Advances in Finance and Stochastics

Springer-Verlag Berlin Heidelberg GmbH

Klaus Sandmann

Philipp J. Schonbucher (Eds.)

Advances in

Finance and

Stochastics

Essays in Honour

of Dieter Sondermann

With 32 Figures

Springer

Klaus Sandmann

Johannes Gutenberg-Universităt Mainz

Lehrstuhl fiir Bankbetriebslehre

Jakob Welder-Weg 9

55128 Mainz

Germany

e-mail: sandmann@forex.bwl.uni-mainz.de

Philipp J. Schonbucher

Rheinische Friedrich- Wilhelms-Universităt Bonn

Inst. f. Gesellschafts- u. Wirtschaftswissenschaften

Statistische Abteilung

Adenauerallee 24-42

53113 Bonn

Germany

e-mail: P.Schonbucher@finasto.uni-bonn.de

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Advances in finance and stochastics: essays in honour of Dieter Sondermann/

Klaus Sandmann; Philipp J. Schonbucher (ed.).

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Preface

Finance and Stochastics and Dieter Sondermann are directly and inextricably

linked to each other. The recognition and the success of this journal would

not have been possible without his untiring commitment, his sensitivity for

scientific quality and originality as well as his trustworthiness when dealing

with the authors. One could almost say: Finance and Stochastics is Dieter

Sondermann, since without him this journal would not be.

In the preface of the first issue of Finance and Stochastics in January

1997, Dieter Sondermann referring to the significance of the thesis of Louis

Bachelier, stated: 'Thus, the year 1900 may be considered as the birth date

of both Finance and Stochastics'. Further on he wrote: 'The journal Finance

and Stochastics is devoted to the fruitful interface of these two disciplines'.

What is there to add? It was important to identify and to articulate such

a goal, yet to translate it into action and to make it possible was crucial. It

is due to Dieter Sondermann's initiative and constant work that the idea of

Finance and Stochastics has turned into a highly reputable and successful

project. His unfailing commitment as founder and chief editor has made this

journal an important publication forum of international renown. A publication in Finance and Stochastics is a guarantee of originality and quality for

scientific papers.

Thus, what could have been more natural than the idea of honouring Dieter Sondermann on the occasion of his 65th birthday with a collection of

research papers entitled Advances in Finance and Stochastics? Those who

know him would surely agree that especially Dieter Sondermann, in his modest and undemonstrative way, would never have approved of such an honour.

Luckily, the person to be honoured does not have a say in the matter. However, if he had had one and had not been able to prevent it happening, it is

likely that he would have warned us emphatically against a conception that

was one-sided and looked back upon his own contributions. He might even

have considered the exercise quite superfluous. Instead, his one and only concern would have been for the reader interested in scientific knowledge and

the solution of problems.

'The future has more Futures'. This 'bon mot' of the financial market

also holds good for Dieter Sondermann's scientific work and his involvement

which has always been diverse and with a clear focus on the future. Dieter

VI

Preface

Sondermann was among the first scientists in Germany to apply themselves

to the study of Mathematical Finance. Influenced by the seminal work of

Fisher Black, Myron Scholes and Robert C. Merton and by virtue of his own

profound understanding of the theory of general equilibrium, his contributions often mark the starting point for further development. In 1985, with

Hedging of Non-Redundant Contingent Claims, Dieter Sondermann, together

with Hans Follmer, paved the way for the pricing and hedging of options in

an incomplete financial market. At an early stage he recognises the importance of the theory of arbitrage for the evaluation of insurance risk, which

he demonstrates in Reinsurance in Arbitrage Free Markets (1991). With a

similar feel for new ground, he proved the market model approach to the

term structure of interest rates in his work Closed Form Solutions for Term

Structure Derivatives with Log-Normal Interest Rates (1997, together with

Kristian R. Miltersen and Klaus Sandmann).

Dieter Sondermann's academic career might be considered surprising and

unusual, especially in its initial phase. Yet, if one looks at it from today's

perspective, one can see easily how each step and each stage form an integral part of a consistent whole. He was born on May 10th, 1937 in Duisburg,

Germany. His early years do not directly point to an academic career: his employment as a forwarding agent in the 'Rhenania Allgemeine Speditions AG'

in 1953, his examination in 1956 as a business assistant and finally his activity until 1958 as an expedient in the shipping company 'Vereinigte Stinnes

Reederei GmbH' in Duisburg Ruhrort. During those years, two notions must

have become rooted in his mind, his love of the Rhine and of shipping and

his love of pursuing promising ideas. After his Abitur in 1960, Dieter Sondermann embarked on his studies of Mathematics, Physics and Economics at the

University of Bonn. Little did he know (or even hope), when leaving Bonn in

1962 with a Vordiplom and heading for Hamburg, that he was to return as a

professor of Economics and Statistics not quite 17 years later. Many stages

and formative encounters awaited him still. After his Diplom in Mathematics in 1966, Prof. Dr. Heinz Bauer who had noticed this promising young

mathematician from Hamburg, invited him to the University of Erlangen.

Here, after only two years, Dieter Sondermann obtained his Ph.D. in 1968.

There was no respite for the young academic with such diverse interests. No

sooner had he obtained his doctoral degree than his interest in Economics

was kindled - and this with lasting effect, not least because of Theory of

Value, the 'magic' book by Gerard Debreu. It fascinated him and filled him

with enthusiasm. With his innate capacity for sound judgement he clearly

grasped the opportunities and perspectives contained in the work - and he

used them. In 1970 Dieter Sondermann was appointed lecturer in Mathematical Economics at the University of Saarbriicken. Yet, curiosity deriving from

fascination requires scientific discussion. Thus his path led to the Center of

Operations Research and Econometrics, CORE, in Louvain, Belgium, where

from 1970 to 1972 he was a visiting research professor. Here at CORE a

Preface

VII

great number of committed young scientists met up, and it was here that the

foundations were laid for many scientific and personal friendships which were

to last until the present day. In 1972 Dieter Sondermann returned to Bonn,

this time as Visiting Associate Professor, in 1973 he was in Berkeley, USA,

and a year later he accepted a full professorship in Economics at the University of Hamburg, Germany. The same year he joined the editorial board

of the Journal of Mathematical Economics, founded by Werner Hildenbrand,

on which he served until 1985. At the same time, from 1973 until 1980, he

was a member of the editorial board of the Journal of Economic Theory,

and from 1983 until 1992 of that of the Applicandae Mathematicae (Acta). In

1979 he became Fellow of the lAS at the Hebrew University, Jerusalem. This

is also the year in which he accepted a chair in Economics and Statistics at

the University of Bonn.

Dieter Sondermann, Bonn, the Rheinische Friedrich-Wilhelms University

and the Rhine are intertwined in so many different ways. His house by the

Rhine serves as a refuge for him, his wife, his family and their friends. Even

the perennial threat of high water cannot mar his lifelong attachment to the

Rhine and Rhine shipping. Instead, with a calmness that is so typical of

him, he will contemplate such a phenomenon of nature in statistical terms.

With the same calmness, full of determination, and most successfully, Dieter

Sondermann manages, from 1985 until 1999, Stochastics of Financial Markets, the subproject B 3 of the Sonderforschungsbereich 303. During these

15 years, this research team, under his leadership, gains recognition at home

and abroad and makes a lasting contribution towards the development and

importance of Mathematical Finance. His open and problem-oriented style

of discussion deeply influences work methods and fosters an atmosphere of

curiosity. To bring into accord both research and teaching has always been

for him - and still is - a constant matter of concern. In a personal and human

manner that is so characteristic of him, Dieter Sondermann has, throughout the years, supported and influenced the career of his numerous members

of staff. Many of his students, themselves now in responsible positions at

universities or in industry, remain deeply indebted to him.

There are as many reasons for showing our gratitude to Dieter Sondermann as there are possibilities for expressing this. With Advances in Finance

and Stochastics we simply want to say: Thank you!

The future has more Futures, Dieter!

February 2002, Bonn

Klaus Sandmann

Philipp Schonbucher

Introduction

In many areas of finance and stochastics, significant advances have been made

since this field of research was opened by Black, Scholes and Merton in 1973.

The collection of contributions in Advances in Finance and Stochastics reflects this variety. Necessarily, a selection of topics had to be made, and we

endeavoured to choose those that are currently in the focus of active research

and will remain so in future. This selection spans risk management, portfolio theory and multi-asset derivatives, market imperfections, interest-rate

modelling and exotic options.

Since Follmer and Sondermann (1986) published one of the first mathematical finance papers on risk management in incomplete markets, quantitative research has developed rapidly in this area. The first three papers of this

volume represent the recent developments in this area.

In the first paper on risk management, Delbaen extends the fundamental

notion of a coherent risk measure in two directions from the original definition

in Artzner et.al. (1999): the underlying probability space is now be a general

probability space (and not finite) and the class of risks that are measured

is extended to encompass all random variables on this space. Using methods

from the theory of convex games he is able to prove the analogies of the results

of Artzner et.al. (1999) in this much more general setup. But not everything

carries through identically from the discrete setup: Delbaen shows that now a

coherent risk measure has to be allowed to assume infinite values, representing

completely unacceptable risks. The following contribution by Follmer and

Schied also treats coherent risk measures, but only as a special case of a more

general class of risk measures: the convex risk measures. The authors show

that convex risk measures can be represented as a supremum of expectations

under different measures, corrected by a penalty function that depends on the

probability measure alone. They also connect these risk measures to utilitybased risk measurement. The third article on risk management is authored

by Embrechts and Novak who give a survey of recent developments in the

modelling and measurement of extremal events. While the first two articles

are concerned with the question of a consistent allocation of risk capital to a

given set of risks, this article gives asymptotic answers to the question of the

probability with which this level of risk capital will be exceeded.

X

Introduction

The part on portfolio theory opens with a paper by Werner in which he

develops a multi-period extension to the CAPM, the APT and similar factor

pricing models. By measuring the risk of the assets in terms of the risk of the

underlying dividend streams (instead of the one-period returns), the author

is able to give conditions under which exact factor pricing relationships hold.

In contrast to this portfolio-selection problem, Duan and Pliska consider the

pricing of options on multiple co-integrated assets. Apart from providing necessary conditions for cointegration of a set of assets with GARCH-stochastic

volatilities, they also study the effect that cointegrating relationships under

the physical measure have on the dynamics of the assets under the equilibrium pricing measure and on the dynamics of risk premia. In the following

paper, Madan, Milne and Elliott study the effects that arise when several investors use different, individual factor pricing models, and these models are

aggregated. While Werner took the factor structure as given in his model,

Madan et.al. want to understand where economy-wide risk factors and riskpremia arise from, they shift the focus from asset-returns to identifying and

explaining investor-specific risk exposures.

Market imperfections are the theme of the next three contributions.

Kabanov and Stricker consider super-hedging strategies under transaction

costs. They characterise the hedging-set (the set of initial endowments that

allow a self-financing super-replication) of a contingent claim in a general

setup with non-constant transaction costs. In the following paper, Frey and

Patie address the problem of hedging options in illiquid markets. In a simulation study they show that a hedging strategy based upon a nonlinear partial

differential equation that includes liquidity effects can significantly improve

the performance of the hedge. In Frey and Patie's contribution illiquidity

takes the form of market impact, Le. the transactions of a large trader move

prices, but he is able to trade at any time he chooses. Rogers and Zane consider a different kind of illiquidity in the third paper of this group: Here,

traders are only allowed to trade at Poisson arrival times which they cannot

influence. The traders' objective is a consumption/investment problem similar to Merton (1969). Rogers and Zane establish that Merton's investment

rule (investing a fixed proportion of wealth in the risky asset) is still optimal,

and characterize the modification of the optimal consumption process. Using

an asymptotic expansion, they assess the cost of illiquidity to the investor.

The two contributions on interest-rate modelling both build upon the

market-modelling approach for observed effective interest rates by Miltersen,

Sandmann and Sondermann (1997). Bhar et.al. provide an estimation methodology for a short-rate model which explicitly recognizes the fact that the

short term interest-rate is unobservable. Their approach aims to connect the

stochastic models for the continuously compounded short rate with the observed effective, discretely compounded rates.

Introduction

XI

Schlogl analyses this connection in the other direction and shows that

every market model implies a model for the continuously compounded short

rate that is uniquely determined by the interpolation method used for rates

maturing between tenor dates. He provides an interpolation method which

preserves the Markovian properties of discrete-tenor models but allows for

continuous stochastic dynamics of the short rate.

The final set of contributions has its focus on specific pricing problems

that arise in the pricing of exotic options, in particular the connection between

insurance and financial markets, optimal stopping, and barrier features which

all affect the payoff of the option in a nonlinear way.

The connection between the markets for insurance and financial risks

has been a long-standing area of interest to Dieter Sondermann. Nielsen and

Sandmann analyse in their contribution one example where this connection is

particularly evident: equity-linked life and pension insurance contracts. The

authors give results for the existence of a fair periodic premium and provide

approximate and numerical results for their magnitude.

Optimal stopping is the theme of the contributions by Schweizer; Shepp,

Shiryaev and Sulem; and Peskir and Shiryaev. Schweizer analyses the optimal stopping problems posed by Bermudan options. As Bermuda options

can only be exercised in a subset of the lifetime of the option, the early exercise strategies are subject to this additional restriction. Schweizer shows

under which conditions the problem can be reduced to a modified American

(unrestricted) optimal stopping problem, and how super-replication strategies

can be derived in this setup.

Shepp, Shiryaev and Sulem consider an option that combines American

early exercise, a knockout barrier and lookback-features: the barrier version

of the Russian option. Here, the early exercise strategies are restricted by

the knockout barrier of the option. Despite the complicated structure of the

option, they are able to provide the optimal exercise strategy and the value

function of this derivative.

The following contribution by Schiirger contains an analysis of the distribution, moments and Laplace transforms of the suprema of several stochastic

processes - a problem with immediate applications for the pricing of barrier

and lookback options. Schiirger gives explicit formulae for these quantities for

Bessel processes as well as for strictly stable Levy processes with no positive

jumps. For this he uses an elegant transformation from the maximum of a

stochastic process to its first hitting time.

The final contribution again addresses the question of optimal stopping.

Peskir and Shiryaev analyse the Poisson disorder problem, the problem of

detecting a change in the intensity of a Poisson process. In this context they

show that the smooth-pasting condition is not always valid for the optimal

value function if the state vector can be discontinuous.

XII

Introduction

All authors are leading experts in their fields and we are very grateful to

them for their contributions to this volume. Special thanks also go to Anne

Ruston for expert advice in language questions, Catriona Byrne and Susanne

Denskus from Springer and to Florian Schroder.

Through the input of all these people this book has become a fitting

present to mark the occasion of Dieter Sondermann's 65 th birthday: a volume

of up-to-date research on honour of a creative researcher and the editor of a

leading journal, who has helped shape the subject of mathematical finance.

1. Follmer, Hans and Dieter Sondermann (1986): "Hedging of Non-Redundant Contingent Claims" in: W. Hildenbrandt and A. Mas-Colell (eds.) Contributions to

Mathematical Economics, in Honor of Gerard Debreu, North-Holland.

2. Sondermann, Dieter (1991) "Reinsurance in Arbitrage Free Markets", Insurance:

Mathematics and Economics 10, 191-202.

3. Miltersen, Kristian, Klaus Sandmann and Dieter Sondermann (1997), "Closed

Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates" ,

Journal of Finance 52(1), 409-430.

Table of Contents

Coherent Risk Measures on General Probability Spaces

Freddy Delbaen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Robust Preferences and Convex Measures of Risk

Hans Follmer, Alexander Schied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39

Long Head-Runs and Long Match Patterns

Paul Embrechts, Sergei Y. Novak ................................. 57

Factor Pricing in Multidate Security Markets

Jan Werner ................................................... 71

Option Pricing for Co-Integrated Assets

Jin- Chuan Duan, Stanley R. Pliska ............................... 85

Incomplete Diversification and Asset Pricing

Dilip B. Madan, Frank Milne, Robert J. Elliott . .................... 101

Hedging of Contingent Claims under Transaction Costs

Yuri M. Kabanov, Christophe Stricker . ............................ 125

Risk Management for Derivatives in Illiquid Markets:

A Simulation Study

Rudiger Frey, Pierre Patie ....................................... 137

A Simple Model of Liquidity Effects

L.-C.-G. Rogers, Omar Zane . .................................... 161

Estimation in Models of the Instantaneous Short Term

Interest Rate by Use of a Dynamic Bayesian Algorithm

Ramaprasad Bhar, Carl Chiarella, W ol/gang J. Runggaldier . . . . . . . . .. 177

Arbitrage-Free Interpolation in Models of Market Observable

Interest Rates

Erik Schlogl .................................................... 197

XIV

Table of Contents

The Fair Premium of an Equity-Linked Life and Pension

Insurance

J. Aase Nielsen, Klaus Sandmann . ................................ 219

On Bermudan Options

Martin Schweizer . .............................................. 257

A Barrier Version of the Russian Option

Larry A. Shepp, Albert N. Shiryaev, Agnes Sulem ................... 271

Laplace Transforms and Suprema of Stochastic Processes

Klaus Schurger ................................................. 285

Solving the Poisson Disorder Problem

Goran Peskir, Albert N. Shiryaev ................................. 295

List of Contributors

BHAR, RAMAPRASAD

School of Banking and Finance,

The University of New South Wales

Sydney 2052, Australia

r.bhar~unsw.edu.au

CHIARELLA, CARL

School of Finance and Economics,

University of Technology, Sydney

PO Box 123, Broadway, NSW 2007, Australia

carl.chiarella~uts.edu.au

DELBAEN, FREDDY

Departement fiir Mathematik,

Eidgenossische Technische Hochschule Ziirich

ETH-Zentrum, CH 8092 Ziirich, Switzerland

delbaen@math.ethz.ch

DUAN, JIN-CHUAN

J.L. Rotman School of Management, University of Toronto

105 St. George Street, Toronto M5S 3E6, Canada

jcduan~rotman.utoronto.ca

ELLIOTT, ROBERT J.

Department of Mathematical Sciences, University of Alberta

Edmonton, Alberta T6G 2G1, Canada

relliott@gpu.srv.ualberta.ca

XVI

List of Contributors

EMBRECHTS, PAUL

Departement fUr Mathematik,

Eidgenossische Technische Hochschule Zurich

ETH-Zentrum, CH 8092 Zurich, Switzerland

embrecht@math.ethz.ch

FOLLMER, HANS

Institut fUr Mathematik, Humboldt-Universitiit Berlin

Unter den Linden 6, 10099 Berlin, Germany

foellmer@mathematik.hu-berlin.de

FREY, RUDIGER

Mathematisches Institut, Fakultiit fUr Mathematik und Informatik,

Universitiit Leipzig

Augustusplatz 10/11, D-04109 Leipzig, Germany

frey@mathematik.uni-leipzig.de

KABANOV, YURI M.

Laboratoire de Mathematiques, Universite de Franche-Comte

Universite de Franche-Comte, 16 Route de Gray, F-25030 Besanc;on

Cedex, France

youri.kabanov@math.univ-fcomte.fr

MADAN, DILIP

B.

Department of Finance, Robert H. Smith School of Business,

University of Maryland

4409 Van Munching Hall, College Park, MD 20742-1815, USA

dmadan@rhsmith.umd.edu

MILNE, FRANK

Department of Economics Queen's University

Kingston, Ontario K7L 3N6, Canada

milnef@qed.econ.queensu.ca

NIELSEN,

J.

AASE

Aarhus University, Dept. of Operations Research

Bldg. 530, Ny Munkegade, DK-8000 Aarhus, Denmark

atsjan@imf.au.dk

List of Contributors

NOVAK, SERGEI

Y.

Department of Mathematical Sciences, Brunei University

Uxbridge UB8 3PH, United Kingdom

mastssn~brunel.ac.uk

PATIE, PIERRE

RiskLab, Departement fiir Mathematik,

Eidgenossische Technische Hochschule Zurich

ETH-Zentrum, CH 8092 Ziirich, Switzerland

patie~math.ethz.ch

PESKIR, GORAN

Institute of Mathematics, University of Aarhus

Ny Munkegarde, 8000 Aarhus, Denmark

goran~imf.au.dk

PLISKA, STANLEY R.

Department of Finance, University of Illinois at Chicago

601 S. Morgan Street, Chicago, IL 60607-7124, USA

srpliska~uic.edu

ROGERS,

L. C.

G.

University of Bath, Department of Mathematical Sciences

University of Bath, Bath BA2 7AY, United Kingdom

L.C.G.Rogers@bath.ac.uk

RUNGGALDIER, WOLFGANG

J.

Dipartimento di Matematica Pura ed Applicata,

Universita di Padova

Via Belzoni 7, 35131-Padova, Italy

runggal@math.unipd.it

SANDMANN, KLAUS

Johannes Gutenberg-University Mainz,

Lehrstuhl fiir Allgemeine BWL und Bankbetriebslehre

Jakob Welder Weg 9, 55099 Mainz, Germany

sandmann@forex.bwl.uni-mainz.de

XVII

XVIII List of Contributors

SCHIED, ALEXANDER

Institut fUr Mathematik, Technische Universitiit Berlin

MA 7-4, StraBe des 17. Juni 136, 10623 Berlin, Germany

schied~mathematik.hu-berlin.de

SCHLOGL, ERIK

University of Technology, Sydney

PO Box 123, Broadway NSW 2007, Australia

Erik.Schlogl~uts.edu.au

SCHONBUCHER, PHILIPP

Universitiit Bonn, Institut fur

Gesellschafts- und Wirtschaftswissenschaften,

Statistische Abteilung

Adenauerallee 24-26, D-53113 Bonn, Germany

p~schonbucher.de

SCHURGER, KLAUS

Universitiit Bonn, Institut fUr

Gesellschafts- und Wirtschaftswissenschaften,

Statistische Abteilung

Adenauerallee 24-26, D-53113 Bonn, Germany

schuerger~finasto.uni-bonn.de

SCHWEIZER, MARTIN

Ludwigs-Maximilians University Munchen,

Mathematisches Institut

TheresienstraBe 39, D-80333 Munchen, Germany

mschweiz~mathematik.uni-muenchen.de

SHEPP, LARRY

A.

Rutgers University, Department of Statistics

Piscataway, NJ 08854-8091, USA

shepp~stat.rutgers.edu

List of Contributors

SHIRYAEV, ALBERT

N.

Steklov Mathematical Institute (MIRAN)

Gubkina 8,117966 GSP-1, Moscow, Russia

shiryaev~mi.ras.ru

STRICKER, CHRISTOPHE

Laboratoire de Mathematiques, Universite de Franche-Comte

16 Route de Gray, F-25030 Besan<;on Cedex, France

Christophe.Stricker~ath.univ-fcomte.fr

SULEM, AGNES

Institut National de Recherche en Informatique et en Automatique

(INRIA)

Domaine de Voluceau, Rocquencourt - B.P. 105,

78153 Le Chesnay Cedex, France

agnes.sulem~inria.fr

WERNER, JAN

Department of Economics, University of Minnesota

1151 Heller Hall, Minneapolis, MN 55455 U.S.A.

jwerner~atlas.socsci.umn.edu

ZANE,OMAR

Warburg Dillon Read, London,

Quantitative Risk, Models and Statistics

UBS, 1 Finsbury Avenue London EC2M 2PG, United Kingdom

Omar.Zane~wdr.com

XIX

Coherent Risk Measures on General

Probability Spaces

Freddy Delbaen*

Department of Mathematics, Eidgenossische Technische Hochschule, Zurich,

Switzerland

Summary. We extend the definition of coherent risk measures, as introduced by

Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how

to define such measures on the space of all random variables. We also give examples

that relates the theory of coherent risk measures to game theory and to distorted

probability measures. The mathematics are based on the characterisation of closed

convex sets P u of probability measures that satisfy the property that every random

variable is integrable for at least one probability measure in the set P u.

Key words: capital requirement, coherent risk measure, capacity theory, convex

games, insurance premium principle, measure of risk, Orlicz spaces, quantile, scenario, shortfall, subadditivity, submodular functions, value at risk

1 Introduction and Notation

The concept of coherent risk measures together with its axiomatic characterization was introduced in the paper Artzner et.al. [1] and further developed

in [2]. Both these papers supposed that the underlying probability space was

finite. The aim of this paper is twofold. First we extend the notion of coherent risk measures to arbitrary probability spaces, second we deepen the

relation between coherent risk measures and the theory of cooperative games.

In many occasions we will make a bridge between different existing theories.

In order to keep the paper self contained, we sometimes will have to repeat

known proofs. In March 2000, the author gave a series of lectures at the Cattedra Galileiana at the Scuola Normale di Pisa. The subject of these lectures

was the theory of coherent risk measures as well as applications to several

problems in risk management. The interested reader can consult the lecture

notes Delbaen [8]. Since the original version of this paper (1997), proofs have

undergone a lot of changes. Discussions with colleagues greatly contributed to

* The author acknowledges financial support from Credit Suisse for his work and

from Societe Generale for earlier versions of this paper. Special thanks go to

Artzner, Eber and Heath for the many stimulating discussions on risk measures

and other topics. I also want to thank Maafi for pointing out extra references to

related work. Discussions with Kabanov were more than helpful to improve the

presentation of the paper. Part of the work was done during Summer 99, while

the author was visiting Tokyo Institute of Technology. The views expressed are

those of the author.

K. Sandmann et al. (eds.), Advances in Finance and Stochastics

© Springer-Verlag Berlin Heidelberg 2002

2

Freddy Delbaen

the presentation. The reader will also notice that the theory of convex games

plays a special role in the theory of coherent risk measures. It was Dieter

Sondermann who mentioned the theory of convex games to the author and

asked about continuity properties of its core, see Delbaen [7]. It is therefore

a special pleasure to be able to put this paper in the Festschrift.

Throughout the paper, we will work with a probability space (J?, F, 1P).

With L 00 (J?, F, 1P) (or L 00 (1P) or even L 00 if no confusion is possible), we mean

the space of all equivalence classes of bounded real valued random variables.

The space LO(J?, F, 1P) (or LO(IP) or simply LO) denotes the space of all equivalence classes of real valued random variables. The space LO is equipped

with the topology of convergence in probability. The space LOO(lP'), equipped

with the usual LOO norm, is the dual space of the space of integrable (equivalence classes of) random variables, L1(J?,F,lP') (also denoted by L1(lP') or

L1 if no confusion is possible). We will identify, through the Radon-Nikodym

theorem, finite measures that are absolutely continuous with respect to lP',

with their densities, i.e. with functions in L1. This may occasionally lead to

expressions like lip. - ill where p. is a measure and i E L1. If Q is a probability defined on the a-algebra F, we will use the notation EQ to denote the

expected value operator defined by the probability Q. Let us also recall, see

Dunford and Schwartz [12] for details, that the dual of LOO(lP') is the Banach

space ba(J?, F, lP') of all bounded, finitely additive measures p. on (J?, F) with

the property that lP'(A) = 0 implies p.(A) = O. In case no confusion is possible

we will abbreviate the notation to ba(lP'). A positive element p. E ba(lP') such

that p.(1) = 1 is also called a finitely additive probability, an interpretation

that should be used with care. To keep notation consistent with integration

theory we sometimes denote the action p.(f) of p. E ba(lP) on the bounded

function i, by EJLU], The Yosida-Hewitt theorem, see [32], implies for each

p. E ba(lP'), the existence of a uniquely defined decomposition p. = P.a + p.p,

where P.a is a a-addtive measure, absolutely continuous with respect to lP',

i.e. an element of L1(lP'), and where p.p is a purely finitely additive measure.

Furthermore the results in Yosida and Hewitt [32] show that there is a countable partition (An)n of J? into elements of F, such that for each n, we have

that p.p(An) = O.

The paper is organised as follows. In section 2 we repeat the definition

of coherent risk measure and relate this definition to submodular and supermodular functionals. We will show that using bounded finitely additive

measures, we get the same results as in Artzner et.al. [2]. This section is a

standard application of the duality theory between Loo and its dual space ba.

The main purpose of this section is to introduce the notation. In section 3 we

relate several continuity properties of coherent risk measures to properties

of a defining set of probability measures. This section relies heavily on the

duality theory of the spaces £1 and Loo. Examples of coherent risk measures

are given in section 4. By carefully selecting the defining set of probability

measures, we give examples that are related to higher moments of the random

Coherent Risk Measures on General Probability Spaces

3

variable. Section 5 studies the extension of a coherent risk measure, defined

on the space Loo to the space LO of all random variables. This extension to

LO poses a problem since a coherent risk measure defined on LO is a convex

function defined on LO. Nikodym's result on LO, then implies that, at least

for an atomless probability IP', there are no coherent risk measures that only

take finite values. The solution given, is to extend the risk measures in such

a way that it can take the value +00 but it cannot take the value -00. The

former (+00) means that the risk is very bad and is unacceptable for the

economic agent (something like a risk that cannot be insured). The latter

(-00) would mean that the position is so safe that an arbitrary amount of

capital could be withdrawn without endangering the company. Clearly such

a situation cannot occur in any reasonable model. The main mathematical

results of this section are summarised in the following theorem

Theorem 1.1. If Puis a norm closed, convex set of probability measures,

all absolutely continuous with respect to IP', then the following properties are

equivalent:

(1) For each f E L~ we have that

lim inf EQ[f!\ n]

n QEP"

(2) There is a 'Y

< +00

> 0 such that for each A with IP'[A] :::; 'Y we have

inf Q[A] = O.

QEP"

(3) For every f E L~ there is Q E P u such that EQ[J] < 00.

(4) There is a () > 0 such that for every set A with IP'[A] < () we can find an

element Q E P u such that Q[A] = O.

(5) There is a J > 0, as well as a number K such that for every set A

with IP'[A] < () we can find an element Q E P u such that Q[A] = 0 and

II~IIDO :::;K.

In the same section 5, we also give extra examples showing that, even

when the defining set of probability measures is weakly compact, the Beppo

Levi type theorems do not hold for coherent risk measures. Some of the

examples rely on the theory of non-reflexive Orlicz spaces. In section 6 we

discuss, along the same lines as in Artzner et.al. [2], the relation with the

popular concept, called Value at Risk and denoted by VaR. Section 7 is

devoted to the relation between convex games, coherent risk measures and

non-additive integration. We extend known results on the sigma-core of a

game to cooperative games that are defined on abstract measure spaces and

that do not necessarily fulfill topological regularity assumptions. This work

is based on previous work of Parker, [25] and of the author [7]. In section 8

we give some explicit examples that show how different risk measures can be.

4

Freddy Delbaen

2 The General Case

In this section we show that the main theorems of the papers Artzner et.al. [1]

and [2] can easily be generalised to the case of general probability spaces. The

only difficulty consists in replacing the finite dimensional space 1R'o by the

space of bounded measurable functions, Loo(r). In this setting the definition

of a coherent risk measure as given in Artzner et.al. [1] can be written as:

Definition 2.1. A mapping p : L 00 (n, F, IP')

-t

IR is called a coherent risk

measure if the following properties hold

(1)

(2)

(3)

(4)

If X 2: 0 then p(X) ::; O.

Subadditivity: p(Xl + X 2) ::; p(Xd + P(X2)'

Positive homogeneity: for ,X 2: 0 we have p('xX) = 'xp(X).

For every constant function a we have that p(a + X) = p(X) - a.

Remark: We refer to Artzner et.al. [1] and [2] for an interpretation and

discussion of the above properties. Here we only remark that we are working

in a model without interest rate, the general case can "easily" be reduced to

this case by "discounting".

Although the properties listed in the definition of a coherent risk measure

have a direct interpretation in mathematical finance, it is mathematically

more convenient to work with the related submodular function, 'IjJ, or with

the associated supermodular function, ¢. The definitions we give below differ

slightly from the usual ones. The changes are minor and only consist in the

part related to positivity, i.e. to part one of the definitions.

Definition 2.2. A mapping 'IjJ: LOO

-t

IR is called submodular if

(1) For X ::; 0 we have that 'IjJ(X) ::; O.

(2) If X and Y are bounded random variables then 'IjJ(X + Y) ::; 'IjJ(X) +'IjJ(Y).

(3) For'x 2: 0 and X E LOO we have 'IjJ('xX) = ,X'IjJ(X)

The submodular function is called translation invariant if moreover

(4) For X E L oo and a E IR we have that 'IjJ(X

Definition 2.2'. A mapping ¢: L oo

-t

+ a) = 'IjJ(X) + a.

IR is called supermodular if

(1) For X 2: 0 we have that ¢(X) 2: O.

(2) If X and Yare bounded random variables then ¢(X + Y) 2: ¢(X) + ¢(Y).

(3) For'x 2: 0 and X E Loo we have ¢('xX) = 'x¢(X)

The supermodular function is called translation invariant if moreover

(4) For X E Loo and a E IR we have that ¢(X

+ a) = ¢(X) + a.

Coherent Risk Measures on General Probability Spaces

5

Remark: If p is a coherent risk measure and if we put 1jJ(X) = p( -X) we

get a translation invariant submodular functional. If we put cj>(X) = -p(X),

we obtain a supermodular functional. These notations and relations will be

kept fixed throughout the paper.

Remark: Submodular functionals are well known and were studied by Choquet in connection with the theory of capacities, see [6]. They were used by

many authors in different applications, see e.g. section 7 of this paper for

a connection with game theory. We refer the reader to [30] for the development and the application of the theory to imprecise probabilities and belief

functions. These concepts are certainly not disjoint from risk management

considerations. In [29], P. Walley gives a discussion of properties that may

also be interesting for risk measures. In [21], MaaB gives an overview of existing theories. The following properties of a translation invariant supermodular

mappings cj>, are immediate

(1) cj>(0) = 0 since by positive homogeneity: cj>(0) = cj>(2 x 0) = 2cj>(0).

(2) If X::; 0, then cj>(X) ::; o. Indeed 0 = cj>(X + (-X)) ~ cj>(X) + cj>(-X)

and if X ::; 0, this implies that cj>(X) ::; -cj>( - X) ::; O.

(3) If X ::; Y then cj>(X) ::; cj>(Y). Indeed cj>(Y) ~ cj>(X) + cj>(Y - X) ~ cj>(X).

(4) cj>(a) = a for constants a E JR.

(5) If a ::; X ::; b, then a ::; cj>(X) ::; b. Indeed X - a ~ 0 and X - b ::; o.

(6) cj> is a convex norm-continuous, even Lipschitz, function on VXJ. In other

words 1cj>(X - Y)I ::; IIX - Ylloo.

(7) cj> (X - cj>(X)) = O.

The following theorem is an immediate application of the bipolar theorem

from functional analysis.

Theorem 2.3. Suppose that p: Loo(lP') --+ JR is a coherent risk measure with associated sub(super)modular function 1jJ (cj». There is a convex

a( ba(lP') , L oo (IP')) -closed set Pba of finitely additive probabilities, such that

1jJ(X) = sup EJL[X]

and

JLEPba

Proof. Because -p(X) = cj>(X) = -1jJ( -X) for all X E Loo, we only have to

show one of the equalities. The set C = {X I cj>(X) ~ O} is clearly a convex

and norm closed cone in the space Loo(IP'). The polar set Co = {JL I 'rIX E C :

EJL[X] ~ O} is also a convex cone, closed for the weak* topology on ba(IP'). All

elements in Co are positive since L,+ C C. This implies that for the set Pba,

defined as Pba = {JL I JL E Co and JL(I) = I}, we have that Co = UA>OAPba.

The duality theory, more precisely the bipolar theorem, then implies that

C = {X I 'rIJL E Pba : EJL[X] ~ O}. This means that cj>(X) ~ 0 if and

only if EJL[X] ~ 0 for all JL E Pba. Since ¢(X - cj>(X)) = 0 we have that

X - cj>(X) E C and hence for all JL in Pba we find that EJL[X - cj>(X)] ~ O.

This can be reformulated as

6

Freddy Delbaen

Since for arbitrary € > 0, we have that ¢>(X - ¢>(X) - €) < 0, we get that

X -¢>(X)-€ fI. C. Therefore there is a I-L E Pba such that EI'[X -¢>(X)-€] < 0

which leads to the opposite inequality and hence to:

o

Remark on notation: From the proof of the previous theorem we see that

there is a one-to-one correspondence between

(1) coherent risk measures p,

(2) the associated supermodular function ¢>(X) = -p(X),

(3) the associated submodular function 'I/J(X) = p(-X),

(4) weak* closed convex sets of finitely additive probability measures

Pba

C ba(lP') ,

(5) 11.1100 closed convex cones C C Loo such that L,+ C C.

The relation between C and p is given by

p(X)

= inf {a 1X + a E C}.

The set C is called the set of acceptable positions, see Artzner et.al. [2]. When

we refer to any of these objects it will be according to these notations.

Remark on possible generalisations: In the paper by Jaschke and Kuchler,

[18] an abstract ordered vector space is used. Such developments have interpretations in mathematical finance and economics. In a private discussion

with Kabanov it became clear that there is a way to handle transactions costs

in the setting of risk measures. In order to do this, one should replace the

space Loo of bounded real-valued random variables by the space of bounded

random variables taking values in a finite dimensional space IRn. By replacing

n by {I, 2, ... , n} x n, part of the present results can be translated immediately. The idea to represent transactions costs with a cone was developed by

Kabanov, see [19].

Remark on the interpretation of the probability space: The set n and the IIalgebra:F have an easy interpretation. The lI-algebra:F for instance, describes

all the events that become known at the end of an observation period. The

interpretation of the probability IP' seems to be more difficult. The measure

IP' describes with what probability events might occur. But in economics and

finance such probabilities are subjective. Regulators of the finance industry

might have a completely different view on probabilities than the financial

institutions they control. Inside one institution there might be a different view

between the different branches, trading tables, underwriting agents, etc .. An

Coherent Risk Measures on General Probability Spaces

7

insurance company might have a different view than the reinsurance company

and than their clients. But we may argue that the class of negligible sets

and consequently the class of probability measures that are equivalent to lI"

remains the same. This can be expressed by saying that only the knowledge

of the events of probability zero is important. So we only need agreement

on the "possibility" that events might occur, not on the actual value of the

probability.

In view of this, there are two natural spaces of random variables on which

we can define a risk measure. Only these two spaces remain the same when

we change the underlying probability to an equivalent one. These two spaces

are £OO(n,F,lI") and £O(n,F,lI"). The space £0 cannot be given a norm

and cannot be turned into a locally convex space. E.g. if the probability lI"

is atomless, i.e. supports a random variable with a continuous cumulative

distribution function, then there are no nontrivial (i.e. non identically zero)

continuous linear forms on £0, see Nikodym [24]. The extension of coherent

risk measures from £00 to £0 is the subject of section 5.

3 The u-Additive Case

The previous section gave a characterisation of translation invariant submodular functionals (or equivalently coherent risk measures) in terms of finitely

additive probabilities. The characterisation in terms of a-additive probabilities requires additional hypotheses. E.g. if JL is a purely finitely additive

measure, the expression measure cannot be described by a a-additive probability measure. So we need

extra conditions.

Definition 3.1. The translation invariant supermodular mapping IR is said to satisfy the Fatou property if sequence, (X n )n>l' of functions, uniformly bounded by 1 and converging to

X in probability.Remark:

Equivalently we could have said that the coherent risk measure

p associated with the supermodular function

if for the said sequences we have p(X) ::; liminf p(Xn). Using similar ideas

as in the proof of theorem 2.3 and using a characterisation of weak* closed

convex sets in £00, we obtain:

Theorem 3.2. For a translation invariant supermodular mapping

(1) There is an £l(lI")-closed, convex set of probability measures P u , all of

them being absolutely continuous with respect to lI" and such that for X E

£00:

Springer-Verlag Berlin Heidelberg GmbH

Klaus Sandmann

Philipp J. Schonbucher (Eds.)

Advances in

Finance and

Stochastics

Essays in Honour

of Dieter Sondermann

With 32 Figures

Springer

Klaus Sandmann

Johannes Gutenberg-Universităt Mainz

Lehrstuhl fiir Bankbetriebslehre

Jakob Welder-Weg 9

55128 Mainz

Germany

e-mail: sandmann@forex.bwl.uni-mainz.de

Philipp J. Schonbucher

Rheinische Friedrich- Wilhelms-Universităt Bonn

Inst. f. Gesellschafts- u. Wirtschaftswissenschaften

Statistische Abteilung

Adenauerallee 24-42

53113 Bonn

Germany

e-mail: P.Schonbucher@finasto.uni-bonn.de

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Advances in finance and stochastics: essays in honour of Dieter Sondermann/

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Preface

Finance and Stochastics and Dieter Sondermann are directly and inextricably

linked to each other. The recognition and the success of this journal would

not have been possible without his untiring commitment, his sensitivity for

scientific quality and originality as well as his trustworthiness when dealing

with the authors. One could almost say: Finance and Stochastics is Dieter

Sondermann, since without him this journal would not be.

In the preface of the first issue of Finance and Stochastics in January

1997, Dieter Sondermann referring to the significance of the thesis of Louis

Bachelier, stated: 'Thus, the year 1900 may be considered as the birth date

of both Finance and Stochastics'. Further on he wrote: 'The journal Finance

and Stochastics is devoted to the fruitful interface of these two disciplines'.

What is there to add? It was important to identify and to articulate such

a goal, yet to translate it into action and to make it possible was crucial. It

is due to Dieter Sondermann's initiative and constant work that the idea of

Finance and Stochastics has turned into a highly reputable and successful

project. His unfailing commitment as founder and chief editor has made this

journal an important publication forum of international renown. A publication in Finance and Stochastics is a guarantee of originality and quality for

scientific papers.

Thus, what could have been more natural than the idea of honouring Dieter Sondermann on the occasion of his 65th birthday with a collection of

research papers entitled Advances in Finance and Stochastics? Those who

know him would surely agree that especially Dieter Sondermann, in his modest and undemonstrative way, would never have approved of such an honour.

Luckily, the person to be honoured does not have a say in the matter. However, if he had had one and had not been able to prevent it happening, it is

likely that he would have warned us emphatically against a conception that

was one-sided and looked back upon his own contributions. He might even

have considered the exercise quite superfluous. Instead, his one and only concern would have been for the reader interested in scientific knowledge and

the solution of problems.

'The future has more Futures'. This 'bon mot' of the financial market

also holds good for Dieter Sondermann's scientific work and his involvement

which has always been diverse and with a clear focus on the future. Dieter

VI

Preface

Sondermann was among the first scientists in Germany to apply themselves

to the study of Mathematical Finance. Influenced by the seminal work of

Fisher Black, Myron Scholes and Robert C. Merton and by virtue of his own

profound understanding of the theory of general equilibrium, his contributions often mark the starting point for further development. In 1985, with

Hedging of Non-Redundant Contingent Claims, Dieter Sondermann, together

with Hans Follmer, paved the way for the pricing and hedging of options in

an incomplete financial market. At an early stage he recognises the importance of the theory of arbitrage for the evaluation of insurance risk, which

he demonstrates in Reinsurance in Arbitrage Free Markets (1991). With a

similar feel for new ground, he proved the market model approach to the

term structure of interest rates in his work Closed Form Solutions for Term

Structure Derivatives with Log-Normal Interest Rates (1997, together with

Kristian R. Miltersen and Klaus Sandmann).

Dieter Sondermann's academic career might be considered surprising and

unusual, especially in its initial phase. Yet, if one looks at it from today's

perspective, one can see easily how each step and each stage form an integral part of a consistent whole. He was born on May 10th, 1937 in Duisburg,

Germany. His early years do not directly point to an academic career: his employment as a forwarding agent in the 'Rhenania Allgemeine Speditions AG'

in 1953, his examination in 1956 as a business assistant and finally his activity until 1958 as an expedient in the shipping company 'Vereinigte Stinnes

Reederei GmbH' in Duisburg Ruhrort. During those years, two notions must

have become rooted in his mind, his love of the Rhine and of shipping and

his love of pursuing promising ideas. After his Abitur in 1960, Dieter Sondermann embarked on his studies of Mathematics, Physics and Economics at the

University of Bonn. Little did he know (or even hope), when leaving Bonn in

1962 with a Vordiplom and heading for Hamburg, that he was to return as a

professor of Economics and Statistics not quite 17 years later. Many stages

and formative encounters awaited him still. After his Diplom in Mathematics in 1966, Prof. Dr. Heinz Bauer who had noticed this promising young

mathematician from Hamburg, invited him to the University of Erlangen.

Here, after only two years, Dieter Sondermann obtained his Ph.D. in 1968.

There was no respite for the young academic with such diverse interests. No

sooner had he obtained his doctoral degree than his interest in Economics

was kindled - and this with lasting effect, not least because of Theory of

Value, the 'magic' book by Gerard Debreu. It fascinated him and filled him

with enthusiasm. With his innate capacity for sound judgement he clearly

grasped the opportunities and perspectives contained in the work - and he

used them. In 1970 Dieter Sondermann was appointed lecturer in Mathematical Economics at the University of Saarbriicken. Yet, curiosity deriving from

fascination requires scientific discussion. Thus his path led to the Center of

Operations Research and Econometrics, CORE, in Louvain, Belgium, where

from 1970 to 1972 he was a visiting research professor. Here at CORE a

Preface

VII

great number of committed young scientists met up, and it was here that the

foundations were laid for many scientific and personal friendships which were

to last until the present day. In 1972 Dieter Sondermann returned to Bonn,

this time as Visiting Associate Professor, in 1973 he was in Berkeley, USA,

and a year later he accepted a full professorship in Economics at the University of Hamburg, Germany. The same year he joined the editorial board

of the Journal of Mathematical Economics, founded by Werner Hildenbrand,

on which he served until 1985. At the same time, from 1973 until 1980, he

was a member of the editorial board of the Journal of Economic Theory,

and from 1983 until 1992 of that of the Applicandae Mathematicae (Acta). In

1979 he became Fellow of the lAS at the Hebrew University, Jerusalem. This

is also the year in which he accepted a chair in Economics and Statistics at

the University of Bonn.

Dieter Sondermann, Bonn, the Rheinische Friedrich-Wilhelms University

and the Rhine are intertwined in so many different ways. His house by the

Rhine serves as a refuge for him, his wife, his family and their friends. Even

the perennial threat of high water cannot mar his lifelong attachment to the

Rhine and Rhine shipping. Instead, with a calmness that is so typical of

him, he will contemplate such a phenomenon of nature in statistical terms.

With the same calmness, full of determination, and most successfully, Dieter

Sondermann manages, from 1985 until 1999, Stochastics of Financial Markets, the subproject B 3 of the Sonderforschungsbereich 303. During these

15 years, this research team, under his leadership, gains recognition at home

and abroad and makes a lasting contribution towards the development and

importance of Mathematical Finance. His open and problem-oriented style

of discussion deeply influences work methods and fosters an atmosphere of

curiosity. To bring into accord both research and teaching has always been

for him - and still is - a constant matter of concern. In a personal and human

manner that is so characteristic of him, Dieter Sondermann has, throughout the years, supported and influenced the career of his numerous members

of staff. Many of his students, themselves now in responsible positions at

universities or in industry, remain deeply indebted to him.

There are as many reasons for showing our gratitude to Dieter Sondermann as there are possibilities for expressing this. With Advances in Finance

and Stochastics we simply want to say: Thank you!

The future has more Futures, Dieter!

February 2002, Bonn

Klaus Sandmann

Philipp Schonbucher

Introduction

In many areas of finance and stochastics, significant advances have been made

since this field of research was opened by Black, Scholes and Merton in 1973.

The collection of contributions in Advances in Finance and Stochastics reflects this variety. Necessarily, a selection of topics had to be made, and we

endeavoured to choose those that are currently in the focus of active research

and will remain so in future. This selection spans risk management, portfolio theory and multi-asset derivatives, market imperfections, interest-rate

modelling and exotic options.

Since Follmer and Sondermann (1986) published one of the first mathematical finance papers on risk management in incomplete markets, quantitative research has developed rapidly in this area. The first three papers of this

volume represent the recent developments in this area.

In the first paper on risk management, Delbaen extends the fundamental

notion of a coherent risk measure in two directions from the original definition

in Artzner et.al. (1999): the underlying probability space is now be a general

probability space (and not finite) and the class of risks that are measured

is extended to encompass all random variables on this space. Using methods

from the theory of convex games he is able to prove the analogies of the results

of Artzner et.al. (1999) in this much more general setup. But not everything

carries through identically from the discrete setup: Delbaen shows that now a

coherent risk measure has to be allowed to assume infinite values, representing

completely unacceptable risks. The following contribution by Follmer and

Schied also treats coherent risk measures, but only as a special case of a more

general class of risk measures: the convex risk measures. The authors show

that convex risk measures can be represented as a supremum of expectations

under different measures, corrected by a penalty function that depends on the

probability measure alone. They also connect these risk measures to utilitybased risk measurement. The third article on risk management is authored

by Embrechts and Novak who give a survey of recent developments in the

modelling and measurement of extremal events. While the first two articles

are concerned with the question of a consistent allocation of risk capital to a

given set of risks, this article gives asymptotic answers to the question of the

probability with which this level of risk capital will be exceeded.

X

Introduction

The part on portfolio theory opens with a paper by Werner in which he

develops a multi-period extension to the CAPM, the APT and similar factor

pricing models. By measuring the risk of the assets in terms of the risk of the

underlying dividend streams (instead of the one-period returns), the author

is able to give conditions under which exact factor pricing relationships hold.

In contrast to this portfolio-selection problem, Duan and Pliska consider the

pricing of options on multiple co-integrated assets. Apart from providing necessary conditions for cointegration of a set of assets with GARCH-stochastic

volatilities, they also study the effect that cointegrating relationships under

the physical measure have on the dynamics of the assets under the equilibrium pricing measure and on the dynamics of risk premia. In the following

paper, Madan, Milne and Elliott study the effects that arise when several investors use different, individual factor pricing models, and these models are

aggregated. While Werner took the factor structure as given in his model,

Madan et.al. want to understand where economy-wide risk factors and riskpremia arise from, they shift the focus from asset-returns to identifying and

explaining investor-specific risk exposures.

Market imperfections are the theme of the next three contributions.

Kabanov and Stricker consider super-hedging strategies under transaction

costs. They characterise the hedging-set (the set of initial endowments that

allow a self-financing super-replication) of a contingent claim in a general

setup with non-constant transaction costs. In the following paper, Frey and

Patie address the problem of hedging options in illiquid markets. In a simulation study they show that a hedging strategy based upon a nonlinear partial

differential equation that includes liquidity effects can significantly improve

the performance of the hedge. In Frey and Patie's contribution illiquidity

takes the form of market impact, Le. the transactions of a large trader move

prices, but he is able to trade at any time he chooses. Rogers and Zane consider a different kind of illiquidity in the third paper of this group: Here,

traders are only allowed to trade at Poisson arrival times which they cannot

influence. The traders' objective is a consumption/investment problem similar to Merton (1969). Rogers and Zane establish that Merton's investment

rule (investing a fixed proportion of wealth in the risky asset) is still optimal,

and characterize the modification of the optimal consumption process. Using

an asymptotic expansion, they assess the cost of illiquidity to the investor.

The two contributions on interest-rate modelling both build upon the

market-modelling approach for observed effective interest rates by Miltersen,

Sandmann and Sondermann (1997). Bhar et.al. provide an estimation methodology for a short-rate model which explicitly recognizes the fact that the

short term interest-rate is unobservable. Their approach aims to connect the

stochastic models for the continuously compounded short rate with the observed effective, discretely compounded rates.

Introduction

XI

Schlogl analyses this connection in the other direction and shows that

every market model implies a model for the continuously compounded short

rate that is uniquely determined by the interpolation method used for rates

maturing between tenor dates. He provides an interpolation method which

preserves the Markovian properties of discrete-tenor models but allows for

continuous stochastic dynamics of the short rate.

The final set of contributions has its focus on specific pricing problems

that arise in the pricing of exotic options, in particular the connection between

insurance and financial markets, optimal stopping, and barrier features which

all affect the payoff of the option in a nonlinear way.

The connection between the markets for insurance and financial risks

has been a long-standing area of interest to Dieter Sondermann. Nielsen and

Sandmann analyse in their contribution one example where this connection is

particularly evident: equity-linked life and pension insurance contracts. The

authors give results for the existence of a fair periodic premium and provide

approximate and numerical results for their magnitude.

Optimal stopping is the theme of the contributions by Schweizer; Shepp,

Shiryaev and Sulem; and Peskir and Shiryaev. Schweizer analyses the optimal stopping problems posed by Bermudan options. As Bermuda options

can only be exercised in a subset of the lifetime of the option, the early exercise strategies are subject to this additional restriction. Schweizer shows

under which conditions the problem can be reduced to a modified American

(unrestricted) optimal stopping problem, and how super-replication strategies

can be derived in this setup.

Shepp, Shiryaev and Sulem consider an option that combines American

early exercise, a knockout barrier and lookback-features: the barrier version

of the Russian option. Here, the early exercise strategies are restricted by

the knockout barrier of the option. Despite the complicated structure of the

option, they are able to provide the optimal exercise strategy and the value

function of this derivative.

The following contribution by Schiirger contains an analysis of the distribution, moments and Laplace transforms of the suprema of several stochastic

processes - a problem with immediate applications for the pricing of barrier

and lookback options. Schiirger gives explicit formulae for these quantities for

Bessel processes as well as for strictly stable Levy processes with no positive

jumps. For this he uses an elegant transformation from the maximum of a

stochastic process to its first hitting time.

The final contribution again addresses the question of optimal stopping.

Peskir and Shiryaev analyse the Poisson disorder problem, the problem of

detecting a change in the intensity of a Poisson process. In this context they

show that the smooth-pasting condition is not always valid for the optimal

value function if the state vector can be discontinuous.

XII

Introduction

All authors are leading experts in their fields and we are very grateful to

them for their contributions to this volume. Special thanks also go to Anne

Ruston for expert advice in language questions, Catriona Byrne and Susanne

Denskus from Springer and to Florian Schroder.

Through the input of all these people this book has become a fitting

present to mark the occasion of Dieter Sondermann's 65 th birthday: a volume

of up-to-date research on honour of a creative researcher and the editor of a

leading journal, who has helped shape the subject of mathematical finance.

1. Follmer, Hans and Dieter Sondermann (1986): "Hedging of Non-Redundant Contingent Claims" in: W. Hildenbrandt and A. Mas-Colell (eds.) Contributions to

Mathematical Economics, in Honor of Gerard Debreu, North-Holland.

2. Sondermann, Dieter (1991) "Reinsurance in Arbitrage Free Markets", Insurance:

Mathematics and Economics 10, 191-202.

3. Miltersen, Kristian, Klaus Sandmann and Dieter Sondermann (1997), "Closed

Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates" ,

Journal of Finance 52(1), 409-430.

Table of Contents

Coherent Risk Measures on General Probability Spaces

Freddy Delbaen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Robust Preferences and Convex Measures of Risk

Hans Follmer, Alexander Schied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39

Long Head-Runs and Long Match Patterns

Paul Embrechts, Sergei Y. Novak ................................. 57

Factor Pricing in Multidate Security Markets

Jan Werner ................................................... 71

Option Pricing for Co-Integrated Assets

Jin- Chuan Duan, Stanley R. Pliska ............................... 85

Incomplete Diversification and Asset Pricing

Dilip B. Madan, Frank Milne, Robert J. Elliott . .................... 101

Hedging of Contingent Claims under Transaction Costs

Yuri M. Kabanov, Christophe Stricker . ............................ 125

Risk Management for Derivatives in Illiquid Markets:

A Simulation Study

Rudiger Frey, Pierre Patie ....................................... 137

A Simple Model of Liquidity Effects

L.-C.-G. Rogers, Omar Zane . .................................... 161

Estimation in Models of the Instantaneous Short Term

Interest Rate by Use of a Dynamic Bayesian Algorithm

Ramaprasad Bhar, Carl Chiarella, W ol/gang J. Runggaldier . . . . . . . . .. 177

Arbitrage-Free Interpolation in Models of Market Observable

Interest Rates

Erik Schlogl .................................................... 197

XIV

Table of Contents

The Fair Premium of an Equity-Linked Life and Pension

Insurance

J. Aase Nielsen, Klaus Sandmann . ................................ 219

On Bermudan Options

Martin Schweizer . .............................................. 257

A Barrier Version of the Russian Option

Larry A. Shepp, Albert N. Shiryaev, Agnes Sulem ................... 271

Laplace Transforms and Suprema of Stochastic Processes

Klaus Schurger ................................................. 285

Solving the Poisson Disorder Problem

Goran Peskir, Albert N. Shiryaev ................................. 295

List of Contributors

BHAR, RAMAPRASAD

School of Banking and Finance,

The University of New South Wales

Sydney 2052, Australia

r.bhar~unsw.edu.au

CHIARELLA, CARL

School of Finance and Economics,

University of Technology, Sydney

PO Box 123, Broadway, NSW 2007, Australia

carl.chiarella~uts.edu.au

DELBAEN, FREDDY

Departement fiir Mathematik,

Eidgenossische Technische Hochschule Ziirich

ETH-Zentrum, CH 8092 Ziirich, Switzerland

delbaen@math.ethz.ch

DUAN, JIN-CHUAN

J.L. Rotman School of Management, University of Toronto

105 St. George Street, Toronto M5S 3E6, Canada

jcduan~rotman.utoronto.ca

ELLIOTT, ROBERT J.

Department of Mathematical Sciences, University of Alberta

Edmonton, Alberta T6G 2G1, Canada

relliott@gpu.srv.ualberta.ca

XVI

List of Contributors

EMBRECHTS, PAUL

Departement fUr Mathematik,

Eidgenossische Technische Hochschule Zurich

ETH-Zentrum, CH 8092 Zurich, Switzerland

embrecht@math.ethz.ch

FOLLMER, HANS

Institut fUr Mathematik, Humboldt-Universitiit Berlin

Unter den Linden 6, 10099 Berlin, Germany

foellmer@mathematik.hu-berlin.de

FREY, RUDIGER

Mathematisches Institut, Fakultiit fUr Mathematik und Informatik,

Universitiit Leipzig

Augustusplatz 10/11, D-04109 Leipzig, Germany

frey@mathematik.uni-leipzig.de

KABANOV, YURI M.

Laboratoire de Mathematiques, Universite de Franche-Comte

Universite de Franche-Comte, 16 Route de Gray, F-25030 Besanc;on

Cedex, France

youri.kabanov@math.univ-fcomte.fr

MADAN, DILIP

B.

Department of Finance, Robert H. Smith School of Business,

University of Maryland

4409 Van Munching Hall, College Park, MD 20742-1815, USA

dmadan@rhsmith.umd.edu

MILNE, FRANK

Department of Economics Queen's University

Kingston, Ontario K7L 3N6, Canada

milnef@qed.econ.queensu.ca

NIELSEN,

J.

AASE

Aarhus University, Dept. of Operations Research

Bldg. 530, Ny Munkegade, DK-8000 Aarhus, Denmark

atsjan@imf.au.dk

List of Contributors

NOVAK, SERGEI

Y.

Department of Mathematical Sciences, Brunei University

Uxbridge UB8 3PH, United Kingdom

mastssn~brunel.ac.uk

PATIE, PIERRE

RiskLab, Departement fiir Mathematik,

Eidgenossische Technische Hochschule Zurich

ETH-Zentrum, CH 8092 Ziirich, Switzerland

patie~math.ethz.ch

PESKIR, GORAN

Institute of Mathematics, University of Aarhus

Ny Munkegarde, 8000 Aarhus, Denmark

goran~imf.au.dk

PLISKA, STANLEY R.

Department of Finance, University of Illinois at Chicago

601 S. Morgan Street, Chicago, IL 60607-7124, USA

srpliska~uic.edu

ROGERS,

L. C.

G.

University of Bath, Department of Mathematical Sciences

University of Bath, Bath BA2 7AY, United Kingdom

L.C.G.Rogers@bath.ac.uk

RUNGGALDIER, WOLFGANG

J.

Dipartimento di Matematica Pura ed Applicata,

Universita di Padova

Via Belzoni 7, 35131-Padova, Italy

runggal@math.unipd.it

SANDMANN, KLAUS

Johannes Gutenberg-University Mainz,

Lehrstuhl fiir Allgemeine BWL und Bankbetriebslehre

Jakob Welder Weg 9, 55099 Mainz, Germany

sandmann@forex.bwl.uni-mainz.de

XVII

XVIII List of Contributors

SCHIED, ALEXANDER

Institut fUr Mathematik, Technische Universitiit Berlin

MA 7-4, StraBe des 17. Juni 136, 10623 Berlin, Germany

schied~mathematik.hu-berlin.de

SCHLOGL, ERIK

University of Technology, Sydney

PO Box 123, Broadway NSW 2007, Australia

Erik.Schlogl~uts.edu.au

SCHONBUCHER, PHILIPP

Universitiit Bonn, Institut fur

Gesellschafts- und Wirtschaftswissenschaften,

Statistische Abteilung

Adenauerallee 24-26, D-53113 Bonn, Germany

p~schonbucher.de

SCHURGER, KLAUS

Universitiit Bonn, Institut fUr

Gesellschafts- und Wirtschaftswissenschaften,

Statistische Abteilung

Adenauerallee 24-26, D-53113 Bonn, Germany

schuerger~finasto.uni-bonn.de

SCHWEIZER, MARTIN

Ludwigs-Maximilians University Munchen,

Mathematisches Institut

TheresienstraBe 39, D-80333 Munchen, Germany

mschweiz~mathematik.uni-muenchen.de

SHEPP, LARRY

A.

Rutgers University, Department of Statistics

Piscataway, NJ 08854-8091, USA

shepp~stat.rutgers.edu

List of Contributors

SHIRYAEV, ALBERT

N.

Steklov Mathematical Institute (MIRAN)

Gubkina 8,117966 GSP-1, Moscow, Russia

shiryaev~mi.ras.ru

STRICKER, CHRISTOPHE

Laboratoire de Mathematiques, Universite de Franche-Comte

16 Route de Gray, F-25030 Besan<;on Cedex, France

Christophe.Stricker~ath.univ-fcomte.fr

SULEM, AGNES

Institut National de Recherche en Informatique et en Automatique

(INRIA)

Domaine de Voluceau, Rocquencourt - B.P. 105,

78153 Le Chesnay Cedex, France

agnes.sulem~inria.fr

WERNER, JAN

Department of Economics, University of Minnesota

1151 Heller Hall, Minneapolis, MN 55455 U.S.A.

jwerner~atlas.socsci.umn.edu

ZANE,OMAR

Warburg Dillon Read, London,

Quantitative Risk, Models and Statistics

UBS, 1 Finsbury Avenue London EC2M 2PG, United Kingdom

Omar.Zane~wdr.com

XIX

Coherent Risk Measures on General

Probability Spaces

Freddy Delbaen*

Department of Mathematics, Eidgenossische Technische Hochschule, Zurich,

Switzerland

Summary. We extend the definition of coherent risk measures, as introduced by

Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how

to define such measures on the space of all random variables. We also give examples

that relates the theory of coherent risk measures to game theory and to distorted

probability measures. The mathematics are based on the characterisation of closed

convex sets P u of probability measures that satisfy the property that every random

variable is integrable for at least one probability measure in the set P u.

Key words: capital requirement, coherent risk measure, capacity theory, convex

games, insurance premium principle, measure of risk, Orlicz spaces, quantile, scenario, shortfall, subadditivity, submodular functions, value at risk

1 Introduction and Notation

The concept of coherent risk measures together with its axiomatic characterization was introduced in the paper Artzner et.al. [1] and further developed

in [2]. Both these papers supposed that the underlying probability space was

finite. The aim of this paper is twofold. First we extend the notion of coherent risk measures to arbitrary probability spaces, second we deepen the

relation between coherent risk measures and the theory of cooperative games.

In many occasions we will make a bridge between different existing theories.

In order to keep the paper self contained, we sometimes will have to repeat

known proofs. In March 2000, the author gave a series of lectures at the Cattedra Galileiana at the Scuola Normale di Pisa. The subject of these lectures

was the theory of coherent risk measures as well as applications to several

problems in risk management. The interested reader can consult the lecture

notes Delbaen [8]. Since the original version of this paper (1997), proofs have

undergone a lot of changes. Discussions with colleagues greatly contributed to

* The author acknowledges financial support from Credit Suisse for his work and

from Societe Generale for earlier versions of this paper. Special thanks go to

Artzner, Eber and Heath for the many stimulating discussions on risk measures

and other topics. I also want to thank Maafi for pointing out extra references to

related work. Discussions with Kabanov were more than helpful to improve the

presentation of the paper. Part of the work was done during Summer 99, while

the author was visiting Tokyo Institute of Technology. The views expressed are

those of the author.

K. Sandmann et al. (eds.), Advances in Finance and Stochastics

© Springer-Verlag Berlin Heidelberg 2002

2

Freddy Delbaen

the presentation. The reader will also notice that the theory of convex games

plays a special role in the theory of coherent risk measures. It was Dieter

Sondermann who mentioned the theory of convex games to the author and

asked about continuity properties of its core, see Delbaen [7]. It is therefore

a special pleasure to be able to put this paper in the Festschrift.

Throughout the paper, we will work with a probability space (J?, F, 1P).

With L 00 (J?, F, 1P) (or L 00 (1P) or even L 00 if no confusion is possible), we mean

the space of all equivalence classes of bounded real valued random variables.

The space LO(J?, F, 1P) (or LO(IP) or simply LO) denotes the space of all equivalence classes of real valued random variables. The space LO is equipped

with the topology of convergence in probability. The space LOO(lP'), equipped

with the usual LOO norm, is the dual space of the space of integrable (equivalence classes of) random variables, L1(J?,F,lP') (also denoted by L1(lP') or

L1 if no confusion is possible). We will identify, through the Radon-Nikodym

theorem, finite measures that are absolutely continuous with respect to lP',

with their densities, i.e. with functions in L1. This may occasionally lead to

expressions like lip. - ill where p. is a measure and i E L1. If Q is a probability defined on the a-algebra F, we will use the notation EQ to denote the

expected value operator defined by the probability Q. Let us also recall, see

Dunford and Schwartz [12] for details, that the dual of LOO(lP') is the Banach

space ba(J?, F, lP') of all bounded, finitely additive measures p. on (J?, F) with

the property that lP'(A) = 0 implies p.(A) = O. In case no confusion is possible

we will abbreviate the notation to ba(lP'). A positive element p. E ba(lP') such

that p.(1) = 1 is also called a finitely additive probability, an interpretation

that should be used with care. To keep notation consistent with integration

theory we sometimes denote the action p.(f) of p. E ba(lP) on the bounded

function i, by EJLU], The Yosida-Hewitt theorem, see [32], implies for each

p. E ba(lP'), the existence of a uniquely defined decomposition p. = P.a + p.p,

where P.a is a a-addtive measure, absolutely continuous with respect to lP',

i.e. an element of L1(lP'), and where p.p is a purely finitely additive measure.

Furthermore the results in Yosida and Hewitt [32] show that there is a countable partition (An)n of J? into elements of F, such that for each n, we have

that p.p(An) = O.

The paper is organised as follows. In section 2 we repeat the definition

of coherent risk measure and relate this definition to submodular and supermodular functionals. We will show that using bounded finitely additive

measures, we get the same results as in Artzner et.al. [2]. This section is a

standard application of the duality theory between Loo and its dual space ba.

The main purpose of this section is to introduce the notation. In section 3 we

relate several continuity properties of coherent risk measures to properties

of a defining set of probability measures. This section relies heavily on the

duality theory of the spaces £1 and Loo. Examples of coherent risk measures

are given in section 4. By carefully selecting the defining set of probability

measures, we give examples that are related to higher moments of the random

Coherent Risk Measures on General Probability Spaces

3

variable. Section 5 studies the extension of a coherent risk measure, defined

on the space Loo to the space LO of all random variables. This extension to

LO poses a problem since a coherent risk measure defined on LO is a convex

function defined on LO. Nikodym's result on LO, then implies that, at least

for an atomless probability IP', there are no coherent risk measures that only

take finite values. The solution given, is to extend the risk measures in such

a way that it can take the value +00 but it cannot take the value -00. The

former (+00) means that the risk is very bad and is unacceptable for the

economic agent (something like a risk that cannot be insured). The latter

(-00) would mean that the position is so safe that an arbitrary amount of

capital could be withdrawn without endangering the company. Clearly such

a situation cannot occur in any reasonable model. The main mathematical

results of this section are summarised in the following theorem

Theorem 1.1. If Puis a norm closed, convex set of probability measures,

all absolutely continuous with respect to IP', then the following properties are

equivalent:

(1) For each f E L~ we have that

lim inf EQ[f!\ n]

n QEP"

(2) There is a 'Y

< +00

> 0 such that for each A with IP'[A] :::; 'Y we have

inf Q[A] = O.

QEP"

(3) For every f E L~ there is Q E P u such that EQ[J] < 00.

(4) There is a () > 0 such that for every set A with IP'[A] < () we can find an

element Q E P u such that Q[A] = O.

(5) There is a J > 0, as well as a number K such that for every set A

with IP'[A] < () we can find an element Q E P u such that Q[A] = 0 and

II~IIDO :::;K.

In the same section 5, we also give extra examples showing that, even

when the defining set of probability measures is weakly compact, the Beppo

Levi type theorems do not hold for coherent risk measures. Some of the

examples rely on the theory of non-reflexive Orlicz spaces. In section 6 we

discuss, along the same lines as in Artzner et.al. [2], the relation with the

popular concept, called Value at Risk and denoted by VaR. Section 7 is

devoted to the relation between convex games, coherent risk measures and

non-additive integration. We extend known results on the sigma-core of a

game to cooperative games that are defined on abstract measure spaces and

that do not necessarily fulfill topological regularity assumptions. This work

is based on previous work of Parker, [25] and of the author [7]. In section 8

we give some explicit examples that show how different risk measures can be.

4

Freddy Delbaen

2 The General Case

In this section we show that the main theorems of the papers Artzner et.al. [1]

and [2] can easily be generalised to the case of general probability spaces. The

only difficulty consists in replacing the finite dimensional space 1R'o by the

space of bounded measurable functions, Loo(r). In this setting the definition

of a coherent risk measure as given in Artzner et.al. [1] can be written as:

Definition 2.1. A mapping p : L 00 (n, F, IP')

-t

IR is called a coherent risk

measure if the following properties hold

(1)

(2)

(3)

(4)

If X 2: 0 then p(X) ::; O.

Subadditivity: p(Xl + X 2) ::; p(Xd + P(X2)'

Positive homogeneity: for ,X 2: 0 we have p('xX) = 'xp(X).

For every constant function a we have that p(a + X) = p(X) - a.

Remark: We refer to Artzner et.al. [1] and [2] for an interpretation and

discussion of the above properties. Here we only remark that we are working

in a model without interest rate, the general case can "easily" be reduced to

this case by "discounting".

Although the properties listed in the definition of a coherent risk measure

have a direct interpretation in mathematical finance, it is mathematically

more convenient to work with the related submodular function, 'IjJ, or with

the associated supermodular function, ¢. The definitions we give below differ

slightly from the usual ones. The changes are minor and only consist in the

part related to positivity, i.e. to part one of the definitions.

Definition 2.2. A mapping 'IjJ: LOO

-t

IR is called submodular if

(1) For X ::; 0 we have that 'IjJ(X) ::; O.

(2) If X and Y are bounded random variables then 'IjJ(X + Y) ::; 'IjJ(X) +'IjJ(Y).

(3) For'x 2: 0 and X E LOO we have 'IjJ('xX) = ,X'IjJ(X)

The submodular function is called translation invariant if moreover

(4) For X E L oo and a E IR we have that 'IjJ(X

Definition 2.2'. A mapping ¢: L oo

-t

+ a) = 'IjJ(X) + a.

IR is called supermodular if

(1) For X 2: 0 we have that ¢(X) 2: O.

(2) If X and Yare bounded random variables then ¢(X + Y) 2: ¢(X) + ¢(Y).

(3) For'x 2: 0 and X E Loo we have ¢('xX) = 'x¢(X)

The supermodular function is called translation invariant if moreover

(4) For X E Loo and a E IR we have that ¢(X

+ a) = ¢(X) + a.

Coherent Risk Measures on General Probability Spaces

5

Remark: If p is a coherent risk measure and if we put 1jJ(X) = p( -X) we

get a translation invariant submodular functional. If we put cj>(X) = -p(X),

we obtain a supermodular functional. These notations and relations will be

kept fixed throughout the paper.

Remark: Submodular functionals are well known and were studied by Choquet in connection with the theory of capacities, see [6]. They were used by

many authors in different applications, see e.g. section 7 of this paper for

a connection with game theory. We refer the reader to [30] for the development and the application of the theory to imprecise probabilities and belief

functions. These concepts are certainly not disjoint from risk management

considerations. In [29], P. Walley gives a discussion of properties that may

also be interesting for risk measures. In [21], MaaB gives an overview of existing theories. The following properties of a translation invariant supermodular

mappings cj>, are immediate

(1) cj>(0) = 0 since by positive homogeneity: cj>(0) = cj>(2 x 0) = 2cj>(0).

(2) If X::; 0, then cj>(X) ::; o. Indeed 0 = cj>(X + (-X)) ~ cj>(X) + cj>(-X)

and if X ::; 0, this implies that cj>(X) ::; -cj>( - X) ::; O.

(3) If X ::; Y then cj>(X) ::; cj>(Y). Indeed cj>(Y) ~ cj>(X) + cj>(Y - X) ~ cj>(X).

(4) cj>(a) = a for constants a E JR.

(5) If a ::; X ::; b, then a ::; cj>(X) ::; b. Indeed X - a ~ 0 and X - b ::; o.

(6) cj> is a convex norm-continuous, even Lipschitz, function on VXJ. In other

words 1cj>(X - Y)I ::; IIX - Ylloo.

(7) cj> (X - cj>(X)) = O.

The following theorem is an immediate application of the bipolar theorem

from functional analysis.

Theorem 2.3. Suppose that p: Loo(lP') --+ JR is a coherent risk measure with associated sub(super)modular function 1jJ (cj». There is a convex

a( ba(lP') , L oo (IP')) -closed set Pba of finitely additive probabilities, such that

1jJ(X) = sup EJL[X]

and

JLEPba

Proof. Because -p(X) = cj>(X) = -1jJ( -X) for all X E Loo, we only have to

show one of the equalities. The set C = {X I cj>(X) ~ O} is clearly a convex

and norm closed cone in the space Loo(IP'). The polar set Co = {JL I 'rIX E C :

EJL[X] ~ O} is also a convex cone, closed for the weak* topology on ba(IP'). All

elements in Co are positive since L,+ C C. This implies that for the set Pba,

defined as Pba = {JL I JL E Co and JL(I) = I}, we have that Co = UA>OAPba.

The duality theory, more precisely the bipolar theorem, then implies that

C = {X I 'rIJL E Pba : EJL[X] ~ O}. This means that cj>(X) ~ 0 if and

only if EJL[X] ~ 0 for all JL E Pba. Since ¢(X - cj>(X)) = 0 we have that

X - cj>(X) E C and hence for all JL in Pba we find that EJL[X - cj>(X)] ~ O.

This can be reformulated as

6

Freddy Delbaen

Since for arbitrary € > 0, we have that ¢>(X - ¢>(X) - €) < 0, we get that

X -¢>(X)-€ fI. C. Therefore there is a I-L E Pba such that EI'[X -¢>(X)-€] < 0

which leads to the opposite inequality and hence to:

o

Remark on notation: From the proof of the previous theorem we see that

there is a one-to-one correspondence between

(1) coherent risk measures p,

(2) the associated supermodular function ¢>(X) = -p(X),

(3) the associated submodular function 'I/J(X) = p(-X),

(4) weak* closed convex sets of finitely additive probability measures

Pba

C ba(lP') ,

(5) 11.1100 closed convex cones C C Loo such that L,+ C C.

The relation between C and p is given by

p(X)

= inf {a 1X + a E C}.

The set C is called the set of acceptable positions, see Artzner et.al. [2]. When

we refer to any of these objects it will be according to these notations.

Remark on possible generalisations: In the paper by Jaschke and Kuchler,

[18] an abstract ordered vector space is used. Such developments have interpretations in mathematical finance and economics. In a private discussion

with Kabanov it became clear that there is a way to handle transactions costs

in the setting of risk measures. In order to do this, one should replace the

space Loo of bounded real-valued random variables by the space of bounded

random variables taking values in a finite dimensional space IRn. By replacing

n by {I, 2, ... , n} x n, part of the present results can be translated immediately. The idea to represent transactions costs with a cone was developed by

Kabanov, see [19].

Remark on the interpretation of the probability space: The set n and the IIalgebra:F have an easy interpretation. The lI-algebra:F for instance, describes

all the events that become known at the end of an observation period. The

interpretation of the probability IP' seems to be more difficult. The measure

IP' describes with what probability events might occur. But in economics and

finance such probabilities are subjective. Regulators of the finance industry

might have a completely different view on probabilities than the financial

institutions they control. Inside one institution there might be a different view

between the different branches, trading tables, underwriting agents, etc .. An

Coherent Risk Measures on General Probability Spaces

7

insurance company might have a different view than the reinsurance company

and than their clients. But we may argue that the class of negligible sets

and consequently the class of probability measures that are equivalent to lI"

remains the same. This can be expressed by saying that only the knowledge

of the events of probability zero is important. So we only need agreement

on the "possibility" that events might occur, not on the actual value of the

probability.

In view of this, there are two natural spaces of random variables on which

we can define a risk measure. Only these two spaces remain the same when

we change the underlying probability to an equivalent one. These two spaces

are £OO(n,F,lI") and £O(n,F,lI"). The space £0 cannot be given a norm

and cannot be turned into a locally convex space. E.g. if the probability lI"

is atomless, i.e. supports a random variable with a continuous cumulative

distribution function, then there are no nontrivial (i.e. non identically zero)

continuous linear forms on £0, see Nikodym [24]. The extension of coherent

risk measures from £00 to £0 is the subject of section 5.

3 The u-Additive Case

The previous section gave a characterisation of translation invariant submodular functionals (or equivalently coherent risk measures) in terms of finitely

additive probabilities. The characterisation in terms of a-additive probabilities requires additional hypotheses. E.g. if JL is a purely finitely additive

measure, the expression measure cannot be described by a a-additive probability measure. So we need

extra conditions.

Definition 3.1. The translation invariant supermodular mapping IR is said to satisfy the Fatou property if sequence, (X n )n>l' of functions, uniformly bounded by 1 and converging to

X in probability.Remark:

Equivalently we could have said that the coherent risk measure

p associated with the supermodular function

if for the said sequences we have p(X) ::; liminf p(Xn). Using similar ideas

as in the proof of theorem 2.3 and using a characterisation of weak* closed

convex sets in £00, we obtain:

Theorem 3.2. For a translation invariant supermodular mapping

(1) There is an £l(lI")-closed, convex set of probability measures P u , all of

them being absolutely continuous with respect to lI" and such that for X E

£00:

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