73
Monetary Risk Measures -Representations & V@R by Mikael D. Novak M.Sc. Thesis in Mathematics by Research Department of Mathematics National University of Singapore 2003
Monetary Risk Measures-Representations & V@R
byMikael D. Novak
M.Sc. Thesis in Mathematics by ResearchDepartment of Mathematics
National University of Singapore 2003
AbstractIn Chapter 1, we start out by introducing V @Rλ and AV @Rλ. We then continue
by giving a proper definition of the monetary risk measure and introducing the notionacceptance set. Then a discussion of convex and coherent risk measures follows. Weend the chapter with two representations in term of finitely additive measures, one forconvex risk measures and one for coherent risk measures.
The financial market model will, to some extent, be discussed in Chapter 2. We alsoinvestigate how different continuity properties affect convex and coherent risk measures.Due to some continuity criteria we get representations of coherent and convex riskmeasures in term of probability measures. We end Chapter 2 by giving a representationof convex risk measure in a financial market.
In Chapter 3, we return to V @Rλ and AV @Rλ. This time we thoroughly derivesome relations between these risk measures.
Keywords
Monetary risk measureValue at RiskAcceptance setConvexCoherentContinuity from above\below
i
ii
AcknowledgmentsThere is one person that I never can thank enough for all the help I received to defeatthe obstacles in my way, the support and valuable advice he always gave me and thepatience he had with me. It is of course no one else but my supervisor and mentor,Jiann-Hua Lou, that I owe my thesis to.
Next, I would like to thank Lee Seng Luan and the Department of Mathematics atThe National University of Singapore for the hospitality. For the support from Sweden,I thank Harald Lang and Lars Holst. Finally, I would like to thank Henrik GoranssonSandvikens stipendiefond for the scholarship and Kansli I at The Royal Institute ofTechnology (KTH) for helping me financially with the travel expenses to Singapore.
The main parts of the theory on risk measures has been taken from [4]. Some ofthe examples and remarks in the thesis are influenced by [7] as well as [10].
iii
iv
ContentsChapter 1 - Monetary Risk Measures 11.1 - What is Risk? 11.2 - Value at Risk 21.3 - Average Value at Risk 61.4 - The Acceptance Set 141.5 - Representations 21
Chapter 2 - Convex Risk Measures 292.1 - The Financial Market Model 292.2 - Continuity properties 372.3 - Convex Risk Measures in a Financial Market 48
Chapter 3 - V @R and AV @R 573.1 - V @R and AV @R 57
Bibliography 68
v
v
Chapter 1 - Monetary Risk Measures
To most people who have studied finance or financial mathematics, Value at Risk(V @R) is a well-known concept. One reason why V @R has become something of astandard is that it is rather easy to understand and it allows us to quantify risk in asingle number. With V @R we can answer questions like “How bad can it get 95% ofthe time?”.
In this first chapter we will define two risk measures, V @R and Average Valueat Risk (AV @R). The second one is defined in terms of V @R. We will then proceedto look at some properties that convex and coherent risk measures satisfy. The mainresults of this chapter will be the derivation of representation for the coherent case andfor the convex case. In Chapter 3, we will return to V @R and AV @R and look furtherat some relationships between these risk measures. Right after answering the followingquestion, we will giving some examples on V @R.
1.1 - What is Risk?
The original meaning of the Italian word risco was ”cut of like a rock” and reflectsthe danger sailors were facing while navigating around dangerous rocks. The risk thatwe encounter in financial markets is, however, of a less dramatic nature.
Risk is related to randomness, the unknown future, and can be described as thevolatility of outcomes, e.g, payoff from a portfolio. Risk can be human created and takethe form of inflation or have its origin in a natural phenomenon like floods. Financialrisk can be those which are related to possible losses in financial markets, e.g, losses dueto defaults on financial obligations or movement in currencies. Whatever the risk maybe, it will always be important for the growth of the economy, that there are peoplewilling to take it on.
1
1.2 - Value at Risk
We now define V @R.
Definition 1.2.1.The Value at Risk at level λ for a financial position X is defined as
V @Rλ(X) := infm|P [X < −m] ≤ λ = − supm|P [X < m] ≤ λ.
♦In the above definition, X(ω) denotes the discounted net worth of the financial
position at the end of the trading period for the outcome ω ∈ Ω.In some cases, the payoff of a portfolio can have something that remind us of
an exponential distribution. This could be the case if we have the position of a longstraddle, which involves the purchase of a call and a put. The worst payoff will happenwhen the spot price does not move. The V @R for a payoff will in this case look likeFigure 1.2.2 below.
Figure 1.2.2.
Remark 1.2.3.From the figure above we see that V @R does not take into account how the distri-
bution looks like below the point of −V @R and this, of course, is non-desirable in somecases. This is one reason why just a single number, V @R, is not enough for a decisionmaker in the choice of a portfolio.
♦Value at Risk was developed to handle one type of financial risk, market risk. A
less mathematical definition could be that V @R summarizes the worst loss that canoccur with probability λ. From Figure 1.2.2, we see that V @R is nothing more but theλ-quantile of the lower end of the distribution. In the literature, we can find some smallvariations of the definition of V @R. However, we will always assume that we talk aboutV @R as defined in Definition 1.2.1.
2
Example 1.2.4.One way to determine the Value at Risk for a portfolio, is to look at the historical
values of the revenues that the portfolio generate. If we choose λ to be 0.05 we wouldlike to find the five worst performing days out of 100 business days. In this example,we have 250 recorded revenues. Therefore we would like to find the revenue X∗ so thatthe observations to its left is 250× 5% = 12.5. The recorded revenues are displayed inthe histogram in Figure 1.2.6. From the figure, we can see that V @R0.05 ∈ [−0.1, 0].To get more information about V @R we would have to make smaller intervals in thehistogram or approximate the historic values to a parametric distribution. The latterwill be done in Example 1.2.5.
♦
Example 1.2.5.Let us now estimate the empirical values in Example 1.2.4 with a parametric ran-
dom variable. Looking at Figure 1.2.6, we choose the parametric random variable tohave an Exp(α) distribution. The Value at Risk at level λ for X ∈ Exp(α) is
V @Rλ(X) := − supx | 1− e−1α x ≤ λ.
Since the random variable X has a continuous distribution we can write
V @Rλ(X) = − supx | 1− e−1α x ≤ λ = −x | 1− e−
1α x = λ = α · ln(1− λ).
We approximate α to be 1.004 by applying the predefined function expfit() in MATLABto the historical revenues. Thus, with the approximated random variable we get theValue at Risk at level 0.05 to be
V @R0.05(X) = α · ln(1− 0.05) = 1.004 · ln(0.95) = −0.0515.
This value is in the interval [−0.1, 0] as predicted from the empirical distribution inExample 1.2.4.
♦
3
Figure 1.2.6.
V @R is by far the most used risk measure, however it is just another memberof a family of mappings called monetary measures of risk, for which we give a properdefinition.
Definition 1.2.7.Let X be a linear space of bounded functions containing the constants. A mapping
ρ : X −→ IR is called a monetary measure of risk if it satisfies the following conditionfor all X, Y ∈ X .
(1) Monotonicity: If X ≤ Y, then ρ(X) ≥ ρ(Y ).
(2) Translation invariance: If n ∈ IR, then ρ(X + n) = ρ(X)− n.♦
That a risk measure should be monotone is rather obvious; if a financial positionY has a larger net worth than X we would consider Y as a less risky position. Letus interpret ρ(X) as the needed capital to make X acceptable, i.e., ρ(X + ρ(X)) = 0.Then, translation invariance simply means that if we invest n in a risk-free manner thiswould of course lower the needed capital to get the position X risk-free.
4
Remark 1.2.8.Let us show that V @Rλ(X) is a monetary measure of risk, i.e., it satisfies conditions
(1) and (2) stated above.Monotonicity: Assume that X ≤ Y .
V @Rλ(X) = infm ∈ IR | P [X + m < 0] ≤ λ
V @Rλ(Y ) = infm ∈ IR | P [Y + m < 0] ≤ λFor each m1 ∈ m ∈ IR | P [X + m < 0] ≤ λ, since Y + m1 ≥ X + m1, we have
P [Y + m1 < 0] ≤ P [X + m1 < 0] ≤ λ
and thus,m1 ∈ m ∈ IR | P [Y + m < 0] ≤ λ.
As a result,
m ∈ IR | P [X + m < 0] ≤ λ ⊆ m ∈ IR | P [Y + m < 0] ≤ λ
andV @Rλ(X) ≥ infm2 ∈ IR | P [Y + m2 < 0] ≤ λ = V @Rλ(Y ).
Translation invariance:
V @Rλ(X + n) = infm ∈ IR | P [(X + n) + m < 0] ≤ λ= infn + m ∈ IR | P [X + (n + m) < 0] ≤ λ − n
=V @Rλ(X)− n.
♦
Remark 1.2.9.There are some properties that are sometimes desirable for a monetary measure of
risk. The first property is convexity of a monetary measure of risk and the second oneis positive homogeneity.
(1) Convexity: ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ), for 0 ≤ λ ≤ 1.
(2) Positive Homogeneity: If λ ≥ 0, then ρ(λX) = λρ(X).
A monetary measure that satisfies property (1) is called a convex monetary measureof risk. If both (1) and (2) are satisfied we say that the risk measure is a coherentmonetary measure of risk. If we assume (1) and (2) then
ρ(X + Y ) = ρ(λ
λX +
(1− λ)(1− λ)
Y ) ≤ λρ(1λ
X) + (1− λ)ρ(1
(1− λ)Y ) = ρ(X) + ρ(Y ).
This gives us a third property:
5
(3) Subadditivity: ρ(X + Y ) ≤ ρ(X) + ρ(Y ).
Conversely, if we now assume (1) and (3) then
ρ(λX + (1− λ)Y ) ≤ ρ(λX) + ρ((1− λ)Y ) = λρ(X) + (1− λ)ρ(Y ).
To sum up, if property (2) is satisfied by a risk measure, (1) and (3) will be equivalent.The convexity property (1) is a desirable property among financial measures of risk.The reason for this is that it allows the risk to be diversified if we add more assets toour portfolio. If property (3) applies to a risk measure used by a manager for a tradinggroup she/he can with ease control risk by delegating risk factors to the traders. Themanager can be certain that the total risk from the traders will not add up to morethan the sum of all the risk factors. Positive homogeneity is however not always a gooddescription of real life situation, e.g., it does not take into account factors like liquidityrisk which can be a non-linear risk. We also notice that if ρ is positive homogeneous,then ρ(0) = 0 and we say that ρ is normalized.
♦
1.3 - Average Value at Risk
It is time to introduce a risk measure that satisfies all the properties in Remark 1.2.9.
Definition 1.3.1.The Average Value at Risk at level λ ∈ (0, 1) of a position X ∈ X is given by
AV @Rλ(X) =1λ
∫
0
λ
V @Rγ(X)dγ.
♦
Remark 1.3.2.Let us show that AV @Rλ(X) satisfies monotonicity and translation invariance.Monotonicity: Assume X ≤ Y. Then V @Rλ(X) ≥ V @Rλ(Y ) and
AV @Rλ(X) =1λ
∫
0
λ
V @Rγ(X)dγ ≥ 1λ
∫
0
λ
V @Rγ(Y )dγ = AV @Rλ(X).
Translation invariance:
AV @Rλ(X + n) =1λ
∫
0
λ
V @Rγ(X + n)dγ
=1λ
∫
0
λ
V @Rγ(X)dγ − 1λ
∫
0
λ
ndγ
=AV @Rλ(X)− n.
♦
6
We will show that AV @R is a coherent measure of risk. This will be done in twosteps. First, we have to show that a measure of risk is coherent if and only if it can takeon a special representation. This will be one of the main results of this chapter. Thesecond step consists of showing that AV @R actually takes this representation. This willbe done in Chapter 3. Meanwhile we will assume that AV @R is a coherent monetarymeasure of risk.
Example 1.3.3.Now we derive an expression for AV @Rλ(X) in the case when X ∈ Exp(α)
AV @Rλ(X) :=1λ
∫ λ
0
V @Rγ(X)dγ
=α
λ
∫ λ
0
ln(1− γ)dγ
=− α− (1− λ)λ
V @Rλ(X).
When X ∈ Exp(1) we get
V @R0.05(X) = −0.0513AV @R0.05(X) = −0.0254.
Notice that, in this case, V @R dominates AV @R. In Chapter 3, we will investigate thisrelations in more detail.
♦In this text, AV @R will have different appearance. After showing the following
two lemmas, we will in Proposition 1.3.6 show that
AV @Rλ(X) =1λ
E[(q −X)+]− q. (1.1)
This is nothing but the expectation of X when X is below the λ-quantile q. In Remark1.2.3 we said that V @R do not give any information about how the probability is dis-tributed below the point −V @Rλ(X), i.e., we can have two different financial positionswith the same V @R, but one of the positions is much more favorable than another.Representation (1.1) of AV @R gives us a better understanding why this measure doesnot suffer from the same weakness.
For an increasing function F : IR −→ [0, 1], define
q(s) := infx ∈ IR|F (x) > s.
We will need the following technical lemma.
7
Lemma 1.3.4.q : (0, 1) −→ IR is increasing and right-continuous. Moreover, F and q satisfy
F (q(s)−) ≤ s ≤ F (q(s)+) (1.2)
andq(F (x)−) ≤ x ≤ q(F (x)) (1.3)
for all x ∈ IR and s ∈ (0, 1). Moreover,
F (x+) = infs ∈ (0, 1)|q(s) > x. (1.4)
Proof. For s < t,x | F (x) > t ⊆ x | F (x) > s.
Thus, q(s) is increasing.The set x | F (x) > s is the union of the sets x | F (x) > s + ε for ε > 0. This
implies that q is right-continuous.Let us look at the first pair of inequality (1.2). We have that x < q(s) implies
F (x) ≤ s. Thus, F (q(s)−) ≤ s. If x > q(s) then F (x) > s. Consequently, F (q(s)+) ≥ s.Now we look at (1.3). F (y) > F (x) implies y ≥ x. Hence, q
(F (x)
) ≥ x. If s < F (x)then q(s) ≤ x. This implies q
(F (x)−) ≤ x.
To show (1.4) we observe that
q(s) > x + ε =⇒ F (x + ε) ≤ s.
Thus, F (x+) ≤ infs ∈ (0, 1)|q(s) > x. On the other hand
q(F (x + ε)
)> x + ε > x,
which implies F (x + ε) ∈ s ∈ (0, 1)|q(s) > x.♦
Denote by FX the distribution function of a random variable X. In the next lemma,we will show how we can generate a random variable with the same distribution as Xby using the inverse of FX and a random variable with a uniform distribution.
Lemma 1.3.5.Suppose X is a real-valued random variable on (Ω,F , P ) with distribution function
FX(x) = P [X ≤ x]
and let qX denote the right-continuous inverse of FX , i.e., for s ∈ (0, 1),
qX(s) := infx ∈ IR|FX(x) > s.
8
(a) Let U be a uniform random variable on a probability space (Ω, F , P ) with dis-tribution on (0, 1), i.e., P [U ≤ s] = s for all s ∈ (0, 1). Then
X(ω) := qX(U(ω))
has the same distribution as X.(b) If FX is continuous, then FX(X) is uniformly distributed on (0, 1).
Proof. (a) We will show (a) by showing
P [X ≤ x] = P [X ≤ x].
Let F := FX and q := qX . From Lemma 1.3.4 we know that q(s) ≤ x implies F (x) ≥ s.This yields that
s ∈ (0, 1)|q(s) ≤ x ⊆ (0, F (x)]. (1.5)
From (1.4) we have that s < F (x) implies q(s) ≤ x and therefore
(0, F (x)) ⊆ s ∈ (0, 1)|q(s) ≤ x (1.6)
It follows that
F (x) = P[U ∈ (
0, F (x))] ≤ P [U ∈ s ∈ (0, 1)|q(s) ≤ x] ≤ P
[U ∈ (0, F (x)]
]= F (x).
SinceU ∈ s ∈ (0, 1)|q(s) ≤ x ⇐⇒ X := q(U) ≤ x,
we have that
P [X ≤ x] = F (x) = P [U ∈ s ∈ (0, 1)|q(s) ≤ x] = P [X ≤ x].
(b) Since we are assuming that F is continuous, we can write
F(q(s)
)= F (infx ∈ IR|F (x) > s) = s.
This together with (1.5) give us
(0, F (x)] = s ∈ (0, 1) | q(s) ≤ x.
Thus we have that
P [F (X) < s] = P [X < q(s)] = F (q(s)) = s,
and the lemma is proven.♦
We have so far worked a great deal with quantiles. Both V @R and AV @R are basedon quantiles. Since all the examples given above have been dealing with continuous
9
random variables, the λ-quantile has been unambiguous. Let us instead consider a non-contiguous random variable X on (Ω,F , P ). In this case, any real number q with theproperty
P [X ≤ q] ≥ λ and P [X < q] ≤ λ,
is a λ-quantile for the random variable X. We denote the set of all λ-quantiles for therandom variable X as the interval [q−λ (X), q+
λ (X)], where
q−λ (X) := infx | P [X ≤ x] ≥ λ,and
q+λ (X) : = infx | P [X ≤ x] > λ
= supx | P [X < x] ≤ λ.Observe that both q−λ and q+
λ are λ-quantiles. A short proof that q+λ is a λ-quantile
follows; P [X < q+λ (X)] ≤ λ is obvious. For ε > 0, we have
P [X < q+λ (X) + ε] ≥ λ.
Then, ε ↓ 0 yields P [X ≤ q+λ (X)] ≥ λ. In similar manner, we can show that q−λ is a
λ-quantile.Moreover, note that −q+
λ (·) =: V @Rλ(·). However, as Acerbi and Tasche point outin [10], it is not obvious to choose this definition of V @Rλ since the λ-quantile for arandom variable X can be ambiguous.
We will now show representation (1.1) for AV @R.
Proposition 1.3.6.Suppose that X ∈ X and that q is a λ-quantile for X, i.e., q ∈ [qλ
−(X), qλ+(X)].
Then
AV @Rλ(X) =1λ
E[(q −X)+]− q (1.7)
=1λ
infs∈IR
(E[(q −X)+]− λs). (1.8)
Proof. We will first show (1.7), then we will show that the set of minimizers for(1.8) is in fact [qλ
−(X), qλ+(X)]. Let us start by fixing a λ-quantile q ∈ [q−λ (X), q+
λ (X)]and define
q(γ) :=
q, if γ = λq+γ (X), if γ 6= λ
We also define P as the Lebesgue measure on Ω := [0, 1] and define U(ω) := ω. Weknow from Lemma 1.3.5 that X and X := q(U) have the same distribution since q isthe right-continuous inverse of the distribution function of X. Thus,
−λ ·AV @Rλ(X) = −∫ λ
0
V @Rγ(X)dγ =∫ λ
0
q(γ) · 1dγ
= E[q(U)IU≤λ] = E[X;U ≤ λ].
10
Since γ 7→ q(γ) is increasing and q(λ) = q, we have that
U ≤ λ ⊆ X ≤ q.
It follows that
E[X; U ≤ λ] = E[X; X ≤ q]− E[X; X ≤ q, U > q].
Since X ≤ q, U > λ ⊆ X = q, E[X; X ≤ q, U > λ] will degenerate to
E[X; X ≤ q, U > λ] = E[q · IX≤q,U>λ] = q · P [X ≤ q, U > λ].
We rewrite P [X ≤ q, U > λ] as
P [X ≤ q, U > λ] =P [X ≤ q]− P [X ≤ q, U ≤ λ]
=P [X ≤ q]− P [U ≤ λ]=F (q)− λ
and getE[X; X ≤ q, U > λ] = q · (F (q)− λ),
where F is the distribution function of X under P . If we summarize the result above
AV @Rλ =1λ
∫
(0,λ)
V @Rγ(X)dγ
= − 1λ
E[q(U); U ≤ λ]
=1λ
(E[−X; X ≤ q] + q(F (q)− λ)
), (1.9)
where we have used that
E[X; X ≤ q] = E[X;X ≤ q].
Now we can replace qF (q)− E[X; X ≤ q] with E[(q −X)+] and (1.7) is proven. Indeed,
E[(q −X)+] =∫
(x≤q)
(q − x)f(x)dx
=∫
(x≤q)
q · f(x)dx−∫
(x≤q)
x · f(x)dx
=qF (q)− E[X;X ≤ q]
where f(x) = F ′(x). We will show
AV @Rλ(X) =1λ
infs∈IR
(E[(q −X)+]− λs)
11
by showing that the set of minimizers for
g(s) := E[(s−X)+]− λs, s ∈ IR,
coincides with the set [q−λ (X), q+λ (X)]. From Fubini’s lemma, see for instance Theorem
8.8 in [8], we see that
E[(s−X)+] =∫
(s− z)+f(z)dz
=∫ ∫
I(z≤x≤s)dxf(z)dz
=∫
(−∞,s)
∫
(−∞,x)
f(z)dzdx
=∫
(−∞,s)
F (x)dx
Fix any q ∈ [qλ−(X), qλ
+(X)] and let s ≥ q. Since F (x) is increasing, we have
∫
(q,s)
F (x)dx ≥ F (q) · (s− q).
Thus,
g(q)− g(s) = E[(q −X)+]− λq − E[(s−X)+] + λs ≤ (λ− F (q)
)(s− q).
From the fact that q ∈ [q−λ (X), q+λ (X)] if and only if F (q) ≥ λ and F (q−) ≤ λ, we can
write g(q)− g(s) ≤ 0. If we now let s ≤ q, then
g(q)− g(s) =∫
(s,q)
F (x)dx− λ(q − s) ≤ (F (q−)− λ)(q − s) ≤ 0.
Moreover, if s /∈ [qλ−(X), qλ
+(X)], then g(q)− g(s) < 0 for both cases.♦
We end this section with an alternative approach to the method derived in Example1.3.3 on how to approximate AV @Rλ.
Example 1.3.7.Assume that we have 200 registered revenues (x1, ..., x200) from a portfolio X and
we want to derive AV @R0.05(X). We assume, as before, that X have an exponentialdistribution. One approach is to first approximate the parameter α in Exp(α) as in Ex-ample 1.2.5 and then use the formula that we derived in Example 1.3.3 for AV @Rλ(X).With this method, the result is
α1 = 0.9117 and AV @R10.05(X) = −0.0232.
12
Another approach is to choose the 10 revenues (x1:200, ..., x10:200) with the lowest valueand use the formula
AV @R1
0.05(X) =∑10
i=1 xi:200
10= −0.0211.
If we add data so we have 2000 registered revenues we get the following result with thefirst method:
α2 = 1.0005 and AV @R20.05(X) = −0.0254.
With the second method we get
AV @R2
0.05(X) =∑100
i=1 xi:2000
100= −0.0253.
Proposition 4.1 in [10] states that with probability 1
AV @Rλ(X) = − limn→∞
∑bnλci=1 Xi:n
bnλc = AV @Rλ(X)
where the elements in the ordered sequence (Xi:n) have the same distribution as X.♦
1.4 - The Acceptance SetFor a monetary measure of risk, ρ, there is a set, Aρ, called the acceptance set,
defined byAρ := X ∈ X | ρ(X) ≤ 0.
We call the elements in Aρ acceptable positions.
Remark 1.4.1.The acceptance set is the set of all financial positions which does not require any
extra capital to become acceptable. In the next example, we will assume that thereexists a financial market. If we are allowed to hedge some of our risk with a portfoliofrom the financial market, we can actually get a large acceptance set without additionalcost.
♦
Example 1.4.2.Assume that there exists a financial market. Let πi denote the price of asset i at
time 0. The price at time 1 will be denoted by Si. We can denote the discount profitfrom asset i as
Y i =Si
1 + r− πi,
where r denotes the discounting rate. A portfolio in the financial market can be denotedby a vector ξ where the component ξi in ξ denotes the number of shares of asset i. Ifwe assume that we can loan the needed capital for the hedge position, there is nothing
13
stopping us from actually hedging some of our initial risk. We write the new positionas X + ξ · Y . This means that we can have a situation where
ρ(X) > 0
butρ(X + ξ · Y ) ≤ 0.
Thus, we can treat X as an acceptable position if there exists a portfolio ξ such thatρ(X + ξ · Y ) ≤ 0. We write the extended acceptance set as
A := X ∈ X | ∃ξ ∈ IRd with ρ(X + ξ · Y ) ≤ 0.
In Section 2.1, we will discuss the financial market in more detail. In particular, we willuse the financial market to get an acceptable position from an unacceptable position interms of V @R.
♦
14
Lemma 1.4.3.Any monetary measure of risk ρ is Lipschitz continuous with respect to the supre-
mum norm:|ρ(X)− ρ(Y )| ≤ ||X − Y ||.
Proof. Since ||X − Y || ≥ X − Y , monotonicity gives that ρ(Y + ||X−Y ||) ≤ ρ(X)and by translation invariance
ρ(Y )− ||X − Y || ≤ ρ(X).
But we can also write ||X − Y || ≥ Y −X, gives that ρ(X + ||X − Y ||) ≤ ρ(Y ) and
ρ(X)− ||X − Y || ≤ ρ(Y ).
Putting these together, we get the result.♦
In the following two propositions, we will derive some relations between the mone-tary risk measure and the acceptance set.
Proposition 1.4.4.Suppose that ρ is a monetary measure of risk with acceptance set
Aρ := X ∈ X |ρ(X) ≤ 0.
Then,(a) Aρ is non-empty, and satisfies the following two conditions:
infm ∈ IR |m ∈ Aρ > −∞
X ∈ Aρ, Y ∈ X , Y ≥ X ⇒ Y ∈ Aρ.
Moreover, Aρ has the following closure property: For X ∈ Aρ and Y ∈ X ,
λ ∈ [0, 1]|λX + (1− λ)Y ∈ Aρ is closed in [0, 1]. (1.10)
(b) ρ can be recovered from Aρ:
ρ(X) = infm ∈ IR|m + X ∈ Aρ.
(c) ρ is a convex risk measure if and only if Aρ is convex.(d) ρ is positively homogeneous if and only if Aρ is a cone. In particular, ρ is
coherent if and only if Aρ is a convex cone.
15
Proof. (a) For any m ∈ IR, m + ρ(m) ∈ Aρ. Thus, Aρ 6= ∅. If X ∈ Aρ then
ρ(X) ≤ 0
and from translation invariance we know that if Y ≥ X then ρ(Y ) ≤ ρ(X). Thus
ρ(Y ) ≤ 0,
which implies thatY ∈ Aρ.
The function λ 7→ ρ(λX + (1− λ)Y ≤ 0) is, by Lemma 1.4.3, Lipschitz continuous andtherefor continuous. Consequently, the continuity implies that the set of λ ∈ [0, 1] suchthat
ρ(λX + (1− λ)Y ) ≤ 0
is closed.(b) Translation invariance implies that for every X ∈ X we have :
infm ∈ IR|m + X ∈ Aρ = infm ∈ IR|ρ(m + X) ≤ 0= infm ∈ IR|ρ(X) ≤ m = ρ(X)
(c) If ρ is convex we can write
ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ) ∀λ ∈ [0, 1].
Accordingly, it is obvious that the left-hand side is less or equal to zero if ρ(X) ≤ 0 andρ(Y ) ≤ 0. Thus Aρ is convex. Conversely, if Aρ is convex
ρ
(λ(X + ρ(X)
)+ (1− λ)
(Y + ρ(Y )
)) ≤ 0 for λ ∈ [0, 1],
since for any X,Y ∈ X the sets X + ρ(X), Y + ρ(Y ) will be contained in Aρ. Withtranslation invariance we can take out the constants from ρ and the convexity of ρfollows.
(d) If ρ is positively homogeneous, i.e., ρ(λX) = λρ(X) for λ > 0, we must havethat
ρ(λX) = λρ(X) ≤ 0 if X ∈ Aρ
Thus, λX ∈ Aρ and therefore Aρ must be a cone. Conversely, let Aρ be a cone. Hence,X + ρ(X) ∈ Aρ implies that for λ ≥ 0,
λ(X + ρ(X)
) ∈ Aρ,
i.e.,
ρ
(λ(X + ρ(X)
)) ≤ 0.
By translation invariance we get ρ(λX) ≤ λρ(X) for λ > 0. To show the other inequality,we choose m < ρ(X) and notice that m + X /∈ Aρ. Thus, λm + λX /∈ Aρ for any λ ≥ 0and λm < ρ(λX). Let m ↑ ρ(X) and the proof is completed.
16
♦Conversely, given an acceptance set A, we can define the functional ρA by
ρA(X) := infm ∈ IR | m + X ∈ A.
With some assumptions on the setA, the following proposition will show the relationshipbetween A and ρA.
Proposition 1.4.5.Assume that A is a non-empty subset of X which satisfies
infm ∈ IR|m ∈ A > −∞
X ∈ A, Y ∈ X , Y ≥ X ⇒ Y ∈ A.
Then the functional ρA has the following properties:(a) ρA is a monetary measure of risk.(b) If A is a convex set, then ρA is a convex measure of risk.(c) If A is a cone, then ρA is positively homogeneous.(d) A is a subset of AρA . If A satisfies (1.10) then A ≡ AρA .
Proof. (a) We have to show that ρA satisfies monotonicity, translation invarianceand that ρA takes finite values for all X ∈ X . To this end let X, Y ∈ X and X ≤ Y .Choose an m1 ∈ m ∈ IR|m + X ∈ A. We get m1 + X ∈ A and m1 + X ≤ m1 + Y .From one of the assumptions in the statement of the proposition, we have m1 + Y ∈ A.This means that m1 ∈ n ∈ IR|Y + n ∈ A. Thus,
m1 ≥ infn ∈ IR|Y + n ∈ A = ρA(Y )
and we have monotonicity. To show translation invariance, observe
ρA(X + m1) : = infm ∈ IR|X + m1 + m ∈ A= infm + m1 −m1 ∈ IR|X + (m + m1) ∈ A= infm + m1 ∈ IR|X + (m + m1) ∈ A −m1
= ρA(X)−m1,
and the result follows.To show that ρA only takes finite values we fix an Y ∈ A. We then know that
0 ∈ m ∈ IR|m + Y ∈ A
and so0 ≥ infm ∈ IR|m + Y ∈ A.
For X ∈ X given, there exists a finite number m with m + X > Y since X and Y arebounded. By monotonicity and translation invariance, we know that
ρ(X + m) = ρ(X)−m ≤ ρ(Y ) ≤ 0
17
and thatρ(X) ≤ m.
Thus, ρ is bounded from above. We now take another finite number m′ such thatX + m′ ≤ 0. Thus,
ρ(X + m′) = ρ(X)−m′ ≥ ρ(0).
From the assumption, we know that
ρ(0) = infm ∈ IR|m ∈ A > −∞.
(b) Suppose that X1, X2 ∈ X . Choose m1, m2 ∈ IR such that
mi + Xi ∈ A, i = 1, 2.
For any λ ∈ [0, 1], the convexity of A implies that
λ(X1 + m1) + (1− λ)(X2 + m2) ∈ A.
Note that ρA(Xi) ≤ mi implies Xi + mi ≥ 0 for i = 1, 2. By monotonicity and transla-tion invariance
0 ≥ρA(λ(X1 + m1) + (1− λ)(X2 + m2)
)
=ρA(λX1 + (1− λ)X2
)− λm1 − (1− λ)m2
(c) Assume that A is a cone and choose some X ∈ X . As always, X + ρA(X) ∈ A.If λ ≥ 0 then
λ(X + ρA(X) ∈ AThus, we can write
ρA
(λ(X + ρA(X)
)) ≤ 0
andρA(λX) ≤ λρA(X).
To show the converse inequality, we chose a constant m < ρA(X). Then m + X is notin A and the same counts for λ(m + X). We can now write
ρA(λ(m + X)
)= ρA(λX)− λm > 0
ρA(λX) > λm
If we let m ↑ ρA(X) we will get the desired inequality:
ρA(λX) ≥ λρA(X)
(d) We have the following relation:
AρA :=X ∈ X | ρA(X) ≤ 0=X ∈ X | infm ∈ IR | m + X ∈ A ≤ 0⊇X ∈ X | infm ∈ IR | m + X ∈ A = 0=A
18
Let us now assume that A satisfies (1.10).We want to show that ρA(X) ≤ 0 implies X ∈ A.First we show that
X + ρA(X) ∈ A.
Since X + ρA(X) + ε ∈ A for every ε > 0 we have by (1.10) that
Λ =λ ∈ [0, 1] | λ(X + ρA(X) + ε
)+ (1− λ)
(X + ρA(X)
) ∈ A=λ ∈ [0, 1] | X + ρA(X) + λε ∈ A
is closed in [0, 1]. Thus, since Λ is closed and the fact that
X + ρA(X) + λε ∈ A
for every λ > 0 imply that this will be true for λ = 0. Suppose that ρA(X) ≤ 0. Then,
X + ρ(X) ≤ X
and we have that X ∈ A. The proposition is proven.♦
Example 1.4.6.Consider the worst case measure ρmax(X) defined by
ρmax(X) = − infω∈Ω
X(ω) for all X ∈ X .
In this example, X will be the linear space of all bounded measurable functions on somemeasurable space (Ω,F). The value ρmax(X) is the least upper bound for the potentialloss which can occur in any scenario. Eventually, we will show that the correspondingacceptance set A is given by the convex cone of all non-negative functions in X . Wefirst show that ρmax(X) fulfills the monotonicity and translation invariance conditions.Monotonicity: Clearly, if X ≤ Y then
−ρmax(X) = infω∈Ω
X(ω) ≤ infω∈Ω
Y (ω) = −ρmax(Y ).
Translation invariance: If m ∈ IR then,
ρmax(X + m) = − infω∈Ω
(X(ω) + m) = − infω∈Ω
(X(ω))−m = ρmax(X)−m.
Let us now look at the corresponding acceptance set
Aρmax := X ∈ X |ρmax(X) ≤ 0.
Note thatAρmax : = X ∈ X |ρmax(X) ≤ 0
= X ∈ X | infω∈Ω
X(ω) ≥ 0= X ∈ X |X(ω) ≥ 0 for all ω ∈ Ω.
19
Thus, Aρmax is a convex cone. From Proposition 1.4.4 we have that ρmax is a coherentrisk measure. In particular, any normalized monetary measure ρ on X satisfies
ρ(X + ρmax(X)
) ≤ 0.
Thus,ρ(X) ≤ ρmax(X)
and we can say that ρmax is the most conservative, normalized risk measure.♦
The coherent risk measure ρmax, from the example above, can be represented inthe form;
ρmax(X) = supQ∈M1
EQ[−X],
where M1 is the class of all probability measures on (Ω,F). That any mapping in thisform is coherent is rather obvious. We will however in the following section show thatall coherent risk measures can be representation in the form
ρ(X) = supQ∈Q
EQ[−X], X ∈ X
for some subset Q of M1,f (Ω,F). Here we denote M1,f to be the set of finitely additiveset functions Q : F → [0, 1] with Q[Ω] = 1.
1.5 - Representations
In the following theorem, we show that all coherent risk measures can be represen-tation in the form
ρ(X) = supQ∈Q
EQ[−X], X ∈ X
for some subset Q of M1,f (Ω,F). For Q ∈M1,f , we define
EQ[X] :=∫
XdQ.
In the context of this paper, it is of course interesting to have a representation in termsof probability measures, i.e., Q is contained in the set M1. In Section 2.2, we will derivesome sufficient properties for ρ which guarantee Q ⊂M1.
Proposition 1.5.1.A functional ρ : X → IR is a coherent measure of risk if and only if there exists a
subset Q of M1,f such that
ρ(X) = supQ∈Q
EQ[−X], X ∈ X . (1.11)
Moreover, Q can be chosen as a convex set for which the supremum in (1.11) is at-tained.
20
Proof. If there exists a subset Q of M1,f such that
ρ(X) = supQ∈Q
EQ[−X], X ∈ X
then ρ is obviously a coherent measure of risk. To show the converse we will constructfor any X ∈ X a finitely additive set function QX such that ρ(X) = EQX
[−X] andρ(Y ) ≥ EQX
[−Y ] for all Y ∈ X . Then
ρ(Y ) = maxQ∈Q0
EQ[−Y ]
for all Y ∈ X where Q0 := QX |X ∈ X. This holds even if we replace Q0 with itsconvex hull, defined as
Q := conv(Q0) = n∑
i
αiQXi |QXi ∈ Q0, αi ≥ 0,
n∑
i
αi = 1, n ∈ IN.
To our help, we will need three sets which we define as:
B := Y ∈ X | − ρ(Y ) > 1,C1 := Y ∈ X |Y ≤ 1,C2 := Y ∈ X |Y ≤ X
−ρ(X).
Note that B, C1 and C2 are convex. The convex hull of C1 and C2 is given by
C := ∑
i
αiYi|Yi ∈ C1 ∪ C2, αi ∈ [0, 1], and∑
i
αi = 1
thus, ∑
i
αiYi =∑
i1
αi1Yi1 +∑
i2
αi2Yi2
=∑
i1
αi1
∑i1
αi1Yi1∑i1
αi1
+∑
i2
αi2
∑i2
αi2Yi2∑i2
αi2
= αY1 + (1− α)Y2
where Yk and Yik∈ Ck for k = 1, 2.
So, if Y ∈ C we have that Y = αY1 + (1− α)Y2 for some Yi ∈ Ci and α ∈ [0, 1].Then, by monotonicity ρ(Y ) = ρ(αY1 + (1− α)Y2) ≥ ρ(α + (1− α)Y2). By translationinvariance and positive homogeneity we get ρ(Y ) ≥ −α + (1− α)ρ(Y2). Since Y2 ∈ C2
we know thatρ(Y2) ≥ ρ(
X
−ρ(X)) =
1−ρ(X)
ρ(X) = −1,
i.e., −ρ(Y2) ≤ 1. Hence,
−ρ(Y ) ≤ α + (1− α)(−ρ(Y2)) ≤ α + (1− α) · 1 = 1,
21
which implies that B and C are disjoint sets. Let X be endowed with the supremumnorm
‖ Y ‖:= supω∈Ω
|Y (ω)|.
Then, C1 contains the unit ball B1(0) defined by B1(0) = X ∈ X |‖ X ‖≤ 1. SinceC1 ⊂ C we know that C has a non-empty interior. Theorem A.35 in [4] can be appliedand yield a non-zero continuous linear functional, l(αF + βG) = αl(F ) + βl(G) suchthat
c := supY ∈C
l(Y ) ≤ infZ∈B
l(Z).
Since the unit ball is contained in C and l is linear, c must be strictly positive. Withoutloss of generality we can assume that c = 1. Since 1 ∈ C we know that l(1) ≤ 1. We alsohave that any constant b > 1 is contained in B since
ρ(b) < ρ(1) = −1 + ρ(0) = −1.
Consequently,l(1) = lim
b↓1l(b) ≥ c = 1
and l(1) = 1.If A ∈ F then IAc ∈ C1 ⊂ C. This implies that l(IA) = l(1)− l(IAc) ≥ 1− 1 = 0. By
Theorem A.33 in [4] there exists a finitely additive set function
QX ∈M1,f (Ω,F)
such that l(Y ) = EQX [Y ] for any Y ∈ X . Now it remains to show that
ρ(Y ) ≥ EQX[−Y ] ∀Y ∈ X
with equality for Y = X. Because ρ is translation invariant we only have to consider thecase in which ρ(Y ) < 0. To finish the proof we construct
Yn :=Y
−ρ(Y )+
1n
with
ρ(Yn) =ρ(Y )−ρ(Y )
− 1n
< −1.
Thus, Yn ∈ B. Since Yn → Y−ρ(Y ) uniformly,
EQX[Y ] · 1
−ρ(Y )= lim
n↑∞EQX
[Yn] ≥ 1.
The inequality follows from the fact that l(Y ) = EQX[Y ] for any Y ∈ X and
1 = c := supY ∈C
l(Y ) ≤ infZ∈B
l(Z) ≤ EQX [Yn] ∀Yn ∈ B.
Hence, ρ(Y ) ≥ EQX[−Y ].
22
We also have that,
l
(X
−ρ(X)
)= l(X)
1−ρ(X)
= EQX(X)
1−ρ(X)
≤ supY ∈C
l(Y ) = c = 1,
since X−ρ(X) ∈ C2 ⊂ C.
♦
Remark 1.5.2.Derived by Artzner et al. in [1] 1999, the nice result (1.11) appears, however, in a
different setting in [5] from 1981. In Section 10.2 in [5], Huber defines the upper andlower expectations
E∗(X) = infP
∫XdP, E∗(X) = sup
P
∫XdP,
where P is an arbitrary set of probability measures. He then defines the set
P : = P ∈M1 |∫
XdP ≥ E∗[X] for all X
= P ∈M1 |∫
XdP ≤ E∗[X] for all X.
One of his initial goals is, in fact, to find what conditions P must satisfy so that itis representable by some E∗ and what conditions E∗ must satisfy so that it is repre-sentable by some P. The answer is; P must be closed and convex to be representableby E∗ and E∗ must be monotone, positively affinely homogeneous and subadditive tobe representable by P.
♦
Remark 1.5.3.By now we have completed the first step in showing that AV @Rλ is a coherent
measure of risk, since we have a representation for coherent measures.♦
Follmer et al. extends the theory of the coherent risk measure that can be foundin [3] to the case where we have a convex risk measure. For the convex counterpart ofthe above stated proposition, we need to introduce so called penalty functions
α : M1,f −→ IR ∪ ∞.
withinf
Q∈M1,f
α(Q) ∈ R.
Theorem 1.5.4.Any convex measure of risk ρ on X is of the form
ρ(X) = max(EQ[−X]− αmin(Q)
), X ∈ X , (1.12)
23
where the penalty function αmin is given by
αmin(Q) := supX∈Aρ
EQ[−X] for Q ∈M1,f .
Moreover, αmin is the minimal penalty function which represents ρ, i.e., any penaltyfunction α for which
ρ(X) := supQ∈M1,f
(EQ[−X]− α(Q))
holds satisfies α(Q) ≥ αmin(Q) for all Q ∈M1,f .
Proof. We will start by showing
ρ(X) ≥ supQ∈M1,f
(EQ[−X]− αmin(Q)
)
for all X ∈ X . Since X ′ := ρ(X) + X ∈ Aρ, obviously
αmin(Q) ≥ EQ[X ′] = EQ[−X]− ρ(X).
Thus we have that ρ(X) ≥ EQ[−X]− αmin(Q) for all Q ∈M1,f and in particular forthe supremum
ρ(X) ≥ supQ∈M1,f
(EQ[−X]− αmin(Q)
).
Now we have to show that the inequality can be reversed. To this end, we construct,for a given X, QX ∈M1,f such that
ρ(X) ≤ EQX[−X]− αmin(QX). (1.13)
This, will prove (1.12). By translation invariance we can assume that ρ(X) = 0 we alsoassume, without loss of generality, that ρ is normalized. Since we assume that ρ(X) isequal to zero, X will not be included in the convex set B := Y ∈ X |ρ(Y ) < 0. Wewant to show that B has a non-empty interior so that we can apply Theorem A.35 in[4] on B. Since ρ is normalized, B contains the open ball B1 = Y ∈ X | ‖ Y − 1 ‖< 1where
‖ Y − 1 ‖= supω|Y (ω)− 1|.
Hence B has a non-empty interior and we can now apply Theorem A.35 in [4] to B andthe set consisting of only the given element X i.e., X. Thus we can separate the twodisjoint sets by a non-zero continuous linear functional l on X such that
l(X) ≤ infY ∈B
l(Y ) =: b.
We will next show that there exists QX ∈M1,f such that EQX [Y ] = l(Y )l(1) for all Y ∈ X
and that (1.13) actually holds for QX . We need to show that l(Y ) ≥ 0 if Y ≥ 0: Mono-tonicity and translation invariance of ρ imply that 1 + λY ∈ B since
ρ(1 + λY ) = ρ(λY )− 1 < 0, ∀λ > 0.
24
Hence, if we assume l(Y ) < 0 then from the linearity of l we have that, ∀λ > 0,
l(X) ≤ l(1 + λY ) = l(1) + λl(Y ).
λ could be chosen arbitrarily large so that l(X) would be forced to be −∞, which is acontradiction.
Now we have to show that l(1) > 0: Since l is non-zero, there must be some
Y ∈ B1(0) = Y ∈ X |‖ Y ‖< 1
such that 0 < l(Y ) = l(Y +)− l(Y −). Since l is positive for this Y we know thatl(Y +) > 0 and therefore l(1− Y +) ≥ 0 since Y ∈ B1(0). Hence, linearity of l givesthat
l(1) = l(1− Y +) + l(Y +) > 0.
By the two preceding steps and Theorem A.33 in [4] we conclude that there existssome QX ∈M1,f such that
EQX [Y ] =l(Y )l(1)
for all Y ∈ X . Now we only have to show that (1.13) holds for QX . Since
B := Y ∈ X |ρ(Y ) < 0 and Aρ := Y ∈ X |ρ(Y ) ≤ 0
we have that B ⊂ Aρ. Consequently,
αmin(QX) = supY ∈Aρ
EQX[−Y ] ≥ sup
Y ∈BEQX
[−Y ] =−infY ∈B l(Y )
l(1)=−b
l(1).
Furthermore, observe that Y + ε ∈ B, ∀Y ∈ Aρ, ∀ε > 0, since
ρ(Y + ε) = ρ(Y )− ε < 0.
PutCε := Z | Z = Y + ε, Y ∈ Aρ
for a fixed ε > 0. Then Cε ⊂ B and
αmin(QX) = supY ∈Aρ
EQX [−Y ]− ε + ε
= supY ∈Aρ
EQX [−Y − ε] + ε
= supZ∈Cε
EQX [−Z] + ε
≤ supZ∈B
EQX[−Z] + ε
=−b
l(1)+ ε.
25
Now let ε −→ 0 and we have the equality
αmin(QX) =−b
l(1).
Thus,
EQX[X]− αmin(QX) =
−l(X)l(1)
− −b
l(1)=
b− l(X)l(1)
≥ 0 = ρ(X)
since l(X) ≤ infY ∈B l(Y ) =: b. We have now constructed our set QX with the desiredproperty and thus the proof for (4.13) is completed. Let α be any penalty functionsatisfying ρ(X) := supQ∈M1,f
(EQ[−X]− α(Q)
). We can write
ρ(X) ≥ EQ[−X]− α(Q).
Accordingly, this inequality must hold for all X,
α(Q) ≥ supX∈X
(EQ[−X]− ρ(X))
≥ supX∈Aρ
(EQ[−X]− ρ(X))
≥αmin(Q),
where the second inequality follows from the fact Aρ ⊂ X , and hence α dominates αmin.♦
Note that the statement: α represents ρ, is equivalent to the statement: α(Q) < ∞for some Q ∈M1,f .
As we said in the beginning of this section, we will eventually see representations interms of probability measures. In particular, in Section 2.2, we show that if the convexrisk measure ρ is continuous from below, i.e.,
Xn X =⇒ ρ(Xn) ρ(X),
then, ρ will have a representation in terms of probability measures.The representation derived in Proposition 1.5.1 is in fact a special case of the
representation that we find in Theorem 1.5.4. Indeed, if we have the convex case withthe penalty function only taking the values
α(Q) = 0, ∀Q ∈ Q,
andα(Q) = ∞, otherwise
the representation degenerates to the coherent case. In the following corollary, we showthat the minimum penalty function always take this form in the coherent case.
26
Corollary 1.5.5.The minimal penalty function αmin of a coherent measure of risk ρ takes the values
0 and +∞. In particular,
ρ(X) = maxQ∈Qmax
EQ[−X], X ∈ X ,
for the convex setQmax := Q ∈M1,f |αmin(Q) = 0,
and Qmax is the largest set for which a representation of the form
ρ(X) = supQ∈Q
EQ[−X], X ∈ X ,
holds.
Proof. The acceptance set for a coherent risk measure, Aρ, is a cone. Thus, theminimal penalty function satisfies
αmin(Q) = supX∈Aρ
EQ[−X] = supλX∈Aρ
EQ[−X] = λαmin(Q),
for all Q ∈M1,f and λ > 0. This completes the proof.♦
A financial institute can have more than one risk measure in use. In the nextproposition we will show that the supremum of all convex risk measures in a predefinedset will, itself, be convex.
Proposition 1.5.6.Suppose that for every i in some index set I we are given a convex measure of risk
ρi on X with associated penalty function αi. If supi∈I ρi(0) < ∞ then
ρ(X) := supi∈I
ρi(X), X ∈ X ,
is a convex measure of risk that can be represented with the penalty function
α(Q) := infi=I
αi(Q), Q ∈M1,f .
Proof. The condition ρ(0) = supi∈I ρi(0) < ∞ implies that ρ takes only finitevalues. Moreover,
ρ(X) = supi∈I
supQ∈M1,f
(EQ[−X]− αi(Q)
)= sup
Q∈M1,f
(EQ[−X]− inf
i∈Iαi(Q)
).
♦
27
Chapter 2 - Convex Risk Measures
In this chapter, some continuity properties for convex and coherent risk measureswill be investigated. More specifically, we will in Section 2.2 give some sufficient condi-tions for continuity from below and continuity from above respectively, i.e.,
Xn X =⇒ ρ(Xn) ρ(X)
Xn X =⇒ ρ(Xn) ρ(X)
and investigate what these continuity properties imply if a risk measure satisfies them.These results will be very useful for our further discussion of V @Rλ and AV @Rλ inChapter 3. However, this chapter start out with a thorough discussion of the financialmarket. The reason for doing this is that we in Section 2.3 will derive a representationfor convex risk measures in a financial market.
2.1 - The Financial Market Model
As in Example 1.2.4 we have a financial market consisting of d assets with the priceof a share i denoted by πi at time 0 and after 1-time step, Si. Since the purchase of theshare will happen at time 0 and the sale will happen at time 1, we have to discount Si
to make the buying price and selling price comparable. For this, we use the discountingrate r and derive the present value of Si by
Si
1 + r.
We are now ready to define the discounted net gain from the purchase and the sale ofasset i.
Definition 2.1.1.For share i with price πi at time 0 and with price Si at time 1 we define
Y i :=Si
1 + r− πi,
to be the discounted net gain from share i.♦
We will by ξi denote the numbers of shares i we have in our portfolio. The spaceof discounted net gains which can be generated by some portfolio will be denoted by
K := ξ · Y | ξ ∈ IRd,
where ξ = (ξ1, ..., ξd) and Y = (Y 1, ..., Y d).In all financial models, there is an assumption made about arbitrage, i.e., the
possibility of earning more than the discounting rate (risk-free rate) without increasingthe risk. We will in this section discuss how arbitrage-free conditions vary from situation
28
to situation. We start out this discussion by giving a proper definition of an arbitrageopportunity.
Definition 2.1.2.An arbitrage opportunity is a portfolio ξ := (ξ0, ξ) ∈ IRd+1 such that
ξ · π ≤ 0 and ξ · S ≥ 0 P - a.s. and P [ξ · S > 0] > 0.
♦The reason for the shift in notation from ξ to that of ξ = (ξ0, ξ) is that we here
have included a bond, representing the risk-free asset. If ξ0 < 0 we simply interpret thisas taking out a loan. We write π = (1, π). Thus, the price for one unit of the bond isnormalized to 1. We will assume that
πi > 0 for i = 1, 2, .., d,
and that short selling, i.e., that ξi is negative, is allowed. From Definition 2.1.2, we seethat an arbitrage opportunity can be interpreted as a free lottery game with a positiveprobability to win. By combining the two definitions above, every arbitrage opportunitycan be written as
ξ · ( S
1 + r− π) = ξ · Y ≥ 0 P -a.s. and P [ξ · Y > 0] > 0, (2.1)
where we have to assume that P [S0 > 0] = 1.
Remark 2.1.3.The σ-algebra F1 describes the information available at time 1. By
L0 := L0(Ω,F1, P ; IRd),
we denote the space of IRd-valued random variables which are P -a.s. finite F1- measur-able modulo the equivalent relation
Z ∼ Z :⇐⇒ Z = Z P - a.s.
Thus, by L0+ we will denote the convex cone of all non-negative elements in L0. We will
let the notationK ∩ L0
+ = 0, (2.2)
be a shorthand for the absence of arbitrage i.e., (2.1).♦
The next definition is important in view of eliminating arbitrage opportunities interms of probability measures.
Definition 2.1.4.A probability measure P ∗ is called a risk-neutral measure, or a martingale measure,
if
πi = E∗[Si
1 + r], i = 0, 1, ..., d.
♦
29
Remark 2.1.5.We now state, without proof, the fundamental theorem of asset pricing (FTAP):A
market model is arbitrage-free if and only if P 6= ∅ where
P := P ∗ | P ∗ is a risk-neutral measure with P ∗ ≈ P.
♦
We will in the rest of this section consider two fixed sets of probability measuresand investigate how the no-arbitrage condition can be defined for different acceptancesets. We will also see some relations between the fundamental theorem of asset pricingand the no-arbitrage condition.
We begin with defining the first fixed finite set
Q0 = Q1, ..., Qn,
of equivalent probability measures Qi ≈ P such that |Y | ∈ L1(Qi). The set Q0 can beviewed as the set of probability measures, that we think we are likely to rule under.Define the set
B := X ∈ L0 | EQi [X] ≥ 0 for i = 1, ..., nand the acceptance set
A := B ∩ L∞,
The set A is a convex cone. Instead of considering a financial position X as acceptableif X ∈ A, we define a second acceptance set A and require that the hedged position willbe included in A. Thus, we define
A := X ∈ L∞ | ∃ξ ∈ Rd with X + ξ · Y ∈ B.
PutB0 := X ∈ B | EQi [X] = 0 for i = 1, ..., n
and notice thatB0 ∩ L0
+ = 0.Let us introduce the no-arbitrage condition
K ∩ B = K ∩ B0. (2.3)
Thus, we can interpret (2.3) as follows: there exists no position in B with a strict positiveexpectation, i.e, EQi [X] = 0 for all X in B and all i = 1, ..., n. We see that (2.3), in fact,implies (2.2) since K ∩ L0
+ ⊆ B and
K ∩ L0+ = K ∩ B ∩ L0
+ = K ∩ B0 ∩ L0+ = 0.
Thus, we have so far introduced two no-arbitrage conditions (2.2) and (2.3), where (2.3)is the strongest one.
30
It is more realistic to consider a combination of the probability measures in Q0
than only one them. Therefore, we construct the set
R := n∑
i=1
λiQi | λi > 0,
n∑
i=1
λi = 1
of all probability measures that can be constructed as a strictly convex combination ofthe elements in Q0. We now consider R as the set of possible probability measures thatwe rule under. Since we are only interested in these probability measures (FTAP) couldbe restated in terms of R. After the next lemma, which we state without proof, we willpresent a proposition that shows that the fundamental theorem of asset pricing in thiscase in fact is equivalent to (2.3).
Lemma 2.1.6.Suppose that C ⊂ IRn is a non-empty convex set with 0 /∈ C. Then there exists
η ∈ IRn with η · x ≥ 0 for all x ∈ C, and with η · x0 > 0 for at least one x0 ∈ C. Moreover,if infx∈C |x| > 0, then one can find η ∈ IRn with infx∈C η · x > 0.
For a proof, see Theorem A1 in [4].♦
Proposition 2.1.7.The following two properties are equivalent:(a) K ∩ B = K ∩ B0
(b) P ∩R 6= ∅
Proof. (b)⇒(a): For V ∈ K ∩ B and R ∈ R, we have ER[V ] ≥ 0 since every termλiEQi [V ] in ER[V ] is non-negative. If we can choose R ∈ P ∩R, every term will becomezero, which implies ER[V ] = 0, and hence V ∈ B0.
(a)⇒(b): Consider the convex set
C := ER[Y ] | R ∈ R ⊂ IRd.
We have to show that C contains the origin for we then have ER′ [Y ] = 0 for some R′ ∈ Rand this particular R′ must be contained in P. We will show this part by contradiction.To this end, assume that 0 /∈ C and by the preceding lemma, there exists ξ ∈ IRd suchthat
ξ · x ≥ 0 for x ∈ C, (2.4)
andξ · x∗ > 0 for some x∗ ∈ C.
Define V := ξ · Y ∈ K. Condition (2.4) implies
ER[V ] ≥ 0 for all R ∈ R,
hence V ∈ K ∩ B. Let R∗ ∈ R be such that x∗ = ER∗ [Y ]. Then V satisfies ER∗ [V ] > 0and we have
V /∈ K ∩ B0.
31
This contradicts our assumption K ∩ B = K ∩ B0.♦
The second finite set we will consider, Q1, is a subset of M1(P ) i.e., Q ¿ P ,with | Y |∈ L1(Q) for all Q ∈ Q. Recall that we defined B earlier and subsequentlyconstructed A A. In similar fashion, we now introduce
B1 := X ∈ L0 | EQ[X] ≥ γ(Q) for Q ∈ Q1
and in return A and A1 will be constructed. Here γ will be a functional defined on M1
withγ(Q) < 0 for each Q ∈ Q1.
We can consider γ to have the role as a floor for the expectation. Now, we get moreconservative and only accept the elements in A, which are also included in the set B1.Thus, the initial acceptance set we will consider is
A1 := A ∩ B1 = L∞ ∩ (B ∩ B1).
As before, we allow a suitable hedge to make a position X acceptable. Therefore, wedefine our final acceptance set to be
A1 := X ∈ L∞ | ∃ξ ∈ IRd with X + ξ · Y ∈ B ∩ B1.
We will now show the equivalence relation
K ∩ (B ∩ B1) = K ∩ B0 ⇐⇒ K ∩ B = K ∩ B0.
For X ∈ K ∩ B we can find a ε > 0 such that X1 = εX satisfies
EQ[X1] ≥ γ(Q) for Q ∈ Q1.
Thus, X1 ∈ K ∩ (B ∩ B1). If we assume K ∩ (B ∩ B1) = K ∩ B0, we have X1 ∈ K ∩ B0,and clearly that X = 1
εX1 ∈ K ∩ B0, since K ∩ B0 is a cone.
Definition 2.1.8.A contingent claim is a random variable C defined on the underlying probability
space (Ω,F , P ) such that0 ≤ C < ∞ P -a.s.
♦Let us consider C to be just another asset with the price
πd+1 := πC Sd+1 := C
i.e., we extend the available assets. We will call a real number πC ≥ 0 an arbitrage-freeprice of a contingent claim C if the extended market model is arbitrage free. Let Π(C)denote the set of arbitrage-free prices for C.
32
Theorem 2.1.9.Suppose that the set P of equivalent risk-neutral measures for the original market
model is non-empty. Then the set of arbitrage-free prices of a contingent claim C isgiven by
Π(C) =E∗[ C
1 + r
] | P ∗ ∈ P such that E∗[C] < ∞. (2.5)
Moreover, the lower and upper bounds of Π(C) are given by
π↓(C) = infP∗∈P
E∗[ C
1 + r
]and π↑(C) = sup
P∗∈PE∗[ C
1 + r
].
Proof. We start out by showing, in a straightforward manner, ⊆ in (2.5). Thefundamental theorem for asset pricing states that
E∗[ Si
1 + r
]= πi i = 1, ..., d + 1
for some risk-neutral measure P if and only if πC is an arbitrage-free price for C. Fromthe definition of P, we know that P ∈ P. To get ⊇ in (2.5) we just notice that if
πC = E∗[C/(1 + r)]
for some P ∗ ∈ P, then P ∗ is also a risk-neutral measure for the extended market model,and we have equality in (2.5).
Since C is bounded from below, the formula for π↓ is straightforward. The formulafor π↑ will be straightforward in the case E∗[C] < ∞. However, for the case E∞[C] = ∞for some P∞ ∈ P we will need a special treatment to show that π↑(C) = ∞. We will dothis by way of contradiction. Assume π↑(C) < ∞ and take some real number π > π↑(C).Then π is not an arbitrage-free price for C. Thus there exists a portfolio (ξ, ξd+1) ∈ IRd+2
such that π · ξ + πξd+1 ≤ 0, and such that
ξ · S + ξd+1C ≥ 0, P -a.s. and P [ξ · S + ξd+1C > 0] ≥ 0.
If ξd+1 were negative, then it would follow that
0 ≤ C ≤ ξ · S−ξd+1
=: V P -a.s.
But since V is P∞-integrable while C is not this is impossible. Thus, ξd+1 > 0 andC ≥ V. For n ≥ 1 we define
Cn := C ∧ (V ∨ 0 + n).
Then,ξ · S + ξd+1Cn ≥ 0 P -a.s. and P [ξ · S + ξd+1Cn > 0] ≥ 0,
33
so that π cannot be an arbitrage-free price for Cn. However, if n is large enough, then
π < E∞[ Cn
1 + r
]< ∞.
By assumption, there exists another measure P ∗ ∈ P such that
E∗[ Cn
1 + r
] ≤ E∗[ C
1 + r
]< π.
Then we can find an α ∈ (0, 1) such that
αE∗[ Cn
1 + r
]+ (1− α)E∞[ Cn
1 + r
]= π.
Thus, P := αP ∗ + (1− α)P∞ ∈ P and it is a risk-neutral measure since P is convex.This must imply that π ∈ Π(Cn) and contradicts our assumption that π > π↑(C).
♦
Example 2.1.10.Any constant m > π↑(C) will, obviously, not be an arbitrage-free price for the
contingent claim C. Thus, there exists a portfolio (ξ, 1) ∈ IRd+2 such that
ξ · Y +C
1 + r−m ≤ 0.
On the other hand ifξ · Y +
C
1 + r−m ≤ 0
thenE∗[ξ · Y +
C
1 + r] = E∗[
C
1 + r]
≤ m
for all P ∗ ∈ P. In particular m ≥ π↑(C). Thus,
π↑(C) := infm ∈ IR | ∃ξ′ ∈ IRd with m + ξ′ · Y ≥ C
1 + rP -a.s.
.
This expression can be seen as a dual to
π↑(C) = supP∗∈P
E∗[ C
1 + r
].
♦
In the following example, we use the best hedge method to lower our V @R.Example 2.1.11.
For a financial position X, we have
V @R0.05(X) > 0.
34
We will try to find a suitable hedge, ξ∗ = (ξ0, ξ∗) with ξ∗ · π = 0, to make the positionX acceptable .i.e.,
V @R0.05(X + ξ∗Y ) ≤ 0.
Suppose the financial market consists of only one share with the net gain Y and therisk-free bond. We will use the best hedge method in an attempt to find a suitable hedge.Hence, we have to find ξ = ξ∗ ∈ IR which minimizes
σ2X+ξY = σ2
X + 2ξσX,Y + ξ2σ2Y ,
i.e.,
ξ∗ = −σX,Y
σ2Y
and
σ2X+ξ∗Y = σ2
X − σ2X,Y
σ2Y
.
In our case we, will assume
(XY
)∈ N(
(30
),
(4 0.3
0.3 0.04
)).
Thus, we have the following formula for V @R0.05 :
V @R0.05(X + ξ∗Y ) = −(E[X + ξ∗Y ] + Φ−1(0.05) · σX+ξ∗Y )
After minimization we have
V @R0.05(X + ξ∗Y ) = −E[X] + 1.65(σ2X − σ2
X,Y
σ2Y
)−12 = −0.82 < 0.
With the hedge, we have change our position from being N(3, 4)-distributed and unac-ceptable to becoming an acceptable N(3, 1.75)-distributed position.
♦
The discussion we carried out throughout this section will be used in Section 2.3 ofthis chapter. We will in that section derive the representation for convex and coherentrisk measures in a financial market, i.e., where we allow a suitable hedge to make aposition acceptable.
35
2.2 - Continuity properties
Recall that by M1,f (Ω,F) we denote the set of all finitely additive set functionsQ : F → [0, 1] which are normalized to Q(Ω) = 1 and by M1(Ω,F) the set of all prob-ability measures. Thus,
M1 ⊂M1,f .
So far, we have only derived representations in terms of M1,f . Since we are dealing withmonetary risk measures it would be more suitable, in our context, to derive represen-tations in terms of M1. To this end, we have to take on the discussion of continuousproperties on risk measures. In the next lemma, we will show that if ρ is convex andhas a representation in terms of probability measures, i.e.,
ρ(X) = supQ∈M1,f
(EQ[−X]− α(Q)
)= sup
Q∈M1
(EQ[−X]− α(Q)
),
then, ρ will be continuous from above.
Definition 2.2.1.If we have the implication: if (Xn) is a bounded sequence in X which converges
pointwise to X ∈ X , thenρ(X) ≤ lim inf
n↑∞ρ(Xn),
we say that ρ is lower semicontinuous with respect to bounded pointwise convergence.♦
Sometimes ρ is said to have Fatou’s property if it is lower semicontinuous.
Lemma 2.2.2.A convex measure of risk ρ which admits the representation
ρ(X) = supQ∈M1
(EQ[−X]− α(Q)
)
is continuous from above in the sense that
Xn X =⇒ ρ(Xn) ρ(X).
Moreover, continuity from above is equivalent to lower semicontinuity with respect tobounded pointwise convergence.
Proof. We begin with showing ρ(X) ≤ lim infn↑∞ ρ(Xn) for a bounded pointwiseconvergent sequence (Xn) :
EQ[−Xn]− α(Q) ≤ supQ′∈M1
(EQ′ [−Xn]− α(Q′)
).
This should also hold if we take lim inf on both sides,
lim infn↑∞
(EQ[−Xn]− α(Q)
) ≤ lim infn↑∞
[sup
Q′∈M1
(EQ′ [−Xn]− α(Q′)
)].
36
Dominated convergence implies that
EQ[Xn] −→ EQ[X] when n →∞
for each Q ∈M1, if Xn converges to X. Thus,
supQ∈M1
lim infn↑∞
(EQ[−Xn]− α(Q)
) ≤ lim infn↑∞
[sup
Q′∈M1
(EQ′ [−Xn]− α(Q′)
)]
andρ(X) = sup
Q∈M1
(EQ[−X]− α(Q)
)
≤ lim infn↑∞
(sup
Q′∈M1
(EQ′ [−Xn]− α(Q′)
))
= lim infn↑∞
ρ(Xn).
Now we want to show that continuity from above is equivalent to lower semicontinuitywith respect to bounded pointwise convergence. First, we assume lower semicontinuity.By monotonicity, ρ(Xn) ≤ ρ(X) for each n if Xn X, and therefor ρ(Xn) ρ(X)Next, assume continuity from above. Let (Xn) be a bounded sequence in X whichconverges pointwise to X. Define
Ym := supn≥m
Xn ∈ X .
Then Ym decreases P -a.s. to X. Since Yn ≥ Xn we get by monotonicity ρ(Yn) ≤ ρ(Xn).From the assumption, we get ρ(Yn) ρ(X) and therefore limn↑∞ ρ(Yn) = ρ(X). Hence,
lim infn↑∞
ρ(Xn) ≥ lim infn↑∞
ρ(Yn) = limn↑∞
ρ(Yn) = ρ(X).
♦
37
With the previous proposition in mind we would ask ourselves “What conditionswill be sufficient to get a representation in term of probability measures?”. We will getan answer to this question right after the following lemma.
Lemma 2.2.3.Let ρ be a convex measure of risk on X which is represented by the penalty function
α on M1,f , and consider the level sets
Λc := Q ∈M1,f | α(Q) ≤ c, for c > −ρ(0) = infQ∈M1,f
α(Q).
For any sequence (Xn) in X such that 0 ≤ Xn ≤ 1, the following two conditions areequivalent:
(a) ρ(λXn) −→ ρ(λ) for each λ ≥ 1.(b) infQ∈Λc EQ[Xn] −→ 1 for all c > −ρ(0).
Proof. (a)⇒ (b): First we show that for all Y ∈ X
infQ∈Λc
EQ[Y ] ≥ −c + ρ(λY )λ
for all λ > 0. (2.6)
Sinceρ(λY ) = sup
Q∈M1,f
(E[−λY ]− α(Q)
) ≥ E[−λY ]− α(Q)
we have for Q ∈ Λc
c ≥ α(Q) ≥ EQ[−λY ]− ρ(λY )
where the first inequality follows from the definition of Λc. If we rearrange this expressionand divide with λ we get
EQ[Y ] ≥ −c + ρ(λY )λ
.
This should hold even if we take infimum over all Q in Λc on the right side thus, wehave (2.6). If we now consider a sequence (Xn) which satisfies (a), then we can for allλ ≥ 1 write
lim infn↑∞
infQ∈Λc
EQ[Xn] ≥ − limn↑∞
c + ρ(λXn)λ
= 1− c + ρ(0)λ
.
Thus, (b) follows from the fact that we can choose λ arbitrarily large.(b) ⇒ (a): Since λXn ≤ λ, by monotonicity we have that
ρ(λ) ≤ ρ(λXn) = supQ∈M1,f
(E[−λXn]− α(Q)
).
Let us focus on those Q such that α(Q) > 1− ρ(λ) =: c. In this case
EQ[−λXn]− α(Q) < EQ[−λXn] + ρ(λ)− 1 ≤ ρ(λ)− 1
since EQ[−λXn] ≤ 0. Then,
supQ∈Λ′c
(EQ[−λXn]− α(Q)
) ≤ ρ(λ)− 1,
38
where Λ′c is the complement of Λc. Thus, the following holds
supQ∈M1,f
(EQ[−λXn]− α(Q)
)> sup
Q∈Λ′c
(EQ[−λXn]− α(Q)
).
As a result,
supQ∈M1,f
(EQ[−λXn]− α(Q)
)= sup
Q∈Λc
(EQ[−λXn]− α(Q)
)
and hence,ρ(λ) ≤ ρ(λXn) = sup
Q∈Λc
(EQ[−λXn]− α(Q)
)
for all n.
From condition (b) we know that,
supQ∈Λc
(EQ[−λXn]− α(Q)
)= −λ− inf
Q∈Λc
α(Q) = −λ + ρ(0) = ρ(λ)
as n →∞ and we have proven the lemma.♦
We have already seen that when the penalty function is concentrated on the set ofprobability measures, the convex risk measure ρ will be continuous from above. Thenext proposition, gives us a representation with the penalty function concentrated onthe set of probability measures as soon as the convex risk measure ρ is continuous frombelow. Thus we have the implication: Whenever the convex risk measure ρ is continuousfrom below, ρ will also be continuous from above and have a representation with thepenalty function concentrated on the set M1. This also implies that ρ(Xn) → ρ(X) assoon as Xn is a bounded sequence and converges pointwise to X.
Proposition 2.2.4.Let ρ be a convex risk measure with penalty function α, i.e.,
ρ(X) = supQ∈M1,f
(EQ(−X)− α(Q)
).
If ρ is continuous from below then we have a representation on the class M1 of proba-bility measures.
Proof. Recall that Q is σ-additive if and only if Q[An] 1 for any increasingsequence of events An ⊂ An+1 ∈ F with ∪nAn = Ω. Since ρ is continuous from below,we have that ρ(λIAn) ρ(λ) when n −→∞. Thus, we can use Lemma 2.2.3 withXn := IAn . Hence, for any Q such that α(Q) < ∞ we can find c < ∞ such that Q ∈ Λc.We get
1 ≥ Q[An] ≥ infQ′∈Λc
Q′[An] = infQ′∈Λc
EQ′ [IAn ] −→ 1
for as n −→∞.♦
39
In the last proposition we demanded that ρ should be continuous from below on thewhole space X . This will, however, be relaxed in favor of the weaker continuity propertywhere ρ only needs to be continuous from below on the set Cb(Ω) of continuous boundedfunctions on Ω.
Definition 2.2.5.Ω is called a Polish space if it is a separable topological space admitting a complete
metric.♦
Definition 2.2.6.A convex risk measure ρ on X is called tight if there exists an increasing sequence
K1 ⊂ K2 ⊂ ...
of compact subsets of the Polish space Ω such that
ρ(λIKn) −→ ρ(λ), ∀ λ ≥ 1.
♦For the next theorem and its corollary, we assume that Ω is a Polish space and that
F is the σ-field of Borel sets. In the next theorem, we will show that tightness for aconvex risk measure, in fact, is a sufficient condition to get a representation in terms ofprobability measures.
Theorem 2.2.7.Let ρ be a convex measure of risk on X , and let α be a penalty function on M1,f
representing ρ. Then the following conditions are equivalent:(a) ρ is tight.(b) ρ is continuous from below on Cb(Ω), i.e., if (Xn) is a sequence in Cb(Ω) such
that Xn X ∈ Cb(Ω), then ρ(Xn) ρ(X).(c) Each level set Λc = Q | α(Q) ≤ c consists only of probability measures, and
for any ε > 0 there exists a compact set Kε ⊂ Ω such that for all c > −ρ(0)
infQ∈Λc
Q[Kε] ≥ 1− ε(c + ρ(0) + 1).
Thus, each set Λc is relatively compact for the weak topology on M1,f .In particular, ρ has the representation
ρ(X) = maxQ∈M1
(EQ[−X]− αmin(Q)
)
if one of these equivalent conditions is satisfied.
Proof. We will assume that ρ is normalized, i.e., ρ(0) = 0. The condition (b′)stated below is implied by (b).
(b′) If (Xn) is a sequence in Cb(Ω), which increases to a constant λ > 0, then
ρ(Xn) ρ(λ).
40
We will show the following sequence: (c) =⇒ (a) =⇒ (b′) =⇒ (c) =⇒ (b).(c) ⇒ (a): This implication follows from Lemma 2.2.3, we just have to replace Xn
with IKε.
(a)⇒ (b′): If we show that the condition holds for λ > 1 we can, since ρ is assumedconvex and normalized extend the result to all λ > 0. By translation invariance we canassume that Xn ≥ 0 for all n. We will show that for all ε ∈ (0, λ− 1) there exists an nsuch that
ρ(Xn) ≤ ρ(λ) + 2ε.
Since we assume that ρ is tight we know that there exists an N such that
ρ((λ− ε)IKN
) ≤ ρ(λ− ε) + ε = ρ(λ) + 2ε
By Dini‘s lemma, we have that “On a compact set, a sequence of continuous functionsincreasing to a continuous function converges even uniformly”. Thus, since Xn λ weknow that there exists an n0 ∈ IN such that for all n ≥ n0
λ−Xn ≤ ε
on KN . Now by monotonicity
ρ(Xn) ≤ ρ((λ− ε)IKN
) ≤ ρ(λ) + 2ε.
(b′) ⇒ (c): We will show this part in the following steps: First, we show that anyQ ∈ Λc is σ-additive. Then, we construct a suitable set Kε and show that it is compact.The relative compactness of Λc will then follow from Theorem A.25 in [4].
Take a sequence (Yn) ∈ Cb(Ω) which increases to some Y ∈ Cb(Ω), and choose δ > 0such that Xn := 1 + δ(Yn − Y ) ≥ 0 for all n. Clearly, Xn 1 and 0 ≤ Xn ≤ 1 for all n.So, the assumption (b′) yields the condition (a) of Lemma 2.2.3. Thus, EQ[Xn] −→ 1for all Q ∈ Λc, i.e. EQ[Yn] EQ[Y ]. Daniel-Stone representation theorem, TheoremA.31 in [4], implies that Q ∈ Λc is σ-additive.
If we can construct the set Kε, Λc will be relatively compact for the weak topologyon M1 by Prokhorov’s characterization of weakly compact sets on M1. Let us constructthe set Kε. We fix a countable dense set ω1, ω2, ... ⊂ Ω and a complete metric δ whichgenerates the topology of Ω. For r > 0, we define a continuous functions ∆r
i on Ω by
∆ri (ω) := 1− δ(ω, ωi) ∧ r
r.
The function ∆ri is dominated by the indicator function of the closed metric ball
Br(ωi) := ω ∈ Ω | δ(ω, ωi) ≤ r,
i.e.,∆r
i (ω) ≤ IBr.
LetXr
n(ω) := maxi≤n
∆ri .
41
Then Xrn is continuous and satisfies 0 ≤ Xr
n ≤ 1 as well as Xrn 1 for n ↑ ∞. From
(2.6), we have for all λ > 0
infQ∈Λc
EQ
[I⋃n
i=1Br(ωi)
]= inf
Q∈Λc
Q[ n⋃
i=1
Br(ωi)] ≥ inf
Q∈Λc
EQ[Xrn] ≥ −c + ρ(λXr
n)λ
.
Let λk := 2k
ε and rk := 1k . From condition (b‘) we have that there exists nk ∈ IN such
thatρ(λkXrk
nk) ≤ ρ(λk) + 1 = −λk + 1
and since
infQ∈Λc
Q[ nk⋃
i=1
Brk(ωi)
] ≥ −c + 1− λk
λk
we have that
supQ∈Λc
Q[ nk⋂
i=1
Ω\Brk(ω)
] ≤ 1− (−c + 1− λk
λk) =
c + 1λk
= 2−k · ε · (c + 1).
We now define
Kε :=∞⋂
k=1
nk⋃
i=1
Brk(ωi).
Then, for each Q ∈ Λc
Q[Kε] = 1−Q[ ∞⋃
k=1
nk⋂
i=1
Ω\Brk(ωi)
] ≥ 1−∞∑
k=1
2−k · ε · (c + 1) = 1− ε(c + 1),
where we have used the σ-additivity of Q. To confirm that Kε is compact we have toshow that every sequence (xj) in Kε has a converging subsequence in Kε. Since Kε iscovered by Brk
(ω1), Brk(ω2), ..., Brk
(ωnk) for each k, there exists some ik ≤ nk such that
infinitely many members xj are contained in Brk(ωik
). With a standard diagonalizationargument we get a subsequent (xj‘) which for each k is contained in some Brk
(ωik).
Thus, (xj‘) is a Cauchy sequence, i.e., for every ε > 0 we can find a sufficiently large ksuch that (xj′) is contained in a sufficiently small Brk
(ωik) so that δ(xn, xm) ≤ rk < ε.
(c) ⇒ (b) Suppose Xn is a sequence in Cb(Ω) such that Xn X ∈ Cb(Ω). By trans-lation invariance we can assume that Xn ≥ 0 for all n. Let λ be an upper bound for X.Then, for all n
ρ(λ) ≤ ρ(Xn) = supQ∈M1,f
(EQ[−Xn]− α(Q)
).
Since EQ[−Xn] ≤ 0 for all Q, only those Q can contribute to the supremum on theright-hand side for which α(Q) ≤ 1− ρ(λ) = 1 + λ =: c. (See the proof of Lemma 2.2.3)Hence,
ρ(Xn) = supQ∈Λc
(EQ[−Xn]− α(Q)
)
We will now show that (c) implies that EQ[−Xn] converges to EQ[−X] uniformly onQ ∈ Λc ⊂M1, and this will prove condition (b). To this end, we first notice that bydefinition
M1 3 Q 7→ EQ[Y ], Y ∈ Cb(Ω)
42
is continuous for the weak topology on M1. Hence, the relative compactness of Λc andDini’s lemma show that EQ[−Xn] converges to EQ[−X] uniformly on Q ∈ Λc.
♦In the case of a coherent risk measure, the preceding theorem takes the following
form.
Corollary 2.2.8.Let ρ be a coherent measure of risk on X and let Q be any subset of M1,f such that
ρ(X) = supQ∈Q
EQ[−X], X ∈ X .
Then the following conditions are equivalent:(a) ρ is tight.(b) ρ is continuous from below on Cb(Ω), i.e. if (Xn) is a sequence in Cb(Ω) such
that Xn X ∈ Cb(Ω), then ρ(Xn) ρ(X), X ∈ X .(c) Q is a weakly relatively compact subset of M1.
♦It would be of interest to investigate, what will happen if we relax the continuity
conditions? In this thesis however, we will not concern ourselves with this kind ofquestions.
In the first chapter we defined X as the linear space of bounded functions. We willfrom now on consider X to be the Banach space L∞, i.e., the set of all F-measurablefunctions on (Ω,F , P ) such that
‖ X ‖∞:= infc ≥ 0 | P [ |X| > c] = 0 < ∞.
Lemma 2.2.9.Let ρ be a convex measure of risk that satisfies
ρ(X) = ρ(Y ) if X = Y P -a.s.
and which is represented by a penalty function α as in
ρ(X) := supQ∈M1,f
(EQ[−X]− α(Q)
).
Then α(Q) = +∞ for any probability measure Q which is not absolutely continuous withrespect to P.
Proof. If Q ∈M1(Ω,F) is not absolutely continuous with respect to P , thenthere exists A ∈ F such that Q[A] > 0 but P [A] = 0. Take any X ∈ Aρ, and defineXn := X − nIA. Then ρ(Xn) = ρ(X), i.e., Xn is again contained in Aρ. Recall theminimal penalty function αmin, defined in Section 1.5. We get
α(Q) ≥ αmin(Q) ≥ EQ[−Xn] = EQ[−X] + nQ[A] →∞
43
as n ↑ ∞.♦
By M1(P ) = M1(Ω,F , P ), we denote the set of all probability measures Q ∈M1
that are absolute continuous with respect to P.In the rest of this chapter, we use the notation σ(E, F ) from Section A.6 in [4],
to denote the F -topology on E. Consequently, if E′ is the dual for the space E, theweak-topology will be denoted by σ(E, E′).
Theorem 2.2.10.Suppose ρ : L∞ → IR is a convex measure of risk. Then the following conditions
are equivalent.(a) ρ can be represented by some penalty function on M1(P ).(b) ρ can be represented by the restriction of the minimal penalty function αmin to
M1(P ) :ρ(X) = sup
Q∈M1(P )
(EQ[−X]− αmin(Q)
), X ∈ L∞. (2.7)
(c) ρ is continuous from above: If Xn X P -a.s. then ρ(Xn) ρ(X).(d) ρ has the “Fatou property”: For any bounded sequence (Xn) which converges
P -a.s. to some X,ρ(X) ≤ lim inf
n↑∞ρ(Xn).
(e) The acceptance set Aρ of ρ is weak*-closed in L∞, i.e., Aρ is closed withrespect to the topology σ(L∞, L1).
Proof. (e) ⇒ (b): Fix some X ∈ X and any m ∈ R such that
supQ∈M1(P )
(EQ[−X]− αmin(Q)
)< m. (2.8)
Since M1 ⊂M1,f we know that
supQ∈M1(P )
(EQ[−X]− αmin(Q)
) ≤ supQ∈M1,f
(EQ[−X]− αmin(Q)
)= ρ(X).
We only need to show that ρ(X) ≤ m or, equivalently, that m + X ∈ Aρ. We will showthat the assumption m + X /∈ Aρ leads to a contradiction. Since Aρ is by assumptionclosed in the weak∗-topology, we can apply a version of Hahn-Banach theorem, TheoremA.37 in [4], on the existence of separating hyperplanes to the sets Aρ and the compactset B := m + X. Thus we know that there exists a continuous linear functional l on(L∞, σ(L∞, L1)
)such that
β := infY ∈Aρ
l(Y ) > l(m + X) =: γ > −∞. (2.9)
We know that l is of the form l(Y ) = E[Y Z] for some Z ∈ L1.Now we have to show that Z ≥ 0. To do this, fix Y ≥ 0 and note that ρ(λY ) ≤ ρ(0)
for λ ≥ 0, by monotonicity. Hence ρ(λY + ρ(0)
) ≤ 0 and λY + ρ(0) ∈ Aρ. It followsfrom the linearity of l that
−∞ < γ < l(λY + ρ(0)
)= λl(Y ) + l
(ρ(0)
).
44
Taking λ ↑ ∞, yields that l(Y ) ≥ 0, since the inequality above would not hold otherwise,and thus Z ≥ 0. Since l separates Aρ and m + X, it can not be identical to zero, i.e.,we know that P (Z > 0) > 0. Thus,
dQ0
dP:=
Z
E[Z]defines a probability measure Q0 ∈M1(P ). By (2.9), we see that
αmin(Q0) = supY ∈Aρ
EQ0 [−Y ] = supY ∈Aρ
E[−Y
Z
E[Z]]
= − 1E[Z]
infY ∈Aρ
E[Y Z] = − β
E[Z].
However,
EQ0 [X] + m =l(m + X)
E[Z]=
γ
E[Z]<
β
E[Z]= −αmin(Q0),
in contradiction to (2.8). Hence, m + X must be contained in Aρ, and thus m ≥ ρ(X).(b)⇒ (a): is obvious, and (a)⇒ (c)⇔ (d) follows as in Lemma 2.2.2, if we replace
pointwise convergence by P -a.s. convergence.(c) ⇒ (e): For this part of the proof, we use several theorems from Appendix in
[4] together with theory on topology and functionals from Chapter 10 in [9]. We wantto show that Aρ is closed with respect the topology σ(L∞, L1). Since Aρ is convex andL∞ is the dual of L1, by Theorem A.41 in [4] it is sufficient for us to show that
Cr := Aρ ∩ X ∈ L∞ |‖ X ‖∞≤ ris weak∗-closed for every r > 0. Define
Br := X ∈ L∞ |‖ X ‖∞≤ rSince
Cr ⊂ Br ⊂ L∞ ⊂ L1,
we can consider Cr and Br as subsets in L1. From a corollary of Theorem A.45 in [4],we know that Br is weakly compact in L1. If (Xn) is a sequence in Cr converging in L1
to some random variable X, i.e.,
‖ Xn −X ‖1→ 0 as n →∞,
then, since (c) ⇒ (d) the Fatou’s property of ρ implies
ρ(X) ≤ lim infn↑∞
ρ(Xn).
Thus, X ∈ Cr and Cr is closed in L1. By Theorem A.40 in [4], Cr is also weakly closed inL1. Moreover, since Cr ⊂ Br, Br is weakly compact in L1 and Cr is weakly closed in L1
we have that Cr is weakly compact in L1. Each Z ∈ L∞ defines a linear functional onL1 by Y 7→ lZ(Y ). Since lZ | Z ∈ L∞ separates the points of L1 it must also separateall the points of L∞. Hence, we may define σ(L∞, L∞) on L∞. Let us consider Br as atopological subspace of
(L1, σ(L1, L∞)
), i.e., we get the topology for Br by taking those
open subsets O of Br for which there is an O′ ∈, σ(L1, L∞) such that O = Br ∩O′. Thenatural injection from Br into the topological subspace(L∞, σ(L∞, L∞)
)is then continuous, i.e., every set O ∈ σ(L∞, L∞) is contained in
the topology of Br. It follows that Cr, considered as a subset of(L∞, σ(L∞, L∞)
), is
compact, and hence closed. Since L∞ ⊂ L1, σ(L∞, L∞) will be weaker than σ(L∞, L1).Thus, Cr is closed with respect to the topology σ(L∞, L1), i.e, Cr is weak∗-closed in L∞.
♦
45
2.3 - Convex Risk Measures in a Financial Market
In this section, we will combine the theory of monetary risk measures with thetheory of the financial market model developed in Section 2.1. We will conclude thischapter by giving a representation of a convex risk measure in a financial market, wherewe allow a suitable hedge and assume that there exists no arbitrage opportunities.We begin this section by continuing the discussion we had in Section 2.1, about theacceptance set that arise in the financial market.
Recall, once again, Example 1.4.2 and the acceptance set
A := X ∈ X | ∃ξ ∈ IRd with X + ξ · Y ≥ 0,
where we allow hedging without additional cost, i.e., ξ · π = 0, to create more acceptablepositions. In this section, we will always assume that the space of financial positionswill be the space L∞. So far, we have allowed ξ to be any member of IRd. This is,however, not always a good description of reality. There are several reasons to putlower and upper bounds for ξ. For example, we are never guaranteed a loan to fullycover the amount ξ · π. Market illiquidity is another factor that could want us to putan upper bound on the position ξ, since we do not want to take the risk of not resellingthe shares. Hence, there are reasons to put some upper constrains on ξ. If we do notwant to allow short sales, we might want to put a lower bound on the number of sharesin a portfolio. All this comes down to accepting those ξ from a certain set δ, with thefollowing properties:(i) 0 ∈ δ(ii) δ is convex(iii) Each ξ ∈ δ is admissible in the sense that ξ · Y is P -a.s. bounded from below.
Thus, the initial acceptance set will, with only the admissible positions at hand,take the form
Aδ := X ∈ L∞ | ∃ξ ∈ δ with X + ξ · Y ≥ 0 P -a.s..
Aδ will in fact be non-empty and convex and we will, for the rest of this chapter, assumethat
ρAδ (0) = infm ∈ IR | m ∈ Aδ > −∞.
This assumption was not necessary when we defined X to be the space of boundedfunctions. Now, however, we consider X to be the space L∞. After these preparations,we are ready to apply Proposition 1.4.5 to the set Aδ
ρδ := ρAδ .
Thus, ρδ is a convex risk measure. We will now derive a representation for ρδ in termsof probability measures where we have allowed a suitably hedge from the admissible set.
46
Proposition 2.3.1.The minimal penalty function αδ
min of ρδ is given on M1(P ) by
αδmin(Q) = sup
ξ∈δEQ[ξ · Y ] for Q ∈M1(P ).
In particular, ρδ can be represented as
ρδ(X) = supQ∈M1(P )
(EQ[−X]− supξ∈δ
EQ[ξ · Y ])
if ρδ is continuous from above.
Proof. Fix Q ∈M1(P ). Clearly, the expectation EQ[ξ · Y ] is well defined for eachξ ∈ δ by admissibility. If X ∈ Aδ, there exists η ∈ δ such that −X ≤ η · Y P -a.s. Thus,
EQ[−X] ≤ EQ[η · Y ] ≤ supξ∈δ
EQ[ξ · Y ]
for any Q ∈M1(P ). Hence, the definition of the minimal penalty function yields
αδmin(Q) ≤ sup
ξ∈δEQ[ξ · Y ].
To prove the converse inequality, take ξ ∈ δ. Note that
Xk := −((ξ · Y ) ∧ k
)
is bounded since ξ is admissible. Moreover,
Xk + ξ · Y = (ξ · Y − k)Iξ·Y≥k ≥ 0.
Hence for all k,
EQ[ξ · Y ] =EQ[−Xk] + EQ[(ξ · Y − k)Iξ·Y≥k]
≤ supX∈Aδ
EQ[−X] + EQ[(ξ · Y − k)Iξ·Y≥k]
=αδmin(Q) + EQ[(ξ · Y − k)Iξ·Y≥k].
Finally,EQ[(ξY − k)Iξ·Y≥k] −→ 0
when k →∞. We are done.♦
47
Definition 2.3.2.A convex measure of risk ρ on L∞ is called relevant if
ρ(−IA) > ρ(0)
for all A ∈ F such that P [A] > 0.♦
We now state two corollaries of Theorem 2.2.10. Both concern the case of a coherentrisk measure.
Corollary 2.3.3.A coherent measure of risk on L∞ can be represented by a set Q ⊂M1(P ) if and
only if the equivalent conditions of Theorem 2.2.10 are satisfied. In this case, the max-imal representing subset of M1(P ) is given by
Qmax := Q ∈M1(P ) | αmin(Q) = 0.
Moreover, ρ is relevant if and only if Qmax ≈ P in the sense that for any A ∈ F
P [A] = 0 ⇐⇒ Q[A] = 0 for all Q ∈ Qmax.
Proof. The “if and only if”-part is obvious. That the maximal representing subsetis given by Qmax follows as in Corollary 1.5.5. The last part of the corollary follows ifwe can show for any A ∈ F that
P [A] = 0 ⇐ Q[A] = 0 for all Q ∈ Qmax
if and only if ρ is relevant. To show the “if”-part, assume that ρ is relevant. Then
ρ(−IA) = supQ∈Qmax
EQ[IA] = supQ∈Qmax
Q[A] = 0
implies that P [A] = 0.Finally, if Qmax ≈ P then,
supQ∈Qmax
Q[A] > 0
as soon as P [A] > 0.♦
Corollary 2.3.4.For a coherent risk measure ρ on L∞ the following properties are equivalent:(a) ρ is continuous from below: Xn X ⇒ ρ(Xn) ρ(X).
48
(b) There exists a set Q ⊂M1(P ) representing ρ such that the supremum is at-tained:
ρ(X) = maxQ∈Q
EQ[−X] for all X ∈ X .
(c) There exists a set Q ⊂M1(P ) representing ρ such that the set of densities
D :=
dQ
dP| Q ∈ Q
is weakly compact in L1(Ω,F , P ).
Proof. (c)⇒ (a): This follows from Dini‘s lemma as in the proof of Theorem 2.2.7.(a) ⇒ (b): From Corollary 1.1.5 we know that we can choose Q := Qmax and the
supremum can be attained. By Proposition 2.2.4 we know that Qmax is, in fact, a subsetof M1. Lemma 2.2.9 implies that Q ∈ Qmax ⊂M1(P ).
(b) ⇒ (c): If D is not weakly closed, we can always choose the closure. Hence,without loss of generality, we assume that D is weakly closed in L1. Then, by TheoremA.44 in [4], it suffices to show that every continuous functional attains its infimum onD. Every continuous functional on L1 can be written as the form
JX(Z) := E[XZ]
for some X ∈ L∞. Thus,
infZ∈D
JX(Z) = infZ∈D
E[XZ]
= infQ∈Qmax
E[dQ
dPX]
= infQ∈Qmax
EQ[X]
= minQ∈Qmax
EQ[X]
=− ρ(−X).
♦
Assume that δ is a cone. Then the acceptance set Aδ is also a cone and ρδ is acoherent measure of risk. If ρδ is continuous from above, then Corollary 2.3.3 yields therepresentation
ρδ(X) = supQ∈Qδ
max
EQ[−X]
in terms of the non-empty set
Qδmax = Q ∈M1(P ) | αδ
min(Q) = 0.
49
Lemma 2.3.5.Suppose δ is a cone. If ρδ is relevant, then the set δ cannot contain any arbi-
trage opportunities, and Qδmax contains the set P of all equivalent martingale measures
whenever such measures exists.
Proof. Assume that there is an arbitrage opportunity such that
ξa · Y ≥ 0 P -a.s. and P [ξa · Y > 0] > 0.
Let A = ξa · Y > 0. Since ρδ is relevant we can write
ρδ(−IA) = supQ∈Qδ
max
EQ[IA] = supQ∈Qδ
max
Q[A] = supQ∈Qδ
max
Q[ξa · Y > 0] > ρδ(0) = 0.
By Proposition 2.3.1, we have EQ[ξa · Y ] ≤ 0 when Q ∈ Qδmax. Thus, we have ξa · Y ≥ 0
and EQ[ξa · Y ] ≤ 0. Hence, ξa · Y = 0 which contradicts
supQ∈Qδ
max
Q[ξa · Y > 0] > 0.
♦In the next proposition, we extend the result for the contingent claim in Example
2.1.10 by using some of the theory that we derived in this section.
Proposition 2.3.6.Suppose δ consists of all admissible ξ ∈ L0(Ω,F0, P ; IRd), and ρδ is relevant in the
sense of Definition 2.3.2. Then ρδ is continuous from above if and only if the marketmodel is arbitrage-free. In this case, ρδ can be represented in terms of the set P ofequivalent risk-neutral measures:
ρδ(X) = supP∗∈P
E∗[−X]. (2.10)
Proof. Let us assume that the model is arbitrage-free, then
ρδ(X) = infm ∈ IR | m + X ∈ Aδ= infm ∈ IR | ∃ξ ∈ δ, m + X + ξ · Y ≥ 0= sup
P∗∈PE∗[−X],
where the last equality follows as in Example 2.1.10. Thus, Corollary 2.3.3 implies thatρδ is continuous from above.
Conversely, suppose that ρδ is continuous from above. Since δ is a cone,
ρδ(X) = supQ∈Qδ
max
EQ[−X].
The relevance property of ρδ guarantees the absence of arbitrage opportunities in δ.If there would exist any arbitrage opportunities, they would be bounded from below,
50
hence admissible and contained in δ. Consequently, we can conclude that the marketmodel is arbitrage-free.
♦
Recall the acceptance set A and the risk measure ρ that we derived in Section 2.1.We will next derive the minimal penalty function for this risk measure.
Proposition 2.3.7.If both ρA and ρ are continuous from above, then the minimal penalty function αmin
for ρ is given by
αmin(Q) = αδmin(Q) + αmin(Q), Q ∈M1(P ),
where αδmin is as in Proposition 2.3.1, and αmin is the minimal penalty function for ρA.
Proof. We first show
A = Xδ + X | Xδ ∈ Aδ, X ∈ A. (2.11)
If X ∈ A, we know that there exist X ∈ A and ξ ∈ δ such that X + ξ · Y ≥ X. ThenX −X ∈ Aδ since X −X + ξ · Y ≥ 0. Conversely, if Xδ ∈ Aδ then Xδ + ξ · Y ≥ 0 forsome ξ ∈ δ. Hence, for any X ∈ A, we get Xδ + X + ξ · Y ≥ X ∈ A, i.e.,
X := Xδ + X ∈ A.
In view of (2.11), we have
αmin(Q) = supX∈A
EQ[−X]
= supXδ∈Aδ
( supX∈A
EQ[−Xδ −X])
=αδmin(Q) + αmin(Q).
♦
In the case of a coherent risk measure, the representation will take the followingform in a financial market.
Theorem 2.3.8.Under assumption (2.3) the coherent risk measure ρ := ρA corresponding to the
acceptance set A is given by
ρ(X) = supP∗∈P∩R
E∗[−X].
♦
51
This is a special case of the following theorem. Let us now identify the convex riskmeasure ρ1 induced by the convex acceptance set A1 of Section 2.1. Define
R1 := ∑
Q∈Qλ(Q) ·Q | λ(Q) ≥ 0,
∑
Q∈Qλ(Q) = 1 ⊃ R
as the convex hull of Q := Q0 ∪Q1, and define
γ(R) :=∑
Q∈Qλ(Q)γ(Q)
for R =∑
Q λ(Q)Q ∈ R with γ(Q) := 0 for Q ∈ Q0. The next theorem gives us a repre-sentation of convex risk measures where we have a predefined finite space of probabilitymeasures which are risk-neutral.
Theorem 2.3.9.Under assumption (2.3), the convex risk measure ρ1 induced by the acceptance set
A1 := X ∈ L∞ | ∃ξ ∈ Rd with X + ξ · Y ∈ B ∩ B1
is given byρ1(X) = sup
P∗∈P∩R1
(E∗[−X] + γ(P ∗)
), (2.12)
i.e., ρ1 is determined by the penalty function
α1(Q) =
+∞, for Q /∈ P ∩R1,−γ(Q), for Q ∈ P ∩R1.
Proof. Let ρ∗ denote the convex risk measure defined by the right-hand side of(2.12), and let A∗ denote the corresponding acceptance set:
A∗ = X ∈ L∞ | E∗[−X] + γ(P ∗) ≤ 0 for all P ∗ ∈ P ∩R1.
It is enough to showA∗ = A1.
(a) Let us begin with showing that A∗ ⊃ A1. To get this result, take X ∈ A1 andP ∗ ∈ P ∩R1. There exists ξ ∈ IRd and A1 ∈ A1 = A ∩ B1 such that X + ξ · Y ≥ A1,since A1 ∈ B ∩ B1 and X + ξ · Y could be chosen such that it is included in B ∩ B1.Thus,
E∗[X + ξ · Y ] ≥ E∗[A1] ≥ γ(P ∗),
due to P ∗ ∈ R1, since
E∗[A1] =∑
Q∈Qλ(Q) · EQ[A1] ≥
∑
Q∈Qλ(Q)γ(Q) = γ(P ∗)
where the inequality follows from the fact that A1 ∈ B1. Since E∗[ξ · Y ] = 0 due toP ∗ ∈ P, we obtain E∗[X] ≥ γ(P ∗), hence X ∈ A∗.
52
(b)Next, take X ∈ A∗ and assume that X /∈ A1. This means that the vector
x∗ = (x∗1, ..., x∗N )
with componentsx∗i := EQi
[X]− γ(Qi)
does not belong to the convex cone
C := (EQi [ξ · Y ])i=1,...,N + y | ξ ∈ IRd, y ∈ IRN+ ⊂ IRN ,
where Q = Q0 ∪Q1 = Q1, ..., QN with N ≥ n. Should x∗ belong to C, we could write
EQi [X]− γ(Qi) = EQi [ξ · Y ] + yi
which impliesEQi [X + (−ξ) · Y ]− γ(Qi) ≥ 0.
Thus X ∈ A1, which is a contradiction.We will in part (c) of this proof show that C is closed, but we will use this fact
now. Since C is closed and convex, the set C′ := c− x∗ | c ∈ C is also closed, convex,does not contain 0. Thus, we can apply by Lemma 2.1.6 which states that there existsλ ∈ IRN \ 0 such that
infx∈C
λ · (x− x∗) > 0.
We know that C ⊃ IRN+ , and hence,
infx∈C
λ · x > λ · x∗ (2.13)
can only hold for λi ≥ 0 for i = 1, ..., N , otherwise λi < 0 for some i would lead to
infx∈C
λ · x = −∞.
the term λi · xi → −∞. We may assume that∑N
i=1 λi = 1 since λ 6= 0. Define
R :=N∑
i=1
λiQi ∈ R1.
Since C contains the linear space of vectors (EQi [V ])i=1,...,N with V ∈ K, (2.13) implies
ER[X]− γ(R) < infx∈C
λ · x = infξ∈RN
ER[ξ · Y ].
Assume that ER[ξ · Y ] > 0 for some ξ. Then we could find a constant 0 > k ∈ IR so thatfor ξ′ = k · ξ the first inequality does not hold. The same reasoning goes for the caseER[ξ′ · Y ] < 0 with a constant 0 < k ∈ IR and thus
ER[V ] = 0 for V ∈ K,
53
hence R ∈ P. Moreover we have that
ER[X]− γ(R) < infx∈C
λ · x = infξ∈RN
ER[ξ · Y ] = ER[ξ′ · Y ] = 0
thus, ER[X]− γ(R) < 0 contradicting our assumption X ∈ A∗.(c) To show that C is closed, define for ξ ∈ IRd the vector y(ξ) in IRN with the ith
componentyi(ξ) = EQi
[ξ · Y ].
Any x ∈ C admits a representation
x = y(ξ) + z
with z ∈ IRN+ and ξ ∈ N⊥ where
N := η ∈ IRd | EQi [η · Y ] = 0 for i = 1, ..., N,
andN⊥ := ξ ∈ IRd | ξ · η = 0 for η ∈ N,
since for any ψ ∈ IRN
EQi [ψ · Y ] + zi = EQi [(ξ + η) · Y ] + zi = EQi [ξ · Y ] + zi.
Take a sequencexn = y(ξn) + zn, n = 1, 2, ...,
with ξn ∈ N⊥ and zn ∈ IRN+ , such that xn converges to x ∈ IRN . If lim infn | ξn |< ∞,
then we may assume, passing to a subsequence if necessary, that ξn converges to ξ ∈ IRd.In this case, zn must converge to some z ∈ IRN
+ , and we have x = y(ξ) + z ∈ C. Now,show that the case limn | ξn |= ∞ is in fact excluded. In that case, αn := (1 + |ξn|)−1
converges to 0, and the vectors ζn := αnξn stay bounded. Thus, we can assume that ζn
converges to ζ ∈ N⊥. Thus
0 = limn↑∞
(αn · xn) = limn↑∞
(αn · y(ξn)
)+ lim
n↑∞(αn · zn)
andlimn↑∞
(αn · y(ξn)
)= lim
n↑∞(y(ζn)
)= − lim
n↑∞(αn · zn) ∈ −RN
+ .
Since ζ ∈ N⊥ and | ζ |= limn | ζ |= limn | ζn |= 1, we see that y(ζ) 6= 0. To conclude,y(ζ) ∈ −IRN
+ and y(ζ) 6= 0 thus, the inequality
EQi [(−ζ) · Y ] = −yi(ζ) ≥ 0
holds for every i and is strict for some i, in contradiction to our assumption (2.3) whichstates that if EQi [(−ζ) · Y ] ≥ 0, then the inequality degenerates to equality.
♦
54
Chapter 3 - V@R and AV@R
So far, we have defined the most common risk measures based on percentiles, i.e.,V @R and AV @R. Moreover, we have thoroughly investigated continuity properties ofrisk measures. In this chapter, we will return to the risk measures V @R and AV @R.We are now, however, more prepared to take on the discussion of the relation betweenV @R and AV @R. In particular, we show that there exists no smallest convex riskmeasure, continuous from above, which dominates V @R. However, after defining theproperty of distribution invariant, we show that among all the convex, continuous fromabove, risk measures satisfying this property, there exists one smallest risk measurewhich dominates V @R. It turns out that this risk measure is AV @R.
3.1 - V@R and AV@RIt is easy to find examples where V @R behaves as a coherent risk measure. However,
this is not the general case. For two independent random variables X1, X2 ∈ Exp(1),we can easily check that
−3 = V @R0.80(X1 + X2) ≥ V @R0.80(X1) + V @R0.80(X2) = −3.2.
Thus, V @R fails, in general, to satisfy subadditivity and as a result it does not alwaysencourage diversification. For more on the criticism on V @R, turn to [1]. The aboveexample highlights one of the reasons why we are interested in finding a convex riskmeasure that comes close to V @R.
The following proposition states that for a given X ∈ X we can always find a riskmeasure, continuous from above, which dominates V @Rλ(·). In particular, this can bechosen as a coherent risk measure. The reason for wanting the risk measure to satisfythe continuous-from-above property is obvious; we are interested in a representation ofthe convex risk measure in terms of probability measures. See Lemma 2.2.2.
Proposition 3.1.1.For each X ∈ X , and each λ ∈ (0, 1),
V @Rλ(X) = minρ(X) | ρ is continuous from above and dominates V @Rλ(·).
Proof. For a given X ∈ X , we will show that it is always possible to constructthe coherent risk measure ρX so that V @R coincides with ρX at X but ρX domi-nates V @R in all other positions. To this end, let q := −V @Rλ(X) = q+
λ (X). Thenfrom the definition of V @R we have P [X < q] ≤ λ. If A ∈ F satisfies P [A] > λ, thenP [A ∩ X ≥ q] > 0 since P [X ≥ q] ≥ 1− λ. Thus, we can define a measure QA by
QA := P [ · | A ∩ X ≥ q ].
It follows that EQA [−X] ≤ −q = V @Rλ(X). Let Q := QA | P [A] > λ and define acoherent measure of risk ρX via (1.11), i.e.,
ρX(Y ) := supQ∈Q
EQ[−Y ].
55
Since Q ⊂M1(P ), Corollary 2.3.3 states that ρX is continuous from above and theconstruction of Q implies ρX(X) ≤ V @Rλ(X). Hence, the only thing we have to do toconclude the proof is to show that ρX(Y ) ≥ V @Rλ(Y ) for each Y ∈ X . For ε > 0, wedefine A := Y ≤ −V @Rλ(Y ) + ε. Thus, P [A] > λ and consequently QA ∈ Q. Nowwe have that
ρX(Y ) ≥ EQA[−Y ] ≥ V @Rλ(Y )− ε
since QA[A] = 1. Since we can choose ε > 0 arbitrarily, we have thus constructed a coher-ent risk measure, satisfying the property of being continuous from above and dominatingV @Rλ(·) with equality for the position X.
♦The preceding proposition has an interesting implication. It rejects any belief
that there exists a smallest convex risk measure, which is continuous from above anddominates V @R. We state this result in the following corollary.
Corollary 3.1.2.There exists no smallest convex risk measure, and continuous from above, which
dominates V @Rλ.
Proof. Choose any convex risk measure, ρ, continuous from above, which dom-inates V @R. Since ρ and V @R are not identical, there exists some X ∈ X such thatρ(X) > V @Rλ(X). Let ρX be as the risk measure in the proof of Proposition 3.1.1, i.e.,convex, continuous from above, V @R-dominant and satisfies
ρX(X) = V @Rλ(X).
Thus, ρ(X) > ρX(X). Let ρ = ρ ∧ ρX . It is not difficult to see that ρ < ρ and ρ is aconvex risk measure, continuous from above , which dominates V @Rλ.
♦The result in Corollary 3.1.2 provokes the search for an extra property such that
we can find a distinct, smallest convex risk measure which dominates V @R.
In Chapter 1, we defined AV @R and said that it is a coherent risk measure. More-over, in Proposition 1.5.1 we gave a representation for all coherent risk measures. Weare now equipped with all the necessary tools to complete the proof of the claim thatAV @R is a coherent risk measure. This will be done by showing that AV @R takes onthe representation (1.11).
56
Theorem 3.1.3.For λ ∈ (0, 1), AV @Rλ is a coherent measure of risk which is continuous from
below. It has the representation
AV @Rλ(X) = maxQ∈Qλ
EQ[−X], X ∈ X , (3.1)
where Qλ is the set of all probability measures Q ¿ P whose density dQ/dP is P -a.s.bounded by 1/λ. Moreover Qλ is equal to the maximal set Qmax of Corollary 2.3.3.
Proof. Our first goal is to show that
AV @Rλ(X) ≥ EQ′ [−X],
for all Q′ ∈ Qλ. We then construct a probability measure, Q ∈ Qλ, such that for thisparticular Q, the equality is attained, i.e.,
AV @Rλ(X) = EQ[−X].
Take Q′ ∈ Qλ and denote the density dQ′/dP by ϕ. Let q be a λ-quantile of X, then
AV @Rλ(X)− EQ′ [−X] =EQ′ [X]− 1λ
(E[X : X ≤ q]− q · (P [X ≤ q]− λ)
)
=1λ
E[λϕX −XIX≤q + qIX≤q − qλE[ϕ]
]
=1λ
E[(X − q)(λϕ− IX≤q)],
where the first equality follows from (1.9). Since 0 ≤ λϕ ≤ 1, we conclude that
(X − q)(λϕ− IX≤q) ≥ 0,
and hence AV @Rλ(X) ≥ EQ[−X].We now construct Q ∈ Qλ. Choose a λ-quantile q of X and define
κ := 0, if P [X = q] = 0
λ−P [X<q]P [X=q] , otherwise.
Then 0 ≤ κ ≤ 1, and put
ϕ : =1λ
(IX<q + κIX=q)
=1λ
(IX<q + λ− P [X < q])
≤ 1λ
57
satisfies E[ϕ] = EQ[1] = 1λ (P (X < q) + κP (X = q)) = 1. Thus, dQ := ϕdP defines a
probability measure Q ∈ Qλ, for which
EQ[−X] =1λ
(E[−X : X < q]− qλ + qP [X < q]).
By (1.9), the right-hand side equals AV @Rλ(X), and we obtain the identity (3.1). Thus,AV @Rλ is a coherent measure of risk. Moreover, by Corollary 2.3.4 it is also continuousfrom below.
We now have to show that Qλ is the maximal set of Corollary 2.3.3. It amounts toshowing that for a given Q /∈ Qλ
αmin(Q) = supX∈X
(EQ[−X]− ρ(X)) = supX∈X
(EQ[−X]−AV @Rλ(X)) = +∞.
Let us denote by ϕ the density dQ/dP . There exists λ′ ∈ (0, λ) and k > 1/λ′ such thatP [ϕ ∧ k >= 1/λ′] > 0. For c > 0 define X(c) ∈ X by
X(c) := −c · (ϕ ∧ k) · Iϕ≥1/λ′.
Since,
P [X(c) < 0] =P [ϕ ≥ 1λ′
] = E[Iϕ≥ 1λ′ ]
=∫
Iϕ≥ 1λ′ dP ≤
∫ϕλ′Iϕ≥ 1
λ′ dP
=∫
Ω
ϕλ′dP ≤ λ′ ·∫
Ω
ϕdP
=λ′ · E[ϕ] = λ′
<λ,
we have that(i) P [X(c) < 0] < λ
(ii) P [X(c) ≤ 0] = 1 ≥ λ.
Thus, (i) and (ii) imply V @Rλ(X(c)) = 0. This together with (1.7), where we chooseq = −V @Rλ(X(c)) = 0, give us
AV @Rλ(X(c)) =1λ
E[−X(c)]
=c
λE[ϕ ∧ k;ϕ ≥ 1
λ′].
On the other hand,
EQ[−X(c)] =c · E[ϕ · ϕ ∧ k; ϕ ≥ 1λ′
]
≥ c
λ′E[ϕ ∧ k; ϕ ≥ 1
λ′].
Thus, the difference between EQ[−X(c)] and AV @Rλ(X(c)) becomes arbitrarily largeas c ↑ ∞, i.e.,
αmin(Q) = supX∈X
(EQ[−X]− ρ(X)
)= sup
X∈X
(EQ[−X]−AV @Rλ(X)
)= +∞.
58
♦
For A ∈ F with P [A] > λ, we construct the probability measure
QA[ · ] = P [ · | A].
Let Q denote the set of all QA. With the help of Q we now define a third risk measure,Worst Conditional Expectation (WCE).
Definition 3.1.4.We define the worst conditional expectation at level λ by
WCEλ(X) := supQA∈Q
EQ[−X].
♦We will soon find out that WCEλ and AV @Rλ have more in common than the
coherent properties.
Example 3.1.5.Let us derive the WCE at level 0.05 for the random variable X ∈ Exp(1).
WCE0.05(X) := supE[−X | A] | A ∈ F , P [A] > 0.05=E[−X | X ≤ q+
0.05(X)]
=1
0.05E[(q+
0.05(X)−X)+]− q+0.05(X)
=AV @R0.05(X).
To see the second equality, note that since X is positive the only set A of interest is theone for the outcomes where X is as small as possible. Thus,
A = ω ∈ Ω | X(ω) ≤ q+0.05(X).
♦The next corollary shows that WCE and AV @R coincide if the considered random
variable is continuous.
59
Corollary 3.1.6.For all X ∈ X ,
AV @Rλ(X) ≥WCEλ(X)≥E[−X | −X ≥ V @Rλ(X)]≥V @Rλ(X).
Moreover, the first two inequalities are in fact identities if
P [X ≤ q+λ (X)] = λ. (3.2)
This will be the case when X has a continuous distribution.
Proof. We have that WCEλ(X) = supE[−X | A] | A ∈ F , P [A] > λ. To showthe first inequality, let Q be the measure such that Q[B] = P [B | A]. To complete thispart of the proof, we only need to show that every such Q is included in the set Qλ ofTheorem 3.1.3. To do this, it is sufficient to show that ϕ := dQ
dP ≤ 1λ . For any B ∈ Ω
∫
B
ϕdP =∫
ϕIBdP =∫
IBdQ = EQ[IB ] = Q[B] = P [B | A] =P [B ∩A]
P [A]≤ P [B]
λ.
Put C := ϕ > 1λ and show that P [C] = 0, P -a.s. We see that
C =⋃n
Cn where Cn = ϕ ≥ 1λ
+1n.
It is sufficient to show that P [Cn] = 0 :
(1λ
+1n
)P [Cn] ≤∫
Cn
(1λ
+1n
)dP ≤∫
Cn
ϕdP ≤ P [Cn]λ
Hence, 1nP [Cn] ≤ 0 which forces P [Cn] = 0.
To show the second inequality, we let
A := −X ≥ V @Rλ(X)− ε.
Then,P [A] = P [−X ≥ V @Rλ(X)− ε] > λ
andWCEλ(X) ≥ E[−X | −X ≥ V @Rλ(X)− ε].
The second inequality follows by taking the limit ε ↓ 0.The third inequality is obvious. Moreover, with (1.9) we can write
AV @Rλ(X) =1λ
(E[−X;X ≤ q]− qλ + qP [X < q]) = E[−X | −X ≥ V @Rλ(X)]
60
as soon as (3.2) holds.♦
Remark 3.1.7.In [10], the expression
E[−X | −X ≥ V @Rλ(X)] = E[−X | X ≤ q+λ X)]
is called upper Tail Conditional Expectation (TCEλ(X)) and is another risk measurementioned in the literature. If we choose q−λ (X) instead of q+
λ (X), then we writeTCEλ(X) and consequently call the measure lower Tail Conditional Expectation. Atpage 4 of [10], it is stated that WCE was introduced partly because TCE in generalfails to satisfy subadditivity.
♦
We now define the so-called distribution invariant property.
Definition 3.1.8.A monetary measure of risk ρ on L∞(Ω,F , P ) is called distribution invariant if
ρ(X) = ρ(Y )
whenever X and Y have the same distribution under P .♦
Note that V @R, AV @R and WCE are distribution invariant risk measures.
Definition 3.1.9.A set A ∈ F is called an atom of (Ω,F , P ) if P [A] > 0 and if each B ∈ F with
B ⊆ A satisfies either P [B] = 0 or P [B] = P [A].♦
It is rather intuitive that, if (Ω,F , P ) is atomless then, for every A ∈ F and allδ with 0 ≤ δ ≤ P [A], there exists a measurable set B ⊂ A such that P [B] = δ. See,Proposition 4.44 in [4].
Next, we prove some equivalent relations for a atomless space (Ω,F , P ).
Proposition 3.1.10.For any probability space, the following conditions are equivalent.(a) (Ω,F , P ) is atomless, i.e., it does not contain atoms.(b) There exists an i.i.d. sequence X1, X2, ... of random variables with Bernoulli
distribution:P [X1 = 1] = P [X1 = 0] =
12.
(c) For any µ ∈M1(IR) there exists an i.i.d. sequence Y1, Y2, ... with commondistribution µ
(d) (Ω,F , P ) supports a random variable with a continuous distribution.
61
Proof. (a) ⇒ (b). We will do this part by an induction argument. First take anarbitrary set A ∈ F such that P [A] = 1
2 and define X1 := 1 on A and X1 := 0 on Ac.Now, assume that we already have constructed independent random variables X1, ..., Xn,such that
P [Xi = 0] = P [Xi = 1] =12
for all i = 1, ..., n.
To complete the induction argument we have to show that we can construct Xn+1
with the same properties as the already constructed random variables. To this end, wepartition Ω into 2n subsets, B1, ..., B2n where every set Bi correspond to an event
(xi1 , ..., xin) with xij
= 0 or 1
of a unique sequence. Next, we simply divide every set Bi into two disjoint sets, B1i
and B2i such that P [B1
i ] = P [B2i ]. Thus,
P [∪n1B1
i ] = P [∪n1B2
i ] =12.
If we define Xn+1 = 1 on ∪n1B1
i and Xn+1 = 0 on ∪n1B2
i we will have a Bernoulli dis-tributed random variable. Moreover, X1, ..., Xn+1, will be an i.i.d. sequence.
(b) ⇒ (c). By relabeling the sequence X1, X2, ... we can obtain the sequence(Xi,j)i,j∈IN of independent Bernoulli-distributed random variables. We construct a se-quence consisting of uniformly distributed random variables, U1, U2..., by defining
Ui :=∞∑
n=1
2−nXi,n.
We denote by F and q(s) := infx ∈ IR | F (x) > s, the distribution function of µ andits right-continuous inverse. Lemma 1.3.5 implies that the i.i.d. sequence
Yi := q(Ui), i = 1, 2, ...
has common distribution µ.(c) ⇒ (d) and (d) ⇒ (a) are straightforward.
♦
The following lemma implies that we can lower our risk by replacing X with itsexpectation on a set A ∈ F . This could be seen as a “local” hedge where we choose totake away the randomness for some events A. Alternatively, we can view X as a cashflow that we will receive at a certain time. In view of the lemma below we will lowerour risk if we sign a swap contract so that we get the expected value of the cash flowinstead of the stochastic cash flow.
Lemma 3.1.11.Assume that ρ is a convex risk measure with ρ(0) = 0, continuous from above and
distribution invariant, on an atomless probability space. For X ∈ X and for any A ∈ Fwith P [A] > 0,
ρ(X) ≥ ρ(XIAc + E[X | A]IA).
62
In particular,ρ(X) ≥ ρ(E[X]) = −E[X].
Proof. Denote by µ the distribution of X under P [ · | A]. Since (Ω,F , P ) is atom-less, by Proposition 3.1.10, it supports the random variables X, I
A, and an i.i.d. se-
quence (Zn) with the following properties: (X, IA) has the same distribution as (X, IA),
and each Zn is independent of (X, IA) and has distribution µ under P. Then,
Xn := X · IAc + Zn · IA
has the same distribution under P as X. Kolmogorov’s law of large numbers, appliedto the sequence (Zn), yields that
1n
n∑
k=1
Xk −→ X · IAc + E[X | A] · I
A=: Y, P -a.s. as n ↑ ∞.
Hence, using Theorem 2.2.10 in the first step, convexity in the second, and distributioninvariance in the third,
ρ(Y ) ≤ lim infn↑∞
ρ
(1n
n∑
k=1
Xk
)≤ lim inf
n↑∞1n
n∑
k=1
ρ(Xk) = ρ(X).
Finally,ρ(Y ) = ρ(XIAc + E[X | A]IA)
by distribution invariance.♦
We are now ready for the main result of this chapter, namely to show that AV @Rλ
is the smallest, distribution-invariant, convex risk measure continuous from above, thatdominates V @Rλ. The fact that AV @Rλ is not only convex but also coherent couldraise some questions whether this has something to do with the penalty function and theproperty distribution invariant. However, the reason that it happens to be a coherentrisk measure lies in the fact that V @Rλ is positively homogeneous.
Theorem 3.1.12.On an atomless probability space, AV @Rλ is the smallest distribution invariant
convex measure of risk which is continuous from above and dominates V @Rλ.
Proof. That AV @Rλ dominates V @Rλ is already clear from Corollary 3.1.6. Tofinish the proof, we only have to show that, for every distribution invariant convex riskmeasure, ρ, which dominates V @Rλ and which is continuous from above,
ρ(X) ≥ AV @Rλ(X), (3.3)
for every X ∈ X . To this end, we take ε > 0, and define the set
A := −X ≥ V @Rλ(X)− ε
63
and the random variable
Y := X · IAc + E[X | A] · IA.
SinceY = X · IAc + E[X | A] · IA > q+
λ (X) + ε ≥ E[X | A]
on Ac, we get P[Y < E[X | A]
]= 0. On the other hand,
P[Y ≤ E[X | A]
] ≥ P [A] > λ
where the second inequality follows from the fact that
P [A] = P [X ≤ −V @Rλ(Y )] > λ.
Next,P
[Y ≤ E[X | A]
]> λ
andP
[Y < E[X | A]
]= 0
together imply that V @Rλ(Y ) = E[−X | A]. Since we assume that ρ dominates V @Rλ,we have ρ(Y ) ≥ E[−X | A]. Thus,
ρ(X) ≥ E[−X | −X ≥ V @Rλ(X)− ε],
since ρ(X) ≥ ρ(Y ) by Lemma 3.1.11. Taking ε ↓ 0 we get
ρ(X) ≥ E[−X | −X ≥ V @Rλ(X)].
In the case when X is a continuous random variable, we can apply Corollary 3.1.6 whichstates the right-hand side equals AV @Rλ(X). Thus, we have (3.3). If the distributionof X is not continuous, we approximate X P -a.s. by a decreasing sequence (Yn) ⊂ Xsuch that each Yn has a continuous distribution. The relation (3.3) holds for each Yn
and extends to X by continuity from above. To construct the sequence (Yn), we mayassume, in view of Proposition 3.1.10, that there exists a random variable U which isindependent of X and uniformly distributed on (0,1). The distribution of
Yn := X +1n
U
is absolutely continuous. Moreover, Yn decrease to X P -a.s. when n goes to infinity.♦
We said earlier in this chapter that, if X ∈ X is continuous then
AV @Rλ(X) = WCEλ(X).
The next corollary gives a stronger condition for the equality to be attained betweenAV @Rλ and WCEλ. This condition lies in the probability space and not in the indi-vidual random variable.
Corollary 3.1.13.AV @Rλ and WCEλ coincide if the probability space is atomless.
Proof. We know from Corollary 3.1.6 that WCEλ(X) = AV @Rλ(X) if X hasa continuous distribution. Repeating the approximation argument at the end of thepreceding proof, we obtain WCEλ(X) = AV @Rλ(X) for each X ∈ X .
♦
64
Bibliography[1] Artzner, P., Delbaen, F., Eber, J.-M.,Health, D., Coherent Measures of Risk, Math.
Finance 9(1999), 203-228.[2] Butler, C., Mastering Value at Risk - A step by step guide to understand and
applying VaR. Finansial Times Prentice Hall, 1st edition, 1999.[3] Follmer, H.,Schied, A., Convex measures of risk and trading constraints, Finance
Stoch. 6(4)(2002).[4] Follmer, H., Schied A., Stochastic Finance, An Introduction in Discrete Time, De
Gruyter Stud. Math. 27, Walter de Gruyter, Berlin-New York, 2002.[5] Huber, P., Robust Statistics, Wiley Ser. Probab. Math Statist., Wiley, New York,
1981.[6] Hull, J., Options, futures and other derivatives, 5th edition. Prentice Hall, London,
2002.[7] Jorion, P., Value at Risk - The New Benchmark for Managing Financial Risk,
McGraw-Hill, 2nd edition, 2000.[8] Rudin, W., Real and Complex Analysis, McGraw-Hill, 3rd edition, 1987.[9] Royden, H.L., Real Analysis, Macmillan, New York, 1968.
[10] Tasche, D., Acerbi, C., On the coherence of expected shortfall. To appear in Journalof Banking and Finance 26(7), 2002.
65
