Fixed-income securities, basic portfolio optimization, binomial method for options; Ito process and its applications in stock market, Black-Scholes equation and its solution; random numbers, transformation of random numbers and generating normal variates, Monte Carlo integration, pricing options by Monte Carlo simulation, variance reduction techniques, quasi-random numbers and quasi-Monte Carlo simulation; introduction to finite difference methods, explicit and implicit finite difference schemes, Crank-Nicolson method, free-boundary value problems for American options.

For further information see the academic catalog: IAM749 - Numerical Methods with Financial Applications

Probability, Random Processes, and Statistics; Markov Chains; Sampling and Monte Carlo Methods; Parameter Estimation; Uncertainty Propagation in Models; Stochastic Spectral Methods; Surrogate Models and Advanced Topics.

For further information see the academic catalog: IAM768 - Methods and Applications of Uncertainty Quantification

Real analysis, multivariable calculus and linear algebra (along with probability theory) constitute the mathematical foundations of mathematical finance. The objective of the course is to help the student gain an understanding of these subjects so that s/he can participate in research and learning activities (such as taking courses, reading literature, solving research/ applied problems) in mathematical finance and related fields.

For further information see the academic catalog: IAM527 - Advanced Calculus and Integration

First Variation: computing the first variation, Euler-Lagrange equation, extensions; Applications: brachistochrone, Lagrangian and Hamiltonian dynamics; Second Variation: computing the second variation, Ricatti equation, convexity and minimisers; Multivariable Variational Problems: eigenpairs, minimal surfaces, gradient flows; Optimal Control Theory: time-optimal linear control, Pontryagin Maximum Principle; Applications: linear-quadratic regulator, production-consumption, optimal harvesting; Dynamic Programming: Hamilton-Jacobi-Bellmann equation, general linear-quadratic regulator; Further Topics on Differential Games, Stochastic Control Theory.

For further information see the academic catalog: IAM773 - Dynamic Optimisation - Calculus of Variations and Optimal Control

Brief introduction to Statistical Learning: Regression versus Classification; Linear Regression: simple and multiple Linear Regression; Classification: Logistic Regression, Discriminant Analysis; Resampling Methods: Cross-Validation, the Bootstrap; Regularization: Subset Selection, Ridge Regression, the Lasso, Principle Components and Partial Least Squares Regression; Nonlinear Models: Polynomial; Splines; Generalized Additive Models; Tree-Based Models: Decision Trees, Random Forest, Boosting; Support Vector Machines; Unsupervised Learning: Principle Component Analysis, Clustering Methods.

For further information see the academic catalog: IAM557 - Statistical Learning and Simulation