Mathematics

This course introduces financial and commodity derivatives markets and their most commonly traded securities. Securities such as forwards, futures, and options have been traded on exchanges, as well as "over the counter," for decades. The emphasis of this course is on the pricing of such derivative securities. The course starts by looking at future and forward contracts to understand their properties and differences and determine how to value them. After a detailed study of the different types of options, the valuation method of binomial trees (based on the Cox, Ross, and Rubenstein paper of 1979) is discussed. The course then studies the model of a share price evolution introduced by Black, Scholes, and Merton in 1973, and derives the Black-Scholes model for valuing European call and put options on a non-dividend-paying stock. A brief introduction to probability theory is also provided in the course.

This course covers ordinary differential equations and partial differential equations. Topics include: first-order differential equations; second-order linear equations; systems of first-order linear equations; nonlinear systems and stability; separation of variables; boundary-value problems.

Pre-requisites: Calculus I, Calculus II, and Linear Algebra.

This course studies the principle elements of probability theory. Topics covered include: introduction to probability theory, randomized experiments; probability spaces, conditional probability and independence of events; dimensional random variables (distribution function, discrete and continuous variables, transformations, moments, characteristic function and moment generating function, notable distributions); multidimensional random variables (marginal and conditional distributions, independence, transformations, moments, characteristic function, correlation coefficient, notable distributions); Stochastic convergence (convergence of random variables, relationships and properties, laws of large numbers, central limit theorems). An understanding of analysis and algebra is recommended. Texts: V Quesada & A García Pérez. Lecciones de Cálculo de Probabilidades. Díaz de Santos, 1988. VK Rohatgi. An Introduction to Probability Theory and Mathematical Statistics. Wiley, 1976. R Vélez & V Hernández. Cálculo de Probabilidades 1. UNED, 1995. R Vélez. Cálculo de Probabilidades 2. Ediciones Académicas, 2004.

This course is an introduction to the theory and application of linear algebra. Students learn to draw diagrams that illustrate the basic operations of vector algebra in two and three dimensions, recognize the equations of lines and planes in two and three dimensions, perform the matrix computations outlined in the following, solve a linear system using Gaussian elimination, compute the inverse of an invertible matrix, find the orthogonal projection of a vector onto a hyperplane, compute the determinant of a square matrix, compute the characteristic equation of a matrix, find the eigenvectors corresponding to a given eigenvalue, and to diagonalize a diagonalizable matrix.

The course presents the fundamental statistical methods for extreme value analysis, discusses examples of applications regarding floods, storm damage, human life expectancy, and corrosion, provide practical use of the models, and points to some open problems and possible developments. Extreme value theory concerns mathematical modelling of random extreme events. Recent development has introduced mathematical models for extreme values and statistical methods for them. Extreme values are of interest in economics, safety and reliability, insurance mathematics, hydrology, meteorology, environmental sciences, and oceanography, as well as branches in statistics such as sequential analysis and robust statistics. The theory is used for flood monitoring, construction of oil rigs, and calculation of insurance premiums for re-insurance of storm damage. Often extreme values can lead to very large consequences, both financial and in the loss of life and property. At the same time the experience of really extreme events is always very limited. Extreme value statistics is therefore forced to difficult and uncertain extrapolations, but is, none the less, necessary in order to use available experience in order to solve important problems.

This course presents elementary linear analysis in a concrete setting, emphasizing specific techniques important to analysis and its applications. Topics include normed linear spaces; Banach spaces; Hilbert spaces; convexity; orthogonal expansions and their applications; and Riesz representation theorem for functionals.

Topics in this course include: bilinear mappings; complex differential functions; power series; complex contour integrals; Cauchy's theorem and integral formula; Taylor's theorem; zeros of analytic functions and Rouche's theorem; maximum modulus principle; singularities and Laurent series; poles and residues; and residue calculus.

This course provides an introduction to calculus. Emphasis is on an understanding of the basic concepts and techniques, and on developing the practical, computational skills to solve problems from a wide range of application areas. This course illustrates the methods learned by looking at real problems from different fields where these techniques can be applied and through this applied lens all students explore new facets of calculus and deepen their knowledge. The course discusses: functions, limits, and continuity; derivatives; rules of differentiation; maxima and minima; implicit differentiation and rates Integration; definite integrals; applications of integration; improper integrals; differential equations. Prerequisites for this course include substantial high school experience in Mathematics.