## Advanced Methods in Quantitative Finance (VL)

## Course Outline

This course is designed for students and researchers who want to develop professional skills in modern quantitative finance. It is offered to interested students who have had some experience with probability, statistics and software applications but have not had advanced courses in mathematical finance. Although the course assumes only a modest background it moves quickly between different fields of applications and in the end, the participant can expect to have theoretical and computational tools that are deep enough and rich enough to be relied on throughout future professional careers.

The compulsory textbook is readable for the graduate student in financial engineering as well as for the inexperienced newcomer to quantitative finance who wants to get a grip on modern statistical tools in financial data analysis. The experienced reader with a bright knowledge of mathematical finance will probably skip some sections but will hopefully enjoy the various computational tools of the presented techniques. A graduate student might think that some of the econometric techniques are well known. The mathematics of risk management and volatility dynamics will certainly introduce him into the rich realm of quantitative financial data analysis.

The computer inexperienced user of this course is softly introduced into the interactive course concept and will certainly enjoy the various practical examples. The textbook is an e-book which is designed as an interactive document: a stream of text and information with various hints and links to additional tools and features.

The course "Advanced Methods in Quantitative Finance" consists of four parts: Preliminaries, Value at Risk, Credit Risk and Implied Volatility. The first part of the course is a quick refresher of the most important concepts needed for this course. In the second part we treat the Approximation of the Value at Risk in conditional Gaussian Models, show how the VaR can be calculated using copulae and we discuss techniques of risk assessment beyond VaR. We then quantify the risk of yield spread changes via historical simulations. The third part deals with an analysis of rating migration probabilities. The forth part is devoted to the analysis of implied volatilities and their dynamics. We start with an analysis of the implied volatility surface and show how common PCA can be applied to model the dynamics of the surface. In the next two chapters we estimate the risk neutral state price density from observed option prices and the corresponding implied volatilities. We then calculate implied binomial trees to estimate the SPD, and present a method based on a local polynomial estimation of the implied volatility and its derivatives. The proposed methods are used to develop trading strategies based on the comparison of the historical SPD and the one implied by option prices.

The registration in the respective Moodle course is obligatory.

## Contents

##### I. Preliminaries

- Basic concepts of probability theory
- Basic concepts of stochastic processes
- Diusions
- Maximum Likelihood estimation (MLE)
- Estimation of diusions with maximum likelihood

##### II. Value at Risk

- Value at Risk
- Modeling Dependencies with Copulae
- Risk Assessment beyond Value at Risk
- Quantication of Spread Risk by Means of Historical Simulation
- Inflation Expectation and Risk Premiums
- Value at Risk in High Dimensional Systems

##### III. Credit Risk

- Rating Migrations

##### IV. Implied Volatility

- The Analysis of Implied Volatilities
- State Pricing Density Estimation
- Simulation Based Option Pricing