Statistics

This course examines mathematical concepts and tools used in the finance industry, in particular stochastic models and techniques used for financial modelling and derivative pricing.

This course covers basic elements of stochastic processes such as time, state, increments, stationarity and Markovian property. Students develop properties and limit theorems of discrete-time Markov chain and branching processes and then establish key results for the Poisson process and continuous-time Markov chains, such as the memoryless property, super positioning, thinning, Kolmogorov's equations and limiting probabilities.

This course addresses the most important areas in optimization and studies the most common techniques. First, the optimization of unconstrained continuous functions in several variables is considered. Some notions are: partial derivatives; the gradient and the Hessian; stationary points; minima, maxima and saddle points; local and global optima. Techniques to compute optima range from analytical and algebraic techniques (i.e., solving systems of equations) to iterative and approximate numerical techniques (e.g., gradient methods and hill climbing, Newton and quasi-Newton methods, and several others). The course focuses on a selection of these. An important class of functions to consider is that of least squares criteria. Students consider both linear and nonlinear least squares problems and suitable iterative techniques to solve them. Linear least squares problems are often encountered in the context of fitting a model to measurement data. They also allow one to rephrase the problem of solving a nonlinear system of equations as an optimization problem, while the converse is possible too. Second, optimization problems subject to a given set of constraints are addressed. A well-known such class consists of linear optimization functions subject to linear equality or inequality constraints: the class of linear programs. The problem of fitting a linear model to measurement data using the criterion of least absolute deviations can be reformulated as a linear program. Several methods are available to solve such problems, including active set methods and the simplex algorithm, but also interior point methods and primal-dual methods. The Kuhn-Tucker conditions for optimality are discussed. For the optimization of nonlinear functions subject to nonlinear constraints, the course addresses the Lagrange multiplier method. To demonstrate the various optimization problems and solution techniques, the course provides many examples and exercises. To demonstrate the wide range of applicability, these are taken from different fields of science and engineering. To become acquainted with optimization techniques, one computer class is organized in which the basics of the software package Matlab are presented. Prerequisites for this course are calculus and linear algebra.

This course introduces the main econometric methods and techniques used in empirical finance. The course brings together different type of knowledge: finance theory, statistics, and programming. Students learn to use software to specify, estimate, and simulate model of financial data to be used for asset allocation, risk measurement, and risk management. The course discusses topics including basic knowledge in finance, statistics, and probability; introduction to programming; returns: definitions and interpretation, measurement, data collection, and analysis; modeling and simulating returns; estimating linear models of returns; interpreting regression results; and high-order risk sources. Students are required to have completed a statistics course as a prerequisite.

In this course, students use probability as a formalism for handling uncertainty, design simple probability models for prediction, make basic statistical analyses of data, and critically assess and interpret others' analyses.

This course introduces a set of actionable tools and methods for the redaction of a master thesis. It is more specifically focused on new emerging digital practices that significantly ease the implementation of core tasks. A first series of five sessions are devoted to reference research. They cover the use of general and academic search engines, the creation of structured bibliography with Zotero, the reuse of free-licensed content and the writing of state of the art synthesis. Six sessions focus on the management and analysis of databases and corpora, building on the principles of "tidy data" as implemented in the R language and the Tidyverse extensions. A last focus is given to text mining techniques as a way to map a corpus of scholar references.

This course teaches quantitative research methods in the political and social sciences. The course covers descriptive statistics, data visualization, data access, probability, sampling, hypothesis testing, inferential statistics, and an introduction linear regression. Students are introduced to the R statistical software and work with data used in academic research.