Humboldt-Universität zu Berlin - Statistik

Course Outline

This CASE - Center for Applied Statistics and Economics 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.
"Advanced Methods in Finance" consists of four parts: Value at Risk, Credit Risk, Implied Volatility and Econometrics. In the first part we treat the Approximation of the Value at Risk in conditional Gaussian Models and show how the VaR can be calculated using copulae.
The second part starts with an analysis of rating migration probabilities We then quantify the risk of yield spread changes via historical simulations. This part is completed by an analysis of the sensitivity of risk measures to changes in the dependency structure between single positions of a portfolio.
The third 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.

Contents

Copulae and Value-at-Risk

• Conditional Gaussian Models
• Variance Reduction Techniques in Monte Carlo Simulation
• Copulae - definition, examples, properties
• Computing Value-at-Risk with Copulae

Spread Risk and Historical Simulation

• Descriptive Statistics of YieldSpread Time Series
• Historical simulation and Value-at-Risk
• VaR Estimation and backtesting with XploRe

The Analysis of Implied Volatilities

• The Implied Volatility Surface
• Dynamic Analysis

Simulation Based Option Pricing

• Simulation techniques for option pricing
• Quasi Monte Carlo techniques
• Pricing options with simulation techniques
• Stochastic Differencial Equations

Integrated Time Series Models

• Unit Roots in Time Series Models
• ARIMA Models - Main Properties, Estimation, Prediction
• FARIMA Models - Main Properties, Long-Range Dependence
• SARIMA Models - Main Properties, Estimation, Prediction

ARMA models with finite and infinite variance innovations

• ARMA models with Gaussian innovations - properties dependencestructure
• ARMA models with hyperbolic noise - modeling electricity loads in California
• ARMA models with stable innovations - properties, dependence structure

PARMA Models

• Main Properties
• Estimation and Prediction
• Spectral Analysis of PARMA Models
• Applications

Multivariate Time Series

• Main properties of Multivariate Time Series
• Estimation of the Mean and Covariance function
• Multivariate ARMA Models
• Modelling and Prediction with Multivaraite Processes

Premiums in the individual and collective risk models

• Premium calculation principles
• Properties of premiums
• Premiums in the individual risk model
• Premiums in the collective risk model

Pure risk premiums under deductibles

• Properties of the deductibles
• Limited expected value function
• Franchise deductible - payment function, properties, pure risk premium
• Fix amount deductible - payment function, properties, pure risk premium
• Limited proportional deductible - payment function, properties, pure risk premium
• Disappearing deductible - payment function, properties, pure risk premium
• Premiums under deductibles for given loss distributions
• Premium under a limit of indemnity