# Courses and Curriculum

Students who want to specialize in Financial Mathematics are offered a wide range of different courses (not only from the Institute of Mathematical Finance).

## Prerequisites

- Analysis, Linear Algebra, ODEs
- Measure and Integration Theory (for MSc Finance students this is covered in Introduction to Measure Theoretic Probability)
- Introduction to Probability and Statistics, Stochastics I

Further recommendable other mathematical subjects (not necessary but maybe helpful):

- Functional Analysis
- PDEs
- Numerical Analysis (of ODEs, PDEs)

## Courses Offered Regularly

**Financial Mathematics I**(incl. DAV Supplement, every winter semester)

The first course providing in introduction into the modelling in financial markets in discrete time and the pricing and hedging of derivatives. It also gives a first introduction into the Black Scholes model and several other topics like interest rate models and derivatives, risk measures, portfolio optimization or the capital asset pricing model CAPM.**Stochastics II**(Stochastic Processes, every winter semester)

A comprehensive introduction into (the theory of) stochastic processes which are used in other lectures for modeling.**Financial Mathematics II**(every summer semester)

Thorough introduction into the modeling of financial markets in continuous time and the area of stochastic analysis beyond it. The topics covered are: stochastic integration, stochastic differential equations, (semi-)martingales, continuous-time financial market models, valuation and hedging of derivatives in complete and incomplete financial markets, stochastic volatility, introduction to interest rate markets in continuous time**WiMa-Praktikum II**(annually)

Hands on course using Matlab and the Bloomberg system on topics like hedging in incomplete markets, calibrations of market models to option price data, estimation of market implicit distributions, Fourier pricing, parameter estimation in jump-diffusion models.**Practical Financial Engineering**(MSc Finance, every summer semester)

Hands on course using Matlab or R and the Bloomberg system on topics like pricing and hedging of standard or complex derivative instruments, advanced stochastic simulation or numerical routines.**Seminar**(both at Bachelor and Master level)

various current topics please see under courses for the seminars offered previously.

## Specialized Courses Offered Irregularly (Examples)

**Time Series Analysis**Trends, seasonal effects, and stationarity, autocovariance and autocorrelation functions as a tool for analyzing dependencies in time, ARMA (auto regressive moving average) processes, statistical inference and forecasting methods**Nonlinear Time Series Analysis**

In this course we consider and analyse several classes of non-linear time series models, viz. GARCH, Markov-switching as well as threshold autoregressive time series models. To this end we study their common probabilistic and statistical concepts and theory (Markov chains with uncountable state space, stochastic recurrence equations, ergodicity and mixing). Finally, we derive and apply estimators for the model parameters and study the extremal behaviour of the models.**Statistics of Financial Data****Extreme Value Theory and Statistics**In many applications the biggest or smallest values of a sample are of particular relevance. For example, in finance and insurance – particularly in risk assessment and management – many decisions have to be based on the behaviour of the extreme values. We work out the classical extreme value theory for i.i.d. observations, which includes characterizing the possible limit distribution for extremal events and their domains of attraction.

Based on this we address the points over threshold method, where one is interested in the probability of exceeding a given (high) threshold. Finally, we discuss the related statistical estimation issues.

Moreover, applications of the theory in risk management (of banks and insurance companies) are discussed.**Interest Rate Models**

Different possible interest rates and some related financial contracts, ways of estimating the whole term structure based on the interest rates actually observable, analysis of some models for interest rates (short rate models, LIBOR market models and the Heath-Jarrow-Morton Methodology), forward measures, forward and futures contracts and consistent term structure parametrizations.**Lévy Processes and Financial Modelling with Jump Processes**

Stylized facts of financial data and the use of Lévy processes in finance, distributional and path-wise characterizations of Lévy processes, examples of Lévy processes and infinitely divisible distributions, stochastic integration and stochastic differential equations (SDEs) for general semi-martingales, stability of SDEs, continuous time Markov processes, Lévy-driven SDEs and their associated infinitesimal generators, partial integro-differential equations and their use in derivative pricing, incomplete financial market models in continuous time, exponential Lévy models in finance, stochastic volatility models, option pricing by integral transform techniques**Markov Chains**

Markov chains in discrete and continuous time with countable state space, in particular: Definition and elementary properties, examples, stopping times and strong Markov property, recurrence and transience, invariant distributions and limit distributions, classification of states,the generator in continuous time**Stochastic Modelling of Energy Markets****Stochastic Optimization with Applications****Numerical Finance**(usually taught by the Institute of Numerical Mathematics)

Prices of options cannot be computed by evaluating a simple formula if the market model is only slightly more complex than Black-Scholes or if option structures are more complex. How can you price options in such cases and how can you guarantee accuracy of the computations? Contents: Generation of random numbers, Monte-Carlo and Quasi-Monte-Carlo methods, numerical methods for the computation of European and American options: binomial, finite difference and finite element methods, numerical methods for the simulation of stochastic processes: numerical treatment of stochastic differential equations**Methods of Monte Carlo Simulation**(usually taught by the Institute of Stochastics)