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Advances in finance and stochastics

Advances in Finance and Stochastics

Springer-Verlag Berlin Heidelberg GmbH

Klaus Sandmann
Philipp J. Schonbucher (Eds.)

Advances in
Finance and
Essays in Honour
of Dieter Sondermann

With 32 Figures


Klaus Sandmann

Johannes Gutenberg-Universităt Mainz
Lehrstuhl fiir Bankbetriebslehre
Jakob Welder-Weg 9
55128 Mainz
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
e-mail: P.Schonbucher@finasto.uni-bonn.de

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Die Deutsche Bibliothek- CIP-Einheitsaufnahme
Advances in finance and stochastics: essays in honour of Dieter Sondermann/
Klaus Sandmann; Philipp J. Schonbucher (ed.).
ISBN 978-3-642-07792-o
ISBN 978-3-662-04790-3 (eBook)
DOI 10.1007/978-3-662-04790-3

Mathematics Subject Classification (2ooo): 6o-6, 91B
JEL Classification: G13
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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



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



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


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.



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.



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.



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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


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


Table of Contents

The Fair Premium of an Equity-Linked Life and Pension
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


School of Banking and Finance,
The University of New South Wales
Sydney 2052, Australia


School of Finance and Economics,
University of Technology, Sydney
PO Box 123, Broadway, NSW 2007, Australia


Departement fiir Mathematik,
Eidgenossische Technische Hochschule Ziirich
ETH-Zentrum, CH 8092 Ziirich, Switzerland


J.L. Rotman School of Management, University of Toronto
105 St. George Street, Toronto M5S 3E6, Canada


Department of Mathematical Sciences, University of Alberta
Edmonton, Alberta T6G 2G1, Canada


List of Contributors

Departement fUr Mathematik,
Eidgenossische Technische Hochschule Zurich
ETH-Zentrum, CH 8092 Zurich, Switzerland

Institut fUr Mathematik, Humboldt-Universitiit Berlin
Unter den Linden 6, 10099 Berlin, Germany

Mathematisches Institut, Fakultiit fUr Mathematik und Informatik,
Universitiit Leipzig
Augustusplatz 10/11, D-04109 Leipzig, Germany

Laboratoire de Mathematiques, Universite de Franche-Comte
Universite de Franche-Comte, 16 Route de Gray, F-25030 Besanc;on
Cedex, France


Department of Finance, Robert H. Smith School of Business,
University of Maryland
4409 Van Munching Hall, College Park, MD 20742-1815, USA

Department of Economics Queen's University
Kingston, Ontario K7L 3N6, Canada



Aarhus University, Dept. of Operations Research
Bldg. 530, Ny Munkegade, DK-8000 Aarhus, Denmark

List of Contributors


Department of Mathematical Sciences, Brunei University
Uxbridge UB8 3PH, United Kingdom


RiskLab, Departement fiir Mathematik,
Eidgenossische Technische Hochschule Zurich
ETH-Zentrum, CH 8092 Ziirich, Switzerland


Institute of Mathematics, University of Aarhus
Ny Munkegarde, 8000 Aarhus, Denmark


Department of Finance, University of Illinois at Chicago
601 S. Morgan Street, Chicago, IL 60607-7124, USA


L. C.


University of Bath, Department of Mathematical Sciences
University of Bath, Bath BA2 7AY, United Kingdom


Dipartimento di Matematica Pura ed Applicata,
Universita di Padova
Via Belzoni 7, 35131-Padova, Italy

Johannes Gutenberg-University Mainz,
Lehrstuhl fiir Allgemeine BWL und Bankbetriebslehre
Jakob Welder Weg 9, 55099 Mainz, Germany


XVIII List of Contributors

Institut fUr Mathematik, Technische Universitiit Berlin
MA 7-4, StraBe des 17. Juni 136, 10623 Berlin, Germany


University of Technology, Sydney
PO Box 123, Broadway NSW 2007, Australia


Universitiit Bonn, Institut fur
Gesellschafts- und Wirtschaftswissenschaften,
Statistische Abteilung
Adenauerallee 24-26, D-53113 Bonn, Germany


Universitiit Bonn, Institut fUr
Gesellschafts- und Wirtschaftswissenschaften,
Statistische Abteilung
Adenauerallee 24-26, D-53113 Bonn, Germany


Ludwigs-Maximilians University Munchen,
Mathematisches Institut
TheresienstraBe 39, D-80333 Munchen, Germany



Rutgers University, Department of Statistics
Piscataway, NJ 08854-8091, USA

List of Contributors


Steklov Mathematical Institute (MIRAN)
Gubkina 8,117966 GSP-1, Moscow, Russia


Laboratoire de Mathematiques, Universite de Franche-Comte
16 Route de Gray, F-25030 Besan<;on Cedex, France


Institut National de Recherche en Informatique et en Automatique
Domaine de Voluceau, Rocquencourt - B.P. 105,
78153 Le Chesnay Cedex, France


Department of Economics, University of Minnesota
1151 Heller Hall, Minneapolis, MN 55455 U.S.A.


Warburg Dillon Read, London,
Quantitative Risk, Models and Statistics
UBS, 1 Finsbury Avenue London EC2M 2PG, United Kingdom


Coherent Risk Measures on General
Probability Spaces
Freddy Delbaen*
Department of Mathematics, Eidgenossische Technische Hochschule, Zurich,
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


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


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

(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.


(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.


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')


IR is called a coherent risk

measure if the following properties hold


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


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


+ 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


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]



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


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:

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

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

= 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


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
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

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