First
Passage Time Probability for LevyFeller Type
First passage time distribution for random processes are key quantities
in many fields of sciences. For instance, in actuarial mathematics 

RiskLab/SAM researcher:  Dr.
Pierre Patie 

Project proposal:  Prof. Dr. Christoph Schwab (SAM, Department of Mathematics, ETH Zürich), Dr. Pierre Patie  
Start of project:  April 1, 2005  
Last update: March 6, 2005

Fast Deterministic Computation of Valuations for Assets Driven by Lévy ProcessesThe fast numerical valuation of assets whose prices are driven by Brownian motion has become standard practice worldwide. In a BlackScholes setting, this reduces to the numerical solution of a parabolic partial differential equation with various initial/boundary conditions and possibly constraints. When closed form solutions are not available, numerical methods must be employed; methods for handling such numerical problems have been well developed. 

General Lévy processes have been advocated in recent years for models in option pricing. They offer more flexibility than Brownian motion and appear superior e.g. for modelling shortterm asset returns whose distributions are heavytailed. Lévy models lead to parabolic integrodifferential equations with nonlocal Dynkin operators which, in general, cannot be solved in closed form.  
The objective of this project is to develop computational methods for the fast numerical solution of pricing problems driven by general Lévy processes. The approach is able to handle general processes and types of contracts, thereby allowing to assess the model sensitivity and model risk.  
The infinitesimal generator of the process driving the underlying entails due to its jump component large, illconditioned and densely populated matrices which must be inverted in each timestep. Wavelet based discretizations allow to "compress" these matrices to sparse and wellconditioned ones.  
The resulting algorithm has loglinear complexity comparable to that of the best finitedifference schemes for the usual BlackScholes type models, and can accomodate local and stochastic volatility models. Currently, we extended this technology to pricing American contracts on Lévy driven assets.  
As part of this project we also addressed the pricing problem for options within stochastic volatility models. While for European Vanilla options closed form expressions are available, other types of contracts, as e.g., compoundstyle options have to be priced numerically. We developed a wavelet based method which allows to handle the resulting degenerate parabolic partial differential equation due to the stochastic volatility in an optimal fashion.  
We also plan to assess model risk and perform sensitivity studies, including deterministic evaluation of the model's "greeks" and model parameter fitting to given market data.  
RiskLab/SAM researcher:  Dr. AnaMaria Matache (50% RiskLab, 50% SAM)  
Project proposal:  Prof. Dr. Christoph Schwab (SAM, Department of Mathematics,ETH Zürich)  
Start of project:  July 1, 2001  
Papers:  Fast Deterministic Pricing of Options on Lévy Driven Assets  
Wavelet Galerkin Pricing of American Options on Lévy Driven Assets  
Last update: March 6, 2003

Risk Management for Derivatives with Market IlliquiditiesRecent turbulence on financial markets showed that riskmanagement models, which are based on the assumption of perfectly liquid markets, may perform very poorly if market liquidity dries up. In this RiskLab project we propose to study the hedging of derivatives in the presence of market illiquidities. Moreover, we want to analyze various strategies for the liquidation of a given position in potentially illiquid securities. 

In the recent financial literature several models for markets which are not perfectly liquid have been developed. However, the practical implications of these models have not been sufficiently explored further.  
The focus of our project is to:


Reference:  Rüdiger Frey, Market Illiquidity as a Source of Model Risk in Dynamic Hedging, pp. 125136, in MODEL RISK, Concepts, Calibration and Pricing, edited by Prof. Rajna Gibson,Risk Books (2000), ISBN 1 899 332 898.  
RiskLab researcher:  Pierre Patie (25% from March 2000 until June 2001, 75% since July 2001, additional 25% supported by the Swiss Banking Institute)  
Project proposal:  Prof. Dr. Rüdiger Frey (Swiss Banking Institute,University of Zürich)  
Contact at Swiss Re:  Dr. John Hancock  
Contact at UBS:  George Pastrana  
Related project:  An Empirical Investigation on the Illiquidity in Financial Markets  
Start of project:  Summer 2000  
Paper:  Risk Management for Derivatives in Illiquid Markets: A Simulation Study  
Last update: November 26, 2001

Dependence Modelling in Risk ManagementThis project foresees a contribution to fundamental research in risk management (RM). The ideas will be applicable in all areas of RM (market, credit, insurance, etc.) and will be crucial to questions of risk aggregation. The project aims to establish the fundamentals of dependence and correlation modelling. 

The stochastic modelling of dependent risks in
the static case has essentially been completed in a RiskLab
project during the residency of Daniel Straumann. A research report is available online
and
has been well received by practitioners and academics alike. In this
followup project, we want to look at the following topics:


RiskLab researcher:  Filip Lindskog (100%)  
Project proposal:  Prof. Dr. Alexander McNeil (Department of Mathematics,ETH Zürich)  
Contact at Swiss Re:  Dr. Frank H. Krieter  
Contact at UBS:  John Gavin  
Start of project:  Middle of February 2000  
Papers:  Modelling Dependence with Copulas and Applications to Risk Management  
Linear Correlation Estimation  
Common Poisson Shock Models  
Modelling Dependencies in Credit Risk Management  
Kendall's Tau for Elliptical Distributions  
Risk Management in Credit Risk Portfolios with Correlated Assets  
Multivariate Extremes, Aggregation and Dependence in Elliptical Distributions  
Last update: March 11, 2003

Concluded Research ProjectsVolatility Estimation and Risk Measurement: From Short to LongTime HorizonsMarket risk management, portfolio optimization and option pricing methods can only be as good as the model of the underlying volatility process. The aim of the project is to use intraday highfrequency financial data to improve risk measurement at long time horizons. This possibility exists because (i) volatility estimation on a time horizon of several days can be improved by the use of intraday data, and (ii) a portfolio optimization and risk management system with portfolio reallocation horizons of about three months is possible, if the underlying model works with time steps of the order of one week. The project is highly dataoriented. It covers the range from very short time horizons for volatility estimation to long time horizons for portfolio optimisation and risk management. In particular, the following questions will be investigated:


RiskLab researchers:  PD Dr. Wolfgang Breymann  
Dr. Céline Azizieh (visiting postdoctoral research fellow at RiskLab from Oct. to Dec. 2001)  
David Mac Audière (guest from 22. 4. to 31. 7. 2002)  
Project proposal:  PD Dr. Wolfgang Breymann  
Contact at Swiss Re:  Dr. John Hancock  
Contact at UBS AG:  Dr. Marco Finardi (UBS Warburg)  
Duration of project:  September 2001 until December 2003  
Paper:  Dependence Structures
for Multivariate HighFrequency Data in Finance Intraday Empirical Analysis and Modelling of Diversified World Stock Indices 

Reports:  An Optimisation Algorithm for the Volatility Forecasting of FX Rates with the Stochastic Cascade Model  
Modelling Financial Time Series with a Multifractal Model  
Ph.D. Thesis:  Extreme Values of Gaussian Processes and A Heterogeneous Multi Agents Model (by Christoph M. Schmid)  
Software:  HfFinance  An SPLUS Tool for Deseasonalizing HighFrequency Financial Data  
Forthcoming Papers:  Estimation of the Stylized Facts of a Stochastic Cascade Model  
A Realistic Heterogeneous MultiAgent Model for the FX Market  
Last update: April 29, 2004

Liquidity Shocks in an Evolutionary Portfolio TheoryThe purpose of this project is to introduce liquidity shocks into the new theory of portfolio selection which has recently been suggested by Hens and SchenkHoppé (2001). This theory is based on evolutionary reasoning in simple repeated market situations. According to this new point of view the ultimate success of a portfolio strategy is measured by the wealth share the strategy is eventually able to conquer in an evolutionary process of market selection. For the case of identical and constant savings rates, Evstigneev, Hens and SchenkHoppé (2001) identify a simple portfolio strategy as being the unique strategy that palmost surely will gain the total market wealth. We want to analyze whether this neat result is robust to the more realistic case of exogenous savings that follow some stochastic process. 

Project proposal:  Prof. Dr. Thorsten Hens (Institute for Empirical Economic Research, University of Zürich)  
Enrico De Giorgi (RiskLab)  
RiskLab researcher:  Enrico De Giorgi (50%, additional 50% by IEW)  
Duration of project:  December 2001 until April 2003  
Paper:  RewardRisk Portfolio
Selection and Stochastic Dominance Evolutionary Portfolio Selection with Liquidity Shocks 

Last update: March 6, 2003

An Empirical Investigation on the Illiquidity in Financial MarketsIt is common wisdom that the standard ValueatRisk (VaR) measure of market risk lacks rigor with respect to liquidity risk. Neglecting liquidity risk leads to underestimation of overall risk, undercapitalization, and too many violations of calculated VaR. 

Attempts to quantify liquidity risk have focused both on the price impact of the execution of trades for a given portfolio (endogenous liquidity) and on portfolioindependent liquidity measures that reflect market behavior such as market spread (exogenous illiquidity). In this project, we will focus on methods for empirically quantifying exogenous liquidity risk. Our approach will lead to a liquidity VaR which incorporates a mixture of conditional market VaR and conditional (i.e. time varying) spread risk. Modeling conditional liquidity is required to yield a full integration of market and liquidity risk within a single conditional measure. Finally, we plan to go one step further and try to find variables with predictive power for market illiquidity, which could serve as an "early warning" tool for market participants, telling them when to correct their risk measures upward.  
RiskLab researcher:  Christian Buhl (100%, WWZ, University of Basle)  
Project proposal:  Prof. Dr. Heinz Zimmermann (WWZ, University of Basle)  
Duration of project:  April 2001 until March 2003  
Related project:  Risk Management for Derivatives with Market Illiquidities  
Last update: April 23, 2003

Measures of Multiperiod Risk and Time Allocation of CapitalIt is very difficult to define "acceptability" for a multiperiod risky project, portfolio, or position. In this work, we intend to compare different extensions of the generalized scenarios riskmeasure method and evaluate the consequences in several business applications. In particular, we plan to explore:


Project proposal:  Prof. Dr. Philippe Artzner  
RiskLab researcher:  Prof. Dr. Philippe Artzner (as Visiting Research Professor)  
Related previous projects:  Rules of Capital Allocation (CAPA) and Coherent Measures of Risk  
Coherent Allocation of the Risk Capital  
Duration of project:  March 2001 until February 2003  
Research will continue by other means. A proposal to the Actuarial Education and Research Fund for a research grant on Multiperiod Risk Measurement in Insurance is pending.  
Paper:  Coherent Multiperiod Risk Measurement  
Slides:  Online available are 22 slides used on Nov. 25, 2002, at the Quantitative Finance Seminar, Toronto Fields Institute.  
Last update: March 16, 2003

Strategic LongTerm Financial Risks (SLTFR)The development of a methodology that could be used for the measurement of strategic longterm financial risks is an important task. Existing modelling instruments allow for a relatively good measurement of market risks of trading books over relatively small time intervals. These models, however, have some severe deficiencies if they are applied to longer time periods. 

In this project we investigate models that are proposed to be used to model the evolution of risk factors. We test these models on real data by backtesting expected shortfall predictions.  
Project proposal:  Swiss Re  
Duration of project:  Middle of November 1999 until middle of March 2003  
RiskLab researchers:  Roger Kaufmann (100%)  
Pierre Patie (50% from March 2000 until June 2001)  
Contact at Swiss Re:  Dr. Niklaus Bühlmann  
Dr. John Hancock  
Contact at Credit Suisse Group: 
Elisabeth Bourqui  
Contact at UBS Warburg:  Dr. Marco Finardi  
Papers:  Strategic LongTerm Financial Risks (final report including slides)  
On the Normality of LongTerm Financial LogReturns (diploma thesis)  
Last update: March 15, 2003

Credit Risk Portfolio ModelsTaking into account recent insights into the means of modeling dependence structures (see Modeling Dependent Defaults by R. Frey and A. McNeil or A RiskFactor Model for RatingsBased Bank Capital Rules by M. B. Gordy), this project plans to determine how to design a credit portfolio model integrated in the existing market and credit risk models used by financial institutions. Some particular issues to be addressed are:


RiskLab researcher:  Dr. Dirk Tasche (50%, Oct. 2001 until Feb. 2002)  
Project proposal:  Dr. Giovanni Cesari and Dr. Marco Finardi (both UBS AG) together with RiskLab  
Duration of project:  Oct. 2001 until Feb. 2002  
Contact at UBS:  Dr. Giovanni Cesari  
Contact at Credit Suisse Group: 
Dr. Harry Stordel  
Contact at Swiss Re:  Dr. Stephan Schreckenberg  
Related projects:  Dependence Modelling in Risk Management  
Combined Market and Credit Risk Stress Testing  
Risk Modelling for a Swiss Retail/Middle Market Loan Portfolio  
Last update: March 2, 2003

Banks' Optimal Hedging Decisions Under the Threat of Bank RunsThis project will analyze a bank's risk management decision when the bank is financed with deposits and equity with limited liability. The bank is subject to the risk of depositor runs. Bankruptcy costs and regulatory restrictions are the primary motivations for risk management. Developing a model similar to Froot and Stein (1998), the optimal hedging decision will first be studied in a oneperiod framework. As a second step, it is planned to study the optimal hedging decision in a dynamic framework. Due to superior information, the bank earns a rent from its assets as well as from its deposits. These rents constitute the bank's franchise value. Optimal hedging strategies in the multiperiod case will then be derived taking the loss of the franchise value into account in the case of bank runs. 

RiskLab researcher:  Wolfgang Bauer (50%, other 50% by SBI)  
Project proposal:  Wolfgang Bauer and Prof. Dr. Rajna Gibson (Swiss Banking Institute,University of Zürich)  
Contact at Credit Suisse Group: 
Dr. Christian A. Walter  
Duration of project:  October 2001 until December 2002  
Related previous project:  Capital Allocation under Regulatory Constraints  
Paper:  Risk Management Strategies for Banks  
Last update: January 29, 2003

Risk Modelling for a Swiss Retail/Middle Market Loan PortfolioSo far credit risk modelling has concentrated largely on bonds or externally rated loans, probably mainly because of data availability. But modelling retail loans is a more important issue for loan portfolio management purposes and risk capital allocation at large retail and middle market banks (like Credit Suisse), as well as in view of the upcoming changes in the regulatory environment. The data situation (lack of long historic time series) necessitates other approaches than using market data, and the modelling approaches currently used (e.g. CreditRisk+) might not be the best available. Especially when concerned with dependence problems, there is a need to introduce macroeconomic conditions into the model or linking it with market risk. The project aims to close this gap and make a potentially big contribution to modelling credit risk, also closing a gap between science and practitioners. 

Research focus areas are:


RiskLab researchers:  Enrico De Giorgi (100%)  
Filip Lindskog (partially)  
Project proposal:  Urs Wolf (Credit Suisse) together with RiskLab  
Contact at Credit Suisse:  Urs Wolf and Vlatka Komaric  
Contact at UBS:  Dr. Giovanni Cesari,Dr. Alexandre Kurth and Dr. Armin Wagner  
References:  Web page with credit risk related links  
Duration of project:  November 2000 until November 2001  
Papers:  Common Poisson Shock Models  
Modelling Dependencies in Credit Risk Management  
An Intensity Based NonParametric Default Model for Residential Mortgage Portfolios  
Default Risk for Residential Mortgage Portfolios  
Last update: March 29, 2004

Capital Allocation under Regulatory ConstraintsUsing a stylised model similar to that of Froot and Stein (1998) the project will focus on the capital budgeting, capital structure, and risk management decisions of financial firms under regulatory constraints. The questions and issues that will be addressed in this framework are the following:


RiskLab researcher:  Aydin Akgün (50%)  
Project proposal:  Prof. Dr. Rajna Gibson and Aydin Akgün (Swiss Banking Institute,University of Zürich)  
Duration of project:  March 2001 until August 2001  
Paper:  A Utility Maximization Model of Capital Budgeting with Default Risk and Regulatory Constraints  
Last update: March 7, 2003

Combined Market and Credit Risk Stress TestingVarious crisis events have demonstrated the need for financial institutions such as banks and insurance companies to perform scenario analysis under stress conditions. This goes beyond the VaR framework, which is the standard tool to estimate losses within large but not extreme market movements. Only recently new methods such as Extreme Value Theory* have been applied to model tail events. 

Usually these methods are applied within a market risk environment only. Credit spreads as one of the market risk drivers are captured within the VaR framework, but no other credit effects (such as default probabilities, default correlations, and recovery rates) are taken into account.  
Efforts are underway to link credit and market risk within a common modelling framework (such as models for the short interest rate that also take into account credit effects**) but these concepts have not been well established for risk management yet. The goal of this project is the development of a theoretically wellunderstood and empirically founded conceptual framework for a combined market and credit risk stress test methodology.  
The project can be divided into the following
steps:


* See for example A.J. McNeil, Extreme Value Theory for Risk Managers, ETH Zurich, 1999  
** See for example D. Duffie and K. Singleton, Modeling Term Structures of Defaultable Bonds, 1999  
Project proposal:  Dr. Jörg Behrens and Dr. Marco Finardi (both UBS AG)  
Duration of project:  January 2000 until September 2001  
Contact at Swiss Re:  Dr. John Hancock and Marcel Rüegg  
Contact at UBS:  Dr. Giovanni Cesari and Dr. Christian Hauswirth  
RiskLab researchers:  Dr. Maria Kafetzaki Boulamatsis (60%, Jan. 2000 until Feb. 2001)  
PD Dr. Nicole Bäuerle (3 weeks as visiting postdoctoral research fellow)  
Dr. Dirk Tasche (5 weeks as visiting postdoctoral research fellow, 50% from April until Sep. 2001)  
Report:  Combined Market and Credit Risk Stress Testing Based on the Merton Model  
Paper:  Risk Management in Credit Risk Portfolios with Correlated Assets  
Last update: March 11, 2003

Investigation of the Market Price of Credit Risk for the Swiss Bond MarketCredit risk plays an important role in the valuation and risk management of most financial contracts. This leads to a great interest in problems of valuing credit risk  from a theoretical and practical point of view. In the nineties socalled intensity based (or reduced form) models have been developed, which assume that at each instant there is some probability that a firm defaults on its obligations. Duffie and Singleton (1995, 1997) follow this approach. Duffie (1999) tested the model for US corporations. The aim of this project is to examine how well the model of Duffie and Singleton describes corporate bond yields for the Swiss bond market and how the spread can be separated into the expected and unexpected loss. In a first step, the riskfree term structure on the basis of Swiss treasury bonds is estimated with a twofactor CIR model. Based on these results we estimate the instantaneous probability of default which follows a translated single factor squareroot diffusion process, with a modification that allows for the default process to be correlated with the factors driving the defaultfree term structure. 

RiskLab researcher:  Jacqueline Henn (100%, s/bf, HSG)  
Project proposal:  Prof. Dr. Heinz Zimmermann (s/bf, HSG)  
RiskLab support:  March 2000 until March 2001  
Paper:  Bewertung von Kreditrisiken  empirische Untersuchungen am Schweizer Kapitalmarkt  
Last update: October 9, 2001

Rules of Capital Allocation (CAPA) and Coherent Measures of RiskThe project will compare classical as well as newer allocation rules, and evaluate their consequences for


Contact at UBS:  George Pastrana  
Duration of project:  January 2000  February 2001  
Related concluded project:  Coherent Allocation of the Risk Capital  
Related followup project:  Measures of Multiperiod Risk and Time Allocation of Capital  
Report:  Rules of Capital Allocation and Coherent Measures of Risk (CAPA)  
Related work:  Expected RiskAdjusted Return for Insurance Based Models (diploma thesis)  
Additionl information:  Please contact Prof. Dr. Philippe Artzner or Prof. Dr. Freddy Delbaen.  
Last update: March 28, 2001

Model Risk for Credit and Market Risk Sensitive SecuritiesMost derivative securities are subject to market as well as credit risks. However, very few pricing and risk management models have so far integrated those two sources of risk and their interdependencies in a satisfactory way. The recent financial crises in Asia or in Russia have clearly shown that market and credit risks interact both at the macro as well as at the microeconomic levels and that they should not be managed separately. 

The proposed project aims at developing a
flexible pricing and hedging framework in which the market and credit
risk exposures of creditsensitive derivative securities can be
simultaneously modelled. This framework should further allow us to
quantify the impact of model risk on the pricing and hedging of single
and aggregate derivative positions. Broadly speaking, we intend to
address the following questions:


RiskLab researcher:  Aydin Akgün (50%)  
Project proposal:  Prof. Dr. Rajna Gibson (Swiss Banking Institute, University of Zürich)  
Contact at UBS:  Dr. Giovanni Cesari  
Paper:  Defaultable Security Valuation and Model Risk  
Related previous project:  Model Risk Management for Interest Rate Derivatives  
Duration of project:  April 2000 until February 2001  
Last update: June 18, 2001

Model Risk Management for Interest Rate DerivativesIn the first part of our work, we defined the model risk of hedging strategies as the law of the Profit and Loss (or of functionals of the P&L) of misspecified hedging portfolios, we explicited the P&L corresponding to various models and misspecifications  including discontinuities , and finally we proposed an original tool to manage the model risk, namely the solution of a stochastic game problem. 

For one year we worked on the theoretical and the numerical resolution of the stochastic game problem. We now have most of the desired results for one dimensional problems, and we are ready to face important questions such as the aggregation of contracts. These questions lead to delicate and new modelling problems and multidimensional stochastic control (or game) equations. If solved efficiently, they should be helpful to construct risk management techniques over large amounts of periods.  
People working on the project:


Papers:  Modeling the Term Structure of Interest Rates: A Review of the Literature  
Model Risk Analysis for Bond Options in a HeathJarrowMorton Framework  
Interest Rate Model Risk: What are we Talking About?  
Model Risk with JumpDiffusion Processes  
Volatility Model Risk Measurement and Strategies against Worst Case Volatilities  
Model Risk Management Against Worst Case Volatility Processes for Discount Bond Options  
Related project:  Model Risk for Credit and Market Risk Sensitive Securities  
Duration of project:  June 1997 until June 2000  
Last update: January 10, 2001

Dynamic Financial Analysis in Nonlife InsuranceIn order to analyze the financial effects of different entrepreneurial strategies for nonlife insurance companies over a given time horizon, one finds two primary techniques in use today: The first one, the socalled scenario testing, projects results under specific deterministic scenarios in the future. The disadvantage of this approach is the fact that only a few arbitrary scenarios are tested in order to decide how good a strategy is. The other technique is stochastic simulation, better known as Dynamic Financial Analysis (DFA). Here many different scenarios are generated stochastically with the aim of giving information about the distribution of some important variables, like surplus, written premiums or loss ratio. There are lots of different versions of DFA. Our aim is not to give a comprehensive description of all of them, but rather to show the common ideas of these different models. We show how the modules of such a model can be constructed and related into an efficient complex. 

RiskLab researcher:  Roger Kaufmann (100%)  
Project proposal:  Zurich Financial Services (ZFS) and Prof. Dr. Paul Embrechts  
Contact at ZFS:  Andreas Gadmer and Ralf Klett  
Paper:  Introduction to Dynamic Financial Analysis  
Duration of project:  April until November 1999 (Roger Kaufmann's diploma thesis on DFA already started November 1998)  
Followup work:  Under the guidance of Roger Kaufmann, Sergi Prokopenko developed an C++ program for DFA.  
Last update: September 3, 2000

Correlation in Insurance and FinanceInsurance has traditionally been built on the assumption of independence of claims and the law of large numbers has governed the determination of premiums. An important question is: how can one incorporate dependencies? Linear correlation plays a central role in financial theory. Since the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) are essentially based on means and variances, linear correlation proves to be a natural description of stochastic dependence, and is justified under certain distributional assumptions. However, modern risk management often places us in situations where these distributional assumptions are untenable. Moreover, we are often interested in risk measures other than variance (or standard deviation), e.g. ValueatRisk (VaR). In this context the following question is quite natural:


Project proposal:  Prof. Dr. Paul Embrechts and Dr. Alexander McNeil (both Department of Mathematics, ETH Zürich)  
RiskLab researcher:  Daniel Straumann (100%)  
Paper:  Correlation and Dependence in Risk Management: Properties and Pitfalls  
Duration of project:  January 1998 until August 1999  
Followup project:  Dependence Modelling in Risk Management  
Last update: August 31, 2000

Coherent Allocation of the Risk CapitalMuch effort has been expanded in the recent years to measure the financial risks faced by a firm. Such risk measures aggregate the amounts of risk of the firm's various business units into a single number, the risk capital. However, much less effort has been devoted to the problem of allocating the risk capital back to the business units. Such an allocation is crucial for the comparison of business units on a riskadjusted basis. The goal of this project is therefore to:


Project proposal:  Prof. Dr. HansJakob Lüthi (IFOR, ETH Zürich)  
RiskLab researcher:  Dr. Michel Denault  
Paper:  Coherent Allocation of Risk Capital  
Duration of project:  June 1998 until October 1999  
Related followup projects:  Rules of Capital Allocation (CAPA) and Coherent Measures of Risk  
Measures of Multiperiod Risk and Time Allocation of Capital  
Last update: March 28, 2001

Generalizations of Bessel Processes 

RiskLab researcher:  Dr. Anja GöingJaeschke  
Papers:  Some Generalizations of Bessel Processes (intermediate report)  
Parameter Estimation and Bessel Processes in Financial Models (Part of Ph.D. thesis)  
A Survey and Some Generalizations of Bessel Processes (final version)  
A Clarification Note about Hitting Times Densities for OrnsteinUhlenbeck Processes  
Duration of project:  October 1996 until March 1998  
Last update: March 6, 2003

Estimation in Financial Models 

RiskLab researcher:  Dr. Anja GöingJaeschke  
Papers:  Estimation in Financial Models (intermediate report)  
Parameter Estimation and Bessel Processes in Financial Models (Part of Ph.D. thesis)  
Duration of project:  April 1995 until September 1996  
Last update: September 19, 2000

Risk Aggregation Techniques for Complex Financial PortfoliosThe goal of this project is to develop operational measurements for the aggregated risk of portfolios of financial industries. An increasing number of complex financial instruments are being used in business today. Whereas many quantitative techniques for the analysis of a single instrument are well established, there still is a lack of mathematical methods for estimating the cumulative risk of a portfolio. New concepts, new models, and new quantitative techniques must be developed in order to aggregate the different risks of different instruments in different markets. We aim not only to determine confidence estimates for the profit and loss of a portfolio, but also to quantify these values for extreme (stress) situations. Moreover we hope to get a better understanding of the dynamic nature of risk profiles by making use of advanced mathematical modeling and simulation techniques. 

RiskLab researcher:  Dr. Gerold Studer  
Project proposal:  Prof. Dr. HansJakob Lüthi (IFOR, ETH Zürich)  
RiskLab reports:  Value At Risk and Maximum Loss Optimization  
Quadratic Maximum Loss for Risk Measurement of Portfolios  
Factors at Risk  
Ph.D. Thesis:  Maximum Loss for Measurement of Market Risk  
Papers:  Maximum Loss for Risk Measurement of Portfolios  
Quadratic Maximum Loss  
Risk Measurement with Maximum Loss  
Market Risk Computation for Nonlinear Portfolios  
Last update: March 15, 2003

Approximations of ProfitandLoss Distributions 

Papers:  Approximation of P&L Distributions  
Approximation of P&L Distributions, Part II  
Last update: September 15, 2000
