# Mathematics Refresher

This course reviews the mathematical foundations necessary to absorb the body of knowledge of the ARPM Certificate

Univariate calculus

Derivative and chain rule (instantaneous forward rate)

Monotonicity (trading strategies in the Almgren-Chriss model)

Concavity/convexity (concavity/convexity of an index of satisfaction)

Taylor polynomial (Taylor approximations)

Integral (continuous rebalancing limit)

Fundamental theorem of calculus (transformation of a random variable)

Change of variable (derivation of the Black-Scholes- Merton formula)

Integration by parts formula (the P&L of trading strategy)

Multivariate calculus

Multivariate function (representation of a distribution; multivariate cdf)

Partial derivative, directional derivative, and gradient (Greeks; derivatives)

Iterated integral (marginalization; e.g. uniform bivariate)

Chain rule for multivariate function (Euler decomposition)

Convexity and Hessian matrix (convexity analysis)

Calculus

Change of variables and Jacobian (pdf of an invertible function; pdf of a copula)

Relative extremum (mode)

Unconstrained optimization problem (mode)

Second derivative test (mean-variance trade-off)

Constrained optimization problem (linear programming; convex programming)

The method of Lagrange multiplier (views processing)

Derivative of functional (influence function; influence function of maximum likelihood estimators)

Variational problem (mean-variance optimization problem in the Almgren-Chriss model)

Euler-Lagrange equation (P&L optimization: Almgren-Chriss model)

Linear algebra

Matrices and vectors (portfolio P&L scenarios; stochastic discount factor)

Matrix manipulations (matrix algebra; matrix calculus)

Linear independence, spanning, and basis (market completeness)

Inverse of a matrix (pseudo-inverse)

Trace and determinant of a square matrix (properties of trace; )

Diagonalization of symmetric matrices (spectral theorem example)

The Gram-Schmidt procedure (Gram-Schmidt)

Affine transformations (Taylor expansion of characteristic function)

Positive definite matrices (covariance; modal square-dispersion)

Geometry of random variables

Fourier transform