Marrakech Avril 2007
Monetary Measures of Risk
Monetary Measures of Risk
F.Delbaen, H.Foellmer, A.Schied
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Value at Risk and Average Value at Risk
Value at Risk and Average Value at Risk
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Value at Risk
Value at Risk
Let X a financial position defined on (Ω, F, P). The Value at Risk of
X atlevel λ is defined as the amount that we have to add to X such
that
P[X + V aRλ < 0] ≤ λ ≤ P[X + V aRλ ≤ 0]
Value at Risk is the opposite of the λ-quantile.
Quantile Properties
Let F be a increasing function on (a,b) such that limx7→a F (x) = c,
and limx7→b = d. The left and right continuous inverse functions are
defined as
q − (s) := sup{x | F (x) < s} = inf{x | F (x) ≥ s} F (q + (s)−) ≤ s ≤ F (q − (s)+)
q + (s) := sup{x | F (x) ≤ s} = inf{x | F (x) > s}
q − is left continuous, and q + right continuous. F is an inverse of q,
and F (x+) := inf{x ∈ c, d) |q(s)) > x}
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Value at Risk Drawbacks
Value at Risk Drawbacks
Examples : F.Delbaen, Coherent Risk Measures Pise 2001
⇒ No subadditivity :$ 100 loan to a client with 8.10−3 defaut
probability =⇒ VaRλ ≤ 0 if λ = 10−2
Two loans of each $50 with the same defaut probability assumed
to be independent. =⇒ VaRλ = $50
⇒ Bad to detect concentration A bank lends $ 1 to 150 clients
independent, with the same defaut probability p = 1.2%. Let
PN
Z = i=1 Zi be the total number of defaut= the amount of the
loses.
Given the binomial distribution of Z, with λ = 10− , VaRλ = $5
and tail expectation = 6.287
When the clients are dependent and the distribution given by
2
dQ = ceαZ dQ, choosing α such that Q[Z = 1] = 1.2%, then
VaRλ = 6 and tail expectation =14.5
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Average Value at Risk Averλ (X)
Average Value at Risk Averλ (X)
Assume X to be bounded with a continuous distribution function.
The Average Value at Risk is the tail expectation
Averλ (X) = E[−X |X ≤ qλ ]
Characterisation of Average Value at Risk
Z
1 λ
Averλ (X) =
V aRµ dµ
λ 0
1
= inf {−K + E[(K − X)+ ]}
K
λ
1
= inf {E[−Z X] | 0 ≤ Z ≤ , andE[Z] = 1
Z
λ
1
= E[−Zλ X] whereZλ = 1{ X ≤ qλ }
λ
The two last equations are related to a dual formulation.
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Average Value at Risk Averλ (X)
⇒ Proof : Let G(K) = −K + λ1 E[(K − X)+ ] be the convex
function, derivable w.r. to K since F is continuous.
Then, G0 (K) = −1 + λ1 P[X ≤ K] is ≤ 0 if K ≤ qλ , and ≥ 0 of not
and qλ is a minimum of G function. Moreover
G(qλ ) = −qλ +
1
E[(qλ − X)+ ] = Averλ (X)
P[X ≤ qλ ]
⇒ Since X=qU where U is uniformly distriuted on [0, 1],
+
Rλ
E[(qλ − X) ] = 0 (qλ − qµ)dµ. The first identity follows easily.
⇒ On the other hand, Averλ (X) = E[−X Zλ ] = E[−X λ1 1{X≤qλ } ].
For notational simplicity, make a translation to obtain qλ (X) = 0.
For any Z satisfying above assumptions,
−XZ + XZλ = −X(Z − λ1 )1{X<0} − Z1{X>0}) is negative. Then
Zλ is an optimal variable for the dual problem.
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Average Value at Risk Averλ (X)
Cash Invariance Property
Both VaRλ (X) and Averλ (X) are cash invariante. That is not the
case for the convex functional E[(−X)+ ].
• Averλ (X) is the largest convex risk measure dominated by
E[(−X)+ ].
• More difficult is to show that Averλ (X) is the smallest convex risk
measure dominating VaRλ (X).
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Basic Properties
Coherent and Convex Risk Measures :
Basic Properties
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Coherent Risk measures
Coherent Risk measures
First introduce in a famous paper of Arztner, Delbaen, Eber.(1996 ?)
Definition and properties
• A coherent risk measure ρ is a map Ψ → ρ(Ψ) of the set of the
bounded financial positions into R such that
? decreasing, X ≥ Y =⇒ ρ(X) ≤ ρ(Y )
? sub-addtive, ⇐⇒ diversification effect.
ρ(X + (Y ) ≤ ρ(X) + ρ(Y ).
+
? homogenous : ∀λ ∈ R ρ λ Φ = λρ Φ .
? cash invariant : ρ(Ψ + m) = ρ(Ψ) − m
Average Var is a coherent risk measure.
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Coherent Risk measures
Convex Risk Measure
To take into account that the size of the risky position has some
influence on the risk measure, Foellmer and Schied have
introduced monetary convex risk measures.
• A convex risk measure ρ is a map Ψ → ρ(Ψ) of the set of the
bounded financial positions into R such that
? decreasing, X ≥ Y =⇒ ρ(X) ≤ ρ(Y )
? convex, ⇐⇒ diversification effect.
ρ(αX + (1 − α)Y ) ≤ αρ(X) + (1 − α)ρ(Y ).
? cash invariant :ρ(X + m) = ρ(X) − m
• if ρ(0) = 0, the risk measure is said to be normalized
• ρ(X) ≤ 0 characterizes an acceptable risk
L∞ (P) -Risk Measures
Assume given a reference probability measure P. If ρ(1A ) = 0 as
soon that P(A) = 0, the risk measure is defined on L∞ (P).
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Convex Analysis and Risk measures
Convex Analysis and Risk measures
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Fenchel Transform of Convex function
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Three Fundamental Theorems in
Functional Analysis
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Penalty function, and Duality
Penalty function, and Duality
The dual formulation of this convex functional is a key point for our
study.
Let M1,f be the set of all finitely additive measures on (Ω, F), taken
as the dual space of the space of bounded variables.
Theorem : (FS) There exists a penalty function α taking values
in R ∪ {+∞} s.t.
∀Ψ ∈ X ,
ρ(Ψ) = supQ∈M1,f {EQ (−Ψ) − α(Q)}
∀Q ∈ M1,f ,
α(Q) = supΨ∈X {EQ (−Ψ) − ρ(Ψ)} = supΨ∈Aρ {EQ (−Ψ)}
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Penalty function, and Duality
Sketch of the Proof
Example : Entropic risk measure
1 eγ (X) = γE exp(− X) = sup EQ (−X) − γh(Q|P)
γ
Q∈M1
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Penalty function, and Duality
where h is the entropic function

 E ( dQ ln dQ ) if Q P
P dP
dP
h(Q|P) =
 +∞
otherwise
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Continuity of Convex Risk Measures
Continuity of Convex Risk Measures
We are looking for representation in terms of σ-additive measures
• Any convex risk measure ρ defined on X and satisfying eq :duality
proba is continuous from above, in the sense that
Ψn & Ψ =⇒ ρ Ψn % ρ(Ψ).
• The converse is not true in general, but holds under continuity
from below assumption :
Ψn % Ψ
=⇒
ρ Ψn & ρ(Ψ).
Then any additive measure Q such that α(Q) < +∞ is σ-additive
and eq :duality proba holds true. Moreover, from i), ρ is also
continuous by above.
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Risk Measures on L∞ (P)
Risk Measures on L∞ (P)
Some new notations : M1,ac (P) is the set of finitely additive
measures absolutely continuous w. r. to P and M1,ac (P) is the set of
probability measures absolutely continuous w. r. to P.
Theorem
Let P be a given probability measure.
1. Any convex risk measure ρ on X satisfying eq :riskmeasure
equalite may be considered as a risk measure on L∞ (P). A dual
representation holds true in terms of absolutely continuous
additive measures Q ∈ M1,ac (P).
2. ρ admits a dual representation on M1,ac (P) :
ρ(Ψ) =
sup
EQ [−Ψ]−ρ(Ψ) ,
EQ [−Ψ]−α Q
α(Q) = sup
Ψ∈L∞ (P)
Q∈M1,ac (P)
if and only if one of the equivalent properties holds :
a) ρ is continuous from above (Fatou property) ;
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Risk Measures on L∞ (P)
b) ρ is closed for the weak*-topology σ(L∞ , L1 ) ;
c) the acceptance set {ρ ≤ 0} is weak*-closed in L∞ (P).
3. Assume that ρ is a coherent (homogeneous) risk measure,
satisfying the Fatou property. Then,
EQ [−Ψ] | α(Q) = 0
ρ(Ψ) =
sup
(1)
Q∈M1,ac (P)
The supremum in eq :coherent duality is a maximum iff one of
the following equivalent properties holds :
a) ρ is continuous from below ;
b) the convex set Q = {Q ∈ M1,ac α(Q) = 0} is weakly
compact in L1 (P).
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Risk Measures : Examples
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Indifference (Reservation) price
Indifference (Reservation) price
Super-replication price
Given a family of hedging instruments H, the super-replication seller
price is defined as
π s (X) = inf{x| ∃ H ∈ H such that x + H ≥ X}
For the buyer, π b (X) = −π s (−X).
Reservation price Proposed by Hodges-Neuberger in 1989, the
pricing rule is based on expected utility maximization.
the seller price of contingent claim is the smallest amount
leading the seller between selling the claim or doing nothing.
E(u(X − π)) = u(0)
or E(u(W + X − π)) = E(u(W ))
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Indifference (Reservation) price
In the exponential case
1
1 = EP exp − W
EP exp − (W + X − π(X))
γ
γ
or equivalently :
π(X) = eγ W − eγ W + X
i
h
where eγ Ψ = γ ln EP exp − γ1 Ψ
• The indifference pricing rule for the buyer π b (X) has the desired
property of increasing monotonicity, convexity and translation
invariance as :
π(X + m) = π(X) + m
A large Bid-Ask spread
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Risk measure and Super-hedging
Risk measure and Super-hedging
Risk measure generated by a convex subset H
Let H be a convex set such that
inf{m ∈ R, ∃ ξT ∈ H s.t. m ≥ ξT } > −∞.
• A convex risk measure ν H is defined by :
∀Ψ ∈ X
ν H (Ψ) = inf{m ∈ R, ∃ξT ∈ H s.t. m + Ψ ≥ ξT }
−ν H is the “super buyer price”
πb (Ψ) = sup{x, ∃ξT ∈ H s.t. Ψ ≥ x + ξT }
• The associated penalty function is : κH (Q) = supΨ∈H {EQ (−Ψ)}
• Moreover, if H is a cone, ν H is a coherent risk measure in the
sense of Artzner and alii, and the penalty function is
H
l Q = 0 if Q∈MH ; +∞ otherwise
where MH = {Q ∈ M1,f | ∀ξ ∈ H EQ ξ ≥ 0.}
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Risk measure and Super-hedging
Other interpretation Let lH be the convex indicator on X of the
convex set H.
The risk measure ν H , defined in Equation (??), is the largest convex
risk measure dominated by lH and it can be expressed as :
ν H (Ψ) = inf {ρworst (Ψ − ξ) + lH (ξ)}
ξ∈X
where ρworst (Ψ) = supω∈Ω {−Ψ(ω)} is the worst case risk meassure
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