10-hours course

This course reviews the mathematical foundations necessary to absorb the Body of Knowledge of the ARPM Certificate
Derivative and chain rule (instantaneous forward rate)
Taylor polynomial (Taylor approximations)
Fundamental theorem of calculus (transformation of a random variable)
Integration by parts formula (the P&L of trading strategy)
Partial derivative, directional derivative, and gradient (Derivatives; Greeks)
Iterated integral (marginalization, e.g. uniform bivariate)
Chain rule for multivariate function (Euler decomposition)
Convexity and Hessian matrix (convexity analysis)
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 multipliers (views processing)
Functional (e.g. mean; variance; median; mode)
Euler-Lagrange equation (P&L optimization: Almgren-Chriss model)
Row operations and rank of a matrix
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)
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