4.6 Case study: compounding and continuous ratesCase study35 min
  7. 1.4.7 Case study: simple returns and log returnsCase study30 min
1.5Sequences and series5 lessons
You will learn:
Sequences, Arithmetic and geometric series, Working with Σ, Convergence intuition.
Applied to:
Annuities, perpetuities and bond prices from scratch
In Python:
Price a bond with a loop, then with the closed-form annuity formula
  1. 1.5.1 SequencesConcept25 min
  2. 1.5.2 Arithmetic and geometric seriesConcept20 min
  3. 1.5.3 Infinite series and when they convergeConcept25 min
  4. 1.5.4 Practice set: seriesPractice set20 min
  5. 1.5.5 Case study: annuities, perpetuities and bond pricesCase study35 min
1.6Interpolation4 lessons
You will learn:
Linear interpolation, Cubic-spline interpolation.
Applied to:
Building a zero curve from par yields
In Python:
scipy.interpolate on a realistic yield curve
  1. 1.6.1 Linear interpolationConcept25 min
  2. 1.6.2 Cubic splinesConcept30 min
  3. 1.6.3 Practice set: interpolationPractice set20 min
  4. 1.6.4 Case study: building a zero curveCase study30 min
1.7Trigonometry and complex numbers4 lessons
You will learn:
Unit circle, Sine and cosine, Complex numbers, Euler's formula eiθe^{i\theta}.
Applied to:
The groundwork for Fourier methods in option pricing
In Python:
Plot eiθe^{i\theta} on the unit circle
  1. 1.7.1 Angles, the unit circle, sine and cosineConcept30 min
  2. 1.7.2 Complex numbersConcept25 min
  3. 1.7.3 Euler's formulaConcept25 min
  4. 1.7.4 Practice set: trigonometry and complex numbersPractice set20 min
2

Level 2 · About 30 hours

Single-variable calculus

Limits, derivatives and integrals as rules you can use, then as the language of sensitivity in duration, convexity and the Greeks.

0 of 40 lessons

2.1Limits and continuity5 lessons
You will learn:
Intuitive limits, One-sided limits, Continuity, Limits at infinity.
Applied to:
Continuous compounding as the limit of (1 + r/n)ⁿ
In Python:
Tabulate and plot a limit numerically
  1. 2.1.1 Limits: what a function approachesConcept25 min
  2. 2.1.2 One-sided limits and continuityConcept30 min
  3. 2.1.3 Limits at infinityConcept30 min
  4. 2.1.4 Practice set: limitsPractice set25 min
  5. 2.1.5 Case study: continuous compounding as a limitCase study30 min
2.2The derivative4 lessons
You will learn:
Rate of change, Tangent slope, Definition as a limit, Finite-difference estimate.
Applied to:
Bond price sensitivity to yield; duration as a derivative
In Python:
Estimate dP/dy by bumping the yield
  1. 2.2.1 Rate of change and the slope of a tangentConcept25 min
  2. 2.2.2 The derivative as a limit, and finite differencesConcept35 min
  3. 2.2.3 Practice set: the derivativePractice set20 min
  4. 2.2.4 Case study: bond price sensitivity and durationCase study30 min
2.3Differentiation rules7 lessons
You will learn:
Power, product, quotient and chain rules, Derivatives of exp and log, Derivatives of sine and cosine.
Applied to:
Modified duration; elasticity
In Python:
Check every hand derivative with SymPy
  1. 2.3.1 Power, constant and sum rulesConcept25 min
  2. 2.3.2 Product and quotient rulesConcept25 min
  3. 2.3.3 The chain ruleConcept30 min
  4. 2.3.4 Derivatives of exponentials and logarithmsConcept30 min
  5. 2.3.5 Derivatives of sine and cosineConcept35 min
  6. 2.3.6 Practice set: differentiationPractice set25 min
  7. 2.3.7 Case study: modified duration and elasticityCase study30 min
2.4Using derivatives6 lessons
You will learn:
Maxima and minima, Concavity, Second derivative, Newton's method.
Applied to:
Convexity; risk aversion as the curvature of utility; solving for yield and implied volatility
In Python:
Implement Newton's method for implied volatility
  1. 2.4.1 Maxima and minimaConcept25 min
  2. 2.4.2 Concavity and the second derivativeConcept25 min
  3. 2.4.3 Newton's method with derivativesConcept30 min
  4. 2.4.4 Practice set: using derivativesPractice set25 min
  5. 2.4.5 Case study: convexity and risk aversionCase study30 min
  6. 2.4.6 Case study: implied volatility by Newton's methodCase study30 min
2.5Integration5 lessons
You will learn:
Riemann sums, Antiderivatives, Fundamental Theorem of Calculus.
Applied to:
Present value of a continuous cash-flow stream; area under a density
In Python:
Riemann sums converging to an exact integral
  1. 2.5.1 Area and Riemann sumsConcept30 min
  2. 2.5.2 AntiderivativesConcept25 min
  3. 2.5.3 The fundamental theorem of calculusConcept30 min
  4. 2.5.4 Practice set: integrationPractice set25 min
  5. 2.5.5 Case study: present value of a continuous cash flowCase study35 min
2.6Integration techniques5 lessons
You will learn:
Substitution, Integration by parts, Improper integrals, The Gaussian integral.
Applied to:
The normal-distribution integrals behind Black–Scholes
In Python:
scipy.integrate.quad against closed forms
  1. 2.6.1 Integration by substitutionConcept30 min
  2. 2.6.2 Integration by partsConcept30 min
  3. 2.6.3 Improper integrals and the Gaussian integralConcept35 min
  4. 2.6.4 Practice set: integration techniquesPractice set25 min
  5. 2.6.5 Case study: the normal integrals behind Black–ScholesCase study35 min
2.7Taylor series4 lessons
You will learn:
Taylor polynomials, The size of the error, Standard series for eˣ, ln(1 + x), sin x and cos x.
Applied to:
Duration–convexity and delta–gamma approximations
In Python:
Measure approximation error as the shock grows
  1. 2.7.1 Taylor polynomialsConcept30 min
  2. 2.7.2 Standard series and the size of the errorConcept35 min
  3. 2.7.3 Practice set: Taylor seriesPractice set20 min
  4. 2.7.4 Case study: duration–convexity and delta–gammaCase study35 min
2.8Numerical calculus4 lessons
You will learn:
Forward, backward and central differences, Trapezoid and Simpson rules, Error against step size.
Applied to:
Bump-and-reprice Greeks; numerical present values
In Python:
Plot error against step size on a log-log chart
  1. 2.8.1 Finite differences and their errorsConcept35 min
  2. 2.8.2 Trapezoid and Simpson rulesConcept35 min
  3. 2.8.3 Practice set: numerical calculusPractice set20 min
  4. 2.8.4 Case study: bump-and-reprice GreeksCase study35 min
3

Level 3 · About 22 hours

Linear algebra

Vectors and matrices as objects you can picture and compute with, then as portfolios, covariance and factor models.

0 of 34 lessons

3.1Vectors4 lessons
You will learn:
Addition and scaling, Dot product, Length and angle, Orthogonality.
Applied to:
Portfolio return as w·r; correlation as the cosine of an angle
In Python:
NumPy vectors and the dot product
  1. 3.1.1 Vectors and their arithmeticConcept30 min
  2. 3.1.2 The dot product, length and angleConcept35 min
  3. 3.1.3 Practice set: vectorsPractice set20 min
  4. 3.1.4 Case study: portfolio returns and correlation as an angleCase study35 min
3.2Matrices5 lessons
You will learn:
Multiplication, Transpose, Identity and inverse, A matrix as a linear map.
Applied to:
The covariance matrix; portfolio variance wᵀΣw
In Python:
Portfolio variance three ways
  1. 3.2.1 Matrices and matrix multiplicationConcept30 min
  2. 3.2.2 Matrices as linear mapsConcept30 min
  3. 3.2.3 Transpose, identity and inverseConcept35 min
  4. 3.2.4 Practice set: matricesPractice set25 min
  5. 3.2.5 Case study: the covariance matrix and portfolio varianceCase study35 min
3.3Linear systems4 lessons
You will learn:
Gaussian elimination, Rank, Solution sets.
Applied to:
Replicating a payoff; when a market is complete
In Python:
numpy.linalg.solve and matrix_rank
  1. 3.3.1 Linear systems and Gaussian eliminationConcept30 min
  2. 3.3.2 Rank and the shape of the solution setConcept30 min
  3. 3.3.3 Practice set: linear systemsPractice set20 min
  4. 3.3.4 Case study: replicating a payoff, and complete marketsCase study35 min
3.4Determinants3 lessons
You will learn:
Volume scaling, Invertibility test.
Applied to:
Spotting redundant assets in a payoff matrix
In Python:
Determinants of near-singular covariance matrices
  1. 3.4.1 Determinants as volume scalingConcept35 min
  2. 3.4.2 Practice set: determinantsPractice set20 min
  3. 3.4.3 Case study: redundant assets in a payoff matrixCase study30 min
3.5Spaces and projections5 lessons
You will learn:
Span, Basis, Independence, Orthogonal projection, Least squares.
Applied to:
Regression as a projection; the minimum-variance hedge
In Python:
Least squares by projection and by numpy.linalg.lstsq
  1. 3.5.1 Span, independence and basisConcept35 min
  2. 3.5.2 Orthogonal projectionConcept35 min
  3. 3.5.3 Least squaresConcept30 min
  4. 3.5.4 Practice set: spaces and projectionsPractice set25 min
  5. 3.5.5 Case study: the minimum-variance hedge as a projectionCase study35 min
3.6Eigenvalues and eigenvectors5 lessons
You will learn:
Characteristic polynomial, Diagonalisation, Spectral theorem for symmetric matrices.
Applied to:
Principal components of the yield curve: level, slope, curvature
In Python:
numpy.linalg.eigh on a yield-change covariance matrix
  1. 3.6.1 Eigenvalues and eigenvectorsConcept35 min
  2. 3.6.2 Diagonalisation and matrix powersConcept30 min
  3. 3.6.3 Symmetric matrices and the spectral theoremConcept30 min
  4. 3.6.4 Practice set: eigenvaluesPractice set20 min
  5. 3.6.5 Case study: principal components of the yield curveCase study35 min
3.7Positive definiteness and Cholesky4 lessons
You will learn:
Quadratic forms, Positive definite and semidefinite tests, Cholesky factorisation.
Applied to:
Valid correlation matrices; simulating correlated returns
In Python:
Simulate correlated normals and check the sample correlation
  1. 3.7.1 Quadratic forms and positive definitenessConcept35 min
  2. 3.7.2 The Cholesky factorisationConcept30 min
  3. 3.7.3 Practice set: positive definitenessPractice set20 min
  4. 3.7.4 Case study: valid correlation matrices and correlated simulationCase study35 min
3.8Singular value decomposition (optional)4 lessons
You will learn:
Singular value decomposition, Low-rank approximation.
Applied to:
Compressing a factor model
In Python:
Rank-k approximation of a returns matrix
  1. 3.8.1 The singular value decompositionConcept35 min
  2. 3.8.2 Low-rank approximationConcept25 min
  3. 3.8.3 Practice set: the singular value decompositionPractice set20 min
  4. 3.8.4 Case study: compressing a factor modelCase study35 min
4

Level 4 · About 20 hours

Multivariable calculus and optimisation

Partial derivatives, gradients and constrained optimisation, then the Greeks and portfolio choice.

0 of 32 lessons

4.1Functions of several variables4 lessons
You will learn:
Surfaces and level curves, Partial derivatives.
Applied to:
Option value V(S, σ, t); the Greeks as partial derivatives
In Python:
Surface plot of an option value
  1. 4.1.1 Functions of two variables: surfaces and level curvesConcept30 min
  2. 4.1.2 Partial derivativesConcept35 min
  3. 4.1.3 Practice set: partial derivativesPractice set20 min
  4. 4.1.4 Case study: the Greeks as partial derivativesCase study35 min
4.2Gradient and chain rule4 lessons
You will learn:
Gradient, Directional derivative, Multivariable chain rule, Total differential.
Applied to:
Attributing P&L to risk factors
In Python:
Attribute a P&L move to each factor
  1. 4.2.1 The gradient and directional derivativesConcept30 min
  2. 4.2.2 The multivariable chain rule and the total differentialConcept35 min
  3. 4.2.3 Practice set: gradients and the chain rulePractice set20 min
  4. 4.2.4 Case study: attributing P&L to risk factorsCase study35 min
4.3Second order4 lessons
You will learn:
Hessian, Multivariable Taylor expansion, Saddle points.
Applied to:
Delta–gamma–vega P&L approximation
In Python:
Compare the approximation with full revaluation
  1. 4.3.1 The Hessian and second-order Taylor expansionConcept35 min
  2. 4.3.2 Minima, maxima and saddle pointsConcept30 min
  3. 4.3.3 Practice set: second-order methodsPractice set20 min
  4. 4.3.4 Case study: delta–gamma–vega P&LCase study35 min
4.4Unconstrained optimisation5 lessons
You will learn:
First- and second-order conditions, Gradient descent, Newton's method in several variables.
Applied to:
Fitting a curve by least squares
In Python:
Gradient descent from scratch against scipy.optimize.minimize
  1. 4.4.1 Optimality conditions in several variablesConcept30 min
  2. 4.4.2 Gradient descentConcept30 min
  3. 4.4.3 Newton's method in several variablesConcept30 min
  4. 4.4.4 Practice set: unconstrained optimisationPractice set25 min
  5. 4.4.5 Case study: fitting a curve by least squaresCase study35 min
4.5Constrained optimisation5 lessons
You will learn:
Lagrange multipliers, KKT intuition, Convexity.
Applied to:
Minimum-variance portfolio; efficient frontier; two-fund theorem
In Python:
Closed-form frontier against a numerical optimiser
  1. 4.5.1 Lagrange multipliersConcept30 min
  2. 4.5.2 Inequality constraints and the KKT conditionsConcept30 min
  3. 4.5.3 Convex functions and convex problemsConcept30 min
  4. 4.5.4 Practice set: constrained optimisationPractice set25 min
  5. 4.5.5 Case study: the minimum-variance portfolio and the efficient frontierCase study35 min
4.6Portfolio theory5 lessons
You will learn:
Utility, Mean–variance, CAPM derivation, Sharpe ratio.
Applied to:
The tangency portfolio
In Python:
Derive it, then confirm with scipy.optimize
  1. 4.6.1 Utility and mean–variance preferencesConcept35 min
  2. 4.6.2 The tangency portfolio and the Sharpe ratioConcept30 min
  3. 4.6.3 The capital asset pricing modelConcept30 min
  4. 4.6.4 Practice set: portfolio theoryPractice set25 min
  5. 4.6.5 Case study: the tangency portfolio from dataCase study35 min
4.7Multiple integrals5 lessons
You will learn:
Double integrals, Change of variables, Jacobian, Polar coordinates.
Applied to:
Joint densities; why the Gaussian integral equals √π
In Python:
Monte Carlo against exact double integrals
  1. 4.7.1 Double integralsConcept35 min
  2. 4.7.2 Change of variables and the JacobianConcept35 min
  3. 4.7.3 Polar coordinates and the Gaussian integralConcept30 min
  4. 4.7.4 Practice set: multiple integralsPractice set25 min
  5. 4.7.5 Case study: joint densities and Monte Carlo integrationCase study35 min
5

Level 5 · About 35 hours

Probability and statistics

Random variables, conditional expectation and limit theorems made routine, then estimation and regression on return data.

0 of 55 lessons

5.1Probability basics6 lessons
You will learn:
Sample space and events, Axioms, Counting, Conditional probability, Bayes' rule, Independence.
Applied to:
Default and rating-migration probabilities
In Python:
Simulate and count
  1. 5.1.1 Sample spaces, events and the axioms of probabilityConcept30 min
  2. 5.1.2 CountingConcept30 min
  3. 5.1.3 Conditional probability and independenceConcept30 min
  4. 5.1.4 Bayes' ruleConcept25 min
  5. 5.1.5 Practice set: probabilityPractice set25 min
  6. 5.1.6 Case study: default and rating-migration probabilitiesCase study35 min
5.2Discrete random variables5 lessons
You will learn:
PMF and CDF, Bernoulli, Binomial, Geometric, Poisson.
Applied to:
One-period binomial asset model; counting defaults
In Python:
scipy.stats discrete distributions
  1. 5.2.1 Random variables, PMFs and CDFsConcept25 min
  2. 5.2.2 Bernoulli, binomial and geometric distributionsConcept30 min
  3. 5.2.3 The Poisson distributionConcept30 min
  4. 5.2.4 Practice set: discrete distributionsPractice set25 min
  5. 5.2.5 Case study: the one-period binomial model, and counting defaultsCase study35 min
5.3Expectation and moments5 lessons
You will learn:
Expectation, LOTUS, Variance, Skewness, Kurtosis, Linearity.
Applied to:
Return moments; why kurtosis matters for value at risk
In Python:
Sample moments of simulated and realistic returns
  1. 5.3.1 Expectation and LOTUSConcept30 min
  2. 5.3.2 Variance, and linearity of expectationConcept35 min
  3. 5.3.3 Skewness and kurtosisConcept30 min
  4. 5.3.4 Practice set: expectation and momentsPractice set20 min
  5. 5.3.5 Case study: return moments and value at riskCase study35 min
5.4Continuous distributions6 lessons
You will learn:
PDF, Uniform, Exponential, Normal, Lognormal, Student t, χ².
Applied to:
Lognormal prices; fat-tailed returns
In Python:
Fit and compare densities
  1. 5.4.1 Densities, distribution functions and the uniform distributionConcept30 min
  2. 5.4.2 The exponential distributionConcept25 min
  3. 5.4.3 The normal distributionConcept30 min
  4. 5.4.4 The lognormal, Student t and chi-squared distributionsConcept30 min
  5. 5.4.5 Practice set: continuous distributionsPractice set25 min
  6. 5.4.6 Case study: lognormal prices and fat-tailed returnsCase study35 min
5.5Joint distributions5 lessons
You will learn:
Joint, marginal and conditional distributions, Covariance and correlation, Independence, Multivariate normal.
Applied to:
Portfolio risk; a first look at conditional expectation E[X | Y]
In Python:
Simulate a bivariate normal and estimate E[X | Y]
  1. 5.5.1 Joint, marginal and conditional distributionsConcept30 min
  2. 5.5.2 Covariance, correlation and independenceConcept35 min
  3. 5.5.3 The multivariate normal distributionConcept35 min
  4. 5.5.4 Practice set: joint distributionsPractice set25 min
  5. 5.5.5 Case study: portfolio risk and conditional expectationCase study35 min
5.6Transformations and generating functions5 lessons
You will learn:
Functions of random variables, Jacobian method, Moment generating function, Characteristic function, Sums of random variables.
Applied to:
Lognormal from normal; sums of independent normals
In Python:
Check a transformed density by simulation
  1. 5.6.1 Functions of random variablesConcept35 min
  2. 5.6.2 Moment generating and characteristic functionsConcept30 min
  3. 5.6.3 Sums of independent random variablesConcept25 min
  4. 5.6.4 Practice set: transformationsPractice set20 min
  5. 5.6.5 Case study: the lognormal from the normalCase study30 min
5.7Inequalities and limit theorems5 lessons
You will learn:
Markov, Chebyshev, Jensen, Law of large numbers, Central limit theorem.
Applied to:
Monte Carlo error shrinks like 1/√N
In Python:
Watch the central limit theorem emerge
  1. 5.7.1 The Markov, Chebyshev and Jensen inequalitiesConcept35 min
  2. 5.7.2 The law of large numbersConcept30 min
  3. 5.7.3 The central limit theoremConcept35 min
  4. 5.7.4 Practice set: inequalities and limit theoremsPractice set25 min
  5. 5.7.5 Case study: the error of a Monte Carlo priceCase study35 min
5.8Statistical inference5 lessons
You will learn:
Estimators, Bias and consistency, Sampling distributions, Confidence intervals, Hypothesis tests.
Applied to:
Is this Sharpe ratio significant?
In Python:
Bootstrap against analytic confidence intervals
  1. 5.8.1 Estimators: bias, variance and consistencyConcept35 min
  2. 5.8.2 Sampling distributions and confidence intervalsConcept35 min
  3. 5.8.3 Hypothesis testsConcept30 min
  4. 5.8.4 Practice set: statistical inferencePractice set25 min
  5. 5.8.5 Case study: is this Sharpe ratio significant?Case study35 min
5.9Maximum likelihood4 lessons
You will learn:
Likelihood, Log-likelihood, Maximum likelihood estimator, Fisher information, Standard errors.
Applied to:
Calibrate a Student t to daily returns
In Python:
Write the log-likelihood and maximise it
  1. 5.9.1 Likelihood and the maximum likelihood estimatorConcept35 min
  2. 5.9.2 Fisher information and standard errorsConcept35 min
  3. 5.9.3 Practice set: maximum likelihoodPractice set25 min
  4. 5.9.4 Case study: fitting a Student t to daily returnsCase study35 min
5.10Linear regression5 lessons
You will learn:
OLS in scalar and matrix form, Assumptions, t and F tests, Multiple regression.
Applied to:
Beta estimation; hedge ratios
In Python:
OLS by hand with NumPy and scipy.stats
  1. 5.10.1 Simple linear regressionConcept35 min
  2. 5.10.2 Regression in matrix formConcept30 min
  3. 5.10.3 Assumptions, t tests and F testsConcept35 min
  4. 5.10.4 Practice set: linear regressionPractice set25 min
  5. 5.10.5 Case study: estimating beta and hedge ratiosCase study35 min
5.11Discrete-time processes4 lessons
You will learn:
Random walks, Markov chains, The binomial tree as a process.
Applied to:
Multi-step binomial option pricing
In Python:
Build and price on a recombining tree
  1. 5.11.1 Random walksConcept30 min
  2. 5.11.2 Markov chainsConcept35 min
  3. 5.11.3 Practice set: discrete-time processesPractice set25 min
  4. 5.11.4 Case study: pricing on a multi-step binomial treeCase study35 min
6

Level 6 · About 14 hours

Differential equations

The ODE and PDE essentials, solved by hand and numerically, then used for bond prices and option-pricing grids.

0 of 20 lessons

6.1First-order ODEs4 lessons
You will learn:
Separable equations, Linear equations, Integrating factors.
Applied to:
Bond price under a deterministic short rate; continuous dividends
In Python:
scipy.integrate.solve_ivp against the exact solution
  1. 6.1.1 Differential equations, and separable equationsConcept35 min
  2. 6.1.2 Linear first-order equations and integrating factorsConcept35 min
  3. 6.1.3 Practice set: first-order equationsPractice set25 min
  4. 6.1.4 Case study: bond prices under a deterministic short rateCase study35 min
6.2Second-order linear ODEs4 lessons
You will learn:
Constant coefficients, Characteristic equation, General solution.
Applied to:
The perpetual American put
In Python:
Solve and plot the boundary-value problem
  1. 6.2.1 Constant coefficients and the characteristic equationConcept35 min
  2. 6.2.2 General solutions and boundary conditionsConcept35 min
  3. 6.2.3 Practice set: second-order equationsPractice set20 min
  4. 6.2.4 Case study: the perpetual American putCase study35 min
6.3PDE basics4 lessons
You will learn:
What a PDE is, Boundary, initial and terminal conditions.
Applied to:
Option value V(S, t) pinned down by its payoff at expiry
In Python:
Visualise a solution surface
  1. 6.3.1 What a partial differential equation isConcept30 min
  2. 6.3.2 Initial, boundary and terminal conditionsConcept30 min
  3. 6.3.3 Practice set: partial differential equationsPractice set20 min
  4. 6.3.4 Case study: an option value fixed by its payoffCase study35 min
6.4The heat equation4 lessons
You will learn:
Diffusion intuition, Fundamental solution, Change of variables.
Applied to:
Transform Black–Scholes into the heat equation
In Python:
Simulate diffusion of heat and of probability
  1. 6.4.1 Diffusion and the heat equationConcept30 min
  2. 6.4.2 The fundamental solutionConcept30 min
  3. 6.4.3 Practice set: the heat equationPractice set20 min
  4. 6.4.4 Case study: from Black–Scholes to the heat equationCase study35 min
6.5Finite differences for PDEs4 lessons
You will learn:
Explicit scheme, Implicit scheme, Crank–Nicolson, Stability.
Applied to:
Price a European option on a grid
In Python:
Match the closed form within 1 cent
  1. 6.5.1 The explicit scheme and stabilityConcept35 min
  2. 6.5.2 Implicit schemes and Crank–NicolsonConcept35 min
  3. 6.5.3 Practice set: finite differencesPractice set20 min
  4. 6.5.4 Case study: pricing a European option on a gridCase study35 min
PG

Proof Gym

About 19 hours, one session a week alongside the levels. Proofs live here, not in the lessons. Weekly practice in four formats (reorder the steps, fill the gap, find the error, write it yourself), so a proof-based course feels like a step, not a cliff.

Open the Proof Gym →

  1. PG1 · alongside Levels 0–1

    Logic and language

    Example: Negate "the fund beats the index every year" correctly

    1. PG1.1Statements and truth values
    2. PG1.2Implication, converse and contrapositive
    3. PG1.3Quantifiers: for every and there exists
    4. PG1.4Negating a statement
    5. PG1.5Stage review: for every ε there exists δ

  2. PG2 · alongside Levels 1–2

    Proof techniques

    Example: √2 is irrational; the exponent and log rules from Level 1, proved

    1. PG2.1Direct proof: even and odd numbers
    2. PG2.2The exponent rules, proved
    3. PG2.3The logarithm rules, proved
    4. PG2.4The contrapositive, and proof by cases
    5. PG2.5Proof by contradiction: √2 is irrational
    6. PG2.6Stage review: choosing a technique

  3. PG3 · alongside Levels 1–2

    Induction

    Example: The geometric-series and annuity formulas

    1. PG3.1Why induction works
    2. PG3.2The geometric series, proved
    3. PG3.3The annuity formula, proved
    4. PG3.4Strong induction
    5. PG3.5Stage review: two proofs of your own

  4. PG4 · alongside Level 2

    ε–δ limits

    Example: (1 + 1/n)ⁿ converges

    1. PG4.1What a limit means: ε and N
    2. PG4.2Limits of sums and products
    3. PG4.3Limits of functions: ε and δ
    4. PG4.4Supremum, infimum and completeness
    5. PG4.5Monotone sequences, and why (1 + 1/n)ⁿ converges
    6. PG4.6Stage review: limits from the definition

  5. PG5 · alongside Level 2

    Series and convergence

    Example: The Taylor series of eˣ converges for every x

    1. PG5.1A series is the limit of its partial sums
    2. PG5.2The comparison and ratio tests, proved
    3. PG5.3Absolute and conditional convergence
    4. PG5.4The series for eˣ converges for every x
    5. PG5.5Stage review: proving a series converges

  6. PG6 · alongside Levels 3–5

    Sets and functions

    Example: Why infinitely many coin tosses needs measure theory

    1. PG6.1Sets and set operations, proved
    2. PG6.2Injective, surjective and bijective maps
    3. PG6.3Countable sets
    4. PG6.4Uncountable sets, and infinitely many coin tosses
    5. PG6.5Stage review: sets, maps and sizes

32 lessons in 6 stages. Each one checks a claim with numbers first, then proves it.