Advances in finance and stochastics

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 meas

Advances in Finance and Stochastics

Johannes Gutenberg-Universităt Mainz

Lehrstuhl fiir Bankbetriebslehre

Rheinische Friedrich- Wilhelms-Universităt Bonn

Inst f Gesellschafts- u Wirtschaftswissenschaften

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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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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 publica-

tion 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 eter Sondermann on the occasion of his 65th birthday with a collection of

Di-research papers entitled Advances in Finance and Stochastics? Those who

know him would surely agree that especially Dieter Sondermann, in his est 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 How-ever, 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 con-cern would have been for the reader interested in scientific knowledge and the solution of problems

mod-'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 contribu-tions 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 ance of the theory of arbitrage for the evaluation of insurance risk, which

import-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 integ-ral 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 em-ployment as a forwarding agent in the 'Rhenania Allgemeine Speditions AG'

in 1953, his examination in 1956 as a business assistant and finally his ity 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 Sonder-mann 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

activ-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 Mathemat-ics 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 Mathemat-ical 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

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 Mar- kets, 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, through-out 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 mann as there are possibilities for expressing this With Advances in Finance and Stochastics we simply want to say: Thank you!

Sonder-The future has more Futures, Dieter!

February 2002, Bonn

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 re-flects 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, port-folio theory and multi-asset derivatives, market imperfections, interest-rate modelling and exotic options

Since Follmer and Sondermann (1986) published one of the first atical finance papers on risk management in incomplete markets, quantitat-ive research has developed rapidly in this area The first three papers of this volume represent the recent developments in this area

mathem-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 utility-based 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 ne-cessary 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 equilib-rium pricing measure and on the dynamics of risk premia In the following paper, Madan, Milne and Elliott study the effects that arise when several in-vestors 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 risk-premia 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 simula-tion 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 con-sider 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 sim-ilar 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 meth-odology 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 ob-served 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 op-timal stopping problems posed by Bermudan options As Bermuda options can only be exercised in a subset of the lifetime of the option, the early ex-ercise 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 bution, 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

distri-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 tingent Claims" in: W Hildenbrandt and A Mas-Colell (eds.) Contributions to Mathematical Economics, in Honor of Gerard Debreu, North-Holland

Con-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

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

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

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

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

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 7 AY, United Kingdom

Johannes Gutenberg-University Mainz,

Lehrstuhl fiir Allgemeine BWL und Bankbetriebslehre

Jakob Welder Weg 9, 55099 Mainz, Germany

sandmann@forex.bwl.uni-mainz.de

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,

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

Coherent Risk Measures on General

Key words: capital requirement, coherent risk measure, capacity theory, convex games, insurance premium principle, measure of risk, Orlicz spaces, quantile, scen- ario, shortfall, subadditivity, submodular functions, value at risk

1 Introduction and Notation

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

character-in [2] Both these papers supposed that the underlycharacter-ing probability space was finite The aim of this paper is twofold First we extend the notion of co- herent 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 Cat- tedra 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

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 equi-

valence 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

(equi-valence 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 ility defined on the a-algebra F, we will use the notation EQ to denote the

probab-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 able partition (An)n of J? into elements of F, such that for each n, we have that p.p(An) = O

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

of coherent risk measure and relate this definition to submodular and permodular 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

su-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] < +00

n QEP"

(2) There is a 'Y > 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 Pu 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 Pu 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 Pu 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

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

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

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

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

Definition 2.2' A mapping ¢: L oo -t 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 quet 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

Cho-a connection with gCho-ame theory We refer the reCho-ader to [30] for the ment 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

develop-also be interesting for risk measures In [21], MaaB gives an overview of

exist-ing theories The followexist-ing 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

meas-a( bmeas-a(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

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 1 X + 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 in-terpretations 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 ately The idea to represent transactions costs with a cone was developed by Kabanov, see [19]

immedi-Remark on the interpretation of the probability space: The set n and the

II-algebra:F have an easy interpretation The lI-II-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 ular functionals (or equivalently coherent risk measures) in terms of finitely additive probabilities The characterisation in terms of a-additive probab-

submod-ilities requires additional hypotheses E.g if JL is a purely finitely additive measure, the expression </J(X) = E/L[X] gives a translation invariant sub-modular functional This functional, coming from a purely finitely additive measure cannot be described by a a-additive probability measure So we need extra conditions

Definition 3.1 The translation invariant supermodular mapping </J: £00 t

IR is said to satisfy the Fatou property if </J(X) ~ lim sup </J(Xn), for any

sequence, (Xn)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 </J satisfies the Fatou property

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 </J, the

fol-lowing 4 properties are equivalent

(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:

¢(X) = inf EQ[X]

QEP~

(2) The convex cone C = {X I ¢(X) ~ O} is weak*, i.e a(£oo(JP'),£l(JP'))

closed

(3) ¢ satisfies the Fatou property

(4) If Xn is a uniformly bounded sequence that decreases to X a.s., then

(3) =} (2) If ¢ satisfies the Fatou property, then C is weak* closed This is essentially the Krein-Smulian theorem as used in a remark, due to Grothen-dieck, see [14], Supplementary Exercise 1, Chapter 5, part 4 Translated to our special case of C being a cone, this means that it is sufficient to check that C n Bl is closed in probability (Bl stands for the closed unit ball of

£00) So let Xn be a sequence of elements in C, uniformly bounded by 1 and tending to X in probability The Fatou property then shows that also

¢(X) ~ 0, i.e X E C

(2) =} (1) This is not difficult and is done in exactly the same way as in theorem 2.3 But this time we define the polar Co of C in £1 and we apply the bipolar theorem for the duality pair (£1, £00) Because of C being weak* closed, this poses no problem and we define:

Co = {J I ! E £1 and EIP'[! X] ~ 0 for all X E C} ,

PIT = {f I dQ = ! dJP' defines a probability and! E CO}

Of course we have

Co = U>,~oAPIT

So we find a closed convex set of probability measures PIT such that

¢(X) = inf {EQ[Xll Q E PIT}

(1) =} (2), Fatou's lemma implies that every translation invariant submodular mapping, that is given by the in! over a set of probability measures, satisfies the Fatou property Indeed for each Q E PIT we get

where Xn is a sequence, uniformly bounded by 1 and tending to X in

Corollary 3.3 With the notations and assumptions of theorem 2.3 and 3.2

we get that the set PIT is a(ba(JP'),£oo(JP')) dense in Pba

Coherent Risk Measures on General Probability Spaces 9

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 having the Fatou property,

(2) closed convex sets of probability measures PuC L1 (1P'),

(3) weak* closed convex cones C C L= such that L't' C C

We now give an answer to the question when a coherent risk measure can be given as the supremum of expected values, taken with respect to equivalent probability measures

Definition 3.4 The coherent risk measure p is called relevant if for each set

A E :F with IP'[A] > 0 we have that p( -lA) > O When using the associated submodular function, this means that 1/!(lA) > 0 or when using the associated supermodularfunction: 4>( -lA) < O

It is easily seen, using the monotonicity of p, that the property of being evant is equivalent to p(X) > 0 for each nonpositive X E L= such that

rel-IP'[X < 0] > O The economic interpretation of this property is ward

straightfor-Theorem 3.5 For a coherent risk measure, p, that satisfies the Fatou erty, the following are equivalent

prop-(1) p is relevant

(2) The set P; = {Q E P u I Q ~ 1P'} is non empty

(3) The set P; = {Q E Pu I Q ~ 1P'} is norm (i.e L1 norm) dense in Pu'

(4) There is a set pI C P u of equivalent probability measures such that

of the reader, let us briefly sketch the exhaustion argument Since the set P u

is norm closed and convex, the class of sets

is stable for countable unions It follows that up to lP'-null sets, there is a

maximal element Because p is relevant, the only possible maximal element

is n From this it follows that there is Q E P u such that Q ~ 1P' 0

The following theorem characterises the coherent risk measures that satisfy

a continuity property that is stronger than the Fatou property

Theorem 3.6 For a translation invariant supermodular mapping, 4>, the following properties are equivalent

(1) The set P" is weakly compact in L1

(2) The sets Pba and P" coincide

(3) If (Xn)n>1 is a sequence in L oo , uniformly bounded by 1 and tending to

X in probability, then ¢(Xn) tends to ¢(X)

(4) If (An)n is a increasing sequence whose union is il, then ¢(lAJ tends

to 1

Proof Clearly (1) ¢:> (2) (1) ~ (3) If (1) holds, then the set P" is uniformly integrable (by the Dunford-Pettis theorem, see Dunford and Schwartz [12] or Grothendieck [14]) and it follows that EQ[Xn ] tends to EQ[X] uniformly over

the set P" This implies that ¢(X n ) tends to ¢(X) Clearly (3) ~ (4) To prove (4) ~ (1) observe that (4) implies that the set P" is uniformly integ-rable Indeed, if Bn is a sequence of decreasing sets such that the intersection nnBn is empty, then SUPQEP., Q[Bn] :S 1 - infQEP., Q[B~] and hence tends

to O P" being convex and norm closed, this together with the (easy part of the) Dunford-Pettis theorem, implies that P" is weakly compact 0

Example 3.7 This example shows that the property "¢(lAn) tends to zero

for every decreasing sequence of sets with empty intersection" , does not imply that p satisfies the 4 properties of the preceding theorem It does not even imply that p, or ¢, has the Fatou property Take (il,.1", JP') big enough to

support purely finitely additive probabilities, i.e LOO(JP') is supposed to be

infinite dimensional Take J-t E ba(JP') , purely finitely additive, and let Pba

be the segment (the convex hull) joining the two points J-t and JP' Because

there is a a-additive probability in Pba, it is easily seen that p(lAn) = -¢(lAn) = - infQEPbJAn) tends to zero for every decreasing sequence of

sets with empty intersection But clearly the coherent measure cannot satisfy the Fatou property since P" = {IP'} is not dense in Pba To find "explicitly"

a sequence of functions that contradicts the Fatou property, we proceed as follows The measure J-t is purely finitely additive and therefore, by the Yosida-

Hewitt decomposition theorem (see [32]), there is a countable partition of il into sets (Bn)n>1 such that for each n, we have J-t(Bn) = o Take now X

an element in £00 such that Ep[X] = 0 and such that EJL[X] = -1 This implies that p(X) = 1 For the sequence Xn we take Xn = X lUk<nBk The

properties of the sets Bn imply that J-t = 0 on the union Uk<nBk and hence

we have that p(Xn) = Ep[Xn), which tends to 0 as n tends to 00

The next proposition characterises those coherent risk measures that tend to zero on decreasing sequences of sets

Theorem 3.8 For a coherent risk measure, p, the following are equivalent

(1) For every decreasing sequence of sets (An)n>l with empty intersection,

we have that ¢(lAJ = -p(lAJ tends to zer~

(2) sup {IIJ-talll J-t E Poo} = 1, (where J-t = J-ta + J-tp is the Yosida-Hewitt composition)

de-Coherent Risk Measures on General Probability Spaces 11

(3) The distance from P ba to L1, defined as

inf{llJ.t - 1111 J.t E Pba, I E L1(lp')}, is zero

Proof We start the proof of the theorem with the implication that (2) =>

(1) So we take (An)n>1 a decreasing sequence of sets in F with empty

intersection We have to prove that for every c > ° there is nand J.t E

Pba, such that J.t(An) ::; c In order to do this we take J.t E Pba such that lIJ.tall ~ 1 - c/2 Then we take n so that J.ta(An) ::; c/2 It follows that

J.t(An)::; c/2+ liJ.tpli::; c

The fact that 1 implies 2 is the most difficult one and it is based on the following lemma, whose proof is given after the proof of the theorem

Lemma 3.9 If K is a closed, weak* compact, convex set of finitely additive,

nonnegative measures, such that 8 = inf{lIvplIl v E K} > 0, then there exists

a decreasing (nonincreasing) sequence of sets An, with empty intersection, such that for all v E K, and for all n, v(An) > 8/4

If (2) were false, then

We can therefore apply the lemma in order to get a contradiction to (1) The proof that (2) and (3) are equivalent is left to the reader 0

In the proof of the lemma, as well as in section 5, we will need a minimax orem Since there are many forms of the minimax theorem, let us recall the one we need It is not written in its most general form, but this version will do For a proof, a straightforward application of the Hahn-Banach theorem to-gether with the Riesz representation theorem, we refer to Dellacherie-Meyer, page 404

the-Minimax Theorem Let K be a compact convex subset of a locally convex

space F Let L be a a convex set of an arbitrary vector space E Suppose that

u is a bilinear function u: E x F + lIt For each I E L we suppose that the

partial (linear) function u(l,.) is continuous on F Then we have that

inf sup u(l, k) = sup inf u(l, k)

Proof of Lemma 3.9 Of course, we may suppose that for each J.t E K we have 11J.t1i ::; 1 If A is purely finitely additive then the Yosida-Hewitt theorem implies the existence of a decreasing sequence of sets, say Bn (depending on A!), with empty intersection and such that A(Bn) = IIAII Given J.t E K, it

follows that for every c: > 0, there is a set, A (depending on J.t), such that IP'(A] ::; c and such that J.t(A) ~ 8 For each c > ° we now introduce the convex set, F c , of functions, I E VXJ such that I is nonnegative, I ::; 1 and Ep[J) ::; € The preceding reasoning implies that

Since the set K is convex and weak* compact, we can apply the minimax theorem and we conclude that

It follows that there is a function I E Fe, such that for all I-" E K, we have that EI'[J] 2:: 8/2 We apply the reasoning for € = 2- n in order to find a sequence of nonnegative functions In, such that for each I-" E K we have

EI'[Jn] 2:: 8/2 and such that EIP'[Jn) ::; 2-n We replace the functions In

by Yn = sUPk>n Ik in order to obtain a decreasing sequence Yn such that,

of course, EI'[9n) 2:: 8/2 and such that EP[Ynl ::; 2-n + 1 If we now define

An = {Yn 2:: 8/4}, then clearly An is a decreasing sequence, with a.s empty intersection and such that for each I-" E K we have that I-"(An) 2:: 8/4 0

Example 3.10 In example 3.7, Pba contained a a-additive probability ure The present example is so that the properties of the preceding theorem 3.8 still hold, but there is no a-additive probability measure in Pba In the language of the theorem 3.8 (2), this simply means that the supremum is not

meas-a mmeas-aximum The set n is simply the set of natural numbers The a-algebra

is the set of all subsets of nand lP is a probability measure on n charging all the points in n The space Loo is then Zoo and L1 can be identified with

Z1 The set IF denotes the convex weak* -closed set of all purely finitely ive probabilities 1-" Such measures can be characterised as finitely additive probability measures such that 1-"( { n }) = 0 for all n En This is also a quick way to see that that IF is weak* closed With 8 n we denote the probability

addit-measure (in L1) that puts all its mass at the point n, the so-called Dirac

measure concentrated in n The set Pba is the weak* closure of the set

The set is clearly convex and it defines a coherent risk measure, p Since ously sup {II/-Lalll I-" E Pba} = 1, the properties of theorem 3.8 hold The diffi-culty consists in showing that there is no a-additive measure in the set Pba Take an arbitrary element I-" E Pba By the definition of the set Pba there is

obvi-a generobvi-alised sequence, obvi-also cobvi-alled obvi-a net, I-"a tending to I-" and such that

where each va E IF, where Ln A~ = 1 and each A~ > O We will select subnets, still denoted by the same symbol n, so that

Coherent Risk Measures on General Probability Spaces 13

(1) the sequence I:n >.~ 8 n tends to I:n II':n 8 n for the topology aW, eo) This

is possible since [1 is the dual of co This procedure is the same as selecting

a subnet such that for each n we have that >.~ tends to II':n Of course

II':n ~ 0 and I:n II':n :S 1

(2) from this it follows that, by taking subnets, there is a purely finitely additive, nonnegative measure v' such that

tends to

for the topology a(ba, LOO)

(3) By taking a subnet we may also suppose that the generalised sequence va

converges for a(ba, U'O), to a, necessarily purely finitely additive, element

v E IF

1.x;:-"'2 1

(4) Of course I:n (n+1)n tends to o

As a result we obtain that

If this measure were a-additive, then necessarily for the non absolutely

con-tinuous part, we would have that v' + I:n (n~1)2V = O But, since these measures are nonnegative, this requires that allll':n = 0 and that v' = O This would then mean that IL = v' = 0, a contradiction to ILCD) = 1

4 Examples

The examples of this section will later be used in relation with VaR and

in relation with convex games The coherent measures all satisfy the Fatou property and hence are given by a set of probability measures We do not describe the full set P q , the sets we will use in the examples are not always convex So in order to obtain the set P q we have to take the closed convex hull We recall, see the remark after corollary 3.3, that there is a one-to-one correspondence between norm-closed convex sets of probability measures and coherent risk measures that satisfy the Fatou property

Example 4.1 Here we take

P q = {Q I Q« 1P'}

The corresponding risk measure is easily seen to be p(X) = ess sup( -X),

i.e the maximum loss It is clear that using such a risk measure as capital requirement would stop all financial/insurance activities The corresponding supermodular function is given by ¢(X) = ess inf(X)

Example 4.2 In this example we take for a given 0,0 < 0 < 1:

P u = {I I 0 ~ I, 1111100 ~ ~ and EIP[J] = I} The extreme points of this set are of the form ~ where IP'(A] = 0, see Lindenstrauss [20]

Example 4.3 This example, as well as the next one, shows that although higher moments cannot be directly used as risk measures, there is some way

to introduce their effect For fixed p > 1 and (3 > 1, we consider the weakly compact convex set:

If p = 00 and (3 = 1/0, then we simply find back the preceding example

So we will suppose that 1 < p < 00 If we define q = ~, the conjugate exponent, then we have the following result:

Theorem 4.4 For nonnegative bounded functions X, we have that

where c = min(I,(3 -1)

Proof The right hand side inequality is easy and follows directly from Holder's inequality Indeed for each h E P u we have that

Coherent Risk Measures on General Probability Spaces 15

The left hand side inequality goes as follows We may of course suppose that X is not identically zero We then define Y = 1I~~~~1' As well known and easily checked, we have that IlYllp = 1 Also Ep[XY] = IIXllq We now distinguish two cases:

Case 1: (1 - Ep[Y]) :::; f3 - 1 In this case we put h = Y + 1 - Ep[Y]

Clearly we have that EIl'[h] = 1 and Ilhllp :::; f3 Of course we also have that

Ep[hX] ~ Ep[XY] = IIXllq'

Case 2: (1 - Ep[Y]) ~ f3 - 1 (Of course this can only happen if f3 :::; 2)

In this case we take

(J-l

h = aY + 1- aEp[Y] where a = 1- Ep[Y]'

Clearly Ep[h] = 1 and IIhllp :::; a + 1 - (f3 - I)Ep[Y]/(1 - EIl'[Y]) :::; (J But also Ep[Xh] ~ allXllq ~ ((J - 1)IIXllq, since 1 - Ep[Y] :::; 1 0

Remark: It is easily seen that the constant c has to tend to 0 if (J tends to

1 If we take p = 2 we get a risk measure that is related to the 11.112 norm of the variable More precisely we find that, in the case p = 2 = f3:

In insurance such a risk measure can therefore be used as a substitute for the standard deviation premium calculation principle The use of coherent risk measures to calculate insurance premiums has also been addressed in the paper Artzner et.al [2] For more information on premium calculation principles, we refer to Wang [31]

Remark: In section 5, we will give a way to construct analogous examples

as the one in 4.3, but where the LP space is replaced by an Orlicz space

Example 4.5 Distorted measures In this example we define directly the herent risk measure Section 7 will show that it is a coherent risk measure such that P u is weakly compact We only define pC-X) = 1jJ(X) for nonneg-

co-ative variables X, the translation property is then used to calculate the value for arbitrary bounded random variables The impatient reader can now check that the translation property is consistent with the following definition:

1jJ(X) = p( -X) = 100

(JP'[X > a])i3 da

The number f3 is fixed and is chosen to satisfy 0 < (J < 1 The exponent q is defined as q = 1/ (J The reader can check that f3 = 0, gives us IIXlloo, already

discussed above The value (J = 1 just gives the expected value Ep[X] As

usual the exponent p is defined as p = q / (q - 1) The following theorem gives

the relation between this risk measure and the finiteness of certain moments

We include a proof for the reader's convenience

Theorem 4.6

If X is nonnegative and such that 1000

(IP'[X > a))t3 da < 00, then also X E

U If for some € > 0, X E U+c, then 1000

(IP'[X > a))t3 da < 00

Proof If k = 1000

IP'[X > aj1/qda < 00, then for every x 2: 0 we have the

inequality xlP'[X > xj1/q :=:; k This leads to xq-11P'[X > xj1/p :=:; kq- 1 From

this we deduce that

This gives us IIXllq :=:; ql/qk :=:; e1/ek :=:; 1.5 k The other implication goes as

follows If X E Lq+c for € > 0, then we have that xq+clP'[X > x) :=:; IIXII~!: =

k This implies that

IP[X > x) :=:; kx-(q+c) and hence IP[X > xj1/q :=:; IIXII~~~~)x-(l+~) which gives

We remark that in theorem 4.6, the converse statements do not hold Indeed a

nonnegative variable X such that for x big enough, IP'[X > xl = ex IO~ x)

sat-isfies X E Lq, but nevertheless 1000

IP[X > xj1/q dx = +00 Also a nonnegative random variable such that, again for x big enough, IP[X > x) = x (lO~ x)2 ,

satisfies 1000

IP[X > xj1/q dx < 00 but there is no € > 0 such that X E u+c

Remark: Distorted probability measures were introduced in actuarial

sci-ences by Denneberg [9)

Example 4.7 This example is almost the same as the previous one Instead

of taking a power function x t3 , we can, as will be shown in section 7, take

any increasing concave function f: [0,1) -t [0,1), provided we assume that f

is continuous, that f(O) = 0 and f(l) = 1 The risk measure is then defined, for X bounded nonnegative, as:

Coherent Risk Measures on General Probability Spaces 17

1f;(X) = p( -X) = 100

f (IP'[X > aJ) da

The continuity of f (at 0) guarantees that the corresponding set P u is weakly compact We will not make an analysis of this risk measure Especially the behaviour of f at zero, relates this coherent risk measure to Orlicz spaces, in the same way as the xf3 function was related to Ll/f3 spaces

Example 4.8 This example, not needed in the rest ofthe paper, shows that in order to represent coherent risk measures via expected values over a family of probabilities, some control measure is needed We start with the measurable space ([0,1], F), where F is the Borel cr-algebra A set N is of first category

if it is contained in the countable union of closed sets with empty interior The class of Borel sets of first category, denoted by N, forms a cr-ideal in :F

For a bounded function X defined on [0, 1] and Borel measurable, we define

p(X) as the "essential" supremum of -X More precisely we define

p(X) = min {m I {-X> m} is of first category}

Of course the associated sub modular function is then defined as

1f;(X) = min {m I {X > m} is of first category}

It is clear that p(X) defines a coherent risk measure It even satisfies the Fatou property in the sense that p(X) :::; liminf p(Xn), where (Xn)n>l is a uniformly bounded sequence of functions tending pointwise to X If P were

Q(N) = 0 for each set N of first category

But if Q is a Borel measure that is zero on the compact sets of first category, then it is identically zero However we can easily see that

p(X) = sup E~[-X],

~EP

where P is a convex set of finitely additive probabilities on :F The set P

does not contain any cr-additive probability measure, although p satisfies some kind of Fatou property Even worse, all elements in P are purely finitely

additive

5 Extension to the Space of All Measurable Functions

In this section we study the problem of extending the domain of coherent

risk measures to the space LO of all equivalence classes of measurable

func-tions We will focus on those risk measures that are given by a convex set of probability measures, absolutely continuous with respect to IP' We start with

a negative result The result is well known but for convenience of the reader,

we give more or less full details

Theorem 5.1 If the space (n, .F, IP') is atomless, then there is no real-valued coherent risk measure p on LO This means that there is no mapping

such that the following properties hold:

(1) If X ~ 0 then 1/J(X) ~ O

(2) Subadditivity: 1/J(X 1 + X 2 ) ::; 1/J(Xd + 1/J(X 2 )

(3) Positive homogeneity: for A ~ 0 we have 1/J(AX) = A1/J(X)

(4) For every constant function a we have that 1/J(a + X) = 1/J(X) + a

Proof Suppose that such a mapping would exist, then it can be shown,

ex-actly as in section 2, that 1/J(1) = 1 Also we have that for X ::; 0, necessarily,

1/J(X) ::; O By the Hahn-Banach theorem in its original form, see [3], there exists a linear mapping f: LO + JR, such that f(l) = 1 and f(X) ::; 1/J(X)

for all X E LO If X ~ 0 then also - f(X) = f( -X) ::; 1/J( -X) ::; O It follows that I(X) ~ 0 for all X ~ O So f is a linear mapping on LO that maps non-

negative functions into nonnegative reals By a classic result of Namioka, see [22] (see below, somewhat hidden in the proof of Theorem 5.4, for an outline

of a direct proof), it follows that f is continuous But because the probability space is atomless, there are no non-trivial linear mappings defined on L O, see [24] This contradicts the property that f(l) = 1 and ends the proof 0

Remark: Forerunning the definition of VaR, to be given in section 6, this

theorem shows, indirectly, that VaR cannot be coherent More precisely, since VaR satisfies properties 1,3 and 4 of the definition of coherent measure, VaR

cannot be subadditive

Remark: For completeness and to illustrate why a control in probability (or

in LO-topology) is too crude, we give a sketch of the proof that there are no nontrivial linear mappings defined on L O, see Nikodym [24] for the original

paper More precisely we show that if C c LO is absolutely convex, closed

and has non-empty interior, then C = LO This implies the non-existence

of non-trivial linear mappings The proof can be given an interpretation in risk management Indeed, the idea is to approximate arbitrary (bounded)

Coherent Risk Measures on General Probability Spaces 19

random variables by convex combinations of random variables that are small

in probability We leave further interpretations to the reader Because C has non-empty interior, it follows that 0 is in the interior of C This implies the existence of c > 0 such that for every set A E F, with IP'[A] :s; c, we have that

alA E C and this for all scalars a E R Next, for "1 > 0 and X E L'XJ given,

we take a partition of n into a finite number of sets (Ai)i:S;N such that

(1) each set Ai has measure IP'[Ad :s; c

(2) there is a linear combination Y of the functions lA, such that

IIX - Ylloo :s; "1

Since the convex combinations of functions of the form ailA, exhaust all linear combinations of the functions lAil we find that Y E C Since C was closed we find that X E C Finally since L oo is dense in LO, we find that

LO =C

The previous result seems to end the discussion on coherent measures

to be defined on LO But economically it makes sense to enlarge the range

of a coherent measure The number p(X) tells us the amount of capital to

be added in order for X to become acceptable for the risk manager, the regulator etc If X represents a very risky position, whatever that means,

then maybe no matter what the capital added is, the position will remain unacceptable Such a situation would then be described by the requirement that p(X) = +00 Since regulators and risk managers are conservative it

is not abnormal to exclude the situation that p(X) = -00 Because this would mean that an arbitrary amount of capital could be withdrawn without endangering the company So we enlarge the scope of coherent measures as follows

Definition 5.2 A mapping p: LO ~ RU{+oo} is called a coherent measure defined on LO if

(1) If X 2: 0 then p(X) :s; O

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

(3) Positive homogeneity: for oX> 0 we have p(oXX) = oXp(X)

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

The reader can check that the elementary properties stated in section 2

re-main valid Also it follows from item one in the definition that p cannot be

identically +00 The subadditivity and the translation property have to be interpreted liberally: for each real number a we have that a + (+00) = +00

We can try to construct coherent risk measures in the same way as we did

in theorems 2.2 and 3.2 However this poses some problems The first idea could be to define the risk measure of a random variable X as

sup EQ[-X]

QEP a

This does not work since the random variable X need not be integrable for the measures ij E P u' To remedy this we could try:

sup {EQ[-X] I Q E P q ; X E £l(Q)} or

sup {EQ[-X] I Q E P q ; X+ E £l(Q)} Such definitions have the disadvantage that the set over which the sup is taken depends on the random variable X This poses problems when we try

to compare risk measures of different random variables So we need another definition The idea is to truncate the random variable X from above, say by

n ~ O This means that first, we only take into account the possible future

wealth up to a level n We then calculate the risk measure, using the sup

of all expected values and afterwards we let n tend to infinity By doing so

we follow a conservative viewpoint High future values of wealth playa role, but their effect only enters through a limit procedure Very negative future values of the firm may have the effect that we always find the value +00

This, of course, means that the risk taken by the firm is unacceptable More precisely:

Definition 5.3 For a given, closed convex set, P q, of probability measures, all absolutely continuous with respect to 1P', we define the associated support functional pP", or if no confusion is possible, p as

Of course we need a condition to ensure that p(X) > -00, i e </>(X) < 00,

for all X E £0 This is achieved in the following theorem

Theorem 5.4 With the notation of definition 5.3 we have that the following

properties are equivalent

(1) For each X E £0 we have that p(X) > -00

(2) For each f E £~ we have that

</>U) = lim inf EQ[f t\ n] < +00

Coherent Risk Measures on General Probability Spaces 21

Proof We first prove that (1) => (2) Let f E L~ Clearly

-p(f) = -lim sup EQ [-(f t\ n)] = lim inf EQ [f t\ n]

We next prove that (2) => (3) If 3 would not hold then for each n we

would be able to find An such that IP'[An] ::; 2-n and such that

En = QEPinf Q[An] > o

u

Define now f = L:n nlAn en Because of the Borel-Cantelli lemma, f is well defined Let us also take m = n/E n Then of course we have that for each n

and for the corresponding m:

lim inf EQ [J t\ k] 2 inf EQ[J t\ m] 2 (!:) inf Q[An] 2 n En = n,

which contradicts 2

Let us now show that (3) => (1) For given X, let N be chosen so that

IP'[X 2 N] ::; I, where I is given by (3) Since for each n 2 N we have, by

The proof of the proposition is obvious However we have more:

Theorem 5.6 If Per is a norm closed, convex set of probability measures, all absolutely continuous with respect to IP', then the equivalent properties of theorem 5.4 are also equivalent with:

(4) For every f E L~ there is Q E Per such that EQ[f] < 00

(5) There is a 6 > 0 such that for every set A with IP'[A] < 6, we can find an

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

(6) There is a 6 > 0, as well as a number K such that for every set A with IP'[A] < 6, we can find an element Q E Pu such that Q[A] = 0 and

1I~lloo ::; K

Remark: The proof of this theorem is by no means trivial, so let us first

sketch where the difficulty is We concentrate on (4) Suppose that for a given nonnegative function g, we have that

lim inf EQ[g n 1\ n) = </>(g) < 00

QEP~

This means that for every n we can find In E P u such that

Ep [In (g 1\ n») ~ </>(g) + 1

The problem is that the sequence In does not necessarily have a weakly

convergent subsequence If however we could choose the sequence (fn)n in

such a way that it is uniformly integrable, then it is relatively weakly compact,

a subsequence would be weakly convergent, say to an element I E P u and a direct calculation then shows that also

:S limsupEp [Ik(g 1\ k») :S </>(g) + 1 < +00

k

So in case P u is weakly compact, there is no problem The general case however is much more complicated and requires a careful selection of the sequence In The original proof consisted in constructing a sequence In in

such a way that it became uniformly integrable This was quite difficult and used special features from functional analysis The present proof is much easier but in my viewpoint less transparent It uses the Hahn-Banach theorem directly

Proof It is clear that (6) =} (5) =} (4), which in turn implies the properties

(1),(2) and (3) of Theorem 5.4 So we only have to show that the properties

(1),(2) and (3) imply property (6) Let k > ~ and let A with IP'[A) < ~ be given We will show that 3 implies 6 We suppose the contrary So let us take

Hk = {f I III :S k, I = 0 on A} If Hk and P u were disjoint we could, by the Hahn-Banach theorem, strictly separate the closed convex set P u and

the weakly compact, convex set Hk This means that there exists an element

X E Loo, IIXlioo :S 1 so that

sup {E[X/lI I E Hd < inf {EQ[Xll Q E P u }

We will show that this inequality implies that IIXINlll = o Indeed if not,

we would have lP'[lAclXI > ~IIXIAclhl ~ t and hence for each E > 0 there

is a Q E P u so that Q[A U {IXI > ~IIXIAclldl ~ E This implies that the right side of the separation inequality is bounded by ~IIXIAclll However, the left side is precisely klIXIAclh This implies kliXIAcill < ~IIXIAclh, a contradiction to the choice of k Therefore X = 0 on AC But then property

3 implies that the right side is 0, whereas the left side is automatically equal

to zero This is a contradiction to the strict separation and the implication 3