Multivariate Risk Measures

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Multivariate Risk Measures

Ivar Ekeland and Walter Schachermayer

Zurich, May 5, 2011

Ivar Ekeland and Walter Schachermayer () Multivariate Risk Measures Zurich, May 5, 2011 1 / 16

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Monovariate risk measures

A monovariate risk measure(1-rm) is a functionρ :L^∞(_Ω,F^,^P)!∞ such that:

ρ(0) =0

X Y =)^ρ(X) ρ(Y)

ρ(X +m) =ρ(X) m for m2^R It is

convex ifρ is a convex function

coherent if it is convex and positively homogeneous:

ρ(X+Y) ρ(X) +ρ(Y) ρ(λX) = λρ(X)

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Law-invariant 1-r.m.

We shall write X Y to mean thatX andY have the same law. ρ is law-invariant if X Y implies that ρ(X) =ρ(Y)

De…nition

A 1-rm is strongly coherent if it is convex, law-invariant and:

ρ(X) +ρ(Y) =supn

ρ X+Yê j ^Y ^Yêô There are two fundamental examples of s.c. 1-r.m.

1 Let F 2^L¹ be a probability density, so thatF 0 and E[F] =1.

De…ne

ρ_F :=supn Eh

FXei

j ^X^e ^X^o

2 Set:

ρ_∞(X):=esssup X Both ρ_F and ρ_∞ are strongly coherent.

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Kusuoka’s theorem

Kusuoka (2001) has shown that all strongly coherent risk measures can be built from ρ_F and ρ_∞

Theorem

ρis a strongly coherent 1-r.m. if and only if there is a probability density F and a number s with 0 s 1 such that:

8^X 2^L^∞^, ^ρ(X) =sρ_∞(X) + (1 s)ρ_F(X)

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Multivariate risk measures

A d-dimensional risk measure(d-r.m.) is a function ρ:L^∞ Ω,F^,^P^;^R^d !^R ^{such that:}

ρ(0) =0

X Y =)^ρ(X) ρ(Y)

ρ(X +me) =ρ(X) m for m2^R ^and^e = (1, ...,1) It is

convex ifρ is a convex function

coherent if it is convex and positively homogeneous:

ρ(X+Y) ρ(X) +ρ(Y) ρ(λX) = λρ(X)

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Law-invariant d-r.m.

ρis law-invariant if X Y implies that ρ(X) =ρ(Y) De…nition

(Galichon, Henry) A d-rm is strongly coherentif it is convex, law-invariant and:

ρ(X) +ρ(Y) =supn

ρ X+Ye j ^Y ^Y^e^o

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Examples

There are two fundamental examples of s.c. 1-r.m.

1 Let F = (F1, ...,Fd)2^L¹ ^R^d ^satisfy ^Fⁱ ^0,¹ ⁱ ^d^{, and}

∑iE[F_i] =1. De…ne ρ_F :=supn

Eh FXei

j^X^e ^X^o=sup (

∑

i

Eh FiXei

i j^X^e ^X )

2 Denote byS^d the unit simplex in R^d and let ξ 2^S^d^{, so that}^ξi 0 and ∑ξ_i =1. De…ne

ρ_ξ(X):=esssup Xξ =esssup

∑

^Xⁱ^ξⁱ

3 Let µbe a probability on S^d. Set:

ρ_µ(X):=

Z

S^dρ_ξ(X)dµ(ξ) ρ_F andρ_µ are strongly coherent d-r.m.

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Extension of Kusuoka’s theorem

Schachermayer and IE have shown that all strongly coherent risk measured-r.m. can be built from ρ_F andρ_∞

Theorem

ρis a strongly coherent d-r.m. if and only if there is some F 2^L¹+ R^d with ∑iE[Fi] =1, a probabilityµon S^d and a number s with0 s 1 such that:

8^X 2^L^∞^, ^ρ(X) =sρ_µ(X) + (1 s)ρ_F (X)

This result builds on an earlier result by Galichon, Henry and IE, which we will explain in due course.

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Duality

If ρ:L^∞ R^d !^R is a convex and law-invariant, it is lower

semi-continuous wrt σ L^∞,L¹ (Jouini, Schachermayer, Touzi, 2006). It follows that ρ is coherent (homogeneous) if and only if there exists a closed, convex, law-invariant subsetC of L¹ R^d such that.

ρ(X) = sup

F2^C h^F^,^Xi

We shall set ρ=ρ_C to remember this fact. Denote by T ^{the set of} measure-preserving maps τ fromΩ into itself. Set:

K :=f(F,F τ) j^F 2^C^, ^τ2 T g Since C is law-invariant,K C C. Conversely:

Lemma

ρis strongly coherent i¤ C C is the closed convex hull of K

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Proof

Suppose ρ=ρ_C andC C is the closed convex hull ofK. Then:

ρ_C(X) +ρ_C(Y) = supf h^F^,^Xi h^G,^Xi j (F,G)2^C ^Cg

= supf h^F^,^Xi h^F ^τ,^Yi j ^F 2^C^, ^τ2 T g

= sup h^F^,^X +Y τ ¹i j ^F 2^C^, ^τ2 T sup

n

h^F^,^X+Yei j ^Y^e ^Y^o=ρ_C X+Ye

and the reverse inequality is always true

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Geometry of strongly coherent sets

Theorem

Let C L¹ R^d be strongly coherent. Then there exists a number t, with 0 t 1, and two closed convex law-invariant subsets C^r and C^s of L¹ R^d such that:

1 C = (1 t)C^r+tC^s

2 C^r is σ L¹,L^∞ -compact

3 C^s (L^∞) isσ (L^∞) ,L^∞ -compact, and its extreme points are purely singular

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The regular case

This is the case when C^s =?. In other words, C =C^r is weakly compact in L¹ R^d . For instance, C L^p R^d , so thatρ= ρ_C is continuous in theL^p R^d norm. This case was dealt with in an earlier paper by Galichon, Henry and IE.

SinceC is weakly compact, it is the closed convex hull of its set of strongly exposed points. Let F 2^C be such a point. We claim that ρ=ρ_F.

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Proof

By de…nition, there is some X 2^L^∞ ^R^d such that sup_H₂_C

h^H,^Xi= h^F^,^Xi^{and if} ^Hⁿ 2 ^C is a maximizing sequence, then H_n !^F strongly inLⁱ R^d . Sinceρ is strongly coherent, we have:

ρ(X) +ρ(Y) =supf h^G^,^Xi h^G ^τ,^Yi j ^G 2^C^, ^τ2 T g Let (Gn,τn) be a maximizing sequence. Since we have= instead of , we must have ρ(X) =sup h^Gⁿ^,^Xi^and^ρ(Y) =sup h^Gⁿ ^τ,^Yi^so G_n !^F ^and:

ρ(Y) =sup

τ h^F ^τ,^Yi= sup

τ h^F^,^Y ^τ ¹i= sup

Y Ye

h^F^,^Y^ei so ρ =ρ_F as desired

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The singular case 1

Let G = (A_i) be a …nite partition ofΩ. On eachA_i we are given a point x_i 2^S^d and an element β_i 2(L^∞(R)) . We then build an element β_G 2 ^L^∞ ^R^d by the formula

h^β_G^,^Xi=

∑

^h^βⁱ^,⁽^xⁱ ^X⁾ⁱ

Denote by δi the Dirac mass atx_i. We associate withβ_G the Borel measure µ_G onS^d de…ned by:

µ_G =

∑

^δⁱ^βⁱ⁽^Aⁱ⁾

Lemma

Suppose β_G is law-invariant. Then, for X 2^L^∞ ^R^d ^{we have:}

Z

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Proof

Z

S^d

sup (x X)dµ(x) =

Z

S^d

sup x X τ ¹ dµ(x)

=

∑

^sup ^xⁱ ^X ^τ ¹ ^βⁱ⁽^Aⁱ⁾

h^β_G ^τ,^Xi

and the reverse inequality follows from the fact that β_G is purely singular.

Taking points (ξ_i)and disjoint sets(B_i) with P(B_i)>0 such that

esssup xi X = xi ξ_i andBi fj^X ^ξij<εg

we can …nd a set N with 0<_P(N)<infP(B_i) such that β_G1_N = β_G. We then …nd a measure-preserving transformation which maps N\^Aⁱ ^into B_i and the result follows

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The singular case: simple elements

Let β2 ^L^∞ ^R^d 0 be a purely singular element. We de…ne a …nitely additive measure j^βj ^by:

j^βj[A] =h^e1A,βi Let G = (Ai) be a …nite partition ofΩ. Set:

β_G =

∑

^ξⁱ⁽^j^β^j^[^Gⁱ^])

ξ_i = h^eⁱ¹^Gi,βi h^e1Gi,βi

We can approximate βby the β_G which have a simpler form. We associate with β_G the Borel measure:

µ_G =

∑

^j^β^j^[^Gⁱ^]^δ^ξⁱ