MaPhyStoDanish National Research Foundation Network in Mathematical Physics and Stochastics Funded by The Danish National Research Foundation |
DYNSTOCHStatistical Methods for Dynamical Stochastic Models EU Research Training |
Adept ScientificThe Technical Computing People |

The Concentrated Advanced Course will be given at the Institute
of Mathematical Sciences, University of Copenhagen, **
HC Ørsted Institute, Auditorium 4. See the Infomation below
how to get to the Institute. **
There will be 6 hours of lectures per day, 4 given by Alexander
McNeil. The course is
organized by Søren Asmussen (University of Aarhus),
Jeff Collamore (University of Copenhagen),
Martin Jacobsen (University of Copenhagen),
Thomas Mikosch (University of Copenhagen) and
Michael Sørensen (University of Copenhagen)

**Abstract
** Quantitative methodology is an increasingly important component of
risk management in financial institutions. Financial risk management
presents an extremely interesting area of application for statistics
with many new challenges. Whereas much of traditional statistics
concerns the average, the normal and the expected, risk management has
more to do with the extreme, the abnormal and the unexpected. Central
technical issues will be modelling the volatility of financial return
time series, modelling extreme values and modelling dependent risks.
We will examine methods relevant for both market and credit risk
management.

**The Basics of Quantitative Risk Management**- financial risks and losses, risk measures, VaR, expected
shortfall or conditional VaR, risk factors and mappings

- financial risks and losses, risk measures, VaR, expected
shortfall or conditional VaR, risk factors and mappings
**Standard Statistical Methods**- variance-covariance, historical simulation, Monte Carlo, limits
of standard methods

**Fundamentals of Modelling Dependent Risks**- basic multivariate statistics, multivariate normal
distribution, multivariate normal mixture models, elliptical
distributions, hyperbolic distributions

**Modelling Financial Time Series**- basic time series concepts, empirical properties (stylized
facts) of financial time series, arguments for stochastic volatility,
ARCH and GARCH models

**Basic Topics in Extreme Value Theory**- maxima and worst case losses, extreme value distributions,
generalised Pareto distribution (GPD), peaks-over -thresholds (POT)
method, modelling excess losses and heavy tails, estimation of
quantiles (VaR) and expected shortfall

**Advanced Topics in EVT and Time Series**- Outperforming historical simulation with EVT, EVT for dependent time series, EVT in a stochastic volatility framework
**Copulas, Correlation and Dependent Extreme Values**- introduction to copulas, useful copula families, drawbacks and
fallacies of ordinary correlation, rank correlation, tail dependence

**Multivariate Models: Calibration and Simulation**- efficient correlation estimation, tests for multivariate
normality and ellipticity, fitting copulas to data, Monte Carlo
simulation of dependent risk factors

**Portfolio Credit Risk: Models**- models for dependent defaults (latent variable models and
mixture models), industry examples(KMV/Moodys, CreditMetrics,
CreditRisk+), mapping between models

**Portfolio Credit Risk: Calibration and Model Risk**- understanding sources of model risk, the role of copulas in
standard models, statistical issues in default modelling

**Advanced Multivariate Market Risk Models**- multivariate risk factor properties, multivariate time series models, multivariate GARCH models

The following distinguished researchers have agreed to give supplementary lectures on topics related to finance, risk, insurance mathematics, extremes: Henrik Hult (Stockholm), Thomas Mikosch (University of Copenhagen), Catalin Starica (Chalmers University Gothenburg), Murad Taqqu (Boston)

- We argue that the classical theory of statistical curve estimation offers the right setup for consistent, non-parametric inference of time-changing expected return, volatility and covariance in the analysis of financial returns.
- 1. Risk-return dynamics in stock indexes. An unconditional, non-parametric approach to simultaneous estimation of volatility and expected return is discussed. We show by means of a detailed analysis of the returns of the Standard \& Poors 500 composite stock index over the last fifty years, how theoretical results and methodological recommandations from the statistical theory of non-parametric curve inference allow to consistently estimate both expected return and volatility. Unlike the existing literature, we need not postulate an {\it a priori} relationship risk-return nor specify the evolution of first two moments through covariates. Our analysis gives statistical evidence that the expected excess return (return minus the risk free interest rate) of the Standard \& Poors 500 composite stock index as well as the market price of risk (the ratio excess return over volatility) vary through time both in size and sign. In particular, the periods of negative (positive) excess return and market price of risk seem to coincide with the bear (bull) markets of the index. A complex relationship between risk and expected return emerges, far from the common assumption of a positive, time invariant, linear relation.
- 2. Multivariate dynamics of stock returns. A simple non-stationary model for multivariate returns is proposed. Unlike most of the multivariate econometric models for financial returns, this model supposes the volatility to be exogenous. The vector of returns are independent and have a slowly changing unconditional covariance structure. The methodological frame is that of non-parametric regression with non-random equidistant design points, where the regression function is the evolving unconditional covariance. Special attention is payed to the accurate description of the tails of the innovations. As an application the model is fit to a multivariate data set of returns on three different financial instruments: a foreign exchange rate, an index and an interest rate. The 1-day ahead multivariate distributional forecast performance is evaluated.

- Multivariate regular variation is a theoretical concept which extends the notion of power law tail of a one-dimensional distribution to higher dimensions. It is a useful concept for describing the dependence of multivariate extremal events. In this case, it is not useful to use linear correlations or related dependence measures. Indeed, the correlation of two random variables measures the degree to which they are linearly dependent; it is not suitable for describing the dependence of the components of a vector which assumes values far away from the origin.
- It turns out that various financial time series models such as the popular GARCH model exhibit regular variation in their finite-dimensional distributions. We discuss the consequences for the extremes of such time series and for the extrapolation of the distribution of the random vectors into regions where only a few or no observations are available.

- A natural class of multivariate dynamic stochastic models useful in finance and insurance is the class of continuous-time stochastic processes with independent increments which, at each point in time, satisfies a multivariate regular variation condition. The class includes certain Levy jump processes which are considered as more natural models in finance and insurance than Brownian motion.
- Using the simple structure of these processes one can derive the tail behavior of interesting vectors of functionals of the process. Those include the largest jump of the process in a given time interval and the supremum of the process in this interval.

- 1. An introduction to stable processes. The familiar Gaussian models are named after the German mathematician Carl Friedrich Gauss. Based on the usual bell-shaped probability curve, they do not allow for large deviations and are thus often inadequate for modelling heavy probability tails. In the last decades however, data with heavy "probability tails" have been collected in fields as diverse as economics and telecommunications suggesting the use of non-Gaussian stable processes as possible models. These models always have infinite variance and, in some cases, infinite mean as well. We will provide an introduction to stable processes with particular emphasis on integral representations. These representations provide a physical way for understanding the structure of the stable processes. A number of examples will be presented.
- 2. Local contagion in financial markets. One commonly believes that international markets are more dependent during a crisis than they are are in more tranquil times. This extra dependence is often referred to as contagion between markets. We present a definition of contagion between financial markets based on local correlation and propose a test to detect the presence of contagion. The test does not require the specification of crisis and normal periods. As such, it avoids difficulties associated with testing for correlation breakdown, such as hand picking subsets of the data, and it provides a better understanding of the degree of dependence between financial markets. Using this test, we find evidence of contagion between developed and U.S. equity markets and evidence of flight to quality from the U.S. equity market to the U.S. government bond market.

The Concentrated Advanced Course aims at the graduate student in probability theory, statistics, finance, economics, insurance mathematics and the researcher who wants to get an overview of methods and techniques on modeling, as well as on practitioners from the insurance industry, banks and regulatory authorities.

The course will be accessible for Masters students with a background in statistics and extreme value theory. The interested Masters student can receive 2.5 ECTS for participation in the course and work on a 2-3 day project which consists of a practical piece of data analysis with S-Plus or R, such as fitting a bunch of GARCH models to financial data, or doing some practical extreme value theory or copula fitting. In order to do this, the participants should have access to S-Plus and the S+FinMetrics module. This can be arranged for the period of the course.

Adept Scientific and Alexander McNeil will give a presentation of their software products (S+FinMetrics,...).

** It is preferable if all participants
will bring a laptop.**

**Monday, May 26: Introduction to Risk Management and Financial Time
Series**

09.30-10.15 Registration and Coffee

10.15-11.00 McNeil: Basics of Quantitative Risk Management

11.15-12.00 McNeil: Standard Methods for Risk Management

12.00-14.00 Lunch Break

14.00-14.45 Henrik Hult: Multivariate regular variation for processes with independent increments

14.45-15.15 Coffee

15.15-16.00 McNeil: Financial Time Series

16.15-17.00 Knudsen + McNeil: Short Introduction to
S-Plus/Insightful products + first hands-on experience with S-Plus and
S+Finmetrics

**Tuesday, May 27: Extreme Values in Financial Time Series **

09.15-10.00 McNeil: Basics of EVT

10.15-11.00 McNeil: The POT Method and Tails of Loss distributions

11.00-11.30 Coffee

11.30-12.15 Starica: Risk-return dynamics in stock indexes

12.15-14.00 Lunch Break

14.00-14.45 Taqqu: An introduction to stable processes

14.45-15.15 Coffee

15.15-16.00 McNeil: EVT and Financial Applications

16.15-17.00 McNeil: Practical Session: - Financial Time Series + EVT

**Wednesday, May 28: Multivariate Models and Copulas**

09.15-10.00 McNeil: Basic Multivariate Models

10.15-11.00 McNeil: Copulas and Extremal Dependence

11.00-11.30 Coffee

11.30-12.15 Taqqu: Local contagion in financial markets

12.15-14.00 Lunch Break

14.00-14.45 Starica: Multivariate dynamics of stock returns

14.45-15.15 Coffee

15.15-16.00 McNeil: Fitting Copulas to Data

16.15-17.00 McNeil: Practical Session: Copulas in S+Finmetrics

**Thursday, May 29: Credit Risk **

09.15-10.00 McNeil: Introduction to Portfolio Credit Risk Models

10.15-11.00 McNeil: Modelling Dependent Defaults Credit Risk Models

11.00-11.30 Coffee

11.30-12.15 Mikosch: Multivariate extremes and multivariate regular variation

12.15-14.00 Lunch Break

14.00-14.45 Mikosch: Extremes in financial time series

14.45-15.15 Coffee

15.15-16.00 McNeil: Practical Issues in Credit Risk

16.15-17.00 McNeil: Practical Session - Modelling Default Data

**Friday, May 30: Multivariate Dynamic Models for Market Risk **

09.15-10.00 McNeil: Multivariate Financial Time Series I

10.15-11.00 McNeil: Multivariate Financial Time Series II

11.00-11.30 Coffee

11.30-12.15 McNeil: Practical Session - Multivariate Time Series
Models in S-Plus

12.15 Closing and Lunch Break

There will be a regular registration fee of 500 DKK for all participants, except Masters students, and the participants are expected to have their expenses covered by their home institutions or from other sources.

**The course will also be open to participants
from the financial,
insurance and other industries at an additional fee. For those participants,
the regular fee of 500 DKK and the additonal fee
will be charged by Adept Scientific.
Please register via
Adept Scientific
if you work in industry and are interested in attending this course.
**

**Masters students pay a fee of 100 DKK.
This fee does not include lunches.**

Please register via the registration form
at your earliest convenience **before May 20, 2003**.

The programme of the Course will be on the web after May 20.

The Course starts on Monday, 26 May, 10 a.m. and finishes on Friday, 30 May, 12 a.m.

We have a page with information
on how to get to the HC Ørsteds Institute, where the Course
will be given.

Do not hesitate to contact the MaPhySto secretariat
`(maphysto@maphysto.dk)`

, the secretaries
of the Laboratory of Actuarial Mathematics `(Actuarial@act.ku.dk)`

or the local organizers
Thomas Mikosch `(mikosch@math.ku.dk)`

and
Jeff Collamore
`(collamore@math.ku.dk)`

for more information.

This document was last modified April 29,
2003. Questions or comments to the contents of this document should
be directed to `Actuarial@act.ku.dk`.