A probability metrics approach to financial risk measures by rachev stoyanov and frank j fabozzi

3.4 Probability Metrics and Stochastic Dominance 633.7.2 Stochastic dominance relations of order n 723.7.3 Return versus payoff and stochastic dominance 743.7.4 Other stochastic dominanc

A Probability Metrics Approach to

Financial Risk Measures

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© 2011 Svetlozar T Rachev, Stoyan V Stoyanov and Frank J Fabozzi

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Library of Congress Cataloging-in-Publication Data

Rachev, S T (Svetlozar Todorov)

A probability metrics approach to financial risk measures / Svetlozar T Rachev,

Stoyan V Stoyanov, Frank J Fabozzi, CFA.

p cm.

Includes bibliographical references and index.

ISBN 978-1-4051-8369-7 (hardback)

1 Financial risk management 2 Probabilities I Stoyanov, Stoyan V.

II Fabozzi, Frank J III Title.

HD61.R33 2010

332.01’5192–dc22

2010040519

A catalogue record for this book is available from the British Library.

Set in 10.5/13.5pt Palatino by Thomson Digital, Noida, India

Printed in Malaysia

01 2011

To my grandchildren Iliana, Zoya, and Zari

SVS

To my parents Veselin and Evgeniya Kolevi and

my brother Pavel Stoyanov

FJF

To my wife Donna and

my children Francesco, Patricia, and Karly

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2.4 Definitions of Probability Distances and Metrics 24

3.4 Probability Metrics and Stochastic Dominance 63

3.7.2 Stochastic dominance relations of order n 723.7.3 Return versus payoff and stochastic dominance 743.7.4 Other stochastic dominance relations 76

4 A Classification of Probability Distances 83

4.2 Primary Distances and Primary Metrics 86

4.4 Compound Distances and Moment Functions 99

4.5.1 Interpretation and examples of ideal probability

4.7.1 Examples of primary distances 114

4.7.3 Examples of compound distances 131

5.5 Risk Measures and Dispersion Measures 1795.6 Risk Measures and Stochastic Orders 181

5.8.2 Probability metrics and deviation measures 1845.8.3 Deviation measures and probability

6.2.1 AVaR for stable distributions 200

6.4 Computing Portfolio AVaR in Practice 2076.4.1 The multivariate normal assumption 207

6.7 Risk Measures and Probability Metrics 2236.8 Risk Measures Based on Distortion Functionals 226

7.4.2 The effect of tail truncation 2687.4.3 Infinite variance distributions 2717.5 Asymptotic Distribution, Heavy-tailed Returns 2777.6 Rate of Convergence, Heavy-tailed Returns 2837.6.1 Stable Paretian distributions 283

7.9.1 Proof of the stable limit result 298

8.2 Metrization of Preference Relations 308

8.8.4 Investors with balanced views 3388.8.5 Structural classification of probability distances 339

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The theory of probability metrics is a branch of probability theory

It finds application in different theoretical and applied fields such

as probability theory, queuing theory, insurance risk theory, and nance The theory of probability metrics looks for answers to the fol-lowing basic question: How can one measure the difference betweenrandom quantities? In finance, for example, we assume a stochasticmodel for asset return distributions and, in order to estimate the risk

fi-of a portfolio fi-of assets, we sample from the fitted distribution Then,

we use the generated simulations to calculate portfolio risk In thiscontext, there are two issues arising on two different levels First, theassumed stochastic model should be “close” to the empirical data

In this sense, we say that we need a realistic model in the first place.Second, since the risk estimate is essentially computed from randomscenarios, we have to be aware of the variability of the estimator andhow it depends on the assumed asset return distributions

Although based on universal principles and ideas, the field ofprobability metrics is very specialized Most of the literature ishighly technical and is accessible mostly to specialists in probabilitytheory As far as applications are concerned, apart from our book

Advanced Stochastic Models, Risk Assessment, and Portfolio tion: Ideal Risk, Uncertainty, and Performance Measures (John Wiley &

Optimiza-Sons, 2008), we are unaware of other literature describing tions in finance

This book has two goals The first goal is to describe applications

in finance and extend them where possible The second goal is topresent the theory of probability metrics in a more accessible formwhich would be appropriate for non-specialists in the field Topicsrequiring more mathematical rigor and detail are included in tech-nical appendices to chapters

The book is organized in the following way Chapter 1 provides

a conceptual description of the method of probability metrics andreviews direct and indirect applications in the field of finance Chap-ter 2 provides an introduction to the theory of probability metrics.The classical theory describing investor choice under uncertainty

is provided in Chapter 3 Chapter 4 discusses the classification ofprobability distances to primary, simple, and compound types Theinformation in Chapter 2 is a prerequisite Chapters 5, 6, and 7 aredevoted to risk and uncertainty measures and discuss in detail AVaRand the Monte Carlo method for AVaR estimation Chapter 6 is a pre-requisite to Chapter 7 Finally, Chapter 8 considers the problem ofquantifying stochastic dominance relations and takes advantage ofthe terms introduced in Chapter 3

Svetlozar T RachevStoyan V StoyanovFrank J Fabozzi

About the Authors

Svetlozar (Zari) T Rachevcompleted his Ph.D degree in 1979 fromMoscow State (Lomonosov) University, and his Doctor of ScienceDegree in 1986 from Steklov Mathematical Institute in Moscow Cur-rently he is Chair-Professor in Statistics, Econometrics and Mathe-matical Finance at the University of Karlsruhe in the School of Eco-nomics and Business Engineering He is also Professor Emeritus atthe University of California, Santa Barbara in the Department ofStatistics and Applied Probability He has published seven mono-graphs, eight handbooks and special-edited volumes, and over 300research articles His recently coauthored books published by Wiley

in mathematical finance and financial econometrics include

Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk agement, Portfolio selection, and Option Pricing (2005), Operational Risk:

Man-A Guide to Basel II Capital Requirements, Models, and Man-Analysis (2007), Financial Econometrics: From Basics to Advanced Modeling Techniques

(2007), and Bayesian Methods in Finance (2008) Professor Rachev is

cofounder of Bravo Risk Management Group, specializing in cial risk-management software Bravo Group was recently acquired

finan-by FinAnalytica, for which he currently serves as Chief-Scientist

Stoyan V Stoyanovis a Professor of Finance at EDHEC BusinessSchool and Scientific Director for EDHEC-Risk Institute in Asia.Prior to joining EDHEC, he was the Head of Quantitative Research

ABOUT THE AUTHORS

at FinAnalytica, specializing in financial risk management software

He completed his Ph.D degree with honors in 2005 from the School

of Economics and Business Engineering (Chair of Statistics, metrics and Mathematical Finance) at the University of Karlsruheand is author and co-author of numerous papers His research in-terests include probability theory, heavy-tailed modeling in the field

Econo-of finance, and optimal portfolio theory His articles have recently

appeared in Economics Letters, Journal of Banking and Finance, Applied

Mathematical Finance, Applied Financial Economics, and International Journal of Theoretical and Applied Finance He is a co-author of the

mathematical finance book Advanced Stochastic Models, Risk

Assess-ment and Portfolio Optimization: The Ideal Risk, Uncertainty and mance Measures (2008) published by Wiley.

Perfor-Frank J Fabozziis Professor in the Practice of Finance in the School

of Management at Yale University Prior to joining the Yale faculty,

he was a Visiting Professor of Finance in the Sloan School ofManagement at MIT Professor Fabozzi is a Fellow of the Interna-tional Center for Finance at Yale University and on the AdvisoryCouncil for the Department of Operations Research and Financial

Engineering at Princeton University He is the editor of the Journal

of Portfolio Management His recently co-authored books published

by Wiley in mathematical finance and financial econometrics

in-clude The Mathematics of Financial Modeling and Investment

Manage-ment (2004), Financial Modeling of the Equity Market: From CAPM to Cointegration (2006), Robust Portfolio Optimization and Management

(2007), Financial Econometrics: From Basics to Advanced Modeling

Tech-niques (2007), and Bayesian Methods in Finance (2008) He earned a

doctorate in economics from the City University of New York in

1972 In 2002 Professor Fabozzi was inducted into the Fixed IncomeAnalysts Society’s Hall of Fame and he is the 2007 recipient of the

C Stewart Sheppard Award given by the CFA Institute He earnedthe designation of Chartered Financial Analyst and Certified PublicAccountant

Chapter 1

Introduction

In this chapter, we provide a conceptual description of the method

of probability metrics and discuss direct and indirect applications inthe field of finance, which are described in more detail throughoutthe book

The development of the theory of probability metrics started with the

investigation of problems related to limit theorems in probabilitytheory Limit theorems occupy a very important place in probabilitytheory, statistics, and all their applications A well-known example

is the celebrated central limit theorem (CLT) but there are many otherlimit theorems, such as the generalized CLT, the max-stable CLT,functional limit theorems, etc In general, the applicability of thelimit theorems stems from the fact that the limit law can be regarded

as an approximation to the stochastic model under considerationand, therefore, can be accepted as an approximate substitute Thecentral question arising is how large an error we make by adopt-ing the approximate model and this question can be investigated by

A Probability Metrics Approach to Financial Risk Measures by Svetlozar T Rachev,

Stoyan V Stoyanov and Frank J Fabozzi

© 2011 Svetlozar T Rachev, Stoyan V Stoyanov and Frank J Fabozzi

CHAPTER 1 INTRODUCTION

studying the distance between the limit law and the stochastic model

It turns out that this distance is not influenced by the particular lem Rather, it can be studied by a theory based on some universalprinciples

Generally, the theory of probability metrics studies the lem of measuring distances between random quantities On onehand, it provides the fundamental principles for building probabilitymetrics – the means of measuring such distances On the other, itstudies the relationships between various classes of probability met-rics Another realm of study concerns problems which require aparticular metric while the basic results can be obtained in terms

prob-of other metrics In such cases, the metrics relationship is prob-of primaryimportance

Certainly, the problem of measuring distances is not limited torandom quantities only In its basic form, it originated in differentfields of mathematics Nevertheless, the theory of probability metricswas developed due to the need of metrics with specific properties.Their choice is very often dictated by the stochastic model underconsideration and to a large extent determines the success of theinvestigation Rachev (1991) provides more details on the methods

of the theory of probability metrics and its numerous applications inboth theoretical and more practical problems

There are no limitations in the theory of probability metrics cerning the nature of the random quantities This makes its methodsfundamental and appealing Actually, in the general case, it is more

con-appropriate to refer to the random quantities as random elements.

They can be random variables, random vectors, random functions

or random elements in general spaces For instance, in the context

of financial applications, we can study the distance between tworandom stocks prices, or between vectors of financial variables thatare used to construct portfolios, or between yield curves which aremuch more complicated objects The methods of the theory remain

1.2 APPLICATIONS IN FINANCE

the same, irrespective of the nature of the random elements Thisrepresents the most direct application of the theory of probabilitymetrics in finance: that is, it provides a method for measuring howdifferent two random elements are We explain the axiomatic con-struction of probability metrics and provide financial interpretations

The theory describing investor choice under uncertainty, thefundamentals of which we discuss in Chapter 3, uses a differentapproach Various criteria were developed for first-, second-, andhigher-order stochastic dominance based on the distributions them-selves As a consequence, investment opportunities are compareddirectly through their distribution functions, which is a superiorapproach from the standpoint of the utilized information

As another example, consider the problem of building a sified portfolio The investor would be interested not only in themarginal distribution characteristics (i.e., the characteristics of theassets on a stand-alone basis), but also in how the assets depend

diver-on each another This requires an additidiver-onal piece of informatidiver-onwhich cannot be recovered from the distribution functions of theasset returns The notion of stochastic dependence can be described

by considering the joint behavior of assets returns

The theory of probability metrics offers a systematic approachtowards such a hierarchy of ways to utilize statistical information

CHAPTER 1 INTRODUCTION

It distinguishes between primary, simple, and compound types of

distances which are defined on the space of characteristics, the space

of distribution functions, and the space of joint distributions, tively Therefore, depending on the particular problem, one canchoose the appropriate distance type and this represents anotherdirect application of the theory of probability metrics in the field offinance This classification of probability distances is explained inChapter 4

respec-Besides direct applications, there are also a number of indirectones For instance, one of the most important problems in risk esti-mation is formulating a realistic hypothesis for the asset returndistributions This is largely an empirical question because no argu-ments exist that can be used to derive a model from some generalprinciples Therefore, we have to hypothesize a model that bestdescribes a number of empirically confirmed phenomena aboutasset returns: (1) volatility clustering, (2) autoregressive behavior,(3) short- and long-range dependence, and (4) fat-tailed behavior ofthe building blocks of the time-series model which varies depend-ing on the frequency (e.g., intra-day, daily, monthly) The theory ofprobability metrics can be used to suggest a solution to (4) Thefact that the degree of heavy-tailedness varies with the frequencymay be related to the process of aggregation of higher-frequencyreturns to obtain lower frequency returns Generally, the residualsfrom higher-frequency return models tend to have heavier tails and

this observation together with a result known as a pre-limit theorem

can be used to derive a suggestion for the overall shape of the returndistribution Furthermore, the probability distance used in the pre-limit theorem indicates that the derived shape is most relevant forthe body of the distribution As a result, through the theory of prob-ability metrics we can obtain an approach to construct reasonablemodels for asset return distributions We discuss in more detail limitand pre-limit theorems in Chapter 7

Another central topic in finance is quantification of risk anduncertainty The two notions are related but are not synonymous

Functionals quantifying risk are called risk measures and als quantifying uncertainty are called deviation measures or dispersion

function-1.2 APPLICATIONS IN FINANCE measures Axiomatic constructions are suggested in the literature for

all of them It turns out that the axioms defining measures of tainty can be linked to the axioms defining probability distances,however, with one important modification The axiom of symmetry,which every distance function should satisfy, appears unnecessarilyrestrictive Therefore, we can derive the class of deviation measuresfrom the axiomatic construction of asymmetric probability distances

uncer-which are also called probability quasi-distances The topic is discussed

in detail in Chapter 5

As far as risk measures are concerned, we consider in detailadvantages and disadvantages of value-at-risk, average value-at-risk (AVaR), and spectral risk measures in Chapter 5 and Chapter 6.Since Monte Carlo-based techniques are quite common among prac-titioners, we discuss in Chapter 7 Monte Carlo-based estimation ofAVaR and the problem of stochastic stability in particular The dis-cussion is practical, based on simulation studies, and is inspired bythe classical application of the theory of probability metrics in esti-mating the stochastic stability of probabilistic models We apply theCLT and the Generalized CLT to derive the asymptotic distribution

of the AVaR estimator under different distributional hypotheses and

we discuss approaches to improve its stochastic stability

We mentioned that adopting stochastic dominance rules forprospect selection rather than rules based on certain characteris-tics leads to a more efficient use of the information contained in thecorresponding distribution functions Stochastic dominance rules,however, are of the type “X dominates Y” or “X does not dominateY”: that is, the conclusion is qualitative As a consequence, computa-tional problems are hard to solve in this setting A way to overcomethis difficulty is to transform the nature of the relationship fromqualitative to quantitative We describe how this can be achieved

in Chapter 8, which is the last chapter in the book Our approach isfundamental and is based on asymmetric probability semidistances,

which are also called probability quasi-semidistances.

The link with probability metrics theory allows a classification ofstochastic dominance relations in general They can be primary, sim-ple, or compound but also, depending on the underlying structure,

CHAPTER 1 INTRODUCTION

they may or may not be generated by classes of investors, which

is a typical characterization in the classical theory of choice underuncertainty This is also a topic discussed in Chapter 8

References

Rachev, S T (1991), Probability Metrics and the Stability of Stochastic Models,

Wiley, New York

Chapter 2

Probability Distances and

Metrics

The goals of this chapter are the following:

• To provide examples of metrics in probability theory and pretations from a financial economics perspective

inter-• To introduce formally the notions of a probability metric and aprobability distance

• To consider the general setting of random variables defined on

a given probability space (,A, Pr) taking values in a

separa-ble metric space U, allowing a unified treatment of prosepara-blemsinvolving one-dimensional random variables, random vectors orstochastic processes, for example

• To consider the alternative setting of probability distances on thespace of probability measures P2 defined on the -algebras ofBorel subsets of U2= U × U where U is a separable metric space

• To examine the equivalence of the notion of a probability distance

on the space of probability measuresP2and on the space of jointdistributionsLX2 generated by pairs of random variables (X, Y)taking values in a separable metric space U

A Probability Metrics Approach to Financial Risk Measures by Svetlozar T Rachev,

Stoyan V Stoyanov and Frank J Fabozzi

© 2011 Svetlozar T Rachev, Stoyan V Stoyanov and Frank J Fabozzi

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

Notation introduced in this chapter:

Xp The space of real-valued r.v with E|X|p<∞

X= X(R) The space of real-valued r.v.s

θp The Lp-metric between distribution functions

r(C1, C2) The Hausdorff metric (semimetric between

sets)

K=K  Parameter of a distance space

(U, d) Separable metric space with metric d

Bk= Bk(U) The Borel -algebra on Uk

Pk = Pk(U) The space of probability laws onBk

T˛,ˇ, ,P The marginal of P∈ Pkon the coordinates ˛,

ˇ, , 

X:= X(U) The set of U-valued random variables

LX2:= LX2(U) The space of PrX,Y, X, Y∈ X(U)

u.m.s.m.s Universally measurable separable metric

space

2.2 SOME EXAMPLES OF PROBABILITY METRICS

Important terms introduced in this chapter:

(semi)metric function A special function satisfying properties

making it uniquely positioned forcomputing distances

(semi)metric space A space equipped with a (semi)metric

function for measuring distances betweenspace elements

probability (semi)metric A (semi)metric function designed to measure

distances between random elements

Generally speaking, a functional which measures the distance

between random quantities is called a probability metric These

ran-dom quantities can be of a very general nature For instance, theycan be random variables, such as the daily returns of equities, thedaily change of an exchange rate, etc., or stochastic processes, such

as a price evolution in a given period, or much more complex objects,such as the daily movement of the shape of the yield curve

In this chapter, we provide examples of probability metrics andinterpretations from the perspective of financial economics, limit-ing the discussion to one-dimensional random variables Then weproceed with the axiomatic definition of probability metrics In theappendix, we provide a more technical discussion of the axiomaticconstruction in a much more general context

Below is a list of various metrics commonly found in probability andstatistics In this section, we limit the discussion to one-dimensionalvariables only

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

2.2.1 Engineer’s metric

The engineer’s metric is

EN(X, Y) := |E(X) − E(Y)| X, Y ∈ X1

(2.2.1)where Xpis the space of all real-valued random variables (r.v.s) withE|X|p <∞ In the case of the engineer’s metric, we measure the dis-tance between the random variables X and Y only in terms of thedeviation of their means For example, if X and Y describe the return

on two common stocks, then the engineer’s metric computes thedistance between their expected returns

2.2.2 Uniform (or Kolmogorov) metric

The uniform (or Kolmogorov) metric is

ρ(X, Y) := sup{|FX(x)− FY(x)| : x ∈R} X, Y ∈ X = X(R) (2.2.2)

where FXis the distribution function (d.f.) of X,R= (−∞, +∞), and

Xis the space of all real-valued r.v.s

Figure 2.1 illustrates the Kolmogorov metric The c.d.f.s of tworandom variables are plotted on the top plot and the bottom plotshows the absolute difference between them,|FX(x)− FY(x)|, as afunction of x The Kolmogorov metric is equal to the largest abso-lute difference between the two c.d.f.s A arrow shows where it isattained

If the random variables X and Y describe the return distribution

of the common stocks of two corporations, then the Kolmogorovmetric has the following interpretation The distribution function

FX(x) is by definition the probability that X loses more than a level

x, FX(x)= P(X ≤ x) Similarly, FY(x) is the probability that Y loses

more than x Therefore, the Kolmogorov distance ρ(X, Y) is the

max-imum deviation between the two probabilities that can be attained

2.2 SOME EXAMPLES OF PROBABILITY METRICS

by varying the loss level x If ρ(X, Y)= 0, then the probabilities that

Xand Y lose more than a loss level x coincide for all loss levels.Usually, the loss level x, for which the maximum deviation isattained, is close to the mean of the return distribution, i.e the meanreturn Thus, the Kolmogorov metric is completely insensitive to thetails of the distribution which describe the probabilities of extremeevents – extreme returns or extreme losses

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

0 0.2

Figure 2.2: Illustration of the L´evy metric L(X, Y)√

2 is the maximum tance between the graphs of FX and FYalong a 45 degrees direction Thearrow indicates where the maximum is attained

dis-application in probability theory as it metrizes the weak gence It can be viewed as measuring the closeness between thegraphs of the distribution functions while the Kolmogorov metric

conver-is a uniform metric between the dconver-istribution functions The generalrelationship between the two is

For example, suppose that X is a random variable describing thereturn distribution of a portfolio of stocks and Y is a determinis-tic benchmark with a return of 2.5% (Y= 2.5%) (The deterministicbenchmark in this case could be either the cost of funding over a spec-ified time period or a target return requirement to satisfy a liabilitysuch as a guaranteed investment contract.) Assume also that the port-folio return has a normal distribution with mean equal to 2.5% and

a volatility , X∈ N(2.5%, 2) Since the expected portfolio return isexactly equal to the deterministic benchmark, the Kolmogorov dis-tance between them is always equal to 1/2 irrespective of how small

2.2 SOME EXAMPLES OF PROBABILITY METRICS

0 0.2

Figure 2.3: Illustration of the relationship between the L´evy and

Kolmogorov metrics The length of the vertical arrow equals ρ(X, Y), while

the length of the tilted indicate equals L(X, Y)√

in Figure 2.3

Remark 2.2.1 We see that ρ and L may actually be considered as

metrics on the space of all distribution functions However, this

can-not be done for EN simply because EN(X, Y)= 0 does not imply thecoincidence of FXand FY, while ρ(X, Y)= 0 ⇐⇒ L(X, Y) = 0 ⇐⇒

FX= FY The L´evy metric metrizes weak convergence (convergence

in distribution) in the spaceF, whereas ρ is often applied in the CLT,

cf Hennequin and Tortrat (1965)

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

we explained, FX(x) and FY(x) are the probabilities that X and Y,respectively, lose more than the level x The Kantorovich metricsums the absolute deviation between the two probabilities for allpossible values of the loss level x Thus, the Kantorovich metric pro-vides aggregate information about the deviations between the twoprobabilities This is illustrated in Figure 2.4

2.2 SOME EXAMPLES OF PROBABILITY METRICS

In contrast to the Kolmogorov metric, the Kantorovich metric

is sensitive to the differences in the probabilities corresponding toextreme profits and losses but to a small degree This is becausethe difference|FX(x)− FY(x)| converges to zero as the loss level (x)increases or decreases and, therefore, the contribution of the termscorresponding to extreme events to the total sum is small As a result,the differences in the tail behavior of X and Y will be reflected in

κ(X, Y) but only to a small extent

The Lp-metrics between distribution functions is

pincreases, only the largest absolute differences|FX(x)− FY(x)| start

to matter At the limit, as p approaches infinity, only the largest ference|FX(x)− FY(x)| becomes significant and the metric θ∞(X, Y)turns into the Kolmogorov metric Therefore, if we would like

dif-to accentuate on the differences between the two return tions in the body of the distribution, we can choose a large value

distribu-of p

Remark 2.2.2 Clearly the Kantorovich metric arises as a special

case, κ = θ1 Moreover, we can extend the definition of θpwhen p= ∞

by setting θ∞= ρ One reason for this extension is the following dual

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

K∗(X, Y) := E |X − Y|

Both metrics metrize convergence in probability on X= X(R), thespace of real random variables (Lukacs (1968), Chapter 3, and Dudley(1976), Theorem 3.5)

Assume that X is a random variable describing the return bution of a portfolio of stocks and Y describes the return distribution

distri-of a benchmark portfolio The probability

P(|X − Y| > ε) = P {X < Y − } {X > Y + }

concerns the event that either the portfolio will outperform thebenchmark by or it will underperform the benchmark by There-fore, the quantity 2 can be interpreted as the width of a performance

2.2 SOME EXAMPLES OF PROBABILITY METRICS

band The probability 1− P(|X − Y| > ε) is actually the probabilitythat the portfolio stays within the performance band, i.e it does notdeviate from the benchmark more than in an upward or downwarddirection

As the width of the performance band decreases, the probabilityP(|X − Y| > ε) increases because the portfolio returns will be more

often outside a smaller band The metric K(X, Y) calculates the width

of a performance band such that the probability of the event that theportfolio return is outside the performance band is smaller than half

L1(X, Y)= E|X − Y|

Suppose that X describes the returns of a stock portfolio and Ydescribes the returns of a benchmark portfolio Then the mean abso-lute deviation is a way to measure how closely the stock portfoliotracks the benchmark If p is equal to 2, we obtain

L2(X, Y)= E(X− Y)2

which is a quantity very similar to the tracking error between thetwo portfolios

Remark 2.2.3 Certain relations can be obtained between the Ky

Fan metric, the Lp-metric, and a metric which is similar in nature to

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

the engineer’s metric Define

corre-it is called the absolute moments metric For example, if p= 2 then

MOM2(X, Y) calculates the distance between the standard tions of X and Y From a financial economics perspective, if we

devia-adopt standard deviation as a proxy for risk, MOM2(X, Y) can beinterpreted as measuring the deviation between the risk profiles of

Xand Y

The relationship betweenLp(X, Y), K(X, Y), and MOMp(X, Y)can be summarized in the following way Choose a sequence ofrandom variables, X0, X1, ∈ Xp Then,

Lp(Xn, X0)→ 0 ⇐⇒ K(Xn, X0)→ 0

See, for example, Lukacs (1968), Chapter 3

All of the (semi-)metrics on subsets of X mentioned above may bedivided into three main groups:

• primary;

• simple;

A metric  is primary if (X, Y)= 0 implies that certain moment

characteristics of X and Y agree As examples, we have EN (2.2.1) and MOMp(2.2.11) For these metrics

EN(X, Y)= 0 ⇐⇒ E X = E Y

2.3 DISTANCE AND SEMIDISTANCE SPACES

A metric  is simple if (X, Y)= 0 implies complete coincidencebetween the distribution functions FXand FY,

The third group, the compound (semi-)metrics have the property

Some examples are K (2.2.7), K∗ (2.2.8), andLp (2.2.9) Compoundmetrics imply a stronger form of identity than primary metrics If Xand Y are two random variables which coincide in all states of theworld, possibly except for some states of the world with total prob-ability equal to zero, their distributions functions agree completely.Later on, precise definitions of these classes are given, and westudy the relationships between them Now we begin with a com-mon definition of probability metric which will include the typesmentioned above

In section 2.2, we considered examples of probability metrics andprovided interpretations from a financial economics perspective Allexamples concerned one-dimensional random variables and, there-fore, the probability metric was regarded as an object related to thespace of one-dimensional random variables In a certain sense, therandom variables were considered points in an abstract space andthe probability metric appears as a function measuring the distancebetween these abstract points

In fact, we considered one-dimensional random variables but thegeneral idea of using a special function which can measure distances

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

between abstract points belonging to a certain space does not depend

on this assumption We can consider random elements which could

be of very general nature, such as multivariate variables and tic processes, without changing much the general framework From apractical viewpoint, by extracting the general principles, we are able

stochas-to treat equally easy one-dimensional random variables describing,for example, stochastic returns on investments, multivariate randomvariables describing, for instance, the multi-dimensional behavior

of positions participating in two different portfolios, or much morecomplex objects such as yield curves

To this end, we begin with a slightly more technical discussionconcerning metric spaces, metric and semimetric functions withoutinvolving the notion of random elements The discussion is extendedfurther in section 2.4

We begin with the notions of metric and semimetric space alizations of these notions will be needed in the Theory of ProbabilityMetrics (TPM)

Gener-Definition 2.3.1 A set S : = (S, ) is said to be a metric space with the

metric  if  is a mapping from the product S× S to [0, ∞) havingthe following properties for each x, y, z∈ S:

(1) Identity property: (x, y)= 0 ⇐⇒ x = y;

(2) Symmetry: (x, y)= (y, x);

(3) Triangle inequality: (x, y)≤ (x, z) + (z, y)

Some well-known examples of metric spaces are the following

(a) The n-dimensional vector space Rn endowed with the metric

2.3 DISTANCE AND SEMIDISTANCE SPACES

(b) The Hausdorff metric between closed sets

(c) The H-metric Let D(R) be the space of all bounded functions

f :R→R, continuous from the right and having limits from the left,

of the sets{(x, y) : x ∈R, y= f (x)} and {(x, y) : x ∈R, y= f (x−)} The

H-metric H(f, g) in D(R) is defined by the Hausdorff distance betweenthe corresponding graphs, H(f, g) : f g) Note that in the space

F(R) of distribution functions, H metrizes the same convergence as

the Skorokhod metric:

s(F, G)= inf ε >0 : there exists a strictly increasing continuousfunction :R→R, such that (R)=R, sup

t∈R |(t) − t| < εand sup

t∈R |F((t)) − G(t)| < ε



Moreover, H-convergence inF implies convergence in distributions

(the weak convergence) Clearly, ρ-convergence (see (2.2.2)) implies

H-convergence

If the identity property in Definition 2.3.1 is weakened by changing(1) to

then S is said to be a semimetric space (or pseudometric space) and  a

semimetric (or pseudometric) in S For example, the Hausdorff metric

ris only a semimetric in the space of all Borel subsets of a boundedmetric space (S, )

Obviously, in the space of real numbers EN (see (2.2.1)) is the

usual uniform metric on the real lineR, i.e EN(a, b) := |a − b|, a, b ∈

CHAPTER 2 PROBABILITY DISTANCES AND METRICS

R For p≥ 0, define Fp as the space of all distribution functions Fwith0

−∞F(x)pdx+0∞(1− F(x))pdx <∞ The distribution functionspace F = F0 can be considered as a metric space with metrics ρ and L, while θp(1≤ p < ∞) is a metric in Fp The Ky-Fan metrics(see (2.2.7), (2.2.8)) [resp.Lp-metric (see (2.2.9))] may be viewed assemimetrics in X (resp X1) as well as metrics in the space of allPr-equivalence classes:



X:= {Y ∈ X : Pr(Y = X) = 1} ∀X ∈ X[resp Xp] (2.3.1)

EN , MOMp, θp,Lpcan take infinite values in X so we shall assume,

in the next generalization of the notion of metric, that  may takeinfinite values; at the same time we shall extend also the notion oftriangle inequality

Definition 2.3.2 The set S is called a distance space with distance 

and parameterK=Kif  is a function from S× S to [0, ∞],K≥ 1and for each x, y, z∈ S the identity property (1) and the symmetryproperty (2) hold as well as the following version of the triangleinequality: (3∗) (Triangle inequality with parameterK)

If, in addition, the identity property (1) is changed to (1∗) then S

is called a semidistance space and  is called a semidistance (withparameterK)

Here and in the following we shall distinguish the notions ric’ and ‘distance’, using ‘metric’ only in the case of ‘distance withparameterK= 1, taking finite or infinite values’

‘met-Remark 2.3.1 It is not difficult to check that each distance

 generates a topology in S with a basis of open sets B(a, r) :={x ∈ S; (x, a) < r}, ∈ S, r > 0 We know, of course, that every met-ric space is normal and that every separable metric space has acountable basis In much the same way, it is easily shown that thesame is true for distance space Hence, by Urysohn’s Metrization

2.3 DISTANCE AND SEMIDISTANCE SPACES

Theorem (Dunford and Schwartz (1988), 1.6.19), every separabledistance space is metrizable

Actually, distance spaces have been used in functional analysis for

a long time, as is seen by the following examples

Example 2.3.1 Let H be the class of all nondecreasing continuous

functions H from [0,∞) onto [0, ∞) which vanish at the origin andsatisfy Orlicz’s condition

KH := sup

t>0

H(2t)

Then:= H() is a distance in S for each metric  in S andK= KH

Example 2.3.2 (Birnbaum–Orlicz distance space, Birnbaum and Orliz

(1931), and Dunford and Schwartz (1988), p 400.)

The Birnbaum–Orlicz space LH(H ∈ H) consists of all integrable

functions on [0, 1] endowed with Birnbaum–Orlicz distance:

H(f1, f2) :=

 1

Obviously,K H = KH

Example 2.3.3 Similarly to (2.3.4), Kruglov (1973) introduced the

following distance in the space of distribution functions:

where the function satisfies the following conditions:

(a) is even and strictly increasing on [0,∞), (0) = 0;

(b) for any x and y and some fixed A≥ 1

Obviously,KKr= A

devia-adopt standard deviation as a proxy for risk, MOM2(X, Y) can beinterpreted... Lp-metric, and a metric which is similar in nature to

CHAPTER PROBABILITY