Risk Measures and
Interest Rate Ambiguity
Nicole El Karoui, (Ecole Polytechnique-CMAP),
Claudia Ravanelli, (ISB, Univ of Zurich),
Princeton Workshop - October 6-7, 2006
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Introduction and Motivation
Introduction and Motivation
Risk management
• New developments, new regulatory rules.
• Credit risk regulation : Bâle II
Question : How to take into account ambiguity on interest rates, or time to
default ? Classical monetary risk measures are not adapted tools to do
that.
Design of new financial products in this context
• Credit derivatives
• How to design financial contracts when ambiguity on time to default.
• Which optimal risk transfer in this context ?
Main question
How to separate the specific risk of the future exposure of the
discount (defautable) risk
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Introduction and Motivation
Some related works (among many others... ! !)
? Insurance literature on optimal policy design : Borch (1962), Bühlmann (1970),
Raviv (1979), Gerber (1980)...
? Indifference pricing (often in an exponential framework) :
Hodges-Neuberger(1989), Davis(1997), Rouge-El Karoui(2000), Becherer (2001),
Delbaen et al (2002), Musiela-Zariphopoulou (2004)....
? Risk measures
⇒ (seminal papers) : Deprez-Gerber(1985), Artzner et al.(1999), Delbaen(2000),
Carr-Géman-Madan(2001), Föllmer-Schied(2002), Frittelli-Gianin(2002)...
⇒ (multi-period setting) : Scandolo (2003), Artzner et al.(2004), Cheridito et
al.(2004), Riedel (2004), Weber(2005), Detlefsen-Scandolo (2005), Frittelli-Scandolo
(2005), Klöppel-Schweizer (2005),Föllmer Fo&alii(2006), Bion-Nadal(2006) ? BSDEs
and finance : El Karoui-Quenez(1996), El Karoui-Peng-Quenez(1997), Peng(1997,
2003), Mania-Schweizer(2004), Barrieu-El Karoui(2005)... and many others.
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Convex Risk measures : Basic properties
Convex Risk measures : Basic properties
Definition : Let (Ω, F) be a standard measurable space and X the linear
space of bounded functions (including constant functions).
The functional ρ is a monetary risk measure if it satisfies :
• Convexity, and Decreasing monotonicity ;
• Translation invariance : ∀X ∈ X , ∀m ∈ R, ρ(X + m) = ρ(X) − m.
In particular, ρ(X + ρ(X)) = 0.
Dual representation : The convexity of the framework leads to an
”explicit” representation. There exists a penalty function α taking values in
R ∪ {+∞} such that :
∀Ψ ∈ X ,
ρ(Ψ) = supQ∈M1,f {EQ [−Ψ] − α(Q)}
∀Q ∈ M1,f ,
α(Q) = supΨ∈X {EQ [−Ψ] − ρ(Ψ)}
where M1,f is the set of all additive measures on (Ω, F).
Moreover, the supremum is attained in the first equation in M1,f
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Cash-Invariance and Discounting
Cash-Invariance and Discounting
Definition of monetary risk measure implicitly assumes that future risk
position and risk measure are expressed in the same numéraire.
Convention : 1T = 1 unit of cash available at time T , and D0,T is a random
discount factor.
Spot risk measure (Foellmer-Schied,2004) et Cash-invariance
ρ0 (D0,T XT + m10 ) = ρ0 (D0,T XT ) + ρ0 (m10 ),
ρ0 (m10 ) = −m 10
Forward risk measure (Rouge-El Karoui(2000))
ρT (XT ) = ρT (XT + m1T ) = ρT (XT ) + ρT (m1T ) and ρT (1T ) = −1
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Cash-Invariance and Discounting
Forward versus Spot risk measure
Suppose a zero-coupon bond with maturity T is traded on the market at
B(0, T ). As in interest rates framework, define
R0 D0,T XT := B(0, T)ρT (XT )
• Then, R0 is a cash-invariante risk measure iff ρT satisfies the calibration
−1
. Then, for any QT in the domain of
constraint : ρT (λD−1
0,T ) = −λ B(0, T)
−1
) = B(0, T )−1 .
αT , EQT (D0,T
• Characterisation of R0 domain : Q0 belongs to dom(α0 ) iff the forward
measure QT defined as usual by
dQT
dQ0
= D0,T /B(0, T ) belongs to dom(αT ).
What happens when zero-coupon bonds are non quoted on the
market, and more generally when ambiguity on interest rates
occures
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Cash sub-additive Risk measure
Cash sub-additive Risk measure
Motivation : Given a risk measure ρ and a discount factor D0,T ∈ [0, 1],
observe that, if R(XT ) = ρ(XT D0,T ), then for ∀XT ∈ XT
∀m ≥ 0, ,
R(XT + m1T ) = ρ(XT D0,T + mD0,T )
≥ ρ(XT D0,T + m10 ) = ρ(XT D0,T ) − m10 = R(XT ) − m10 .
We take this property as the natural one to replace the cash invariant axiom,
for risk functional expressed as today cash but directly defined of future
position.
Def : Any convex, decreasing, functional R defined on XT is called Cash
sub-additive Risk measure iff for any XT ∈ XT the function
m ∈ R 7→ R(XT + m1T ) + m10 is nondecreasing.
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Cash sub-additive Risk measure
Example : Ambiguous discount factors
• Assume that the classical risk measure used by the regulator is ρ, a
spot-cash invariant risk measure on XT . But in fact only a confidence
interval for the discount factors is available, that is unknown DT are
ranging between two constants : 0 ≤ dL ≤ dH ≤ 1.
• To give a price to discount factor ambiguity, Regulator assesses risk of
future payoff XT in the worst case interest rates scenario :
ρ,D
R (XT ) := sup ρ(DT XT ) |DT ∈ X , 0 ≤ dL ≤ DT ≤ dH ≤ 1 .
Rρ,D is obviously a cash sub-additive risk measure. Moreover,
R ρ,D (XT ) = ρ(−v(XT )), v(x) = −(dL x+ − dH (−x)+ )
• The classical measure of risk used in global risk management is the Put
premium risk measure (Jarrow (2002)), defined as the expected loss
1 +
R(XT ) = E (−XT ) , r ≥ 1.
r
• Same result still holds, if dH and dL are given FT -r.v in [0, 1].
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Cash sub-additive Risk measure
Proposition
• Let vT be a FT family of real, convex, decreasing functions such : vT (0) = 0,
and (vT )0x ∈ [−1, 0]. Let βT (y) = supx∈R {xy − v(x)} be the FT -measurable
convex Fenchel transform of vT , defined on [-1,0]. For instance, for
ambiguous discount factors, βT (y) = l(−y) where l is the convex indicator
function of the random interval DT = {DT ∈ XT |0 ≤ DL ≤ DT ≤ DH ≤ 1}
• ρ is a cash invariant r.m. on X with minimal penalty function αρ .
⇒ Rρ,v (XT ) := ρ(−v(XT )) is a cash sub-additive r.m., such that
R
ρ,v
(XT ) = sup ρ (DT XT + β(−DT )) | 0 ≤ DT ≤ 1
DT ∈X
⇒ Dual representation.
ρ,v
R
(XT ) = sup
Q, DT
where
n
EQ [−DT XT ] − α
ρ,v
(Q, DT )| 0 ≤ DT ≤ 1
o
αρ,v (Q, DT ) := αρ (Q) + EQ [βT (−DT )]
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Dual Representation
Dual Representation
The dual representation can be obtained
• either using convex analysis tools
• or extending cash sub-additive r.m. to cash invariant r.m. and then
exploiting their dual representation
We are looking for a “minimal” extension of R, based on the key
property : If ρ̂(XT , x) := R(XT 1T − x 1T ) − x, then this functional is cash
invariant as function of (XT , x),
ρ̂(XT + m, x + m) = R(XT + m − (x + m)1T ) − (x + m)
= ρ̂(XT , x) − m
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Minimal enlarged space
Minimal enlarged space
(XT , x) may be viewed as the “coordinates” of some r.v. X̂T on the extended
b = Ω × {1, 0} such that
space Ω
X̂T (ω, θ) = XT (ω)1{θ=1} + x1{θ=0} ,
so that
{θ = 0}is a atome
The correspondence is one to one.
Extended Risk measure
Let R be a cash sub-additive risk measure defined on XT . R generates a
cT of the bounded FbT -r.v.
cash-additive risk-measure ρb on the space X
ρb XT 1{θ=1} + x1{θ=0}
b
= R XT , x := R XT − x 1T − x 10 ,
ρb(XT (ω)1{θ=1} ) can be identified with R(XT ).
b T , y) − y = R(Y
b T , y).
b T , x) = R(XT − x) − x ≤ R(XT − y) − y ≤ R(Y
R(X
In terms of “economic default time”, {θ = 1} may be viewed as {T < τ }
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Dual representation of R
Dual representation of R
Any probability measure Q̂ on Ω̂ can be decomposed into
Q̂(XT 1{θ=1} + x1{θ=1} = Q̂({θ = 1})Q̂(XT |{θ = 1}) + x(1 − Q̂({θ = 1}))
= q̂ Q1 (XT ) + x (1 − q̂) = µ(XT ) + x(1 − µ(1))
Characterization The minimal penalty function α̂(Q̂) = α(µ) depends only
on µ = q̂ Q1 , and

n
o

α̂(Q̂)
= supXT ∈XT ,x∈R EQ̂ [−(XT − x)1θ=1 ] − R(XT − x)



= supYT ∈XT {µ(−YT ) − R(YT )}



 R(X ) = sup
s {µ(−X ) − α(µ)}
T
µ∈M1
T
Remark The risk measure R̂ is not defined on the space XT
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More natural extension
More natural extension
The aim of this second extension is to define a risk measure ρ̃ on a spaceX̃T of
bounded r.v. containing XT . We need
⇒ to define the r.v. X̃T = XT1 1{θ=1} + XT0 1{θ=0}
⇒ to introduce a arbitrary(normalized) monetary risk measure ρ0 on order
to specify a “a priori” risk measure for XT0 .
Two monetary risk measures The functional ρ̃ defined by
ρ̃(XT1
1{θ=1} +
XT0
1{θ=0 }) = R
XT1
+
ρ0 (XT0 )
+ ρ0 (XT0 )
is a monetary risk measure, with minimal penalty function, for any
Q̃ = q̃ 1 Q̃1 + q̃ 0 Q̃0
1
α̃ Q̃ = αR q̃ Q̃
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1
0
+ q̃ α0 Q̃
0
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More natural extension
• The functional ρR (XT ) = ρ̃(XT , XT ) = R (XT + ρ0 (XT )) + ρ0 (XT ) is a
monetary risk measure with penalty functional
αρR (Q) =
inf
(q̃1 ,Q̃1 ,q̃0 ,Q̃0 |Q=q̃1 Q̃1 +q̃0 Q̃0 ))
αR (q̃ 1 Q̃1 ) + q̃ 0 α0 (Q̃0 )
• ρR does not allow us to recover R.
• Given Q̃ ∈ Dom(α̃), and Q the induced probability measure on FT . Then,Q
belongs to Dom(αρR ). Let us denoted by D0,T the Q̃-conditional expectation
of 1θ=1 given FT . Then, Q̃(XT 1θ=1 ) = Q(XT D0,T ), and
(1 − D0,T )
dQ
Q̃(XT 1θ=0 ) = Q(XT (1 − D0,T )), ⇔ dQ =
q̃ 0
0
0
0
R(XT ) = sup(Q,D0,T )∈A {Q(−XT D0,T ) − (αR D0,T .Q + q̃ α Q }
A = {(Q, D0,T )s.t.Q ∈ Dom(αρR ), D0,T .Q ∈ Dom(αR ), Q̃0 ∈ Dom(α0 )}
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Conditional Risk measures
Conditional Risk measures
A natural question is to try to factorize r̃ho in terms of ρR and some
conditional risk measure between FeT and FT .
Convex function and Risk measure on {0, 1} Let ρ be a monetary risk
measure on λ1{θ=1} + µ1{θ=0} . Then
ρ(λ1{θ=1} + µ1{θ=0} ) = ρ((λ − µ)1{θ=1} ) − µ
Put v(x) = ρ(x1{θ=1} ).
Then v is a real convex decreasing function such that v 0 (x) ≥ −1.
Natural extension to previous filtrations, with FT -random VT (XT ).
• Natural way to generate cash subadditive r.m : Rρ, (XT ) := ρ(−VT (XT ))
• In general, such decomposition does not hold
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Optimal Derivative Design
Optimal Derivative Design
Agent A
Exposure XTA
Agent B
Exposure XTB
FT
←−−−−−
−−−−−
π−
0
⇒ We want to determine the optimal transaction (F, π).
Transaction feasibility
? Agent A looks for a hedge of her exposure : inf F ∈X ,π0 RA (XTA − FT ) − π0 .
? Agent B wants to improve her risk measure : RB (XTB + FT ) + π0 ≤ ρB (XTB ).
⇒ Optimal pricing rule :
Princeton-Oct 2006
∗
(πB
)0 (FT )
=
RB XTB
−
RB XTB
+F .
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Inf-convolution
Inf-convolution
Theorem : Let RA and RB be two cash sublinear risk measures with
respective penalty functions αA and αB . Let RA,B be the inf-convolution of
RA and RB
Ψ → RA,B (Ψ) ≡ RA RB (Ψ) = inf
H∈X
RA (Ψ − H) + RB (H)
and assume that RA,B (0) > −∞.
• Then RA,B is a cash sub-linear convex risk measure .
• The associated penalty function is given by
∀µ ∈ Ms1,f ,
αRA,B (µ) = αA (µ) + αB (µ).
• RA,B is continuous from below as soon as this property holds for RA or RB
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Infinitesimal Risk Measures and BSDE’s
Infinitesimal Risk Measures and BSDE’s
Let (Ω, (Ft ), P) be the natural space associated with a Brownian motion
W = (Wt , t ≤ T ). Following BSDE’s point of view (NEK-Peng-Quenez,
Pengg-expectation, Barrieu-NEK, Schweitzer, Delbaen....), we define
−dYt = g(t, Yt , Zt )dt− < Zt , dWt > ,
YT = −XT
The solution is a pair of adapted processes (Rgt , Zt ) in some L2 space.
• (Peng,Barrieu-NEK) if g does not depend on y and convex in z, then
ρt (XT ) = Yt is a cash invariante convex risk measure (+ g with quadratic
growth)
• Only assume g to be convex in both variables (y,z), and decreasing w.r to
y+(linear growth in y quadratic in z). Then Rgt (X) = Yt is a cash
sub-additive risk measure.
• The proof is based on : Rgt (X + m) + m = Ytm is solution of the same
BSDE with coefficient g m (t, y, z) = g(t, y − m, z)+ comparison theorem.
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Infinitesimal Risk Measures and BSDE’s
Dual representation
−dYt = g(t, Yt , Zt )dt− < Zt , dWt > ,
YT = −XT
⇒ Let G(t, β, µ) be the Fenchel transform of g(t, y, z), (β ≥ 0)
⇒ Any Qµ equivalent probability measure w.r. to P
dQµ
:= ΓµT ,
dP
dΓµt = Γµt µ∗t dWt ,
Γµ0 = 1
RT
⇒ DT := e 0 βs ds discount factor (β ≥ 0)h
i
R T − R r β ds
⇒ penalty functions : G(t, β, µ) := EQµ t e t s G(β r , µr )drFt ;
−
Then Yt = Rgt has the following dual representation
Rgt (XT )
h
= ess sup(β, µ)∈A EQµ − e
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−
RT
t
βs ds
XT −
Z
T
e
t
−
Rr
t
βs ds
i
G(r, µ)dr F t
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Summary and conclusions
Summary and conclusions
⇒ We introduce a new class of cash sub-additive risk measures R
⇒ R suitable to asses the risk of financial positions under uncertain interest
rates and other risks (e.g. default risk)
⇒ we extend R to cash invariant r.m. defined on enlarged spaces
• we extend R to cash invariant r.m. defined on enlarged spaces
• Dual representation of R
• we identify R from cash invariant r.m. in term of FT -measures and
FT -stochastic discount factors
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