Suchergebnis: Katalogdaten im Frühjahrssemester 2021

Nummer	Titel	Typ	ECTS	Umfang	Dozierende
Mathematik Master
Wahlfächer Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Weitere Gebiete
401-3502-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	2 KP	4A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	3 KP	6A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-21L	Reading Course To start an individual reading course, contact an authorised supervisor Link and register your reading course in myStudies.	W	4 KP	9A	Betreuer/innen
Kurzbeschreibung	In diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel

Wahlfächer aus Bereichen der angewandten Mathematik ... vollständiger Titel: Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Auswahl: Numerische Mathematik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4658-00L	Computational Methods for Quantitative Finance: PDE Methods	W	6 KP	3V + 1U	C. Marcati, A. Stein
Kurzbeschreibung	Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB and Python programming and knowledge of numerical mathematics at ETH BSc level.
Lernziel	Introduce the main methods for efficient numerical valuation of derivative contracts in a Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility models. Develop implementation of pricing methods in MATLAB and Python. Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Inhalt	1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic volatility models. 2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees. European contracts. 3. Finite Difference methods for Asian, American and Barrier type contracts. 4. Finite element methods for European and American style contracts. 5. Pricing under local and stochastic volatility in Black-Scholes Markets. 6. Finite Element Methods for option pricing under Levy processes. Treatment of integrodifferential operators. 7. Stochastic volatility models for Levy processes. 8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and stochastic volatility models in Black Scholes and Levy markets. 9. Introduction to sparse grid option pricing techniques.
Skript	There will be english lecture notes as well as MATLAB or Python software for registered participants in the course.
Literatur	Main reference (course text): N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013. Supplementary texts: R. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004. Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005. D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008. J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.
Voraussetzungen / Besonderes	Knowledge of Numerical Analysis/ Scientific Computing Techniques corresponding roughly to BSc MATH or BSc RW/CSE at ETH is expected. Basic programming skills in MATLAB or Python are required for the exercises, and are _not_ taught in this course.
401-4656-21L	Deep Learning in Scientific Computing Aimed at students in a Master's Programme in Mathematics, Engineering and Physics.	W	6 KP	2V + 1U	S. Mishra
Kurzbeschreibung	Machine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs.
Lernziel	The objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods.
Inhalt	A selection of the following topics will be presented in the lectures. 1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces. 2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets. 3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors. 4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error. 5. Uncertainty Quantification for PDEs with supervised learning algorithms. 6. Deep Neural Networks as Reduced order models and prediction of solution fields. 7. Active Learning algorithms for PDE constrained optimization. 8. Recurrent Neural Networks and prediction of time series for dynamical systems. 9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs. 10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation. All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others.
Skript	Lecture notes will be provided at the end of the course.
Literatur	All the material in the course is based on research articles written in last 1-2 years. The relevant references will be provided.
Voraussetzungen / Besonderes	The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications. Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful.
401-4652-21L	Nonlocal Inverse Problems	W	4 KP	2V	J. Railo
Kurzbeschreibung	This course is an introduction to the Calderón problem and nonlocal inverse problems for the fractional Schrödinger equation. These are examples of nonlinear inverse problems. The classical Calderón problem models electrical impedance tomography (EIT) and fractional operators appear, for example, in some mathematical models in finance.
Lernziel	Students become familiar with the Calderón problem and some nonlocal phenomena related to the fractional Laplacian. Advanced students should be able to read research articles on the fractional Calderón problems after the course.
Inhalt	In the beginning of the course, we will introduce some basic theory for the classical Calderón problem. The focus of the course will be in the study of nonlocal inverse problems for the fractional Schrödinger equation with lower order perturbations. We discuss necessary preliminaries on Sobolev spaces, Fourier analysis, functional analysis and theory of PDEs. Our scope will be in the uniqueness properties. Classical Calderón problem (about 1/3): Conductivity and Schrödinger equations, Dirichlet-to-Neumann maps, Cauchy data, and related boundary value inverse problems. The methods include, for example, complex geometric optics (CGO) solutions. Fractional Calderón problem (about 2/3): Nonlocal unique continuation principles (UCP), Runge approximation properties, and uniqueness for the fractional Calderón problem. The methods include, for example, Caffarelli-Silvestre extensions, the fractional Poincaré inequality and Riesz transforms.
Skript	Lecture notes and exercises
Literatur	1. M. Salo: Calderón problem. Lecture notes, University of Helsinki (2008). (Available at http://users.jyu.fi/~salomi/index.html.) 2. T. Ghosh, M. Salo, G. Uhlmann: The Calderón problem for the fractional Schrödinger equation. Analysis & PDE 13 (2020), no. 2, 455-475. 3. A. Rüland, M. Salo: The fractional Calderón problem: low regularity and stability. Nonlinear Analysis 193 (2020), special issue "Nonlocal and Fractional Phenomena", 111529. 4. Other literature will be specified in the course.
Voraussetzungen / Besonderes	Functional Analysis I & II or similar knowledge. Any additional knowledge of Fourier analysis, Sobolev spaces, distributions and PDEs will be an asset.
401-3426-21L	Time-Frequency Analysis	W	4 KP	2G	R. Alaifari
Kurzbeschreibung	This course gives a basic introduction to time-frequency analysis from the viewpoint of applied harmonic analysis.
Lernziel	By the end of the course students should be familiar with the concept of the short-time Fourier transform, the Bargmann transform, quadratic time-frequency representations (ambiguity function and Wigner distribution), Gabor frames and modulation spaces. The connection and comparison to time-scale representations will also be subject of this course.
Inhalt	Time-frequency analysis lies at the heart of many applications in signal processing and aims at capturing time and frequency information simultaneously (as opposed to the classical Fourier transform). This course gives a basic introduction that starts with studying the short-time Fourier transform and the special role of the Gauss window. We will visit quadratic representations and then focus on discrete time-frequency representations, where Gabor frames will be introduced. Later, we aim at a more quantitative analysis of time-frequency information through modulation spaces. At the end, we touch on wavelets (time-scale representation) as a counterpart to the short-time Fourier transform.
Literatur	Gröchenig, K. (2001). Foundations of time-frequency analysis. Springer Science & Business Media.
Voraussetzungen / Besonderes	Functional analysis, Fourier analysis, complex analysis, operator theory

Auswahl: Wahrscheinlichkeitstheorie, Statistik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-4611-21L	Rough Path Theory	W	4 KP	2V	A. Allan, J. Teichmann
Kurzbeschreibung	The aim of this course is to provide an introduction to the theory of rough paths, with a particular focus on their integration theory and associated rough differential equations, and how the theory relates to and enhances the field of stochastic calculus.
Lernziel	Our first motivation will be to understand the limitations of classical notions of integration to handle paths of very low regularity, and to see how the rough integral succeeds where other notions fail. We will construct rough integrals and establish solutions of differential equations driven by rough paths, as well as the continuity of these objects with respect to the paths involved, and their consistency with stochastic integration and SDEs. Various applications and extensions of the theory will then be discussed.
Skript	Lecture notes will be provided by the lecturer.
Literatur	P. K. Friz and M. Hairer, A course on rough paths with an introduction to regularity structures, Springer (2014). P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths, Cambridge University Press (2010).
Voraussetzungen / Besonderes	The aim will be to make the course as self-contained as possible, but some knowledge of stochastic analysis is highly recommended. The course “Brownian Motion and Stochastic Calculus” would be ideal, but not strictly required.
401-4626-00L	Advanced Statistical Modelling: Mixed Models Findet dieses Semester nicht statt.	W	4 KP	2V	M. Mächler
Kurzbeschreibung	Mixed Models = (*\| generalized\| non-) linear Mixed-effects Models, extend traditional regression models by adding "random effect" terms. In applications, such models are called "hierarchical models", "repeated measures" or "split plot designs". Mixed models are widely used and appropriate in an aera of complex data measured from living creatures from biology to human sciences.
Lernziel	- Becoming aware how mixed models are more realistic and more powerful in many cases than traditional ("fixed-effects only") regression models. - Learning to fit such models to data correctly, critically interpreting results for such model fits, and hence learning to work the creative cycle of responsible statistical data analysis: "fit -> interpret & diagnose -> modify the fit -> interpret & ...." - Becoming aware of computational and methodological limitations of these models, even when using state-of-the art software.
Inhalt	The lecture will build on various examples, use R and notably the `lme4` package, to illustrate concepts. The relevant R scripts are made available online. Inference (significance of factors, confidence intervals) will focus on the more realistic unbalanced situation where classical (ANOVA, sum of squares etc) methods are known to be deficient. Hence, Maximum Likelihood (ML) and its variant, "REML", will be used for estimation and inference.
Skript	We will work with an unfinished book proposal from Prof Douglas Bates, Wisconsin, USA which itself is a mixture of theory and worked R code examples. These lecture notes and all R scripts are made available from https://github.com/mmaechler/MEMo
Literatur	(see web page and lecture notes)
Voraussetzungen / Besonderes	- We assume a good working knowledge about multiple linear regression ("the general linear model') and an intermediate (not beginner's) knowledge about model based statistics (estimation, confidence intervals,..). Typically this means at least two classes of (math based) statistics, say 1. Intro to probability and statistics 2. (Applied) regression including Matrix-Vector notation Y = X b + E - Basic (1 semester) "Matrix calculus" / linear algebra is also assumed. - If familiarity with [R](https://www.r-project.org/) is not given, it should be acquired during the course (by the student on own initiative).
401-4627-00L	Empirical Process Theory and Applications	W	4 KP	2V	S. van de Geer
Kurzbeschreibung	Empirical process theory provides a rich toolbox for studying the properties of empirical risk minimizers, such as least squares and maximum likelihood estimators, support vector machines, etc.
Lernziel
Inhalt	In this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
401-4632-15L	Causality	W	4 KP	2G	C. Heinze-Deml
Kurzbeschreibung	In statistics, we are used to search for the best predictors of some random variable. In many situations, however, we are interested in predicting a system's behavior under manipulations. For such an analysis, we require knowledge about the underlying causal structure of the system. In this course, we study concepts and theory behind causal inference.
Lernziel	After this course, you should be able to - understand the language and concepts of causal inference - know the assumptions under which one can infer causal relations from observational and/or interventional data - describe and apply different methods for causal structure learning - given data and a causal structure, derive causal effects and predictions of interventional experiments
Voraussetzungen / Besonderes	Prerequisites: basic knowledge of probability theory and regression
401-6102-00L	Multivariate Statistics Findet dieses Semester nicht statt.	W	4 KP	2G	keine Angaben
Kurzbeschreibung	Multivariate Statistics deals with joint distributions of several random variables. This course introduces the basic concepts and provides an overview over classical and modern methods of multivariate statistics. We will consider the theory behind the methods as well as their applications.
Lernziel	After the course, you should be able to: - describe the various methods and the concepts and theory behind them - identify adequate methods for a given statistical problem - use the statistical software "R" to efficiently apply these methods - interpret the output of these methods
Inhalt	Visualization / Principal component analysis / Multidimensional scaling / The multivariate Normal distribution / Factor analysis / Supervised learning / Cluster analysis
Skript	None
Literatur	The course will be based on class notes and books that are available electronically via the ETH library.
Voraussetzungen / Besonderes	Target audience: This course is the more theoretical version of "Applied Multivariate Statistics" (401-0102-00L) and is targeted at students with a math background. Prerequisite: A basic course in probability and statistics. Note: The courses 401-0102-00L and 401-6102-00L are mutually exclusive. You may register for at most one of these two course units.
401-4637-67L	On Hypothesis Testing	W	4 KP	2V	F. Balabdaoui
Kurzbeschreibung	This course is a review of the main results in decision theory.
Lernziel	The goal of this course is to present a review for the most fundamental results in statistical testing. This entails reviewing the Neyman-Pearson Lemma for simple hypotheses and the Karlin-Rubin Theorem for monotone likelihood ratio parametric families. The students will also encounter the important concept of p-values and their use in some multiple testing situations. Further methods for constructing tests will be also presented including likelihood ratio and chi-square tests. Some non-parametric tests will be reviewed such as the Kolmogorov goodness-of-fit test and the two sample Wilcoxon rank test. The most important theoretical results will reproved and also illustrated via different examples. Four sessions of exercises will be scheduled (the students will be handed in an exercise sheet a week before discussing solutions in class).
Literatur	- Statistical Inference (Casella & Berger) - Testing Statistical Hypotheses (Lehmann and Romano)

Auswahl: Finanz- und Versicherungsmathematik
Nummer	Titel	Typ	ECTS	Umfang	Dozierende
401-3629-00L	Quantitative Risk Management	W	4 KP	2V + 1U	P. Cheridito
Kurzbeschreibung	This course introduces methods from probability theory and statistics that can be used to model financial risks. Topics addressed include loss distributions, risk measures, extreme value theory, multivariate models, copulas, dependence structures and operational risk.
Lernziel	The goal is to learn the most important methods from probability theory and statistics used in financial risk modeling.
Inhalt	1. Introduction 2. Basic Concepts in Risk Management 3. Empirical Properties of Financial Data 4. Financial Time Series 5. Extreme Value Theory 6. Multivariate Models 7. Copulas and Dependence 8. Operational Risk
Skript	Course material is available on https://people.math.ethz.ch/~patrickc/qrm
Literatur	Quantitative Risk Management: Concepts, Techniques and Tools AJ McNeil, R Frey and P Embrechts Princeton University Press, Princeton, 2015 (Revised Edition) http://press.princeton.edu/titles/10496.html
Voraussetzungen / Besonderes	The course corresponds to the Risk Management requirement for the SAA ("Aktuar SAV Ausbildung") as well as for the Master of Science UZH-ETH in Quantitative Finance.
401-3923-00L	Selected Topics in Life Insurance Mathematics	W	4 KP	2V	M. Koller
Kurzbeschreibung	Stochastic Models for Life insurance 1) Markov chains 2) Stochastic Processes for demography and interest rates 3) Cash flow streams and reserves 4) Mathematical Reserves and Thiele's differential equation 5) Theorem of Hattendorff 6) Unit linked policies
Lernziel
401-3917-00L	Stochastic Loss Reserving Methods	W	4 KP	2V	R. Dahms
Kurzbeschreibung	Loss Reserving is one of the central topics in non-life insurance. Mathematicians and actuaries need to estimate adequate reserves for liabilities caused by claims. These claims reserves have influence all financial statements, future premiums and solvency margins. We present the stochastics behind various methods that are used in practice to calculate those loss reserves.
Lernziel	Our goal is to present the stochastics behind various methods that are used in prctice to estimate claim reserves. These methods enable us to set adequate reserves for liabilities caused by claims and to determine prediction errors of these predictions.
Inhalt	We will present the following stochastic claims reserving methods/models: - Stochastic Chain-Ladder Method - Bayesian Methods, Bornhuetter-Ferguson Method, Credibility Methods - Distributional Models - Linear Stochastic Reserving Models, with and without inflation - Bootstrap Methods - Claims Development Result (solvency view) - Coupling of portfolios
Literatur	M. V. Wüthrich, M. Merz, Stochastic Claims Reserving Methods in Insurance, Wiley 2008.
Voraussetzungen / Besonderes	The exams ONLY take place during the official ETH examination periods. This course will be held in English and counts towards the diploma "Aktuar SAV". For the latter, see details under www.actuaries.ch. Basic knowledge in probability theory is assumed, in particular conditional expectations.
401-3956-00L	Economic Theory of Financial Markets	W	4 KP	2V	M. V. Wüthrich
Kurzbeschreibung	This lecture provides an introduction to the economic theory of financial markets. It presents the basic financial and economic concepts to insurance mathematicians and actuaries.
Lernziel	This lecture aims at providing the fundamental financial and economic concepts to insurance mathematicians and actuaries. It focuses on portfolio theory, cash flow valuation and deflator techniques.
Inhalt	We treat the following topics: - Fundamental concepts in economics - Portfolio theory - Mean variance analysis, capital asset pricing model - Arbitrage pricing theory - Cash flow theory - Valuation principles - Stochastic discounting, deflator techniques - Interest rate modeling - Utility theory
Voraussetzungen / Besonderes	The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch. Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3936-00L	Data Analytics for Non-Life Insurance Pricing	W	4 KP	2V	C. M. Buser, M. V. Wüthrich
Kurzbeschreibung	We study statistical methods in supervised learning for non-life insurance pricing such as generalized linear models, generalized additive models, Bayesian models, neural networks, classification and regression trees, random forests and gradient boosting machines.
Lernziel	The student is familiar with classical actuarial pricing methods as well as with modern machine learning methods for insurance pricing and prediction.
Inhalt	We present the following chapters: - generalized linear models (GLMs) - generalized additive models (GAMs) - neural networks - credibility theory - classification and regression trees (CARTs) - bagging, random forests and boosting
Skript	The lecture notes are available from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2870308
Voraussetzungen / Besonderes	This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch Good knowledge in probability theory, stochastic processes and statistics is assumed.
401-4920-00L	Market-Consistent Actuarial Valuation Findet dieses Semester nicht statt.	W	4 KP	2V	M. V. Wüthrich
Kurzbeschreibung	Introduction to market-consistent actuarial valuation. Topics: Stochastic discounting, full balance sheet approach, valuation portfolio in life and non-life insurance, technical and financial risks, risk management for insurance companies.
Lernziel	Goal is to give the basic mathematical tools for describing insurance products within a financial market and economic environment and provide the basics of solvency considerations.
Inhalt	In this lecture we give a full balance sheet approach to the task of actuarial valuation of an insurance company. Therefore we introduce a multidimensional valuation portfolio (VaPo) on the liability side of the balance sheet. The basis of this multidimensional VaPo is a set of financial instruments. This approach makes the liability side of the balance sheet directly comparable to its asset side. The lecture is based on four sections: 1) Stochastic discounting 2) Construction of a multidimensional Valuation Portfolio for life insurance products (with guarantees) 3) Construction of a multidimensional Valuation Portfolio for a run-off portfolio of a non-life insurance company 4) Measuring financial risks in a full balance sheet approach (ALM risks)
Literatur	Market-Consistent Actuarial Valuation, 3rd edition. Wüthrich, M.V. EAA Series, Springer 2016. ISBN: 978-3-319-46635-4 Wüthrich, M.V., Merz, M. Claims run-off uncertainty: the full picture. SSRN Manuscript ID 2524352 (2015). England, P.D, Verrall, R.J., Wüthrich, M.V. On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins. Insurance: Mathematics and Economics 85 (2019), 74-88. Wüthrich, M.V., Embrechts, P., Tsanakas, A. Risk margin for a non-life insurance run-off. Statistics & Risk Modeling 28 (2011), no. 4, 299--317. Financial Modeling, Actuarial Valuation and Solvency in Insurance. Wüthrich, M.V., Merz, M. Springer Finance 2013. ISBN: 978-3-642-31391-2 Cheridito, P., Ery, J., Wüthrich, M.V. Assessing asset-liability risk with neural networks. Risks 8/1 (2020), article 16.
Voraussetzungen / Besonderes	The exams ONLY take place during the official ETH examination period. This course will be held in English and counts towards the diploma of "Aktuar SAV". For the latter, see details under www.actuaries.ch. Knowledge in probability theory, stochastic processes and statistics is assumed.
401-3888-00L	Introduction to Mathematical Finance Ein verwandter Kurs ist 401-3913-01L Mathematical Foundations for Finance (3V+2U, 4 ECTS-KP). Obwohl beide Kurse unabhängig voneinander belegt werden können, darf nur einer ans gesamte Mathematik-Studium (Bachelor und Master) angerechnet werden.	W	10 KP	4V + 1U	D. Possamaï
Kurzbeschreibung	This is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation. We prove the fundamental theorem of asset pricing and the hedging duality theorems, and also study convex duality in utility maximization.
Lernziel	This is an introductory course on the mathematics for investment, hedging, portfolio management, asset pricing and financial derivatives in discrete-time financial markets. We discuss arbitrage, completeness, risk-neutral pricing and utility maximisation, and maybe other topics. We prove the fundamental theorem of asset pricing and the hedging duality theorems in discrete time, and also study convex duality in utility maximization.
Inhalt	This course focuses on discrete-time financial markets. It presumes a knowledge of measure-theoretic probability theory (as taught e.g. in the course "Probability Theory"). The course is offered every year in the Spring semester. This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II. For an overview of courses offered in the area of mathematical finance, see Link.
Skript	The course is based on different parts from different textbooks as well as on original research literature. Lecture notes will not be available.
Literatur	Literature: Michael U. Dothan, "Prices in Financial Markets", Oxford University Press Hans Föllmer and Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time", de Gruyter Marek Capinski and Ekkehard Kopp, "Discrete Models of Financial Markets", Cambridge University Press Robert J. Elliott and P. Ekkehard Kopp, "Mathematics of Financial Markets", Springer
Voraussetzungen / Besonderes	A related course is "Mathematical Foundations for Finance" (MFF), 401-3913-01. Although both courses can be taken independently of each other, only one will be given credit points for the Bachelor and the Master degree. In other words, it is also not possible to earn credit points with one for the Bachelor and with the other for the Master degree. This course is the first of a sequence of two courses on mathematical finance. The second course "Mathematical Finance" (MF II), 401-4889-00, focuses on continuous-time models. It is advisable that the present course, MF I, is taken prior to MF II. For an overview of courses offered in the area of mathematical finance, see Link.

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