Cherny 2006

Finance and Stoch (2006) 10: 367–393
DOI 10.1007/s00780-006-0009-1
O R I G I NA L PA P E R
A. S. Cherny
Weighted V@R and its properties
Received: 21 July 2005 / Accepted: 14 December 2005 / Published online: 21 April 2006
© Springer-Verlag 2006
Abstract The paper deals with the study of a coherent risk measure, which we
call Weighted V@R. It is a risk measure of the form
ρµ (X ) =
TV@Rλ (X )µ(dλ),
[0,1]
where µ is a probability measure on [0, 1] and TV@R stands for Tail V@R. After
investigating some basic properties of this risk measure, we apply the obtained
results to the financial problems of pricing, optimization, and capital allocation. It
turns out that, under some regularity conditions on µ, Weighted V@R possesses
some nice properties that are not shared by Tail V@R. To put it briefly, Weighted
V@R is “smoother” than Tail V@R. This allows one to say that Weighted V@R is
one of the most important classes (or maybe the most important class) of coherent
risk measures.
Keywords Capital allocation · Coherent risk measures · Determining set ·
Distorted measures · Minimal extreme measure · No-good-deals pricing · Spectral
risk measures · Strict diversification · Tail V@R · Weighted V@R
Mathematics Subject Classification (2000) 91B16 · 91B30
JEL Classification G10 · G11 · G12
A. S. Cherny
Faculty of Mechanics and Mathematics, Department of Probability Theory,
Moscow State University, 119992 Moscow, Russia
E-mail: [email protected]
Tail Var = ES
368
A. S. Cherny
1 Introduction
Historic overview. The theory of coherent risk measures is a very new, important,
and rapidly evolving branch of modern financial mathematics. This concept was
introduced by Artzner et al. [4,5]. Since then, many papers on the topic have followed; surveys of the modern state of the theory are given in [19], [22, chapter 4],
and [30]. In some sources, the theory of coherent risk measures and related topics
is already called the “third revolution in finance” (see [33]).
A very important class of coherent risk measures is given by Tail V@R (the L^\infty
terms Average V@R, Conditional V@R, and Expected Shortfall are also used). Tail
V@R of order λ ∈ [0, 1] is a map ρλ : L ∞ → R (we have a fixed probability
space (Ω, F , P)) defined by
A Importancia do Tail Var
ρλ (X ) = − inf E X,
Q∈Dλ
Q
Riqueza
where Dλ is the set of probability measures Q that are absolutely continuous with descontada
respect to P with ddQ
≤ λ−1 . (From the financial point of view, X is the discounted
P
P & L of some financial transaction.) The importance of Tail V@R is seen from a
result of Kusuoka [25], who proved that ρλ is the smallest law invariant coherent
risk measure that dominates V@Rλ (we recall the precise formulation in section 2).
This suggests that Tail V@R is one of the best coherent risk measures. For more
information on Tail V@R, see [3], [18, section 6], [19, section 7], [22, section 4.4],
[30, section 1.3].
However, there exists a risk measure, which is, in our opinion, much better than
Tail V@R. It is given by
ρµ (X ) =
ρλ (X )µ(dλ),
(1.1)
[0,1]
where µ is a probability measure on [0, 1]. We call this risk measure Weighted
V@R and its study is the goal of this paper. First of all, let us give two arguments
in favor of Weighted V@R over Tail V@R:
• (Financial argument) Tail V@R of order λ takes into consideration only the
λ-tail of the distribution of X ; thus, two distributions with the same λ-tail will
be assessed by this measure in the same way, although one of them might
clearly be better than the other (see Figure 1). On the other hand, if the right
endpoint of supp µ is 1, then ρµ depends on the whole distribution of X .
Por isso que ao usar o W VaR,
conseguimos ter uma ideia
melhor da distribuicao como um
todo.
Na ES olhamos ate um determinado
quantil \lambda. Ja nessa W VaR,
iremos `somar todos os quantis no
intervalo [0,1]`.
Fig. 1 These two distributions have the same λ-tails (qλ is the λ-quantile), so that TV@Rλ coincides on them. However, the distribution on the right stochastically dominates the distribution
on the left
Weighted V@R and its properties
369
• (Mathematical argument) If the weighting measure µ satisfies the condition
supp µ = [0, 1], then ρµ possesses some nice properties that are not shared
by ρλ . In particular, various optimization problems have a unique solution
(see section 5).
The paper [21] provides some further financial arguments in favor of Weighted
V@R.
It might seem rather surprising that the risk measure, which we call here
Weighted V@R, was considered by actuaries already in the early 1990s, i.e., before the papers of Artzner et al.; see, for example, [20,34] (see also the paper [35],
which appeared at the same time as [4]). These papers are related to an object
termed distorted measure. This is a functional on random variables defined as
0
ρ(X ) =
∞
Ψ (F(x))dx + (Ψ (F(x)) − 1)dx,
−∞
0
(1.2)
Relacao do W VaR com as
medidas distorcidas.
where Ψ : [0, 1] → [0, 1] is an increasing concave function with the properties
Ψ (0) = 0, Ψ (1) = 1, and F is the distribution function of X . It turns out that the
class of these functionals (with different Ψ ) is exactly the class of Weighted V@Rs
(with different µ). For risk measures on L ∞ , this equivalence can be found in [22,
Theorem 4.64] or [30, Theorem 1.51]; for risk measures on L 0 , this equivalence is
proved in section 3 of this paper.
The first appearance of ρµ in the framework of coherent risk measures is in the
paper of Kusuoka [25]. He proved that any law invariant comonotonic coherent
risk measure is of this form. In the same paper, he proved that any law invariant
risk measure has the form supµ∈M ρµ , where M is a set of probability measures
on [0, 1] (we recall the precise formulations in section 2).
Some further considerations of ρµ can be found in the papers of Acerbi [1,2],
who uses the term spectral risk measures for this class.
Furthermore, Carlier and Dana [11] provided a representation of the determining set of Weighted V@R (the definition of this notion is given below). This
Discussão interessante
representation is recalled in section 4.
sobre os espaços
Structure of the paper. According to the classical definition, a coherent risk
measure is defined on bounded random variables. However, for financial applications, it is almost necessary to extend this notion to the space of all random
variables. Indeed, most distributions used in theory (for example, the lognormal
one) are unbounded. In this paper, we consider coherent risk measures defined on
the space L 0 of all random variables. This is done as follows. According to the
basic representation theorem, a functional ρ : L ∞ → R is a coherent risk measure
Fatou Property é usada quando o
if and only if it admits a representation
meu espaço amostral é arbitrário.
ρ(X ) = − inf EQ X,
Q∈D
X ∈ L∞
(1.3)
with some set D of probability measures that are absolutely continuous with respect
to P. For finite Ω, this was proved in [5]; for arbitrary Ω, this was proved by
Delbaen [18]; in the latter case, an additional continuity assumption called the
Fatou property should be imposed on ρ. Following [15,16], we take the representation (1.3) with L ∞ replaced by L 0 as the definition of a coherent risk measure
370
A. S. Cherny
on L 0 . The expectation EQ X is understood as EQ X + − EQ X − with the convention
∞ − ∞ = −∞ so that EQ X is well defined for any Q and X . Thus, ρ takes on
values in the extended real line [−∞, ∞].
Section 2 contains some basic definitions as well as known results on Tail V@R
and Weighted V@R.
In section 3, we provide two representations of Weighted V@R. These are the
extensions of (1.1) and (1.2) to L 0 .
Clearly, different sets D might define the same coherent risk measure. However, among all the sets that define the same risk measure ρ, there exists a largest
one (it has the form D = {Q P : EQ X ≥ −ρ(X ) for any X }). We call it the
determining set of ρ. Finding the structure of this set is important. For example,
in [15, section 2], we considered the no-good-deals pricing technique based on
coherent risk measures. According to this technique, the interval of fair prices of
a contingent claim F is {EQ F : Q ∈ D ∩ R}, where R is the set of risk-neutral
measures and D is the determining set of a risk measure ρ, which serves as an input
to this technique. Furthermore, the solutions of various other financial problems
given in [15,16] are expressed through the determining set. In section 4, we provide
two representations of the determining set of Weighted V@R, both of which are
given in different terms than the one in [11].
The main result of section 5 is that
ρµ (X + Y ) < ρµ (X ) + ρµ (Y )
provided that supp µ = [0, 1] and X , Y are not comonotone (in particular, the
latter condition is satisfied if the distribution of (X, Y ) has a joint density). We
call this the strict diversification property. This property is very important from
the viewpoint of financial mathematics because it leads to the uniqueness of the
solutions of various optimization problems based on the coherent risk measures.
In [15], we introduced the notion of an extreme measure. The class of extreme
measures for a coherent risk measure ρ and a random variable X is defined as
Xρ (X ) = Q ∈ D : EQ X = inf EQ X ∈ (−∞, ∞) ,
Q ∈D
where D is the determining set of ρ. This notion was found to be very convenient
and important. In particular, the solutions of several optimality pricing problems,
the solution of the equilibrium pricing problem, and the solution of the capital allocation problem are expressed through extreme measures (see [15,16]). Moreover,
the risk contribution introduced in [15] is expressed through extreme measures.
In general, the set Xρ (X ) can contain more than one point. However, as shown
in section 6, for ρ = ρµ with µ({0}) = 0, there exists a unique element of Xρ
that is the smallest in the convex order. We call it the minimal extreme measure.
This notion is of importance for financial mathematics as it allows one to select a
(unique) distinguished solution of the problems like capital allocation or optimality
pricing, which possesses some nice properties. We call it the central solution.
One of the most important goals of modern financial mathematics is to narrow the no-arbitrage price intervals of contingent claims as they are known to
be unacceptably wide in most incomplete models. Several ways to do that have
Weighted V@R and its properties
371
been proposed in the literature. One of them consists in considering actively traded
derivatives as basic assets. In particular, a popular model is based on treating as
basic assets the European call options on a fixed asset with a fixed maturity and
different strike prices. The corresponding model was first studied by Breeden and
Litzenberger [9] and Banz and Miller [7]. A literature review of this model is given
in [23]. Let us also mention the paper [14, section 6], in which this model was
analyzed from the general viewpoint of fundamental theorems of asset pricing.
Recently, another (very promising) way to narrow fair price intervals has been
proposed. It is known as no-good-deals pricing. This technique was first considered by Cochrane and Saá-Requejo [17] and Bernardo and Ledoit [8]. An important feature of this theory is that there exists no canonical definition of a good
deal (in particular, [17] and [8] employ different definitions). Carr et al. [12] (see
also the review paper [13]) and Jaschke and Küchler [24] proposed variants of
no-good-deals pricing based on coherent risk measures. These techniques were
further developed in [15] and [32].
In section 7, we combine the two ways of narrowing fair price intervals described above. Namely, we apply the no-good-deals pricing technique from [15] to
the model with European options as basic assets. This leads to a “double reduction”
of fair price intervals. The risk measure employed is Weighted V@R. In fact, the
results of section 7 provide a description of risk-neutral densities that correctly
price traded call options and are “not far” from the real-world density, the “distance” being measured with the help of a coherent risk measure. Let us remark
that the study of the interplay between risk-neutral and real-world densities is a
very popular topic of modern financial mathematics (see, in particular, the papers
[6,10,26]).
Section 8 deals with a Markowitz-type optimization problem of the form
EP X −→ max,
X ∈ A, ρ(X ) ≤ c,
(1.4)
where ρ is a coherent risk measure, c ≥ 0, and A is the set of possible discounted
P & Ls that can be obtained by using various trading strategies in the model under
consideration. Let us remark that in [27], Markowitz proposed to consider an
alternative of his mean-variance optimization problem with variance replaced by
semivariance S(X ) = E((X − EX )− )2 (clearly, variance is not a good measure
of risk because it penalizes profits in the same way as losses). In (1.4), the risk is
measured in a coherent way. Let us remark that problems of type (1.4) were considered in [2,16,28,29]. Here we provide a solution of (1.4) for the case, where ρ
is Weighted V@R and the model is complete, i.e., A = {X ∈ L 1 (Q) : EQ X = 0},
where Q is a given measure (the unique risk-neutral measure). Classical examples
of a complete model are the Black-Scholes model and the Cox–Ross–Rubinstein
model. One more example is the option-based model considered in section 7 (if
we assume that the options on a basic asset with a fixed maturity and all the
positive strikes are traded, then this model is complete). Our solution shows that
the
optimalstrategy typically consists in buying binary options with the payoff
dQ
I dP ≤ c∗ , where c∗ is the optimal threshold explicitly calculated in section 8
(see Figure 5).
372
A. S. Cherny
2 Basic definitions
Let (Ω, F , P) be a probability space. It will be convenient for us to deal not with
coherent risk measures, but with their opposites called coherent utility functions
(this enables one to get rid of numerous minus signs).
Definition 2.1 A coherent utility function on L ∞ is a map u : L ∞ → R with the
properties:
(a) (Superadditivity) u(X + Y ) ≥ u(X ) + u(Y );
(b) (Monotonicity) If X ≤ Y , then u(X ) ≤ u(Y );
(c) (Positive homogeneity) u(λX ) = λu(X ) for λ ∈ R+ ;
(d) (Translation invariance) u(X + m) = u(X ) + m for m ∈ R;
P
→ X , then u(X ) ≥ lim supn u(X n ).
(e) (Fatou property) If |X n | ≤ 1, X n −
The corresponding coherent risk measure is ρ(X ) = −u(X ).
The theorem below was established in [5] for the case of a finite Ω (in this case
the axiom (e) is not needed) and in [18] for the general case. We denote by P the
set of probability measures on F that are absolutely continuous with respect to P.
Throughout the paper, we identify measures from P (these are typically denoted
by Q) with their densities with respect to P (these are typically denoted by Z ).
Theorem 2.2 A function u satisfies conditions (a)–(e) if and only if there exists a
nonempty set D ⊆ P such that
u(X ) = inf EQ X, X ∈ L ∞ .
Q∈D
(2.1)
Now, we use the representation (2.1) to extend coherent utility functions to the
space L 0 of all random variables.
Definition 2.3 A coherent utility function on L 0 is a map u : L 0 → [−∞, ∞]
defined as
u(X ) = inf EQ X, X ∈ L 0 ,
Q∈D
(2.2)
where D is a nonempty subset of P and EQ X is understood as EQ X + − EQ X −
with the convention ∞ − ∞ = −∞.
Clearly, a set D, for which representations (2.1) and (2.2) are true, is not
unique. However, there exists a largest such set given by {Q ∈ P : EQ X ≥
u(X ) for any X }.
Definition 2.4 We call the largest set, for which (2.1) (resp., (2.2)) is true, the
determining set of u.
Remark 2.5 (i) Clearly, the determining set is convex. For coherent utility functions on L ∞ , it is also L 1 -closed. However, for coherent utility functions on
L 0 , it is not necessarily L 1 -closed. As an example, take a positive unbounded
random variable X 0 such that P(X 0 = 0) > 0 and consider D0 = {Q ∈ P :
EQ X 0 = 1}. Clearly, the determining set D of the coherent utility function
u(X ) = inf Q∈D0 EQ X satisfies D0 ⊆ D ⊆ {Q ∈ P : EQ X 0 ≥ 1}. On
the other hand, the L 1 -closure of D0 contains a measure Q0 concentrated on
{X 0 = 0}.
Comb convexa das medidas de prob estão em \mathcal{P}???
Weighted V@R and its properties
373
(ii) Let D be an L 1 -closed convex subset of P . Define a coherent utility function u by (2.1) or (2.2). Then D is the determining set of u. Indeed, assume
is greater than D, i.e., there exists Q0 ∈ D
\D.
that the determining set D
∞
Then, by the Hahn–Banach theorem, we can find X 0 ∈ L such that
EQ0 X 0 < inf Q∈D EQ X , which is a contradiction.
Now, we recall some basic facts related to Tail V@R. The next definition applies
both to L ∞ and to L 0 .
Definition 2.6 Tail V@R is the risk measure corresponding to the coherent utility
function
u λ (X ) = inf EQ X,
Q∈Dλ
where λ ∈ [0, 1] and
dQ
Dλ = Q ∈ P :
≤ λ−1 .
dP
Clearly, u 0 (X ) = essinf ω X (ω). The following well-known proposition provides two representations of Tail V@R with λ > 0. Throughout the paper, we
denote by qλ (X ) the right λ-quantile of X , i.e., qλ (X ) = inf{x : P(X ≤ x) > λ}
(we use the convention inf ∅ = +∞).
Proposition 2.7 (i) Let λ ∈ (0, 1], X ∈ L 0 . Then for any Z ∗ ∈ Dλ such that
λ−1 on {X < qλ (X )},
∗
Z =
(2.3)
0
on {X > qλ (X )},
we have u λ (X ) = EP X Z ∗ . Conversely, if u λ (X ) > −∞, then any Z ∗ ∈ Dλ
such that u λ (X ) = EP X Z ∗ satisfies (2.3).
(ii) Let λ ∈ (0, 1], X ∈ L 0 . Then
xQ(dx) + cqλ (X ),
u λ (X ) = λ−1
(−∞,qλ (X ))
where Q = LawP X and c = 1 − λ−1 Q((−∞, qλ (X ))).
Proof (i) We assume that EP X − < ∞ and λ ∈ (0, 1) (the other cases are analyzed
trivially). Without loss of generality, qλ (X ) = 0. Then, for any Z ∈ Dλ ,
X Z − X Z ∗ = X (Z − λ−1 )I (X < 0) + X Z I (X > 0) ≥ 0.
Furthermore, the a.s. equality here is possible only if Z satisfies (2.3).
(ii) This is an immediate consequence of (i).
The importance of Tail V@R is seen from a result of Kusuoka [25] which
is stated below (its proof can also be found in [22, Theorem 4.61] or [30, Theorem 1.48]). Recall that a coherent utility function u is called law invariant if u(X )
depends only on the distribution of X .
374
A. S. Cherny
Theorem 2.8 Assume that (Ω, F , P) is atomless. Let λ ∈ [0, 1] and u be a law
invariant coherent utility function on L ∞ such that u ≤ qλ . Then u ≤ u λ .
We now introduce the basic object of the paper.
Definition 2.9 (i) Weighted V@R on L ∞ is the risk measure corresponding to
the coherent utility function
u µ (X ) =
u λ (X )µ(dλ), X ∈ L ∞ ,
[0,1]
where µ is a probability measure on [0, 1].
(ii) Weighted V@R on L 0 is the risk measure corresponding to the coherent utility
function
u µ (X ) = inf EQ X, X ∈ L 0 ,
Q∈Dµ
where Dµ is the determining set of u µ on L ∞ .
Thus, we have used the following scheme to define u µ on L 0 :
MUITO IMPORTANTE!!!
u λ on L ∞ −→ u µ on L ∞ −→ Dµ −→ u µ on L 0 .
Let us now recall two results of Kusuoka [25] (the proofs can also
be found in [22, Corollary 4.58, Theorem 4.87] or [30, Corollary 1.45,
Theorem 1.58]), which show the importance of Weighted V@R in view of the
law invariance property. Recall that random variables X and Y are comonotone if
(X (ω2 ) − X (ω1 ))(Y (ω2 ) − Y (ω1 )) ≥ 0 for P × P-a.e. ω1 , ω2 ; a coherent utility
function u is comonotonic if u(X + Y ) = u(X ) + u(Y ) whenever X and Y are
comonotone.
Theorem 2.10 (i) On an atomless probability space, a coherent utility function
u on L ∞ is law invariant and comonotonic if and only if it has the form
u = u µ with some probability measure µ.
(ii) On an atomless probability space, a coherent utility function u on L ∞ is law
invariant if and only if it has the form u = inf µ∈M u µ with some collection
M of probability measures on [0, 1].
Remark 2.11 It is easy to check that Tail V@R is in fact a weighted average of
λ
V@Rs, i.e., u λ (X ) = 0 qs (X )ds. Hence, Weighted V@R is also a weighted average of V@Rs, which supports the term we are using (otherwise, we should have
called it “Weighted Tail V@R”).
3 Representation of Weighted V@R
For X ∈ L 0 and λ ∈ (0, 1], we set

−1 on {X < q (X )},

λ
λ
∗
Z λ (X ) = c
on {X = qλ (X )},

0
on {X > qλ (X )},
where c ∈ [0, λ−1 ] is such that EP Z λ∗ = 1.
(3.1)
Weighted V@R and its properties
375
Lemma 3.1 Let λ ∈ (0, 1], X ∈ L 0 , and f be an increasing function. Then
u λ ( f (X )) = EP f (X )Z λ∗ (X ).
Proof Without loss of generality, qλ (X ) = 0 and f (0) = 0. Then, for any Z ∈ Dλ ,
we can write
f (X )Z − f (X )Z λ∗ (X ) = f (X )(Z − λ−1 )I (X < 0) + f (X )Z I (X > 0) ≥ 0.
Theorem 3.2 (i) Suppose that µ({0}) = 0. Then, for X ∈ L 0 , we have u µ (X ) =
EP X Z µ∗ (X ), where Z µ∗ (X ) =
Z λ∗ (X )µ(dλ).
(0,1]
(ii) We have
u µ (X ) =
u λ (X )µ(dλ), X ∈ L 0 ,
(3.2)
[0,1]
where [0,1] f (λ)µ(dλ) is understood as
with the convention ∞ − ∞ = −∞.
[0,1]
Proof (i) Any Z ∈ Dµ can be represented as
f + (λ)µ(dλ)−
(0,1]
[0,1]
f − (λ)µ(dλ)
Z λ µ(dλ) with Z λ ∈ Dλ (see
Theorem 4.4 below). Due to Lemma 3.1,
EP (m ∨ X ∧ n)Z =
[EP (m ∨ X ∧ n)Z λ ]µ(dλ)
(0,1]
≥
[EP (m ∨ X ∧ n)Z λ∗ (X )]µ(dλ)
(0,1]
= EP (m ∨ X ∧ n)Z µ∗ (X ), m, n ∈ Z.
Obviously,
EP X Z µ∗ (X ) = lim
lim EP (m ∨ X ∧ n)Z µ∗ (X )
n→∞ m→−∞
(3.3)
and the same is true for Z µ∗ (X ) replaced by Z . Thus, EP X Z ≥ EP X Z µ∗ (X ),
so that u µ (X ) = EP X Z µ∗ (X ).
(ii) Suppose first that µ({0}) = 0. Due to Lemma 3.1,
∗
EP (m ∨ X ∧ n)Z µ (X ) =
u λ (m ∨ X ∧ n)µ(dλ), m, n ∈ Z.
(0,1]
Obviously,
u λ (X )µ(dλ) = lim
(0,1]
lim
n→∞ m→−∞
(0,1]
u λ (m ∨ X ∧ n)µ(dλ).
376
A. S. Cherny
Fig. 2 The structure of Ψµ
Combining this with (3.3) and the result of (i), we get (3.2).
Now, let µ({0}) = α > 0. Then µ = αδ0 + (1 − α)
µ and it follows from
Theorem 4.4 that Dµ = α Dδ0 + (1 − α)D
µ . If X is not bounded below, then,
clearly, both sides of (3.2) are equal to −∞. If X is bounded below, then
u µ (X ) = αu 0 (X ) + (1 − α)u µ,
µ (X ) and (3.2) for µ follows from (3.2) for which was proved above.
In order to provide another representation of Weighted V@R, let us consider
the function

x 

µ({0}) +
λ−1 µ(dλ)dy,
x ∈ (0, 1],
Ψµ (x) =
(3.4)

0 (y,1]


0,
x = 0.
Clearly, Ψµ is increasing, concave, Ψµ (0) = 0, and Ψµ (1) = 1 (Figure 2).
In fact, (3.4) establishes a one-to-one correspondence between functions with
these properties and probability measures µ on [0, 1] (for details, see [22,
Lemma. 4.63] or [30, Lemma. 1.50]). Further properties of Ψµ are
Ψµ (0+) =
Ψµ (0+) = µ({0}),
λ−1 µ(dλ),
Ψµ (1−) = µ({1}).
(0,1]
Theorem 3.3 For X ∈ L 0 ,
0
u µ (X ) = −
−∞
∞
Ψµ (F(x))dx + (1 − Ψµ (F(x)))dx,
(3.5)
0
where F is the distribution function of X , and we use the convention ∞−∞ = −∞.
Weighted V@R and its properties
377
Proof It is seen from (3.2) that
u µ (X ) = lim
lim u µ (X mn ),
n→∞ m→−∞
where X mn = m ∨ X ∧n. Similar limit relation holds for the right-hand side of (3.5)
(one should consider the distribution function Fmn of X mn ). For bounded X , the
statement of the theorem is known (see [22, Theorem 4.64] or [30, Theorem 1.51]),
so that the result for general X is obtained by passing to the limit.
Remark 3.4 (i) Some important regularity properties of µ can be expressed in
terms of Ψµ . For example, Ψµ (0+) = 0 if and only if µ({0}) = 0 (this condition will be important in sections 6 and 7; note also that this condition is
equivalent to the lower semi-continuity of u µ on L ∞ ); Ψµ is strictly concave
if and only if supp µ = [0,1] (this condition will be important in sections 5
and 8).
(ii) Integrating (3.5) by parts, we get
u µ (X ) = xdΨµ (F(x)) = EP Y,
R
where Y is a random variable with the distribution function Ψµ ◦ F.
4 Representation of the determining set
We begin with three auxiliary lemmas. The notation µ ν means that ν dominates
µ in the monotone order, i.e., µ((−∞, x]) ≥ ν((−∞, x]) for any x.
Lemma 4.1 If µ ν, then Dµ ⊇ Dν .
, F
P) such
Proof There exist random variables ξ , η on some probability space (Ω,
that Law ξ = µ, Law η = ν, and ξ ≤ η a.s. (see [31, section 1.A]). We can write
u µ (X ) = E
P ϕ(ξ ), u ν (X ) = E
P ϕ(η), where ϕ(λ) = u λ (X ). As ϕ is increasing,
u µ ≤ u ν . Clearly, this implies that Dµ ⊇ Dν .
Lemma 4.2 If µn tend to µ weakly and µn µ, then Dµ = n Dµn .
Proof Suppose that there exists Q0 ∈ n Dµn\Dµ . As Dµ is L 1 -closed, we can
apply the Hahn–Banach theorem, which yields X 0 ∈ L ∞ such that EQ0 X 0 <
inf Q∈Dµ EQ X 0 . Thus, supn u µn (X 0 ) ≤ EQ0 X 0 < u µ (X 0 ). On the other hand,
u µn (X 0 ) → u µ (X 0 ) since u λ (X0 ) is continuous in λ. The obtained contradiction yields the inclusion Dµ ⊇ n Dµn . The reverse inclusion follows from the
previous lemma.
Lemma 4.3 Let µ =
N
Dµ = n=1
an Dλn .
N
n=1 an δλn ,
where λ1 > · · · > λ N ≥ 0. Then
378
A. S. Cherny
N
Proof With no loss of generality, λ N = 0. Denote n=1
an Dλn by D. Clearly, D
is convex. It is seen from Proposition 2.7(i) that, for any X ∈ L ∞ , the minimum of
:= N −1 an Dλn is attained. By James’ theorem,
expectations EP X Z over Z ∈ D
n=1
is weakly compact. We have D = D
+a N P , which is the sum of a convex weakly
D
compact set and a convex weakly closed set. An application of the Hahn–Banach
theorem shows that D is L1 -closed.
Obviously, u µ (X ) = inf Q∈D EQ X for any X ∈ L ∞ . Taking into account
Remark 2.5(ii), we get Dµ = D.
Theorem 4.4 We have



Dµ =
Z λ µ(dλ) : Z (λ, ω) is jointly measurable



[0,1]


and Z λ ∈ Dλ for any λ ∈ [0, 1] .


(4.1)
Proof Denote the right-hand side of (4.1) by D. Set
µn =
n−1
k=1
k −1 k
n−1 µ
,
δ k−1 + µ
, 1 δ n−1 , n ∈ N.
n
n
n
n
n
Due to Lemma 4.3, D ⊆ Dµn , and by Lemma 4.2, D ⊆ Dµ .
Let us prove the reverse inclusion. Clearly, D is convex. Arguing in the same
way as in the previous proof, we conclude that D is L 1 -closed. Take
µn = µ
n
1
k −1 k
0,
µ
δ1 +
,
δ k , n ∈ N.
n
n
n
n
n
k=2
Due to Lemma 4.3, Dµn ⊆ D, and therefore, u µn ≥ u, where u(X ) = inf Q∈D EQ X .
As u µn (X ) → u µ (X ) for any X ∈ L ∞ , we get u µ ≥ u on L ∞ . Employing the
Hahn–Banach argument, we get Dµ ⊆ D.
Now, we describe another representation of Dµ , which was obtained by Carlier
and Dana [11] (the proof can also be found in [22, Theorem 4.73] or [30, Theorem 1.53]).
Theorem 4.5 We have
Dµ = {Q ∈ P : Q(A) ≤ Ψµ (P(A)) for any A ∈ F }


1


qs (Z )ds ≤ Ψµ (x) ∀x ∈ [0, 1] ,
= Z ∈ L 0 : Z ≥ 0, EP Z = 1,


1−x
where qs is the s-quantile and Ψµ is given by (3.4).
Weighted V@R and its properties
379
Fig. 3 The structure of Φµ
For the needs of sections 7 and 8, we now provide one more representation of
Dµ . Let us consider the conjugate to the function Ψµ ,
Φµ (x) = sup [Ψµ (y) − x y], x ∈ R+ .
y∈[0,1]
Clearly, Φµ is decreasing, convex (Figure 3), and has the properties
Φµ (x) = 1 − x, x ≤ µ({1}),
Φµ (x) > 1 − x, x > µ({1}),
λ−1 µ(dλ),
Φµ (x) > µ({0}), x <
(0,1]
Φµ (x) = µ({0}), x ≥
λ−1 µ(dλ),
(0,1]
lim Φµ (x) = µ({0}).
x→∞
Furthermore,
Ψµ (x) = inf [Φµ (y) + x y], x ∈ (0, 1].
y∈R+
Theorem 4.6 We have
Dµ = Z ∈ L 0 : Z ≥ 0, EP Z = 1, and EP (Z − x)+ ≤ Φµ (x) ∀x ∈ R+ .
(4.2)
Proof Let Z ∈ Dµ . Take x ∈ R+ and set y = P(Z > x). Using Theorem 4.5, we
get
+
1
EP (Z − x) =
qs (Z )ds − x y ≤ Ψµ (y) − x y ≤ Φµ (x).
1−y
380
A. S. Cherny
Now, let Z belong to the right-hand side of (4.2). Take x > 0 and set y = q1−x (Z ).
For y > y, we have
EP (Z − y )+ − EP (Z − y)+ ≥ (y − y )P(Z > y) ≥ (y − y )x,
while for y < y, we have
EP (Z − y )+ − EP (Z − y)+ ≥ (y − y )P(Z ≥ y) ≥ (y − y )x.
Thus,
1
qs (Z )ds = EP (Z − y)+ + x y = inf [EP (Z − y )+ + x y ]
y ∈R +
1−x
≤ inf [Φµ (y ) + x y ] = Ψµ (x).
y ∈R +
By Theorem 4.5, Z ∈ Dµ .
5 Strict diversification and optimization
Let us introduce the notation
L 1µ = {X ∈ L 0 : u µ (X ) > −∞ and u µ (−X ) > −∞}.
For more information on the L 1 -spaces related to coherent risk measures, see [15,
subsection 2.2].
Theorem 5.1 Suppose that supp µ = [0, 1]. For X, Y ∈ L 1µ , we have
u µ (X + Y ) > u µ (X ) + u µ (Y )
(5.1)
if and only if X and Y are not comonotone.
Proof The “only if” part for bounded X and Y is a consequence of Theorem 2.10 (i).
The statement for unbounded X and Y is obtained by passing to the limit with the
help of the representation
L 1µ =
X ∈ L 0 : lim sup EQ |X |I (|X | > n) = 0 ,
n→∞ Q∈D
µ
which was proved in [15, subsection 2.2].
Let us prove the “if” part. Suppose that (5.1) is not true. Combining the representation (3.2) with the property u λ (X + Y ) ≥ u λ (X ) + u λ (Y ), we conclude that
u λ (X + Y ) = u λ (X ) + u λ (Y ) for µ-a.e. λ ∈ [0, 1]. As supp µ = [0, 1] and the
functions u λ are continuous in λ, we get u λ (X + Y ) = u λ (X ) + u λ (Y ) for any
λ ∈ [0, 1]. In view of Proposition 2.7(i), for any λ ∈ (0, 1], there exists Z λ∗ ∈ Dλ
such that EP X Z λ∗ = u λ (X ) and EP Y Z λ∗ = u λ (Y ). It is seen from Proposition 2.7(i)
that this is possible only if
P(X < qλ (X ), Y > qλ (Y )) = 0, λ ∈ (0, 1],
P(X > qλ (X ), Y < qλ (Y )) = 0, λ ∈ (0, 1].
From this it is easy to deduce that P((X, Y ) ∈ f ((0, 1])) = 1, where f (λ) =
(qλ (X ), qλ (Y )). Thus, X and Y are comonotone.
Weighted V@R and its properties
381
Remark 5.2 Without the condition supp µ = [0, 1], the theorem does not hold. In
particular, it does not hold for Tail V@R.
Property (5.1) can be called the strict diversification property. It holds in particular if X and Y are independent, or if X and Y have a joint density (with respect
to Lebesgue measure). The strict diversification property leads to the uniqueness
of a solution of several optimization problems based on coherent risk measures
that were considered in [16]. Let us briefly describe two of them.
Let S0 ∈ Rd be the vector of initial prices of several assets and S1 the
d-dimensional random vector of their terminal discounted prices. Let H ⊆ Rd
be a convex cone of possible trading strategies, so that the discounted P & L of a
strategy h ∈ H is h, S1 − S0 . The problem is
EP h, S1 − S0 −→ max,
h ∈ H, ρ(h, S1 − S0 ) ≤ c,
(5.2)
where ρ is a coherent risk measure and c ≥ 0 (this problem is a particular case
of (1.4)). In [16, subsection 2.2], we presented a geometric solution of this problem.
Here we give a sufficient condition for the uniqueness.
Corollary 5.3 Let u = u µ with supp µ = [0, 1]. Suppose that each component
of S1 belongs to L 1µ , S1 has a density with respect to Lebesgue measure, and
sup EP h, S1 −S0 < ∞, where sup is taken over h ∈ H such that ρ(h, S1−S0 ) ≤ c.
Then a solution of (5.2) (if it exists) is unique.
Proof Let h ∗ , h ∗ be different solutions. By Theorem 5.1, h = (h ∗ + h ∗ )/2 satisfies
ρ(h, S1 − S0 ) < c. Taking (1 + ε)h with a small ε > 0, we get a strategy that
performs better than h ∗ , h ∗ .
Remark 5.4 Without the assumption supp µ = [0, 1], the statement above is not
true (see [16, subsection 2.2]).
Consider now a single-agent optimization problem. Thus, in addition to the
objects introduced above, we have a random variable W , which describes the current endowment of some agent. Consider the problem
u(W + h, S1 − S0 ) −−→ max .
h∈H
(5.3)
In [16, subsection 2.5], we gave a geometric solution of this problem. Theorem 5.1
yields
Corollary 5.5 Let u = u µ with supp µ = [0, 1]. Suppose that each component
of S1 belongs to L 1µ , S1 has a density with respect to Lebesgue measure, W ∈ L 1µ ,
and suph∈H u µ (W + h, S1 − S0 ) < ∞. Then a solution of (5.3) (if it exists) is
unique.
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A. S. Cherny
6 Minimal extreme measure and capital allocation
The following definition was introduced in [15].
Definition 6.1 Let u be a coherent utility function on L 0 with the determining
set D. Let X ∈ L 0 . We call a measure Q ∈ D an extreme measure for X if
EQ X = u(X ) ∈ (−∞, ∞). The set of extreme measures for u = u µ is denoted by
Xµ (X ).
It is seen from Theorem 3.2(i) that Xµ (X ) = ∅ provided that µ({0}) = 0 and
X ∈ L 1µ .
Proposition 6.2 Suppose that µ({0}) = 0 and X ∈ L 1µ . Then an element Z =
(0,1] Z λ µ(dλ) ∈ Dµ (here we use the representation of Dµ provided by
Theorem 4.4) belongs to Xµ (X ) if and only if
λ−1
Zλ =
0
a.e. on {X < qλ (X )},
a.e. on {X > qλ (X )}
for µ-a.e. λ.
Proof For Z =
(0,1]
Z λ µ(dλ) ∈ D, we have
EP X Z = lim
lim EP (m ∨ X ∧ n)Z
lim
[EP (m ∨ X ∧ n)Z λ ]µ(dλ)
n→∞ m→−∞
= lim
n→∞ m→−∞
(0,1]
(EP X Z λ )µ(dλ).
=
(6.1)
(0,1]
The inclusion X ∈ L 1µ implies that the function λ → EP X Z λ is µ-integrable. An
application of Proposition 2.7(i) completes the proof.
It is seen from the above proposition that if X has a continuous distribution,
then Xµ (X ) consists of a unique element Z = g(X ), where
g(x) =
λ−1 µ(dλ), x ∈ R
(6.2)
[F(x),1]
and F denotes the distribution function of X .
If Law X has atoms, then clearly Xµ (X ) need not be a singleton. However, it
turns out that there exists a minimal element of Xµ (X ) with respect to the convex
stochastic order.
Weighted V@R and its properties
383
Theorem 6.3 Suppose that µ({0}) = 0 and X ∈ L 1µ . Let
∗
Z µ (X ) =
Z λ∗ (X )µ(dλ),
(0,1]
where Z λ∗ (X ) is defined by (3.1). Then, for any Z ∈ Xµ (X ) and any convex function
f : R+ → R+ , we have EP f (Z µ∗ (X )) ≤ EP f (Z ). Moreover, Z µ∗ (X ) is the unique
element of Xµ (X ) with this property.
Proof For an arbitrary Z = (0,1] Z λ µ(dλ) ∈ Xµ (X ), Proposition 6.2 implies that
Z λ∗ (X ) = EP (Z λ | X ) for µ-a.e. λ. By Fubini’s theorem, Z µ∗ (X ) = EP (Z | X ). An
application of Jensen’s inequality yields the first statement.
Now, suppose that there exists another minimal (in the convex order) element
Z of Xµ (X ). Then Z := (Z µ∗ (X ) + Z )/2 belongs to Xµ (X ) and, for a strictly
Z ) < EP f (Z µ∗ (X )), which is
convex function f with linear growth, we get EP f ( a contradiction.
Definition 6.4 We call the measure Q∗µ (X ) = Z µ∗ (X )P the minimal extreme measure for X .
The minimal extreme measure admits a representation similar to (6.2). Let F
denote the distribution function of X . Then, for any λ ∈ (0, 1], Z λ∗ (X ) = gλ (X ),
where
1 − λ−1 F(x−)
I (F(x−) < λ < F(x)) + λ−1 I (λ ≥ F(x)).
F(x) − F(x−)
gλ (x) =
Hence, Z µ∗ (X ) = g(X ), where
1 − λ−1 F(x−)
µ(dλ) +
g(x) =
F(x) − F(x−)
(F(x−),F(x))
λ−1 µ(dλ).
[F(x),1]
The following statement will be used in financial applications below.
Theorem 6.5 Suppose that µ({0}) = 0 and X, Y ∈ L 1µ . Let (ξn ) be a sequence
of random variables such that ξn ∈ L 1µ , each ξn is independent of (X, Y ), and
P
ξn −
→ 0. Then
EQ∗µ (X +ξn ) Y −−−→ EQ∗µ (X ) Y.
n→∞
Proof Denote qλ = qλ (X ), qλn = qλ (X + ξn ). Then, for λ ∈ (0, 1],


−1 on {X < q },
−1 on {X + ξ < q n },


λ
n
λ
λ
λ
Z λ∗ (X ) = cλ
on {X = qλ }, Z λ∗ (X + ξn ) = cλn
on {X + ξn = qλn },


0
0
on {X > qλ },
on {X + ξn > qλn }.
Fix λ ∈ (0, 1]. By Fubini’s theorem, EP (Z λ∗ (X + ξn ) | X, Y ) = f λn (X ), where
f λn (x) = λ−1 Fn (qλn − x) + cλn Fn (qλn − x), x ∈ R
384
A. S. Cherny
and Fn (x) = P(ξn < x), Fn (x) = P(ξn = x). Obviously, qλn → qλ , and
therefore, f λn → λ−1 on (−∞, qλ ), f λn → 0 on (qλ , ∞). Employing the normalization condition EP f λn (X ) = 1, we conclude that f λn (qλ ) → cλ . Thus,
a.s.
f λn (X ) −−→ Z λ∗ (X ). As 0 ≤ f λn ≤ λ−1 , we get
EP Y Z λ∗ (X + ξn ) = EP Y f λn (X ) −−−→ EP Y Z λ∗ (X ), λ ∈ (0, 1].
n→∞
Note that u λ (Y ) ≤ EP Y Z λ∗ (X + ξn ) ≤ −u λ (−Y ). Furthermore, it follows from
the inclusion Y ∈ L 1µ and the representation (3.2) that the functions λ → u λ (Y )
and λ → u λ (−Y ) are µ-integrable. Applying now (6.1), we complete the proof.
Remark 6.6 Without the assumption that ξn is independent of (X, Y ), the theorem
does not hold. As an example, consider X = 0, ξn = Y /n. Then Q∗µ (X ) = P,
while Q∗µ (ξn ) = Q∗µ (Y ).
Let us now present a financial application of the notion of the minimal extreme
measure. It is related to the capital allocation problem. Delbaen [19, section 9]
proposed the following formulation of this problem. Let X 1 , . . . , X d be random
variables meaning the discounted P & Ls produced by several components of a
firm. Let ρ be a coherent risk measure. A capital allocation between X 1 , . . . , X d
is a vector x 1 , . . . , x d such that
d
d
i
X =
xi ,
(6.3)
ρ
i=1
∀h , . . . , h ∈ R+ , ρ
1
d
i=1
d
i
h X
i
≥
i=1
d
hi x i .
(6.4)
i=1
From the financial point of view, x i is the contribution of the ith component to
the total risk of the firm, or, equivalently, the capital that should be allocated to
this component. In order to illustrate the meaning of (6.4), consider the example
h i = I (i ∈ J ), where J is a subset of {1, . . . , d}. Then (6.4) means that the capital
allocated to a part of the firm does not exceed the risk carried by that part.
It was proved in [15, subsection 2.4] under the assumption u(X i ) > −∞,
u(−X i ) > −∞, i = 1, . . . , d (here u = −ρ) that the set of capital allocations has
the form
d
1
d
i
,
(6.5)
−EQ (X , . . . , X ) : Q ∈ Xρ
X
i=1
where Xρ denotes the set of extreme measures corresponding to ρ.
Suppose now that u = u µ with µ({0}) = 0. It is seen from Proposition 6.2,
that if i X i has a continuous distribution, then a capital allocation is unique.
But in general, this is not the case. For example, if X 2 = −X 1 , then the set of
capital allocations is the interval [a, b] in R2 , where a = (−u µ (X 1 ), u µ (X 1 )),
b = (u µ (−X 1 ), −u µ (−X 1 )).
Weighted V@R and its properties
385
However, if u = u µ with µ({0}) = 0, then there exists a particular element
of (6.5), namely x0 = −EQ∗µ ( X i ) (X 1 , . . . , X d ) (for the example considered
above, x0 = (−EP X 1 , EP X 1 )). We call x0 the central solution of the capital allocation problem. Its role is as follows. Let us disturb X i , i.e., we pass from X i to
X ni = X i + ξni , where each ξni is independent of (X 1 , . . . , X d ), ξni ∈ L 1µ , and
i P
X n is a singleton,
ξni −
→ 0. If i ξni has a continuous distribution, then Xµ i 1
d
so that the capital allocation xn between X n , . . . , X n is unique. By Theorem 6.5,
xn → x0 .
7 Pricing in an option-based model
Let S0 ∈ (0, ∞) be the initial price of some asset and S1 a positive random variable
describing its terminal discounted price. Let K ⊆ R+ be the set of strike prices of
traded European call options on this asset with maturity 1 and let ϕ(K ), K ∈ K be
the price at time 0 of the option that pays (S1 − K )+ at time 1. The set
A=
N
+
h n [(S1 − K n ) − ϕ(K n )] : N ∈ N, K n ∈ K, h n ∈ R
n=1
is the set of P & Ls that can be obtained in the model under consideration (we
assume that 0 ∈ K, which corresponds to the possibility of trading the underlying
asset). We also fix a coherent utility function u.
According to [15], we say that the model satisfies the no-good-deals (NGD)
condition if there exists no X ∈ A with u(X ) > 0.
Now, let F ∈ L 0 be the payoff of some contingent claim. According to [15],
we say that a real number z is an NGD price of F if there exist no X ∈ A, h ∈ R
with u(X + h(F − z)) > 0. The set of NGD prices will be denoted by INGD (F).
It was proved in [15, subsection 3.1] (under some additional conditions that
are automatically satisfied for u = u µ provided that µ({0}) = 0) that the NGD
condition is satisfied if and only if D ∩ R = ∅, where D is the determining set of u
and R is the set of risk-neutral measures, which in this model has the form
R = {Q ∈ P : EQ (S1 − K )+ = ϕ(K ) for any K ∈ K}
(the notation P was introduced in section 2). Furthermore, for F ∈ L 0 such that
u(F) > −∞ and u(−F) > −∞,
INGD (F) = {EQ F : Q ∈ D ∩ R}.
(7.1)
We give below more concrete versions of these results for u = u µ . Let us introduce
the notation
386
A. S. Cherny



Fµ = ψ : ψ is a convex function R+ → R+ , ψ+
(0) ≥ −1,


lim ψ(x) = 0, ψ|K = ϕ|K , ψ ∼ P0 , and
x→∞



+
dψ
(y) − x P0 (dy) ≤ Φµ (x) for any x ∈ R+ .

dP0

R+
denotes the right-hand derivative, ψ is the second derivative taken in
Here ψ+
(b) − ψ (a)) with the conventhe sense of distributions (i.e., ψ ((a, b]) = ψ+
+
tion ψ ({0}) = ψ+ (0) + 1, P0 = LawP S1 , and Φµ is the function introduced in
section 4.
Theorem 7.1 Let u = u µ with µ({0}) = 0.
(i) The NGD condition is satisfied if and only if Fµ = ∅.
(ii) For F = f (S1 ) ∈ L 1µ , we have






f (x)ψ (d x) : ψ ∈ Fµ .
INGD (F) =




R+
Proof (i) Let us prove the “only if” part. By the result mentioned above, there exists
Q ∈ Dµ ∩ R. The function ψ(x) := EQ (S1 − x)+ is convex, lim x→∞ ψ(x) = 0,
and set g(x) = EP (Z | S1 = x). Then
and ψ|K = ϕ|K . Denote Z = ddQ
P
+
ψ(x) = EP (S1 − x) g(S1 ) = (y − x)+ g(y)P0 (dy), x ∈ R.
R+
(0)
ψ+
This representation shows that
≥ −1 and ψ = gP0 . By Theorem 4.6,
(g(y) − x)+ P0 (dy) = EP (g(S1 ) − x)+ ≤ EP (Z − x)+ ≤ Φµ (x), x ∈ R,
R+
so that ψ ∈ Fµ .
Let us prove the “if” part. Take ψ ∈ Fµ and set g = ddψP , Q = g(S1 )P. Then
0
Q ∈ P . The inequality
EQ (g(S1 ) − x)+ = (g(y) − x)+ P0 (dy) ≤ Φµ (x), x ∈ R+ ,
R+
combined with Theorem 4.6, shows that Q ∈ Dµ . Furthermore,
EQ (S1 − K )+ = (y − K )+ ψ (dy) = ψ(K ) = ϕ(K ),
R+
so that Q ∈ R.
K ∈ K,
Weighted V@R and its properties
387
(ii) Take z ∈ INGD (F). By (7.1), z = EQ f (S1 ) with some Q ∈ Dµ ∩ R. The proof
of (i) shows that
z = EP f (S1 )g(S1 ) =
f (x)g(x)P0 (dx) =
f (x)ψ (dx),
R+
R+
where g(x) = EP ddQ
| S1 = x and ψ = EQ (S1 − ·)+ ∈ Fµ .
P
Conversely, take z =
where Q = g(S1 )P, g =
by (7.1), z ∈ INGD (F).
R+ f (x)ψ (dx) with ψ ∈ Fµ . Then z
dψ dP0 . The proof of (i) shows that Q ∈
= EQ f (S1 ),
Dµ ∩ R, and
8 Optimization in a complete model
Let (Ω, F , P) be a probability space. We consider a complete model, in which an
agent can obtain by trading any P & L from the class
A = {X ∈ L 1 (Q) : EQ X = 0},
where Q is a fixed probability measure, which is absolutely continuous with respect
to P. Clearly, problem (1.4) is equivalent to the problem
RAROC(X ) −−−→ max,
X ∈A
where
(8.1)


if EP X > 0 and u(X ) ≥ 0,
+∞
RAROC(X ) =
EP X


otherwise.
−u(X )
We take u = u µ , where µ = δ1 (otherwise u µ (X ) = EP X , and the above problem
is meaningless).
Let Φµ be the function defined in section 4 and set
ϕ(x) = EP (Z 0 − x)+ , x ∈ R+ ,
where Z 0 = ddQ
. We assume that there exists no X ∈ A with u µ (X ) > 0 (indeed,
P
for such X we would have RAROC(X ) = ∞). In view of the results of [15,
subsection 3.2], this is equivalent to the inclusion Q ∈ Dµ . This, in turn, is equivalent to the inequality ϕ ≤ Φµ (see Theorem 4.6). For α ≥ 1, we set
ϕα (x) = αϕ
x − 1
+ 1 , x ∈ R+ ,
α
i.e., the graph of ϕα is the α-stretching of the graph of ϕ with respect to
the point (1, 0) (see Figure 4). Set α∗ = sup{α: ϕα ≤ Φµ }, β = µ({1}),
388
A. S. Cherny
Fig. 4 The structure of Φµ , ϕ, and ϕα
γ = (0,1] λ−1 µ(dλ), and Z = α∗ Z 0 − α∗α−1
, so that ϕα∗ (x) = EP (Z − x)+ .
∗
Note that the left-hand and the right-hand derivatives of ϕα∗ are given by
(ϕα∗ )− (x) = −P(Z ≥ x), x > 0,
(8.2)
= −P(Z > x), x ≥ 0.
(8.3)
(ϕα∗ )+ (x)
Theorem 8.1 Suppose that µ = δ1 , Q = P, and ϕ ≤ Φµ .
(i) We have sup X ∈A RAROC(X ) = (α∗ − 1)−1 .
(ii) If α∗ > 1 and (ϕα∗ )+ (β) > −1, then P(Z = β) > 0 and any random variable of the form X = bI B + cI B c , where B ⊆ {Z = β} and b > 0 > c are
such that EQ X = 0, is optimal for (8.1).
(iii) If α∗ > 1, γ < ∞, and either ϕα∗ (γ ) = Φµ (γ ) > 0 or Φµ (γ ) = 0 and
(ϕα∗ )− (γ ) < 0, then P(Z ≥ γ ) > 0 and any random variable of the form
X = bI B + cI B c , where B ⊆ {Z ≥ γ } and b < 0 < c are such that
EQ X = 0, is optimal for (8.1).
(iv) If α∗ > 1 and there exists x0 ∈ (β, γ ) such that ϕα∗ (x0 ) = Φµ (x0 ), then
P(Z > x0 ) > 0 and any random variable of the form X = bI B +cI B c , where
B = {Z > x0 } and b < 0 < c are such that EQ X = 0, is optimal for (8.1).
If moreover supp µ = [0, 1] and x0 is the unique point of (β, γ ), at which
ϕα∗ = Φµ , then an optimal element of A is unique up to multiplication by a
positive constant.
(v) If α∗ > 1, but neither of conditions (ii)–(iv) is satisfied, then the maximum
in (8.1) is not attained.
Remark 8.2 (i) It is easy to check that if µ has no gap near 1, i.e., µ((1−ε, 1)) > 0
for any ε > 0, then (Φµ )+ (β) = −1. Similarly, if µ has no gap near 0,
i.e., µ((0, ε)) > 0 for any ε > 0, then (Φµ )− (γ ) = 0. Thus, the situation of
(ii) (resp., (iii)) can be realized only if µ has a gap near 1 (resp., near 0). Another
verification of these statements follows from the arguments in the proof of
(v) below.
(ii) In many natural complete models (for instance, in the Black–Scholes
model), we have essinf ω Z 0 (ω) = 0. Then ϕ(ε) > 1 − ε for any ε > 0,
Weighted V@R and its properties
389
and hence, α∗ = 1. By Theorem 8.1 (i), sup X ∈A RAROC(X ) = ∞. A sequence
of elements X n ∈ A with RAROC(X n ) → ∞ is provided by
X n = I (Z ≤ n −1 ) − Q(Z ≤ n −1 ).
Proof of Theorem 8.1 (i) Take R ∈ (0, ∞). It follows from the result in [15, subsection 3.2] that sup X ∈A RAROC(X ) ≤ R if and only if
1
1+ R
Z0 −
∈ Dµ .
R
1+ R
In view of Theorem 4.6, this is equivalent to the conditions Z 0 ≥
∀x ≥ 0, EP
Note that
EP
1+ R
R
1+ R
R
Z0 −
Z0 −
1
1+ R
+
1
1+ R
−x
−x
+
1
1+R
and
≤ Φµ (x).
(8.4)
= ϕα(R) (x), x ∈ R+ ,
where α(R) = 1+R
R . Thus, (8.4) is satisfied if and only if ϕα(R) ≤ Φµ .
Set h = essinf ω Z 0 (ω). Note that ϕ(h) = 1 − h and ϕ(h + ε) > 1 − h − ε for
any ε > 0. As Φµ (0) = 1, we conclude that the condition ϕα(R) ≤ Φµ automati1
. As
cally implies that (1 − h)α(R) ≤ 1, which, in turn, is equivalent to Z 0 ≥ 1+R
a result,
sup RAROC(X ) ≤ R ⇐⇒ ϕα(R) ≤ Φµ ⇐⇒ α(R) ≤ α∗ ⇐⇒ R ≥ (α∗ − 1)−1 .
X ∈A
(ii) The inequality
EP (Z − x)+ = ϕα∗ (x) ≤ Φµ (x), x ∈ R+ ,
(8.5)
combined with Theorem 4.6, shows that Z ∈ Dµ . Consequently, Z ≥ β, and
it follows from (8.3) that P(Z = β) > 0. Due to Theorem 4.4, we can write
Z = [0,1] Z λ µ(dλ) with Z λ ∈ Dλ . As Z 1 = 1, we deduce that, for µ-a.e. λ ∈
[0, 1), Z λ = 0 P-a.e. on {Z = β}. Then, due to the structure of X , for µ-a.e.
λ ∈ [0, 1], we have EP X Z λ = u λ (X ). Thus, EP X Z = u µ (X ).
Applying now the equality
0 = EP X Z 0 =
α∗ − 1
1
α∗ − 1
1
EP X +
EP X Z =
EP X +
u µ (X ),
α∗
α∗
α∗
α∗
we deduce that RAROC(X ) = (α∗ − 1)−1 , so that X is optimal.
(iii) Consider first the case ϕα∗ (γ ) = Φµ (γ ) > 0. Due to (8.5), Z ∈ Dµ , so
that we can write Z = [0,1] Z λ µ(dλ) = ξ + η, where ξ = (0,1] Z λ µ(dλ) and
η = µ({0})Z 0 . As Z λ ≤ λ−1 , we get ξ ≤ γ , so that
ϕα∗ (γ ) = EP (Z − γ )+ ≤ EP η = µ({0}) = Φµ (γ ).
390
A. S. Cherny
The fact that this inequality should be an equality means that P(ξ = γ ) > 0 and
−1
η = 0 P-a.e.
outside
{ξ = γ }. This implies that, for µ-a.e.
λ ∈ (0, 1], Z λ = λ
P-a.e. on Z > γ and Z 0 = 0 P-a.e. outside Z > γ . The proof is now completed in the same way as in (ii).
Now, consider the case Φµ (γ ) = 0 and (ϕα∗ )− (γ ) < 0. As Φµ (γ ) = µ({0}),
we get µ({0}) = 0. Thus, Z = (0,1] Z λ µ(dλ). As Z λ ≤ λ−1 , we get Z ≤ γ . It
follows from (8.2) that P(Z = γ ) > 0. This implies that, for µ-a.e. λ ∈ (0, 1],
Z λ = λ−1 P-a.e. on {Z = γ }. The proof is now completed in the same way as in (ii).
(iv) Set
λ0 = inf





x ∈ [0, 1] :
λ−1 µ(dλ) ≤ x0
(x,1]
λ−1 µ(dλ), r =
l=





,
λ−1 µ(dλ).
[λ0 ,1]
(λ0 ,1]
Consider the case µ({λ0 }) > 0 (the other case is simpler). It is easy to see that
x ∈ [l, r ] and Φµ is linear on [l, r ]. In view of the convexity of ϕα∗ , this implies
that ϕα∗ = Φµ on [l, r ]. As (λ0 ,1] Z λ dµ ≤ l, we get
ϕα∗ (l) = EP (Z − l)+ ≤ EP
Z λ µ(dλ) = µ([0, λ0 ]).
(8.6)
[λ0 ,1]
Furthermore,
λ0
Φµ (l) = Ψµ (λ0 ) − lλ0 = (G(x) − l)dx = −
xdG(x) = µ([0, λ0 ]),
[0,λ0 ]
0
(8.7)
where G(x) =
(x,1] λ
−1 µ(dλ).
In a similar way, we check that
ϕα∗ (r ) = EP (Z − r )+ ≤ EP
Z λ µ(dλ) = µ([0, λ0 )) = Φµ (r ).
(8.8)
(λ0 ,1]
As the inequalities in (8.6) and (8.8) should be equalities, we conclude that there
exists a set B such that Z λ = λ−1 a.e. on B for λ ≥ λ0 and Z λ = 0 a.e. on B c for
λ ≤ λ0 . The proof of the first part of (iv) is now completed in the same way as in (ii).
Let us now prove the uniqueness. Let X be optimal for (8.1). Then X is not
degenerate since otherwise it must be equal to 0. Thus, we can find c ∈ R such
that λ0 = P(X ≤ c) belongs to (0, 1). The analysis of the proof of (ii) shows that
u µ (X ) = EP X Z . Consequently, for µ-a.e. λ ∈ [0, 1], u λ (X ) = EP X Z λ , where
Z λ are taken from the representation Z = [0,1] Z λ µ(dλ). This means that, for
µ-a.e. λ ∈ (λ0 , 1], Z λ = λ−1 P-a.e. on {X ≤ c} and, for µ-a.e. λ ∈ [0, λ0 ],
Weighted V@R and its properties
391
Z λ = 0 P-a.e. outside {X ≤ c}. Consequently, for x0 = (λ0 ,1] λ−1 µ(dλ),
we have EP (Z − x0 )+ = µ([0, λ0 ]). Calculations similar to (8.7) show that
µ([0, λ0 ]) = Φµ (x0 ). Moreover, as λ0 ∈ (0, 1) and supp µ = [0, 1], we have
x0 ∈ (β, γ ). Since such an x0 is unique, we conclude that X takes only two
values. Thus, any optimal X has the form bI B + cI B c with some B ∈ F and some
constants b < c. It is clear from the reasoning given above that P(B) is determined
uniquely. Using the same arguments as in the proof of Corollary 5.3, we deduce that
different optimal elements must be comonotone. Consequently, B is determined
uniquely, so that an optimal strategy is unique up to multiplication by a positive
constant.
(v) Assume the contrary, i.e., the existence of an optimal element X . As X is
not degenerate, we can find c ∈ R such that λ0 = P(X ≤ c) belongs to (0, 1).
Arguing in the same way as above, we prove that ϕα∗ (x0 ) = Φµ (x0 ), where
x0 = (λ0 ,1] λ−1 µ(dλ). We now consider three cases.
Case 1. Assume that x0 = β. This means that µ((λ0 , 1)) = 0. The arguments given
above show that, for µ-a.e. λ ∈ [0, λ0 ], Z = 0 P-a.e. outside {X ≤ c}. Consequently, Z = β P-a.e. on {X > c}, so that
(ϕα∗ )+ (β) = −P(Z > β) > −1.
Thus, in this case the conditions of (ii) are satisfied.
Case 2. Assume that x0 = γ . This means that µ((0, λ0 )) = 0. If µ({0}) > 0, then
ϕα∗ (γ ) = Φµ (γ ) = µ({0}) > 0, so that the conditions of (iii) are satisfied.
Now, assume that µ({0}) = 0. Then Φµ (γ ) = 0. The arguments given above
show that, for µ-a.e. λ ∈ [λ0 , 1], Z λ = λ−1 P-a.e. on {X ≤ c}. As µ([0, λ0 )) = 0,
we get Z = (0,1] λ−1 µ(dλ) = γ P-a.e. on {X ≤ c}, so that
(ϕα∗ )− (γ ) = −P(Z = γ ) < 0.
Thus, in this case the conditions of (iii) are satisfied.
Case 3. Assume that x0 ∈ (β, γ ). Then the conditions of (iv) are satisfied.
Corollary 8.3 Suppose that supp µ = [0, 1], Q = P, ϕ ≤ Φµ , and α∗ > 1. There
exists an optimal element of A if and only if there exists x 0 ∈ (β, γ ) such that
ϕα∗ (x0 ) = Φµ (x0 ). An optimal element is unique up to multiplication by a positive
constant if and only if such an x0 is unique.
Proof The proof of point (v) shows that, under the condition supp µ = [0, 1],
the situations of (ii), (iii) are not realized. Now, the statement follows from
Theorem 8.1.
The financial interpretation of the obtained results is as follows. In most natural
situations, the optimal strategy consists in buying the binary option with the payoff
dQ
I Z ≤ x0 = I Z 0 ≤ α∗−1 (x0 − 1) + 1 = I
≤ c∗ .
dP
The geometric recipe for finding c∗ is given in Figure 5.
392
A. S. Cherny
Fig. 5 The form of c∗
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