Pricing, Hedging and Optimally Designing Derivatives
via Minimization of Risk Measures
20th June 2005
Pauline Barrieu 1 and Nicole El Karoui 2
The question of pricing and hedging a given contingent claim has a unique solution in a complete market
framework. When some incompleteness is introduced, the problem becomes however more dicult. Several
approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a
particular choice criterion. Among them, Hodges and Neuberger [72] proposed in 1989 a method based on
utility maximization.
The price of the contingent claim is then obtained as the smallest (resp.
largest)
amount leading the agent indierent between selling (resp. buying) the claim and doing nothing. The price
obtained is the indierence seller's (resp. buyer's) price. Since then, many authors have used this approach,
the exponential utility function being most often used (see for instance, El Karoui and Rouge [51], Becherer
[11], Delbaen et al. [39] , Musiela and Zariphopoulou [92] or Mania and Schweizer [88]...).
In this chapter, we also adopt this exponential utility point of view to start with in order to nd the optimal
hedge and price of a contingent claim based on a non-tradable risk.
But soon, we notice that the right
framework to work with is not that of the exponential utility itself but that of the certainty equivalent which
is a convex functional satisfying some nice properties among which that of cash translation invariance. Hence,
the results obtained in this particular framework can be immediately extended to functionals satisfying the
same properties, in other words to convex risk measures as introduced by Föllmer and Schied [53] and [54]
or by Frittelli and Gianin [57]. Starting with a utility maximization problem, we end up with an equivalent
risk measure minimization in order to price and hedge this contingent claim.
Moreover, this hedging problem can be seen as a particular case of a more general situation of risk transfer
between dierent agents, one of them consisting of the nancial market.
Therefore, we consider in this
chapter the general question of optimal transfer of a non-tradable risk and specify the results obtained in
1 L.S.E., Statistics Department, Houghton Street, London, WC2A 2AE, United Kingdom. (e-mail: [email protected])
Research supported in part by the EPSRC, under Grant GR-T23879/01.
2 C.M.A.P., Ecole Polytechnique, 91128 Palaiseau Cédex, France. (e-mail: [email protected])
1
the particular situation of an optimal hedging problem.
Both static and dynamic approaches are considered in this chapter, in order to provide constructive answers
to this optimal risk transfer problem. Quite recently, many authors have studied dynamic version of static
risk measures (see for instance, among many other references, Cvitanic and Karatzas [35], Scandolo [106],
Weber [111],
Artzner et al.
[3], Cheridito, Delbaen and Kupper [29], Frittelli and Gianin [58], Gianin
[61], Riedel [101] or Peng [97]).
When considering a dynamic framework, our main purpose is to nd a
trade-o between static and very abstract risk measures as we are more interested in tractability issues and
interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore,
after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of
convex risk measures to BSDEs. For the sake of a better understanding, a whole section in the second part
is dedicated to some key results and properties of BSDEs, which are essential to this denition of dynamic
convex risk measures.
Part I: Static Framework
In this chapter, we focus on the question of optimal hedging of a given risky position in an incomplete market
framework. However, instead of adopting a standard point of view, we look at it in terms of an optimal risk
transfer between dierent economic agents, one of them being possibly a nancial market.
The risk that we consider here is not (directly) traded on any nancial market. We may think for instance of
a weather risk, a catastrophic risk (natural catastrophe, terrorist attack...) but also of any global insurance
risk that may be securitized, such as the longevity of mortality risk...
First adopting a static point of view, we proceed in several steps. In a rst section, we relate the notion
of indierence pricing rule to that of transaction feasibility, capital requirement, hedging and naturally introduce convex risk measures. Then, after having introduced some key operations on convex risk measures,
in particular the dilatation and the inf-convolution, we study the problem of optimal risk transfer between
two agents. We see how the risk transfer problem can be reduced to an inf-convolution problem of convex
functionals. We solve it explicitly in the dilated framework and give some necessary and sucient conditions
in the general framework.
1 Indierence Pricing, Capital Requirement and Convex Risk Measures
As previously mentioned in the introduction, since 1989 and the seminal paper by Hodges and Neuberger
[72] indierence pricing based on a utility criterion has been a popular (academic) method to value claims
in an incomplete market. Taking the buyer point of view, the indierence price corresponds to the maximal
amount
π , the agent having a utility function u is ready to pay for a claim X .
2
In other words,
π is determined
as the amount the agent pays such that her expected utility remains unchanged when doing the transaction:
E[u(X − π)] = u(0).
This price is not a transaction price. It gives an upper bound (for the buyer) to the price of this claim so that
a transaction will take place.
π
also corresponds to the certainty equivalent of the claim payo
X.
Certain
properties this indierence price should have are rather obvious: rst it should be an increasing function
of
X
but also a convex function in order to take into account the diversication aspect of considering a
portfolio of dierent claims rather than the sum of dierent individual portfolios. Another property which is
rather interesting is the cash translation invariance property. More precisely, it seems natural to consider the
situation where translating the payo of the claim
of the price
π
X
by a non-risky amount
m
simply leads to a translation
by the same amount. It is the case, as we will see for the exponential utility in the following
subsection.
1.1 The Exponential Utility Framework
First, let us notice that exponential utility functions have been widely used in the nancial literature. Several
facts may justify their relative importance compared to other utility functions but, in particular, the absence
of constraint on the sign of the future considered cash ows and its relationship with probability measures
make them very convenient to use.
1.1.1 Indierence Pricing Rule
In this introductory subsection, we simply consider an agent, having an exponential utility function
−γ exp −
1
γ x , where
γ
is her risk tolerance coecient. She evolves in an uncertain universe modelled by a
standard probability space
is uncertain, since
W
(Ω, =, P)
with time horizon
T.
The wealth
W
of the agent at this future date
the sake of simplicity, we neglect interest rate between
X
T
can be seen as a particular position on a given portfolio or as the book of the agent.
To reduce her risk, she can decide whether or not to buy a contingent claim with a payo
and
U (x) =
0
and
T
X
at time
T.
For
and assume that both random variables
W
are bounded.
In order to decide whether or not she will buy this claim, she will nd the maximum price she is ready to pay
for it, her indierence price
π(X) for the claim X
given by the constraint
EP U (W +X −π(X)) = EP U (W ) .
Then,
1
1 EP exp − (W + X − π(X))
= EP exp − W
γ
γ
⇔ π(X|W ) = eγ W − eγ W + X
where
eγ
is the opposite of the certainty equivalent, dened for any bounded random variable
eγ Ψ , γ ln EP
The indierence pricing rule
translation invariance:
π(X|W )
h
Ψ
1 i
exp − Ψ .
γ
(1)
has the desired property of increasing monotonicity, convexity and
π(X + m|W ) = π(X|W ) + m.
3
Moreover, the functional
eγ (X) = −π(X|W = 0)
has
eγ (Ψ + m) =
similar properties; it is decreasing, convex and translation invariant in the following sense:
eγ (Ψ) − m.
1.1.2 Some Remarks on the "Price" π(X)
π(X) does not correspond to a transaction price but simply gives an indication of the transaction price range
since it corresponds to the maximal amount the agent is ready to pay for the claim
X
and bear the associated
risk given her initial exposure. This dependency seems quite intuitive: for instance, the considered agent can
be seen as a trader who wants to buy the particular derivative
X
without knowing its price. She determines
it by considering the contract relatively to her existing book.
π b (X|W ),
For this reason and for the sake of a better understanding, we will temporarily denote it by
the
upper-script "b" standing for "buyer". This heavy notation underlines the close relationship between the
pricing rule and the actual exposure of the agent.
The considered framework is symmetric since there is
no particular requirement on the sign of the dierent quantities. Hence, it is possible to dene by simple
analogy the indierence seller's price of the claim
X.
Let us denote it by
π s (X|W ),
the upper-script "s"
standing for "seller". Both seller's and buyer's indierence pricing rules are closely related as
π s (X|W ) = −π b (−X|W ).
Therefore, the seller's price of
X
is simply the opposite of the buyer's price of
−X .
Such an axiomatic approach of the pricing rule is not new. This was rst introduced in insurance under the
name of convex premium principle (see for instance the seminal paper of Deprez and Gerber [42] in 1985)
and then developed in continuous time nance (see for instance El Karoui and Quenez [49]).
When adopting an exponential utility criterion to solve a pricing problem, the right framework to work with
seems to be that of the functional
eγ
and not directly that of utility. This functional, called entropic risk
measure, holds some key properties of convexity, monotonicity and cash translation invariance. It is therefore
possible to generalize the utility criterion to focus more on the notion of price keeping in mind these wished
properties. The convex risk measure provides such a criterion as we will see in the following.
1.2 Convex Risk Measures: Denition and Basic Properties
Convex risk measures can have two possible interpretations depending on the representation which is used:
they can be considered either as a pricing rule or as a capital requirement rule. We will successively present
both of them in the following, introducing each time the vocabulary associated with this particular approach.
1.2.1 Risk Measure as an Indierence Price
We rst recall the denition and some key properties of the convex risk measures introduced by Föllmer and
Schied [53] and [54]. The notations, denitions and main properties may be found in this last reference [54].
In particular, we assume that uncertainty is described through a measurable space
(Ω, =),
and that risky
positions belong to the linear space of bounded functions (including constant functions), denoted by
4
X.
Denition 1.1
ρ:X →R
The functional
is a (monetary) convex risk measure if, for any
Φ
and
Ψ
in
X,
it satises the following properties:
ρ λΦ + (1 − λ)Ψ ≤ λρ(Φ) + (1 − λ)ρ(Ψ);
∀λ ∈ [0, 1]
a)
Convexity:
b)
Monotonicity:
c)
Translation invariance:
Φ ≤ Ψ ⇒ ρ(Φ) ≥ ρ(Ψ);
A convex risk measure
d)
Homogeneity :
ρ
∀m ∈ R
ρ(Φ + m) = ρ(Φ) − m.
is coherent if it satises also:
ρ λ Φ = λρ Φ .
∀λ ∈ R+
Note that the convexity property is essential: this translates the natural fact that diversication should not
increase risk. In particular, any convex combination of admissible risks should be admissible. One of the
major drawbacks of the famous risk measure VAR (Value at Risk) is its failure to meet this criterion. This
may lead to arbitrage opportunities inside the nancial institution using it as risk measure as observed by
Artzner, Delbaen, Eber and Heath in their seminal paper [2].
Intuitively, given the translation invariance,
ρ(X)
may be interpreted as the amount the agent has to hold
to completely cancel the risk associated with her risky position
X
since
ρ(X + ρ(X)) = ρ(X) − ρ(X) = 0.
ρ(X)
(2)
can be also considered as the opposite of the "buyer's indierence price" of this position, since when
paying the amount
−ρ(X),
the new exposure
X − (−ρ(X))
does not carry any risk with positive measure,
i.e. the agent is somehow indierent using this criterion between doing nothing and having this "hedged"
exposure.
The convex risk measures appear therefore as a natural extension of utility functions as they can be seen
directly as an indierence pricing rule.
1.2.2 Dual Representation
In order to link more closely both notions of pricing rule and risk measure, the duality between the Banach
space
X
endowed with the supremum norm
k.k
and its dual space
additive set functions with nite total variation on
(Ω, =),
X 0,
identied with the set
Mba
of nitely
can be used as it leads to a dual representation.
The properties of monotonicity and cash invariance allow to restrict the domain of the dual functional to the
set
M1,f
of all nitely additive measures (Theorem
4.12
in [54]). The following theorem gives an "explicit"
formula for the risk measure (and as a consequence for the price) in terms of expected values:
Theorem 1.2
Let
M1,f
function taking values in
be the set of all nitely additive measures on
R∪
(Ω, =),
and
α Q
the minimal penalty
+∞ :
α Q = sup EQ [−Ψ] − ρ(Ψ)
Ψ∈X
Dom(α) = {Q ∈ M1,f | α Q < +∞}
∀Q ∈ M1,f
≥ −ρ(0) .
(3)
(4)
The Fenchel duality relation holds :
∀Ψ ∈ X
ρ(Ψ) =
sup
Q∈M1,f
5
EQ [−Ψ] − α Q
(5)
Moreover, for any
Ψ∈X
there exists an optimal additive measure
QΨ ∈ M1,f
such that
ρ(Ψ) = EQΨ [−Ψ] − α QΨ = max EQ [−Ψ] − α Q .
Q∈M1,f
Henceforth,
α(Q)
is the minimal penalty function, denoted by
ρ
The dual representation of
αmin (Q)
in
[54].
given in Equation (5) emphasizes the interpretation in terms of a worst case
related to the agent's (or regulator's) beliefs.
Convex Analysis Point of view
penalty function
α
We start with Remark 4.17 and the Appendices 6 and 7 in
dened in (5) corresponds to the Fenchel-Legendre transform on the Banach space
ρ.
of the convex risk-measure
The dual space
set functions with nite total variation.
compact in
[54]. The
0
ba
X =M
and the functional
X0
can be identied with the set
Then the subset
Q → α(Q)
M1,f
Mba
X
of nitely additive
of "nite probability measure" is weak*-
is weak*-lower semi-continuous (or weak*-closed) as
supremum of ane functionals. This terminology from convex analysis is based upon the observation that
lower semi-continuity and the closure of the level sets
{φ ≤ c}
lower semi-continuous (lsc) with respect to the weak topology
strongly closed given that
ρ
are equivalent properties.
σ(X , X 0 )
since any set
{ρ ≤ c}
Moreover
ρ
is
is convex and
is strongly Lipschitz-continuous. Then, general duality theorem for conjugate
functional yields to
ρ(Ψ) = sup (r(Ψ) − ρ∗ (r)),
ρ∗ (r) = sup (r(Ψ) − ρ(Ψ))
Ψ∈X
r∈Mba
with the convention
invariance of
ρ
rQ (Ψ) = EQ [−Ψ]
to prove that when
upper semi-continuous functional
for
Q ∈ M1,f .
We then use the properties of monotonicity and cash
ρ∗ (r) < +∞, −r ∈ M1,f .
EQ [−Ψ] − α(Q)
Moreover by weak*-compacity of
attains its maximum on
M1,f ,
the
M1,f .
In the second part of this chapter, we will intensively used convex analysis point of view when studying
dynamic convex risk measures.
Duality and Probability Measures
representation (5) in terms of
clarity, we use the notation
We are especially interested in the risk measures that admit a
σ -additive probability measures Q.
Q ∈ M1,f
In this paper, for the sake of simplicity and
when dealing with additive measures and
probability measures. So, we are looking for the following representation on
ρ(Ψ) = sup
Q ∈ M1
when considering
X
EQ [−Ψ] − α Q .
(6)
Q∈M1
We can no longer expected that the supremum is attained without additional assumptions. Such representation on
M1
is closely related to some continuity properties of the convex functional
Proposition 4.21 in
Proposition 1.3
i)
ρ
(Lemma 4.20 and
[54]).
Any convex risk measure
ρ
dened on
X
and satisfying (6) is continuous from above,
in the sense that
Ψn & Ψ
ii)
ρ Ψn % ρ(Ψ).
=⇒
The converse is not true in general, but holds under continuity from below assumption:
Ψn % Ψ
ρ Ψn & ρ(Ψ).
=⇒
6
Then any additive measure
Q
such that
α(Q) < +∞
is
σ -additive
and (6) holds true. Moreover, from
i), ρ
is also continuous by above.
1.2.3 Risk Measures on L∞ (P)
The representation theory on
L∞ (P)
was developed in particular by Delbaen [38] and extended by Frittelli
and Gianin [57] and [58] and [54]. When a probability measure
measures
ρ
on
L∞ (P)
instead of on
X
P
is given, it is natural indeed to dene risk
satisfying the compatibility condition:
ρ(Ψ) = ρ(Φ)
Ψ = Φ P − a.s.
if
(7)
Let us introduce some new notations: M1,ac (P) is the set of nitely additive measures absolutely continuous
w. r. to
P
and
M1,ac (P)
is the set of probability measures absolutely continuous w. r. to
Ψn & Ψ P − a.s. =⇒
L∞ (P): ( Ψn % Ψ P − a.s. =⇒ ρ Ψn & ρ(Ψ)).
We also dene natural extension of continuity from below in the space
ρ Ψn % ρ(Ψ)),
P.
or continuity from above in the space
L∞ (P):
(
These additional results on conjugacy relations are given in [54] Theorem 4.31 and in Delbaen [38] Corollary
4.35). Sometimes, as in [38], the continuity from above is called the Fatou property.
Theorem 1.4
Let
P
be a given probability measure.
1. Any convex risk measure
ρ
on
X
satisfying (7) may be considered as a risk measure on
representation holds true in terms of absolutely continuous additive measures
2.
ρ
admits a dual representation on
α(Q) =
sup
L∞ (P).
A dual
Q ∈ M1,ac (P).
M1,ac (P):
EQ [−Ψ] − ρ(Ψ) ,
ρ(Ψ) =
Ψ∈L∞ (P)
sup
EQ [−Ψ] − α Q
Q∈M1,ac (P)
if and only if one of the equivalent properties holds:
a)
ρ
is continuous from above (Fatou property);
b)
ρ
is closed for the weak*-topology
c)
the acceptance set
3. Assume that
ρ
{ρ ≤ 0}
σ(L∞ , L1 );
L∞ (P).
is weak*-closed in
is a coherent (homogeneous) risk measure, satisfying the Fatou property. Then,
ρ(Ψ) =
sup
EQ [−Ψ] | α(Q) = 0
(8)
Q∈M1,ac (P)
The supremum in (8) is a
a)
ρ
b)
the convex set
maximum i one of the following equivalent properties holds:
is continuous from below;
Q = {Q ∈ M1,ac α(Q) = 0}
is weakly compact in
L1 (P).
According to the Dunford-Pettis theorem, the weakly relatively compact sets of
L1 (P)
integrable variables and La Vallée-Poussin gives a criterion to check this property.
A
of
L1 (P)
is weakly relatively compact i it is closed and uniformly integrable.
the La Vallée-Poussin criterion, an increasing convex continuous function
Φ(x)
lim
= +∞
x→∞ x
dQ
sup EP Φ(
) < +∞.
dP
Q∈A
and
7
Therefore, the subset
Moreover, according to
Φ : R+ → R,
function) such that:
are sets of uniformly
(also called Young's
1.3 Comments on Measures of Risk and Examples
1.3.1 About Value at Risk
Risk measures, just as utility functions, go beyond the simple problem of pricing. Both are inherently a choice
or decision criterion. More precisely, when assessing the risk related to a given position in order to dene the
amount of capital requirement, a rst natural approach is based on the distribution of the risky position itself.
In this framework, the most classical measure of risk is simply the variance (or the mean-variance analysis).
However, it does not take into account the whole distribution's features (as asymmetry or skewness) and
especially it does not focus on the real nancial risk which is the downside risk. Therefore dierent methods
have been developed to focus on the risk of losses: the most widely used (as it is recommended to bankers
by many nancial institutions) is the so-called Value at Risk (denoted by
lower tail of the distribution. More precisely, the
V AR
V AR),
based on quantiles of the
associated with the position
X
at a level
ε
is dened
as
V ARε X = inf k : P(X + k < 0) ≤ ε .
The
V AR
corresponds to the minimal amount to be added to a given position to make it acceptable. Such
∀m ∈ R,
∀λ ≥ 0, V ARε λX =
a criterion satises the key properties of decreasing monotonicity, translation invariance since
V ARε X + m = V ARε X − m
λV ARε X .
and nally, the
V AR
is positive homogeneous as
This last property reects the linear impact of the size of the position on the risk measure.
as noticed by Artzner et al.
However,
[2] this criterion fails to meet a natural consistency requirement:
it is not
a convex risk measure while the convexity property translates the natural fact that diversication should
not increase risk. In particular, any convex combination of admissible risks should be admissible. The
absence of convexity of the
V AR
may lead to arbitrage opportunities inside the nancial institution using
such criterion as risk measure. Based on this logic, Artzner et al. [2] have adopted a more general approach
to risk measurement. Their paper is essential as it has initiated a systematic axiomatic approach to risk
measurement. A coherent measure of risk should be convex and satisfy the three key properties of the
Conditional Value at Risk
V AR
For instance, a coherent version of the Value at Risk is the so-called Condi-
tional Value at Risk as observed by Rockafellar and Uryasev [103]. This risk measure is denoted by
CV ARε
and dened as
i
h1
CV ARλ (X) = inf E (X − K)− − K .
K
λ
This coincide with the Expected Shortfall under some assumptions for the
see Corollary 5.3 in Acerbi and Tasche [1]). In this case, the
CV AR
X -distribution
(for more details,
can be written as
CV ARλ (X) = E[−X|X + V ARλ (X) < 0].
Moreover, the
CV AR also coincides with another coherent version of the V AR, called Average
and denoted by
AV AR.
This risk measure is dened as:
AV ARλ (Ψ) =
1
λ
Z
8
λ
V AR (Ψ)d.
0
Value at Risk
For more details, please refer for instance to Föllmer and Schied [54] (Proposition 4.37).
More recently, the axiom of positive homogeneity has been questioned. Indeed, such a condition does not
seem to be compatible with the notion of liquidity risk existing on the market as it implies that the size of
the risky position has simply a linear impact on the risk measure. To tackle this shortcoming, Föllmer and
Schied consider, in [53] and [54], convex risk measures as previously dened.
1.3.2 Risk Measures and Utility Functions
Entropic Risk Measure
The most famous convex risk measure on
measure dened as the functional
work.
eγ
L∞ (P)
is certainly the entropic risk
in the previous section when considering an exponential utility frame-
The dual formulation of this continuous from below functional justies the name of entropic risk
measure since:
∀Ψ ∈ L∞ (P)
where
h(Q|P)
h
1 i
eγ (Ψ) = γ ln EP exp − Ψ = sup EQ [−Ψ] − γh Q|P
γ
Q∈M1
is the relative entropy of
Q
with respect to the prior probability measure
h dQ dQ i
h Q|P = EP
ln
dP
dP
Since
eγ
is continuous from below in
∀Ψ ∈ L∞ (P)
L∞ (P),
if
QP
and
+∞
P,
dened by
otherwise.
by the previous theorem
h
1 i
eγ (Ψ) = γ ln EP exp − Ψ = maxQ∈M1,ac EQ [−Ψ] − γh Q|P
γ
As previously mentioned in Paragraph 1.1.1, this particular convex risk measure is closely related to the
exponential utility function and to the associated indierence price. However, the relationships between risk
measures and utility functions can be extended.
Risk Measures and Utility Functions
More generally, risk measures and utility functions have close
relationships based on the hedging and super-replication problem. It is however possible to obtain a more
general connection between them using the notion of shortfall risk.
More precisely, any agent having a utility function
considered position
Ψ ∈ L∞ (P): EP [U (Ψ)].
L
assesses her risk by taking the expected utility of the
If she focuses on her "real" risk, which is the downside risk,
it is natural to consider instead the loss function
consequence,
U
L
dened by
L(x) = −U (−x)
([54] Section 4.9).
As a
is a convex and increasing function and maximizing the expected utility is equivalent to
minimize the expected loss (also called the shortfall risk ),
EP [L(−Ψ)].
It is then natural to introduce the following risk measure as the opposite of the indierence price:
ρ(X) = inf{m ∈ R EP [L(−Ψ − m)] ≤ l(0)}.
Moreover, there is an explicit formula for the associated penalty function given in terms of the FenchelLegendre transform
L∗ (y) = sup{−xy − lL(x)}
α(Q) = inf
λ>0
of the convex function
n1
L
([54] Theorem 4.106):
dQ o
L(0) + EP L∗ λ
.
λ
dP
9
1.4 Risk Measures and Hedging
In this subsection, we come back to the possible interpretation of the risk measure
ρ(X)
in terms of capital
requirement. This leads also to a natural relationship between risk measure and hedging. We then extend
it to a wider perspective of super-hedging.
1.4.1 Risk Measure and Capital Requirement
Looking back at Equation (2), the risk measure
ρ(X) gives an assessment of the minimal capital
requirement
to be added to the position as to make it acceptable in the sense that the new position (X and the added
capital) does not carry any risk with non-negative measure any more. More formally, it is natural to introduce
Aρ
the acceptance set
related to
ρ
dened as the set of all acceptable positions in the sense that they do not
require any additional capital:
Aρ = Ψ ∈ X ,
Given that the epigraph of the convex risk measure
X × R ρ(Ψ + m) ≤ 0},
the characterization of
ρ
ρ
ρ(Ψ) ≤ 0 .
is
(9)
epi(ρ) = {(Ψ, m) ∈ X × R ρ(Ψ) ≤ m} = {(Ψ, m) ∈
in terms of
Aρ
is easily obtained
ρ(X) = inf m ∈ R; m + X ∈ Aρ .
This last formulation makes very clear the link between risk measure and capital requirement.
From the denition of both the convex risk measure
of the risk measure
ρ,
ρ and the acceptance set Aρ
and the dual representation
it is possible to obtain another characterization of the associated penalty function
α
as:
α Q = sup EQ [−Ψ],
if
Q∈M1,f ,
= +∞,
if not.
(10)
Ψ∈Aρ
α(Q)
is the support function of
−Aρ ,
homogeneous) risk measure, then
By denition, the set
m + X = ξ ∈ Aρ ,
with
H
0.
Aρ
denoted by
α(Q)
ΣAρ (Q).
Aρ
When
only takes the values
0
and
is "too large" in the following sense:
we cannot have an explicit formulation for
ξ
is a cone, i.e.
ρ
is a coherent (positive
+∞.
even if we can write
m + X ∈ Aρ
and in particular cannot compare
Therefore, it seems natural to consider a (convex) class of variables
H
such that
as
m+X
m + X ≥ H ∈ H.
appears as a natural (convex) set from which a risk measure can be generated.
1.4.2 Risk Measures Generated by a Convex Set
Risk Measures Generated by a Convex in X
In this section, we study the generation of a convex risk
measure from a general convex set.
Denition 1.5
the functional
Given a non-empty convex subset
νH
on
H
of
X
such that
inf{m ∈ R ∃ξ ∈ H, m ≥ ξ} > −∞,
X
ν H (Ψ) = inf m ∈ R; ∃ξ ∈ H, m + Ψ ≥ ξ
is a convex risk measure. Its minimal penalty function
10
αH
is given by:
αH (Q) = supH∈H EQ [−H].
(11)
The main properties of this risk measure are listed or proved below:
1. The acceptance set of
νH
H
contains the convex subsets
and
AH =
Ψ ∈ X , ∃ξ ∈ H, Ψ ≥ ξ
Aν H = AH if the last subset is closed in the following sense: For ξ ∈ AH and Ψ ∈ X ,
{λ ∈ [0, 1] > λξ + (1 − λ)Ψ ∈ AH } is closed in [0, 1] (see Proposition 4.6 in [54]).
Moreover,
set
2. The penalty function
αH
associated with
ΣAν H Q = supX∈Aν H EQ [−X].
For any
X ∈ Aν H
there exist
ΣH
Q = supH∈H EQ [−H].
deduce that
3. When
Σ
Aν H
≤Σ
H
is the support function of
Let us show that
>0
respect to the additive measure
νH
ξ∈H
and
Q ∈ M1,f ,
αH
−Aν H
is also nothing else but
such that
−X ≤ −ξ + .
it follows that
dened by
Taking the "expectation" with
X ∈ Aν H
H is a cone, the corresponding risk measure is coherent (homogeneous).
where
on the left hand side, we
; the desired result follows from the observation that
Q∈MH , +∞ otherwise,
if
the
ΣH :
H
is included in
Aν H .
αH
Q =
The penalty function
is the indicator function (in the sense of the convex analysis) of the orthogonal cone
0
.
αH Q =
EQ [−X] ≤ EQ [−ξ] + ε ≤ ΣH Q + ε
Taking the supremum with respect to
MH : lMH
where
MH = Q ∈M1,f ; ∀ξ ∈ H, EQ [−ξ] ≤ 0 .
The dual formulation of
νH
is simply given for
It is natural to associate the convex indicator
lH
Ψ∈X
X
on
ν H (Ψ) = supQ∈MH EQ [−Ψ].
by:
with the set
H, lH (X) = 0 if X ∈ H ; +∞ otherwise.
This convex functional is not translation invariant, and therefore it is not a convex risk measure. Nevertheless,
lH
and
νH
are closely related as follows:
Corollary 1.6
Let
The risk measure
lH
νH,
be the convex indicator on
X
of the convex set
H.
dened in Equation (11), 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)}
Proof:
Let
L = {m ∈ R, ∃ξ ∈ H, m ≥ ξ}.
Moreover, for any
νH
variables in
ν
H
This set is a half-line with lower bound
m0 ∈
/ L, m0 ≤ inf ξ∈H supω ξ(ω).
The same arguments hold for
Therefore,
is the worst case risk measure.
Therefore,
ν (0) = inf ξ∈H supω ξ(ω) = inf ξ ρworst (−ξ).
H
ν (Ψ). may be interpreted as the worst case risk measure
H.
inf ξ∈H supω ξ(ω).
H
ρworst
reduced by the use of (hedging)
This point of view would be generalized in Corollary 3.6 in terms of the inf-convolution
H
= ρworst l .
Risk Measures Generated by a Convex Set in
L∞ (P).
The functional
to be understood in
ν
H
on
L∞ (P),
L∞ (P)
i.e.
L∞ (P)
Assume now
H
to be a convex subset of
is still dened by the same formula (11), in which the inequality has
P − a.s.,
with a penalty function only dened on
H
α (Q) = supH∈H EQ [−H].
11
M1,ac (P)
and given by
H
The problem is then to give condition(s) on the set
M1,ac (P)
M1,ac (P).
and not only on
above of the risk measure
νH
to ensure that the dual representation holds on
By Theorem 1.4, this problem is equivalent to the continuity from
or equivalently to the weak*-closure of its acceptance set
AH .
Properties of
this kind are dicult to check and in the following, we will simply give some examples where this property
holds.
1.5 Static Hedging and Calibration
In this subsection, we consider some examples motivated by nancial risk hedging problems.
1.5.1 Hedging with a Family of Cash Flows
We start with a very simple model where it is only possible to hedge statically over a given period using
a nite family of bounded cash ows
denoted by
{π1 , π2 , ..., πd }.
{C1 , C2 , ..., Cd },
the (forward) price of which is known at time
0
and
All cash ows are assumed to be non-negative and non-redundant. Constants
may be included and then considered as assets.
coherent in the sense that
We assume that the dierent prices are
∃ Q0 ∼ P, s.t. ∀i, EQ0 [Ci ] = πi
Such an assumption implies in particular that any inequality on the cash ows is preserved on the prices.
Gi = Ci − πi .
The quantities of interest are often the gain values of the basic strategies,
We can naturally introduce the non-empty set
Qe
of equivalent "martingale measures" as
Qe = { Q| Q ∼ P, s.t. ∀i, EQ [Gi ] = 0 }
The dierent instruments we consider are very liquid; by selling or buying some quantities
ments, we dene the family
Θ=
Θ
n
of gains associated with trading strategies
G(θ) =
d
X
θ i Gi , θ ∈ R d ,
with initial value
i=1
θi
of such instru-
θ:
d
X
θ i πi
o
i=1
This framework is very similar to Chapter 1 in Föllmer and Schied [54] where it is shown that the assumption
of coherent prices is equivalent to the absence of arbitrage opportunity in the market dened as
(AAO)
G(θ) ≥ 0 P a.s.
These strategies can be used to hedge a risky position
strategy is a par
sell
price π↑ (Y
(m, θ)
such that
m + G(θ) ≥ Y, a.s.
−Y .
So, by setting
In the classical nancial literature, a superhedging
This leads to the notion of superhedging (super-seller)
) = inf{ m | ∃ G(θ) s.t. m + G(θ) ≥ Y }.
static superhedging price of
Y.
⇒ G(θ) = 0 P a.s.
In terms of risk measure, we are concerned with the
H = −Θ,
we dene the risk measure
νH
as
ν H (X) = π↑sell (−X) = inf{m ∈ R, ∃θ ∈ Rd : m + X + G(θ) ≥ 0}
Let us observe that the no arbitrage assumption implies that
−∞.
Moreover, the dual representation of the risk measure
EQ0 [G(θ)] = 0.
νH
Hence,
ν H (0) ≥ EQ0 [−X] >
in terms of probability measures is closely
related to the absence of arbitrage opportunity as underlined in the following proposition (Chapter 4 in [54]):
12
Proposition 1.7
i)
If the market is arbitrage-free, i.e.
(AAO) holds true, the convex risk measure ν H can
be represented in terms of the set of equivalent "martingale" measures
ν H (Ψ) = sup EQ (−X),
where
Qe
as
Qe = {Q ∼ P, EQ (Gi ) = 0, ∀i = 1...d}.
(12)
Q∈Qe
By Theorem 1.2, this
ii)
L∞ (P)-risk
measure is continuous from above.
Moreover, the market is arbitrage-free if
P(X < 0) > 0
that
and
νH
is
sensitive in the sense that ν H (Ψ) > ν H (0) for all Ψ such
P(X ≤ 0) = 1.
1.5.2 Calibration Point of View and Bid-Ask Constraint
This point of view is often used on nancial markets when cash ows depend on some basic assets
(S1 , S2 , ..., Sn ),
whose characteristics will be given in the next paragraph.
We can consider for instance
(Ci ) as payos of derivative instruments, suciently liquid to be used as calibra-
tion tools and static hedging strategies. So far, all agents having access to the market agree on the derivative
prices, and do not have any restriction on the quantity they can buy or sell.
We now take into account some restrictions on the trading.
(forward) price of the dierent cash ows. We denote by
We rst introduce a bid-ask spread on the
πiask (Ci )
the market buying price and by
πibid (Ci )
the market selling price. The price coherence is now written as
∃ Q0 ∼ P,
∀i, πiask (Ci ) ≤ EQ0 [Ci ] ≤ πibid (Ci )
To dene the gains family, we need to make a distinction between cash ows when buying and cash ows
when selling.
Ci ,
both gains
To do that, we double the number of basic gains, by associating, with any given cash-ow
Gbid
= Ci − πibid
i
and
Gask
= πiask (Ci ) − Ci .
i
notation and we still denote any gain by
Gi .
∃ Q0 ∼ P,
Henceforth, we do not make distinction of the
The price coherence is then expressed as
∀ i = 1....2d, EQ0 [Gi ] ≤ 0.
The set of such probability measures, called super-martingale measures, is denoted by
Qse .
Note that the
s
coherence of the prices implies that the set Qe is non empty.
Using this convention, a strategy is dened by a
2d-dimensional
vector
θ,
the components of which are all
non-negative. More generally, we can introduce more trading restriction on the size of the transaction by
constraining
θ
to belong to a convex set
K ⊆ R2d
+
such that
0 ∈ K.
Note that we can also take into account
some limits to the resources of the investor, in such way the initial price
In any case, we still denote the set of admissible strategies by
Θ=
n
G(θ) =
P2d
i=1 θi Gi ,
θ∈K
K
hθ, πi
has an an upper bound.
and the family of associated gains by:
o
.
In this constrained framework, the relationship between price coherence and (
AAO) on Θ has been studied
in details in Bion-Nadal [17] but also in Chapter 1 of [54].
More precisely, as above, the price coherence implies that the risk measure
identically
νH
related to
H = −Θ
is not
−∞.
A natural question is to extend the duality relationship (12) using the subset of super-martingale measures.
13
Using Paragraph 1.4.2, this question is equivalent to show that the minimal penalty function is innite
outside of the set of absolutely continuous probability measures and that
When studying the risk measure
νH
αH (Q) = sup EQ [−ξ] = sup EQ [G(θ)].
ξ∈H
0 ∈ K,
if
is continuous from above.
(Denition 1.5 and its properties), we have proved that:
∀Q ∈ M1,ac (P),
In particular, since
νH
Q ∈ Qse ,
then
α(Q) = 0.
θ∈K
Moreover, if
Θ
is a cone, then
αH
is the indicator
s
function of Qe .
νH
It remains to study the continuity from above of
and especially to relate it with the absence of arbitrage
opportunity in the market. We summarize below the results Föllmer and Schied obtained in Theorem 4.95
and Corollary 9.30 [54].
Proposition 1.8
νH
risk measure
Let the set
K
be a closed subset of
is sensitive. In this case,
νH
ν H (Ψ) =
Rd .
Then, the market is arbitrage-free if and only if the
is continuous from above and admits the dual representation:
sup
EQ [−Ψ] − αH (Q) .
Q∈M1,ac
1.5.3 Dynamic Hedging
A natural extension of the previous framework is the multi-period setting or more generally the continuoustime setting.
We briey present some results in the latter case.
Note that we will come back to these
questions, in the second part of this chapter, under a slightly dierent form, assuming that basic asset prices
are Itô's processes.
We now consider a time horizon
nancial market with
n
Let
Qac
P,
S
[38],
Qac
(Ft ; t ∈ [0, T ])
P.
Q
the
on the probability space
family
of
absolutely
local-martingale}.
continuous
(AAO)
is a closed convex subset of
Q0 ∼ P
such that
martingale
ensures that the set
0
hθu , dSu i = (θ.S)t .
S
is a
measures:
and a
follows a special locally
Qac
Q0 − local-martingale.
Qac
is non empty.
=
{ Q | Q
Then, as in Delbaen
1
L (P).
Let us now introduce dynamic strategies as predictable processes
Rt
S
(Ω, F, P)
To avoid arbitrage, we assume that:
There exists a probability measure
be
is a
a ltration
basic assets, whose (non-negative) vector price process
bounded semi-martingale under
(AAO)
T,
θ
and their gain processes
Gt (θ) =
We only consider bounded gain processes and dene:
ΘST = {GT (θ) = (θ.S)T | θ.S
is bounded}
Delbaen and Schachermayer have established in [40], as in the static case, the following duality relationship,
sup{ EQ [−X] | Q ∈ Qac } = inf{ m | ∃ GT (θ) ∈ ΘST s.t. m + X + GT (θ) ≥ 0}
Putting
H = −ΘST ,
this equality shows that
νH
is a coherent convex risk measure continuous from above.
14
Constrained portfolios
When constraints are introduced on the strategies, everything becomes more
complex. Therefore, we refer to the course held by Schied [108] for more details.
We assume that hedging positions live in the following convex set:
ΘST = {GT (θ) = (θ.S)T | θ.S
is bounded by below, θ
The set of constraints is closed in the following sense: the set
∈ K}
R
{ θdS | θ ∈ K} is closed in the semi-martingale
or Émery topology. The optional decomposition theorem of Föllmer and Kramkov [52] implies the following
dual representation for the risk measure
νH:
ν H (Ψ) =
sup
EQ [−Ψ] − EQ [AQ
T]
Q∈M1,ac
where
AQ
.
is the optional process dened by
The penalty function
αH
AQ
0 =0
of the risk measure
νH
and
dAQ
t = ess supξ∈K EQ [θt dSt |Ft ].
can be described as
EQ [AQ
T]
provided that
Q
satises the
three following conditions:
• Q
•
is equivalent to
Every process
• Q
θ.S
P;
with
θ∈K
is a special semi-martingale under
admits the upper variation process
We can set
αH (Q) = +∞
Remark 1.9
AQ
for the set
Q;
{θ.S | θ ∈ K}.
when one of these conditions does not hold.
Note that there is a fundamental dierence between static hedging with a family of cash ows
and dynamic hedging. In the rst case, the initial wealth is a market data: it corresponds to the (forward)
price of the considered cash ows. The underlying logic is based upon calibration as the probability measures
we consider have to be consistent with the observed market prices of the hedging instruments. In the dynamic
framework, the initial wealth is a given data. The agent invests it in a self-nancing admissible portfolio
which may be rebalanced in continuous time.
The problem of dynamic hedging with calibration constraints is a classical problem for practitioners. This
will be addressed in details after the introduction of the inf-convolution operator. Some authors have been
looking at this question (see for instance Bion-Nadal [16] or Cont [33]).
2 Dilatation of Convex Risk Measures, Subdierential and Conservative Price
2.1 Dilatation: γ -Tolerant Risk Measures
For non-coherent convex risk measures, the impact of the size of the position is not linear. It seems therefore
natural to consider the relationship between "risk tolerance" and the perception of the size of the position.
To do so, we start from a given root convex risk measure
15
ρ.
The risk tolerance coecient is introduced as
a parameter describing how agents penalize compared with this root risk measure. More precisely, denoting
by
γ
the risk tolerance, we dene
ργ
as:
ργ (Ψ) = γρ
ργ
(13)
satises a tolerance property or a dilatation property with respect to the size of the position, therefore it
is called the
ρ
1 Ψ .
γ
γ -tolerant
risk measure associated with
ρ
(also called the dilated risk measure associated with
as in Barrieu and El Karoui [10]). A typical example is the entropic risk measure where
γ -dilated
e1 .
of
The map
(ii)
For any
(iii)
Let
ργ , γ > 0
γ → (ργ − γρ(0))
be the family of
γ -tolerant
risk measures issued of
γ, γ > 0, (ργ )γ 0 = ργ γ 0 .
]0, ∞[×X
by
X
) = ργ (X)
γ
is a homogeneous convex functional, cash-invariant with respect to
We can take
X
inequality to
γ and
0
ρ(0) = 0
γ
h
γ+h and γ+h
(i.e. a coherent risk measure in
X ).
(h > 0),
we have, since
ρ(0) = 0:
X
γ
X
h
γ
X
)≤
ρ( ) +
ρ(0) ≤
ρ( ).
γ+h
γ+h γ
γ+h
γ+h γ
is an immediate consequence of the denition and characterization of tolerant risk measures.
(iii) The perspective functional is clearly homogeneous.
two real coecients, and
of
X
without loss of generality of the arguments. By applying the convexity
with the coecients
ρ(
(ii)
Then,
is non-increasing,
pρ (γ, X) = γρ(
(i)
ρ.
0
The perspective functional dened on
Proof:
is simply the
These dilated risk measures satisfy the following nice property:
Proposition 2.1
(i)
eγ
(γ1 , X1 ), (γ2 , X2 )
To show the convexity, let
two points in the denition space of
β1 ∈ [0, 1] and β2 = 1−β1
pρ .
Then, by the convexity
ρ,
pρ β1 (γ1 , X1 ) + β2 (γ2 , X2 )
The other properties are obvious.
β X + β X 1 1
2 2
β1 γ1 + β2 γ2
h
β1 γ1
X1
β2 γ2
X2 i
≤ (β1 γ1 + β2 γ2 )
ρ(
)+
ρ(
)
β1 γ1 + β2 γ2 γ1
β1 γ1 + β2 γ2 γ2
≤ β1 ργ1 (X1 ) + β2 ργ2 (X2 ).
=
(β1 γ1 + β2 γ2 ) ρ
So, we naturally are looking for the asymptotic behavior of the perspective risk measure when the risk
tolerance either tends to
+∞
or tends to
0.
2.2 Marginal Risk Measures and Subdierential
2.2.1 Marginal Risk Measure
Let us rst observe that
of the family of
ρ
γ -tolerant
is a coherent risk measure if and only if
ργ ≡ ρ.
We then consider the behavior
risk measures when the tolerance becomes innite.
16
Proposition 2.2
a)
Suppose that
The marginal risk measure
ρ(0) = 0,
ρ∞ ,
or equivalently
α∞ = limγ→+∞ (γα)
ρ∞ (Ψ)
If
ρ
ρ
ργ
when
γ
tends to innity, is a
that is:
:= supΨ EQ [−Ψ] − ρ∞ (Ψ) = 0 if α(Q) = 0 ,
= supQ∈M
EQ [−Ψ] α Q = 0 .
α∞ (Q)
Assume now that
∀Q ∈ M1,f .
dened as the non-increasing limit of
coherent risk measure with penalty function
b)
α(Q) ≥ 0
+∞ if not,
1,f
L∞ (P)-risk
is a
is continuous from below, the
measure such that
ρ∞
ρ(0) = 0.
is continuous from below and admits a representation in terms of
absolutely continuous probability measures as:
ρ∞ (Ψ) = maxQ∈M1,ac EQ [−Ψ] α Q = 0 ,
Q ∈ M1,ac α Q = 0
and the set
Proof:
Thanks to Theorem 2.1, for any
a)
−m ≥ ργ (Ψ) ≥ −M
when
L1 (P).
is non empty, and weakly compact in
m ≤ Ψ ≤ M,
Ψ ∈ X ργ (Ψ) & ρ∞ (Ψ)
we also have
when
−m ≥ ρ∞ (Ψ) ≥ −M
γ → +∞.
and
ρ∞
Given the fact that
is nite.
Convexity, monotonicity and cash translation invariance properties are preserved when taking the limit.
ρ∞
Therefore,
is a convex risk measure with
Moreover, given that
(ρδ )γ = ρδγ = (ργ )δ ,
ρ∞ (0) = 0.
we have that
(ρδ )∞ = ρ∞ = (ρ∞ )δ
and
ρ∞
is a coherent risk
measure.
Since
α ≥ 0,
the minimal penalty function is:
= supξ EQ [−ξ] − ρ∞ (ξ)
= supξ supγ>0 EQ [−ξ] − γρ( γξ )
= supγ>0 γα(Q) = 0 if α(Q) = 0 , +∞ if not.
Moreover, α∞ is not identically equal to +∞ since the set Q ∈ M1,f α Q = 0 is not empty given
ρ(0) = 0 = max − α(Q) = −α(Q0 ) for some additive measure Q0 ∈ M1,f , from Theorem 1.2.
α∞ (Q)
Assume now that
ξ ∈ X.
ρ
is continuous from below and consider a non-decreasing sequence
(ξn ∈ X )
that
with limit
By monotonicity,
ρ∞ (ξ) = inf ργ (ξ) = inf inf ργ (ξn ) = inf inf ργ (ξn ) = inf ρ∞ (ξn ).
γ
Then,
b)
ρ∞
When
γ
ξn
ξn
γ
ξn
is also continuous from below.
ρ
is a
L∞ (P)-risk
measure, continuous from below,
ρ
is also continuous from above and the dual
representation holds in terms of absolutely continuous probability measures. Using the same argument as
above, we can prove that
ρ∞
is a coherent
L∞ (P)-risk measure, continuous from below with minimal penalty
function:
α∞ (Q)
Moreover, thanks to Theorem 1.4, the
= 0 if α(Q) = 0
and Q ∈ M1,ac
= +∞ otherwise.
set Q ∈ M1,ac α Q = 0
1
L (P). 17
is non empty and weakly compact in
To have some intuition about the interpretation in terms of marginal risk measure, it is better to refer to
the risk aversion coecient
at
0
in the direction of
Ψ
= 1/γ . ρ∞ (Ψ)
appears as the limit of
of the risk measure
ρ,
1
ρ(Ψ) − ρ(0) ,
i.e. the right-derivative
or equivalently, the marginal risk measure. For instance
e∞ (Ψ) = EP (−Ψ).
In some cases, and in particular when the set
Qα
∞
has a single element, the pricing rule
ρ∞ (−Ψ)
is a linear
pricing rule and can be seen as an extension of the notion of marginal utility pricing and of the Davis price
(see Davis [37] or Karatzas and Kou [75]).
2.2.2 Subdierential and its Support Function
Subdierential
Denition 2.3
Let us rst recall the denition of the subdierential of a convex functional.
Let
φ
be a convex functional on
X.
The subdierential of
φ
at
X
is the set
∂φ(X) = q ∈ X 0 | ∀X ∈ X , φ(X + Y ) ≥ φ(X) + q(−Y )
The subdierential of a convex risk measure
in
Dom(α)
measure
since when
q ∈ ∂ρ(ξ),
then
ρ with penalty function α(q) = supY {q(−Y ) − ρ(Y )} is included
α(q) − (q(−ξ) − ρ(ξ)) ≤ 0.
Q when working with risk measure subdierential.
So, we always refer to nitely additive
In fact, we have the well-known characterization
of the subdierential:
q ∈ ∂ρ(ξ)
if and only if
q ∈ M1,f
is optimal for the maximization program
EQ [−ξ] − α(Q) −→ maxQ∈M1,f .
We can also relate it with the notion of marginal risk measure, when the root risk measure is now centered
around a given element
ξ ∈ X,
ρξ (X) = ρ(X + ξ) − ρ(ξ),
i.e.
ρ∞,ξ (Ψ) ≡ lim γ ρ ξ +
γ→+∞
Using Proposition 2.2, since the
ρξ
penalty function is
ρ∞,ξ (Ψ) =
sup
by dening:
Ψ
− ρ(ξ) .
γ
αξ (Q) ≡ α(Q) − EQ [−ξ] + ρ(ξ), ρ∞,ξ
is coherent and
EQ [−Ψ] ρ(ξ) = EQ [−ξ] − α(Q) .
Q∈M1,f
Proposition 2.4
The coherent risk measure
of the subdierential
∂ρ(ξ)
ρ∞,ξ (Ψ) ≡ limγ→+∞ γ ρ ξ + Ψ
−ρ(ξ)
is the support function
γ
of the convex risk measure
ρ∞,ξ (Ψ) =
sup
ρ
at
ξ:
EQ [−Ψ] ρ(ξ) = EQ [−ξ] − α(Q) =
Q∈M1,f
Proof:
From the denition of the subdierential,
∂ρ(ξ)
= q ∈ X 0 | ∀Ψ ∈ X , ρ(ξ + Ψ) ≥ ρ(ξ) + q(−Ψ)
= q ∈ X 0 | ∀Ψ ∈ X , ρ∞,ξ (Ψ) ≥ q(−Ψ)
= ∂ρ∞,ξ (0).
But
sup EQ [−Ψ]
Q∈∂ρ(ξ)
q ∈ ∂ρ∞,ξ (0)
i
α∞,ξ (Qq ) = 0.
So the proof is complete.
18
The L∞ (P) case:
When working with
L∞ (P)-risk measures, following Delbaen [38] (Section 8), the natural
denition of the subdierential is the following:
∂ρ(ξ) = f ∈ L1 (P) | ∀Ψ ∈ L∞ (P), ρ(ξ + Ψ) ≥ ρ(ξ) + EP [f (−Ψ)]
Using the same arguments as above, we can prove that every
equal to
1.
Since
∂ρ(ξ)
and
ρ(0) = 0
ρ∞,ξ (0),
is also the subdierential of
those of the coherent risk measure
then for any
ξ,
ρ∞,ξ ,
f ∈ ∂ρ(ξ) is non-negative with a P-expectation
the properties of
∂ρ(ξ)
for which we have already shown that if
the eective domain of
α∞,ξ
ρ
may be deduced from
is continuous from below
is non empty. Then, under this assumption,
∂ρ(ξ)
is non empty and we have the same characterization of the subdierential as:
Q ∈ ∂ρ(ξ) ⇐⇒ ρ(ξ) = EQ [−ξ] − α(Q).
We now summarize these results in the following proposition:
Proposition 2.5
Let
ρ
be a
L∞ (P)-risk
measure, continuous from below. Then, for any
is the support function of the non empty subdierential
∂ρ(ξ),
ξ ∈ L∞ (P), ρ∞,ξ
i.e.:
ρ∞,ξ (Ψ) = sup EQ [−Ψ] ; Q ∈ ∂ρ(ξ) .
and the supremum is attained by some
Q ∈ ∂ρ(ξ).
2.3 Conservative Risk Measures and Super-Price
We now focus on the properties of the
γ -tolerant
risk measures when the risk tolerance coecient tends to
+∞.
or equivalently when the risk aversion coecient goes to
0
The conservative risk measures that are then
obtained can be reinterpreted in terms of super-pricing rules. Using vocabulary from convex analysis, these
risk measures are related to recession (or asymptotic) functions.
Proposition 2.6
(a)
When
γ
tends to
0,
the family of
γ -tolerant
which is a coherent risk measure. This conservative risk measure
ρ0+ (Ψ) = lim % (ργ (Ψ) − γρ(0)) =
γ↓0
sup
risk measures
ρ0 +
(ργ )
admits a limit
is simply the super-price of
ρ0 + ,
−Ψ:
EQ [−Ψ] α Q < ∞ .
Q∈M1,f
Its minimal penalty function is
α0+ (Q) = 0 if α(Q) < +∞
(b)
If
ρ
is continuous from above on
L∞ (P),
ρ0+ (Ψ) =
then
ρ0 +
sup
and
= +∞ if not.
is continuous from above and
EQ [−Ψ] α Q < ∞ .
Q∈M1,ac
Proof:
Let us rst observe that
ργ (ξ) = γ ρ( γξ ) − ρ(0) + γρ(0)
monotonic while the second one goes to
The functional
ρ0+
is the sum of two terms. The rst term is
0.
is coherent (same proof as for
ρ∞ )
19
with the acceptance set
Aρ0+ = {ξ, ∀λ ≥ 0, λξ ∈
Aρ − ρ(0)}.
α0+ ≥ γα ≥ 0
On the other hand, by monotonicity, the minimal penalty function
Dom(α),
If
ρ
and
α0+ (Q) = +∞
if not. In other words,
∞
L (P),
is continuous from above on
terms of
M1,ac .
So,
α0+ (Q) = 0
on
α0+
is the convex indicator of
and
α0+ (Q) = +∞
We could have proved directly the continuity from above of
Remark 2.7
eγ .
on
Dom(α).
ρ0 +
but in
if not.
ρ0 + ,
since
ρ0 +
is the non-decreasing limit of
(ργ − γρ(0)). A nice illustration of this result can be obtained when considering the entropic risk measure
In this case, it comes immediately that
ρmax (Ψ)
α0+ (Q) = 0
then the same type of dual characterization holds for
Dom(α),
continuous from above risk measures
; so,
where
ρmax
is here the
e0+ (Ψ) = supQ EQ [−Ψ] h(Q | P) < +∞ = P − ess sup(−Ψ) =
L∞ (P)-worst
case measure.
This also corresponds to the weak super-
replication price as dened by Biagini and Frittelli in [15].
Note that this conservative risk measure
e0+ (Ψ)
cannot be realized as
EQ0 [−Ψ]
for some
Q0 ∈ M1,ac .
It is
a typical example where the continuity from below fails.
3 Inf-Convolution
A useful tool in convex analysis is the inf-convolution operation. While the classical convolution acts on the
Fourier transforms by addition, the inf-convolution acts on Fenchel transforms by addition as we would see
later.
3.1 Denition and Main Properties
The inf-convolution of two convex functionals
φA
and
φB
may be viewed as the functional value of the
minimization program
φA,B (X) = inf
H∈X
φA (X − H) + φB (H) ,
(14)
This program is the functional extension of the classical inf-convolution operator acting on real convex
functions
f g(x) = inf y {f (x − y) + g(y)}.
Illustrative example:
Let us assume that the risk measure
whose the penalty function is the functional
risk measure,
ρB ,
with penalty functional
αA (Q) = 0
αB ,
if
ρA
is the linear one
Q = QA ,
= +∞
qA (X) = EQA [−X],
if not. Given a convex
we deduce from the denition of the inf-convolution that
qA ρB (X) = qA (−X) − αB (QA )
Then,
If it is not the case, the minimal penalty function
qA ρB
is identically
−∞
if
αB (QA ) = +∞.
αA,B
associated with this measure is:
αA,B (Q) = αB (QA ) + αA (Q) = αB (Q) + αA (Q)
20
Moreover, the inmum is attained in the inf-convolution program by any
H∗
such that
αB (QA ) = EQA [−H ∗ ] − ρB (H ∗ )
H∗
that is
is optimal for the maximization program dening the
αB .
QA ∈ ∂ρB (H ∗ ).
In terms of subdierential, we have the rst order condition:
3.1.1 Inf-Convolution and Duality
In our setting, convex functionals are generally convex risk measures, but we have also been concerned by
the convex indicator of convex subset, taking innite values. In that follows, we already assume that convex
functionals
φ
+∞)
we consider are proper (i.e. not identically
(in the sense that the level sets
{X| φB (X) ≤ c}, c ∈ R
measure notations we dene their Fenchel transforms on
and in general closed or lower semicontinuous
are weak*-closed). To be consistent with the risk
X0
as
β(q) = sup {q(−X) − φ(X)}.
X∈X
When the linear form
EQ [X].
q
is related to an additive nite measure
Q ∈ M1,f ,
we use the notation
qQ (X) =
For a general treatment of inf-convolution of convex functionals, the interested reader may refer
to the highlighting paper of Borwein and Zhu [19].
The following theorem extends these results to the
inf-convolution of convex functionals whose one of them at least is a convex risk measure:
Theorem 3.1
Let
ρA
be a convex risk measure with penalty function
functional with Fenchel transform
β.
Let
ρA φB
be the inf-convolution of
X → ρA φB (X) = inf
H∈X
and assume that
• ρA φB
•
ρA φB (0) > −∞.
takes the value
∀Q ∈ M1,f
•
Moreover, if the risk measure
∃Q ∈ M1,f
Proof:
ρA
φB
be a proper closed convex
ρA
and
φB
dened as
(15)
Then,
αA,B
and
and
ρA (X − H) + φB (H)
is a convex risk measure which is nite for all
The associated penalty function
αA
X ∈ X.
+∞
for any
q
outside of
M1,f ,
and
αA,B Q = αA Q + βB qQ ,
s.t.
αA Q + βB qQ < ∞.
is continuous from below, then
ρA φB
is also continuous from below.
We give here the main steps of the proof of this theorem.
The monotonicity and translation invariance properties of ρA φB
are immediate from the denition, since
at least one of the both functionals have these properties.
The convexity property simply comes from the fact that, for any
λ ∈ [0, 1],
the following inequalities hold as
ρA
and
ρB
ρA (λXA + (1 − λ)XB ) − (λHA + (1 − λ)HB )
φB λHA + (1 − λ)HB
21
X A , X B , HA
and
HB
in
X
and any
are convex functionals,
≤ λρA XA − HA + (1 − λ)ρA XB − HB
≤
λφB (HA ) + (1 − λ)φB HB .
By adding both inequalities and taking the inmum in
HA
and in
HB
HA
and
HB
on the left-hand side and separately in
on the right-hand side, we obtain:
ρA φB λXA + (1 − λ)XB ≤ λ ρA φB (XA ) + (1 − λ)ρA ρB (XB ).
Using Equation (3), the associated penalty function is given, for any
αA,B Q
by letting
Q∈M1,f ,
by
= supX∈X EQ [−X] − ρA,B (X)
= supΨ∈X EQ [−X] − inf H∈X ρA (X − H) + φB (H)
= supX∈X supH∈X EQ − (X − H) + EQ [−H] − ρA (X − H) − φB (H)
e ,X −H ∈X
X
e − ρA (X)
e + EQ [−H] − φB (H) = αA Q + βB qQ .
= supX∈X
supH∈X EQ [−X]
e
When
β
q 6∈ M1,f ,
is a proper functional,
holds even when
the same equalities hold true. Since
αA
and
β(q)
βB
ρA
is a convex risk measure,
is dominated from below; so,
αA,B (q) = +∞.
αA (q) = +∞,
This equality
αA,B = αA + βB
they take innite values.
The continuity from below is directly obtained upon considering an increasing sequence of
converging to
X.
(Xn ) ∈ X
Using the monotonicity property, we have
inf ρA φB (Xn )
n
=
inf inf {ρA (Xn − H) + φB (H)}
=
inf inf {ρA (Xn − H) + φB (H)} = inf {ρA (X − H) + φB (H)}
n
H
n
H
H
= ρA φB (X) .
We can now give an inf-convolution interpretation of the convex risk measure
H
and since
νH
as in Corollary 1.6 as the inf-convolution of the convex indicator function of
generated by a convex set
H,
and the worst-case risk
measure. This regularization may be applied at any proper convex functional.
Proposition 3.2
[Regularization by inf-convolution with
ρworst ] Let ρworst (X) = supω (−X(ω)) be the worst
case risk measure.
i) ρworst
ii)
Let
is a neutral element for the inmal convolution of convex risk measures.
H
be a convex set such that
inf{m | ∃ ξ ∈ H} > −∞.
is the inf-convolution of the convex indicator functional of
H
The convex risk measure generated by
H, ν H
with the worst case risk measure,
ν H = ρworst lH
iii)
More generally, let
φ
be a proper convex functional, such that for any
The inmal convolution of
iv)
Let
β
functional
ρworst
and
φ, ρφ = ρworst φ
the penalty functional associated with
αφ ,
restriction of
β
at the set
αφ (q)
φ.
H , φ(H) ≥ − supω H(ω) − c.
is the largest convex risk measure dominated by
Then, the penalty functional associated with
M1,f ,
= β(q) + lM1,f (q)
= β(qQ )
if
qQ ∈ M1,f ,
22
+∞
if not.
ρφ
φ.
is the
v)
Given a general risk measure
ρA
such that
ρA φ(0) > −∞,
then
ρA φ = ρA ρworst φ = ρA ρφ .
Proof:
We start by proving that
ρρworst = ρ.
By denition,
= inf Y {supω (−Y (ω)) + ρ(X − Y )} = inf Y ρ X − (Y − supω (−Y (ω)))
ρρworst (X)
= inf Y ≥0 {ρ(X − Y )} = ρ(X)
To conclude, we have used the cash invariance of
ρ
and the fact that
ρ(X − Y ) ≥ ρ(X)
whenever
Y ≥ 0.
ii) has been proved in Corollary 1.6.
iii) By Theorem 3.1,
ρφ = ρworst φ
is a convex risk measure.
inf-convolution of risk measure, any risk measure
ρ = ρworst ρ ≤ ρworst φ = ρφ .
ρ
Since
dominated by
φ
ρworst
is a neutral element for the
is also dominated by
ρworst φ
since
Hence the result.
Therefore, in the following, we only consider the inmal convolution of convex risk measures. The following
result makes more precise Theorem 3.1 and plays a key role in our analysis.
Theorem 3.3
[Sandwich Theorem] Let
ρA
Under the assumptions of Theorem 3.1 (i.e.
i)
There exists
Q ∈ ∂ρA,B (0)
and
ρB
be two convex risk measures.
ρA,B (0) = ρA ρB (0) > −∞),
such that, for any
X
and any
Y,
ρA ρB (0) ≤ ρA (X) − EQ [−X] + ρB (Y ) − EQ [−Y ] .
ii)
Assume
ρA,B (0) ≥ c.
There is an ane function,
aQ (X) = −EQ [−X] + r,
with
Q ∈ ∂ρA,B (0),
satisfying
ρA (.) ≥ aQ ≥ −ρB (−.) + c.
Moreover, for any
X
such that
is said to be exact at
iii)
ρA (X) + ρB (−X) = ρA,B (0), Q ∈ ∂ρB (−X) ∩ ∂ρA (X).
(16)
The inf-convolution
X.
Interpretation of the Condition
ρA ρB (0) > −∞.
The following properties are equivalent:
ρA ρB (0) > −∞.
The sandwich property (16) holds for some ane function
There exists Q ∈ Dom(αA ) ∩ Dom(αB ).
with
ρA
(resp.
ρB ).
Let ρA
0+ (resp.
aQ (X) = −EQ [−X] + r.
ρB
0+ ) be the conservative risk measure associated
Then
B
ρA
0+ (X) + ρ0+ (−X) ≥ 0.
Before proving this Theorem, let us make the following comment: the inf-convolution risk measure
ρA,B ,
given in Equation (15) may also be dened, for instance, as the value functional of the program
ρA,B (Ψ) = ρA ρB (Ψ) = ρA ν AρB (Ψ) = inf {ρA (Ψ − H) , H ∈ AρB } ,
where
ν AρB
is the risk measure with acceptance set
AρB .
measures generated by a convex set, if needed.
23
This emphasizes again the key role played the risk
Proof:
i)
empty.
More precisely, there exists
ρA,B
By Theorem 3.1, the convex risk measure
Q0 ∈ ∂ρA,B (0)
is nite; so its subdierential
∂ρA,B (0)
ρA,B (X) ≥ ρA,B (0) + EQ0 (−X).
such that
is non
In other
words,
ρA,B (0) ≤ ρA,B (X) + EQ0 (X) ≤ ρA (X − Y ) − EQ0 [−(X − Y )] + ρB (Y ) − EQ0 [−Y ].
ii − a)
Assume that
ρA ρB (0) ≥ c.
Y = −Z ,
Applying the previous inequality at
and
X = U + Y = U − Z,
we have
ρA (U ) − EQ0 [−U ] ≥ −ρB (−Z) − EQ0 [Z] + ρA,B (0).
Then,
−αA (Q0 ) := inf {ρA (U ) − EQ0 [−U ]} ≥ αB (Q0 ) + ρA,B (0) := sup{−ρB (−Z) − EQ0 [Z] + ρA,B (0)}.
U
Z
By Theorem 3.1 this inequality is in fact an equality. Picking
r
yield to an ane function that separates
ρA
and
r = αA (Q0 ), and dening aQ0 (X) = EQ0 [−X]+
−ρB (−.) + c.
ii−b) Finally, when ρA (X)+ρB (−X) = ρA,B (0), by the above inequalities, we obtain −ρB (−X)−EQ0 [−X] ≥
−ρB (−Z) − EQ0 [Z].
iii) In other words,
The implication
(1) ⇒ (2) ⇒ (3)
Very naturally, one obtains
implies that for any
and
(3) ⇒ (1)
(2).
X , ρA ρB (0) > −∞,
ρA
0+ (X) ≥ −EQ0 [−X],
Therefore, we obtain (4) as
diamond
and
ρB
0+ .
ρA
Q0
also belongs to
∂ρA (X).
Q0 ∈ Dom(αA ) ∩ Dom(αB )
ρB (−X) ≥ EQ0 [X] − αB (Q).
Considering
ρA (X) + ρB (−X) ≥ −(αA (Q) + αB (Q))
i.e. the property
(2) ⇒ (4).
and
and taking
(1).
We rst observe that (2), i.e.,
ρB (−X) ≥ EQ0 [−X] − r
implies
ρA (X) ≥
ρB
0+ (−X) ≥ EQ0 [−X].
B
ρA
0+ (X) + ρ0+ (−X) ≥ 0.
The converse implication
(4) ⇒ (2)
is obtained by applying the sandwich property (16) to
ρA
0+
Remark 3.4 (On risk measures on L∞ (P))
sures
as the existence of
and
Moreover,
Let us now look at the following implication
implies
By symmetry,
is clear, using the results i) and ii) of this Theorem.
(3) ⇒ (2)
one obtains
the inmum with respect to
−EQ0 [−X] + r
∂ρB (−X).
belongs to
X , ρA (X) ≥ EQ0 [−X] − αA (Q)
r = sup{αA (Q)); αB (Q)},
Q0
and
ρB ,
Let us consider the inf-convolution between two risk mea-
where one of them, for instance
above) and therefore is dened on
L∞ (P).
ρA ,
is continuous from below (and consequently from
In this case, as the inf-convolution maintains the property of con-
tinuity from below (see Theorem 3.1), the risk measure
ρA ρB
M1,ac (P).
is a risk measure on
L∞ (P),
3.1.2
Risk Measures and Inf-Convolution
γ -Tolerant
having a dual representation on
In this subsection, we come back to the particular class of
is also continuous from below and therefore
γ -tolerant
convex risk measures
ργ
to give an
explicit solution to the exact inf-convolution. Recall that this family of risk measures is generated from a root
risk measure
ρ
by the following transformation
ργ (ξT ) = γργ
ξT
γ
where
γ
is the risk tolerance coecient
with respect to the size of the exposure. These risk measures satisfy the following semi-group property for
the inf-convolution:
24
Proposition 3.5
ργ , γ > 0
Let
be the family of
γ -tolerant
ρ.
risk measures issued of
Then, the following
properties hold:
i)
γA , γB > 0,
For any
ii)
Moreover,
ργA ργB = ργA +γB .
γB
γA +γB X is an optimal structure for the minimization program:
∗
F =
ργA +γB (X) = ργA ργB (X) = inf ργA (X − F ) + ργB (F ) = ργA X − F ∗ + ργB F ∗ .
F
The inf-convolution is said to be exact at
iii)
Let
iv)
Assume
v)
If
and
ρ0
ρ(0) = 0
Both
i)
γ > 0, ργ ρ0γ = (ρρ0 )γ .
be two convex risk measures. Then, for any
and
ρ0+ ρ00+ (0) > −∞,
Proof:
ii)
ρ
F ∗.
and
iii)
ρ0 (0) = 0.
γ = +∞,
When
we also have
this relationship still holds:
ρ0+ ρ00+ = (ρρ0 )0+ .
are immediate consequences of the denition of inmal convolution.
We rst study the stability property of the functional
F ) + ργB (F ) → minF
restricted to the family
a natural candidate becomes
∗
F =
ργ
{αX, α ∈ R}.
by studying the optimization program
ργA (X −
Then, given the expression of the functional
ργ ,
γB
γA +γB X , since
ργA X − F ∗ + ργB F ∗ = (γA + γB )ρ
iv)
ρ∞ ρ0∞ = (ρρ0 )∞ .
1
X = ργC (X).
γA + γB
The asymptotic properties are based on the non increase of the map
innity, pass to the limit is equivalent to take the inmum w.r. of
γ
γ → ργ .
Then, when
γ
goes to
and change the order of minimization,
in such way that pass to the limit is justied.
v)
When
γ
goes to
0,
the problem becomes a minimax problem, and we only obtain the inequality.
When the nite assumption holds, by Theorem 3.1, the minimal penalty function of
By the properties of conservative risk measures,
lDom(α)∩Dom(α)
α0 ).
Since,
α
0
α0+
is the convex indicator of
. On the other hand, the minimal penalty function of
is dominated by the same minimal bound
(ρρ0 )0+
ρ0+ ρ00+
Dom(α).
So,
is
α0+ + α00 + .
α0+ + α00 + =
is the indicator of
Dom(α +
−ρ(0), Dom(α + α0 ) = Dom(α) ∩ Dom(α0 )
risk measures have same minimal penalty functions. This completes the proof.
Both
Pauline:
le point v)
3.1.3 An Example of Inf-Convolution: the Market Modied Risk Measure
est a
We now consider a particular inf-convolution which is closely related to Subsection 1.4 as it also deals with
the question of optimal hedging.
More precisely, the following minimization problem
inf ρ(X − H)
H∈VT
can be seen as an hedging problem, where
consists of restricting the risk measure
VT
convolution
ρν
main role of
ρworst
ρ
VT
corresponds to the set of hedging instruments. It somehow
to a particular set of admissible variables and is in fact the inf-
. Using Proposition 3.2, it can also be seen as the inf-convolution
is to transform the convex indicator
ρlVT ρworst .
The
lVT , which is not a convex risk measure (in particular,
it is not translation invariant), into the convex risk measure
25
ν VT .
revoir
The following corollary is an immediate extension of Theorem 3.1 as it establishes that the value functional
of the problem, denoted by
Corollary 3.6
such that
Let
VT
ρm ,
is a convex risk measure, called market modied risk measure.
be a convex subset of
inf ρ(−ξT ), ξT ∈ VT
> −∞.
L∞ (P)
and
ρ
be a convex risk measure with penalty function
The inf-convolution of
ρ
and
ν
VT
,
ρ
m
≡ ρν
VT
α
, also dened as
ρm (Ψ) ≡ inf ρ(Ψ − ξT ) ξT ∈ VT = ρlVT (Ψ)
(17)
is a convex risk measure, called market modied risk measure, with minimal penalty function dened on
M1,ac (P), αm (Q) = α(Q) + αVT (Q).
Moreover, if
ρ
is continuous from below,
ρm
is also continuous from below.
This corollary makes precise the direct impact on the risk measure of the agent of the opportunity to invest
optimally in a nancial market.
Remark 3.7
Note that the set
VT
is rather general. In most cases, additional assumptions will be added
and the framework will be similar to those described in Subsection 1.5.
Acceptability and market modied risk measure:
The market modied risk measure has to be related
to the notion of acceptability introduced by Carr, Geman and Madan in [28]. In this paper, they relax the
strict notion of hedging in the following way: instead of imposing that the nal outcome of an acceptable
position, suitably hedged, should always be non-negative, they simply require that it remains greater than
an acceptable position. More precisely, using the same notations as in Subsection 1.5 and denoting by
given acceptance set and by
ρA
A
a
its related risk measure, we can dene the convex risk measure:
ν̄ H (X) = inf m ∈ R, ∃θ ∈ K ∃A ∈ A : m + X + G(θ) ≥ A P a.s.
To have a clearer picture of what this risk measure really is, let us rst x
look at
ρA (X + G(θ)).
respect to
Then, the risk measure
ν̄
H
G(θ).
In this case, we simply
is dened by taking the inmum of
ρA (X + G(θ))
with
θ,
ν̄ H (X) = inf ρA (X + G(θ)) = inf ρA (X − H)
H∈H
θ∈K
Therefore, the risk measure
ν̄
H
is in fact the particular market modied risk measure
obtain directly the following result of Föllmer and Schied [54] (Proposition 4.98):
function of this convex risk measure
ν̄
H
ρm = ν H ρA .
We
the minimal penalty
is given by
ᾱH (Q) = αH (Q) + α(Q)
where
αH
is the minimal penalty function of
measure with acceptance set
νH
and
α
is the minimal penalty function of the convex risk
A.
4 Optimal Derivative Design
In this section, we now present our main problem, that of derivative optimal design (and pricing).
The
framework we generally consider involves two economic agents, at least one of them being exposed to a
26
non-tradable risk. The risk transfer between both agents takes place through a structured contract denoted
by
F
for an initial price
structure
F
π.
and its price
The problem is therefore to design the transaction, in other words, to nd the
π.
This transaction may occur only if both agents nd some interest in doing
this transaction. They express their satisfaction or interest in terms of the expected utility of their terminal
wealth after the transaction, or more generally in terms of risk measures.
4.1 General Modelling
4.1.1 Framework
Two economic agents, respectively denoted by
a standard measurable space
(Ω, =, P).
(Ω, =)
A
and
B,
are evolving in an uncertain universe modelled by
or, if a reference probability measure is given, by a probability space
In the following, for the sake of simplicity in our argumentation, we will make no distinction
between both situations. More precisely, in the second case, all properties should hold
P − a.s..
Both agents are taking part in trade talks to improve the distribution and management of their own risk.
The nature of both agents can be quite freely chosen. It is possible to look at them in terms of a classical
insured-insurer relationship, but from a more nancial point of view, we may think of agent A as a market
maker or a trader managing a particular book and of agent B as a traditional investor or as another trader.
T,
More precisely, we assume that at a future time horizon
the value of agent A's terminal wealth, denoted
A
by XT , is sensitive to a non-tradable risk. Agent B may also have her own exposure
that by "terminal wealth", we mean the terminal value at the time horizon
paid or received between the initial time and
T;
π
at time
T.
Note
of all capitalized cash ows
no particular sign constraint is imposed. Agent A wants
to issue a structured contract (nancial derivative, insurance contract...)
price
T
XTB
F
with maturity
T
and forward
A
to reduce her exposure XT . Therefore, she calls on agent B. Hence, when a transaction occurs, the
terminal wealth of the agent A and B are
WTA = XTA − F + π,
WTB = XTB + F − π.
As before, we assume that all the quantities we consider belong to the Banach space
probability measure is given, to
X,
or, if a reference
L∞ (P).
The problem is therefore to nd the optimal structure of the risk transfer
(F, π)
according to a given choice
criterion, which is in our study a convex risk measure. More precisely, assuming that agent A (resp. agent
B) assesses her risk exposure using a convex risk measure
optimal structure
F, π
ρA
(resp.
ρB ),
agent
A's objective is to choose the
in order to minimize the risk measure of her nal wealth
ρA (XTA − F + π) → inf .
F ∈X ,π
Her constraint is then to nd a counterpart. Hence, agent
At least, the
F -structure
B
should have an interest in doing this transaction.
should not worsen her risk measure. Consequently, agent
B
simply compares the
risk measures of two terminal wealth, the rst one corresponds to the case of her initial exposure
the second one to her new wealth if she enters the
F -transaction,
ρB (XTB + F − π) ≤ ρB (XTB ).
27
XTB
and
4.1.2 Transaction Feasibility and Optimization Program
The optimization program as described above as
inf ρA (XTA − F + π)
F ∈X ,π
subject to
ρB (XTB + F − π) ≤ ρB (XTB )
can be simplied using the cash translation invariance property.
More precisely, binding the constraint
imposed by agent B at the optimum and using the translation invariance property of
the optimal pricing rule for a structure
(18)
ρB ,
we nd directly
F:
πB (F ) = ρB XTB − ρB XTB + F .
(19)
This pricing rule is an indierence pricing rule for agent B. It gives for any structure
F
the maximum amount
agent B is ready to pay in order to enter the transaction.
Note also that this optimal pricing rule together with the cash translation invariance property of the functional
ρA
enable us to rewrite the optimization program (18) as follows, without any need for a Lagrangian
multiplier:
inf
F ∈X
or to within the constant
ρB (XTB )
ρA XTA − F + ρB XTB + F − ρB XTB
as:
RAB (XTA , XTB ) = inf {ρA XTA − F + ρB XTB + F }.
(20)
F ∈X
Interpretation in Terms of Indierence Prices
This optimization program (Program (20)) can be
reinterpreted in terms of the indierence prices, using the notations introduced in the exponential utility
framework in Subsection 1.1.2. To show this, we introduce the constants
ρA XTA
and
ρB XTB
in such a
way that Program (20) is equivalent to:
inf
F ∈X
ρA XTA − F − ρA XTA + ρB XTB + F − ρB XTB }.
s
ρA XTA − F − ρA XTA as πA
F |XTA , i.e. the
B
B
A
is
exposure XT , while ρB XT + F − ρB XT
Then, using the previous comments, it is possible to interpret
seller's indierence pricing rule for
F
given agent A's initial
B
b
simply the opposite of πB (F |XT ), the buyer's indierence pricing rule for F given agent B's initial exposure
XTB . For agent A, everything consists then of choosing the structure as to minimize the dierence between
A
her (seller's) indierence price (given XT ) and the (buyer's) indierence price imposed by agent B:
inf
F ∈X
Note that for
F ≡ 0,
s
b
πA
F |XTA − πB
F |XTB ≤ 0.
the spread between both transaction indierence prices is equal to
inmum is always non-positive.
(21)
0.
Hence, the
This is completely coherent with the idea that the optimal transaction
obviously reduces the risk of agent
A.
The transaction may occur since the minimal seller price is less than
the maximal buyer price.
For agent
A,
everything can also be expressed as the following maximization program
b
s
sup πB
F |XTB − πA
F |XTA .
F ∈X
28
(22)
The interpretation becomes then more obvious since the issuer has to optimally choose the structure in order
to maximize the "ask-bid" spread associated with transaction.
Relationships with the Insurance Literature and the Principal-Agent Problem
The relationship
between both agents is very similar to a Principal-Agent framework. Agent A plays an active role in the
transaction. She chooses the "payment structure" and then is the "Principal" in our framework. Agent B,
on the other hand, is the "Agent" as she simply imposes a price constraint to the Principal and in this sense
is rather passive.
Such a modelling framework is also very similar to an insurance problem: Agent A is looking for an optimal
"insurance" policy to cover her risk (extending here the simple notion of loss as previously mentioned). In
this sense, she can be seen as the "insured". On the other hand, Agent B accepts to bear some risk. She
plays the same role as an "insurer" for Agent A. In fact, this optimal risk transfer problem is closely related
to the standard issue of optimal policy design in insurance, which has been widely studied in the literature
(see for instance Borch [18], Bühlman [23], [24] and [25], Bühlman and Jewell [27], Gerber [60], Raviv [98]).
One of the fundamental characteristics of an insurance policy design problem is the sign constraint imposed
on the risk, that should represent a loss. Other specications can be mentioned as moral hazard or adverse
selection problems that have to be taken into account when designing a policy (for more details, among a
wide literature, refer for instance to the two papers on the relation Principal-Agent by Rees [99] and [100]).
These are related to the potential inuence of the insured on the considered risk.
Transferring risk in nance is somehow dierent. Risk is then taken in a wider sense as it represents the
uncertain outcome. The sign of the realization does not a priori matter in the design of the transfer. The
derivative market is a good illustration of this aspect: forwards, options, swaps have particular payos which
are not directly related to any particular loss of the contract's seller.
4.2 Optimal Transaction
This subsection aims at solving explicitly the optimization Program (20):
RAB (XTA , XTB ) = inf {ρA XTA − F + ρB XTB + F }.
F ∈X
The value functional
RAB (XTA , XTB )
can be seen as the residual risk measure after the
F -transaction,
or
equivalently as a measure of the risk remaining after the transaction. It obviously depends on both initial
exposures
XTA
and
XTB
since the transaction consists of an optimal redistribution of the respective risk of
both agents.
Let us denote by
Fe ≡ XTB + F ∈ X .
The program to be solved becomes
RAB XTA , XTB = inf {ρA XTA + XTB − Fe + ρB Fe },
e∈X
F
or equivalently, using Section 3, it can be written as the following inf-convolution problem
RAB XTA , XTB = ρA ρB (XTA + XTB ).
29
(23)
As previously mentioned in Theorem 3.1, the condition
ρA ρB (0) > −∞
∀ξ ∈ X ,
inf-convolution problem. This condition is equivalent to
ρA
0+ (ξ)
is required when considering this
+ ρB
0+ (−ξ) ≥ 0
(Theorem 3.3 iii)).
This property has a nice economic interpretation, since it says that the inf-convolution program makes sense
if and only if for any derivative
ξ,
the conservative seller price of the agent
A, −ρA
0+ (ξ),
is less than the
B
conservative buyer price of the agent B, ρ0+ (−ξ).
In the following, we assume such a condition to be satised. The problem is not to study the residual risk
measure as previously but to characterize the optimal structure
F̃ ∗
or
F∗
such that the inf-convolution is
exact at this point.
To do so, we rst consider a particular framework where the optimal transaction can be explicitly identied.
This corresponds to a well-studied situation in economics where both agents belong to the same family.
4.2.1 Optimal Transaction between Agents with Risk Measures in the Same Family
More precisely, we now assume that both agents have
root risk measure
ρ
with risk tolerance coecients
γ -tolerant
γA
and
γB ,
risk measures
ργ A
and
ργ B
from the same
as introduced in Subsection 2.1.
In this
framework, the optimization program (23) is written as follows:
RAB XTA , XTB = ργA ργB (XTA + XTB )
In this framework, the optimal risk transfer is consistent with the so-called Borch's theorem. In this sense,
the following result can be seen as an extension of this theorem since the framework we consider here is
dierent from that of utility functions.
In his paper [18], Borch obtained indeed, in a utility framework,
optimal exchange of risk, leading in many cases to familiar linear quota-sharing of total pooled losses.
Theorem 4.1 (Borch [18])
The residual risk measure after the transaction is given by:
RAB XTA , XTB = inf ργA XTA − F + ργB XTB + F = ργC (XTA + XTB )
F ∈X
γC = γA + γB .
XTB ,
depending only on the
(to within a constant).
(24)
The optimal structure is given as a proportion of the initial exposures
XTA
with
and
risk tolerance coecients of both agents:
F∗ =
γA
γB
XTA −
XB
γA + γB
γA + γB T
The equality in the equation (24) has to be understood
Proof:
P a.s.
The optimization program (20) to be solved (with
RAB XTA , XTB = inf
F̃ ∈X
Using Proposition 3.5, the optimal structure
replacing
e∗
F
by
∗
F −
XTB .
F̃ ≡ XTB + F ∈ X )
L∞ (P).
is
ργA XTA + XTB − Fe + ργB Fe .
is
Fe∗ =
γB
A
γA +γB (XT
+ XTB ).
The result is then obtained by
Comments and properties:
relative tolerance.
F̃ ∗
if the space of structured products is
i)
Both agents are transferring a part of their initial risk according to their
The optimal risk transfer underlines the symmetry of the framework for both agents.
30
Moreover, even if the issuer, agent A has no exposure, a transaction will occur between both agents. The
structure
F
enables them to exchange a part of their respective risk.
Note that if none of the agents is
initially exposed, no transaction will occur. In this sense, the transaction has a non-speculative underlying
logic.
ii)
Note also that the composite parameter
γA
and
γB .
γC
is simply equal to the sum of both risk tolerance coecients
This may justify the use of risk tolerance instead of risk aversion where harmonic mean has to
be used.
4.2.2 Individual Hedging as a Risk Transfer
In this subsection, we now focus on the individual hedging problem of agent A and see how this problem can
be interpreted as a particular risk transfer problem. The question of optimal hedging has been widely studied
in the literature under the name of hedging in incomplete markets and pricing via utility maximization in
some particular framework. Most of the studies have considered exponential utility functions. Among the
numerous papers, we may quote the papers by Frittelli [55], El Karoui and Rouge [51], Delbaen et al. [39],
Kabanov and Stricker [74] or Becherer [11].
We assume that agent A assesses her risk using a (L∞ (P)) risk measure
initial exposure
X
using instruments from a convex subset
VTA
(of
ρA .
L∞ (P)).
She can (partially) hedge her
Her objective is to minimize the
risk measure of her terminal wealth.
inf ρA XTA − ξ .
(25)
A
ξ∈VT
The
L∞ (P)
framework has been carefully described in Subsection 1.5.
In particular, to have coherent
transaction prices, we assume in the following that the market is arbitrage-free.
As already mentioned in Subsection 3.1.3, the opportunity to invest optimally in a nancial market has a
direct impact on the risk measure of the agent and transforms her initial risk measure
m
modied risk measure ρA
= ρA ν
A
A
ρm
A (0) > −∞
is satised. The hedging problem of
B
being now the nancial market with
is identical to the previous risk transfer problem (20), agent
the associated risk measure
into the market
.
This inf-convolution problem makes sense if the condition
agent
ρA
νA.
Existence of an Optimal Hedge
The question of the existence of an optimal hedge can be answered
using dierent approaches. One of them is based on analysis techniques and we present it in this subsection.
In the following, however, when introducing dynamic risk measures, we will consider other methods leading
to a more constructive answer.
In this subsection, we are interested in studying the existence of a solution for the hedging problem of agent
A (Program (25)) or equivalently for the inf-convolution problem in
L∞ (P).
The following of existence can
be obtained:
Theorem 4.2
Let
VT
from below, such that
be a convex subset of
L∞ (P)
and
inf ξ∈VT ρ(−ξ) > −∞.
31
ρ
be a convex risk measure on
L∞ (P)
continuous
Assume the convex set
VT
L∞ (P).
bounded in
The inmum of the hedging program
ρm X , inf ρ X − ξ
ξ∈VT
is attained for a random variable
ξT∗
L∞ (P),
in
belonging to the closure of
VT
with respect to the a.s.
convergence.
Proof:
First note that the proof of this theorem relies on arguments similar to those used by Kabanov and
Stricker [74]. In particular, a key argument is the Komlos Theorem (Komlos [81]):
Lemma 4.3 (Komlos)
(φn0 )
a subsequence
(φn ),
of
Let
(φn )
(φn )
be a sequence in
and a function
L1 (P)
such that
φ∗ ∈ L1 (P)
supn EP (|φn |) < +∞.
Then there exists
φn00
such that for every further subsequence
the Cesaro-means of these subsequences converge to
ϕ∗ ,
N
1 X
lim
φn00 (ω) = φ∗ (ω)
N →∞ N 00
of
that is
for almost every
ω ∈ Ω.
n =1
We rst show that the set
Sr = {ξ ∈ L∞ (P) ρ(X − ξ) ≤ r}
is closed for the weak*-topology. To do that, by
the Krein-Smulian theorem ([54] Theorem A.63), it is sucient to show that
in
Sr ∩ {ξ; kξk∞ ≤ C}
is closed
∞
L (P).
Let
(ξn ∈ VT )
converges a.s. to
ξ
∗
. Since
ρA
bounded sequences and then
Given the assumption that
subsequence
(ξjk ∈
Note that
ξen
VTA . Since
ρA
C,
be a sequence bounded by
ξ∗
converging in
is continuous from below,
belongs to
(ξn )
is
∞
L
Sr . Sr
VTA
A subsequence still denoted by
ξn
is continuous w.r. to pointwise convergence of
-bounded, we can apply Komlos lemma:
ξen ,
1
n
n
P
ξjk
ρ is continuous w.r.
therefore, there exists a
converges almost surely to
ξ ∗ ∈ L∞ (P).
k=1
as a convex combination of elements of
is continuous from below,
ξ∗.
to
is weak*-closed.
VTA ) such that the Cesaro-means,
belongs to
ρ
L∞
VTA .
So
ξ∗
belongs to the
a.s.
closure of
to pointwise convergence of bounded sequences.
lim sup ρA X − ξen ≤ ρA X − ξ ∗ = ρA lim X − ξen ≤ lim inf ρA X − ξen .
n
Then,
n
∗
ρm
A (X) ≤ ρA (X − ξ ) ≤ limn inf ρA
Finally, given the convergence of
inf ξ∈VTA ρA (X − ξ).
4.2.3
γ -Tolerant
ρA X
1
n
n
P
n
(X − ξjk ) ≤ limn inf
k=1
− ξjk to
1
n
n
P
ρA (X − ξjk ) by Jensen inequality.
k=1
∗
ρm
A (X), the Cesaro-means also converge and ρA (X − ξ ) =
Risk Measures: Derivatives Design with Hedging Opportunities
We now consider the situation where both agents
A
and
B
have a
γ -dilated
risk measure, dened on
L∞ (P)
and continuous from above. Moreover, they may reduce their risk by transferring it between themselves but
also by investing in the nancial market, choosing optimally their nancial investments.
The investment opportunities of both agents are described by two convex subsets
VTA
and
VTB
of
L∞ (P).
In
order to have coherent transaction prices, we assume that the market is arbitrage-free. In our framework,
this can be expressed as the existence of a probability measure which is
32
equivalent
to
P
in both sets of
probability measures
MVTi = Q ∈ M1,e (P); ∀ξ ∈ VTi , EQ [−ξ] ≤ 0
∃Q ∼ P
i = A, B .
for
Equivalently,
Q ∈ MVTA ∩ MVTB .
s.t.
This opportunity to invest optimally in a nancial market reduces the risk of both agents. To assess their
respective risk exposure, they now refer to market modied risk measures
ρm
γA
and
ρm
γB
dened if
J = A, B
as
A
ρm
γA (Ψ) = ργA ν (Ψ)
B
ρm
γB (Ψ) = ργB ν (Ψ).
and
Let us consider directly the optimal risk transfer problem with these market modied risk measures, i.e.
m
A
m
B
RAB
XTA + XTB = inf ρm
γ A X T − F + ργ B X T + F
(26)
F ∈X
The details of this computation will be given in the next subsection, when considering the general framework.
The residual risk measure
m
RAB
XTA +XTB
dened in equation (28) may be simplied using the commutativity
property of the inf-convolution and the semi-group property of
m
RAB
XTA + XTB
where
ργ C
is the
γ -tolerant
γ -tolerant
m
A
B
= ρm
γA ργB XT + XT
risk measures:
= ργA ν A ργB ν B XTA + XTB
= ργA ργB ν A ν B XTA + XTB
= ργC ν A ν B XTA + XTB .
risk measure associated with the risk tolerance coecient
This inf-convolution program makes sense under the initial condition
γC = γA + γB .
m
ρm
γA ργB (0) > −∞.
Such an assump-
tion is made. The following theorem gives the optimal risk transfer in dierent situations depending on the
access both agents have to the nancial markets.
Theorem 4.4
and
Let both agents have
γ -tolerant
γA
risk measures with respective risk tolerance coecients
γB .
(a) If both agents have the same access to the nancial market from a
cone, VT , then an optimal structure,
solution of the minimization Program (28) is given by:
F∗ =
γA
γB
XA −
XB.
γA + γB T
γA + γB T
(b) Assume that both agents have dierent access to the nancial market via
Suppose
∗
ηB
∈
∗
∗
ξ ∗ = ηA
+ ηB
(B)
VT and
is an optimal solution of the Program
(A+B)
VT
=
ξTA
F∗ =
+
ξTB
|
ξTA
∈
(A)
VT ,
ξTB
(B) ∈ VT
two convex sets VTA and VTB .
inf ξ∈V (A+B) ργC XTA + XTB − ξ
T
with
. Then
γB
γA
γB
γA
XA −
XB −
η∗ +
η∗
γA + γB T
γA + γB T
γA + γB A γA + γB B
is an optimal structure. Moreover,
∗
i) ηB
is an optimal hedging portfolio of
1
1
∗
ργB XTB + F ∗ − ηB
=
γB
γB
XTB + F ∗
for Agent
B
1
ργB XTB + F ∗ − ξB =
ργC XTA + XTB − ξ ∗ .
(B)
γC
ξB ∈VT
inf
33
(A)
∗
ηA
∈ VT
,
∗
ii) ηA
XTA − F ∗
is an optimal hedging portfolio of
1
1
∗
ργA XTA − F ∗ + ηA
=
γA
γA
for Agent
A
ργA XTA − F ∗ + ξA
inf
(A)
ξA ∈VT
Pauline: Je ne suis pas sure que le hedging problem avec
l'espace VTA+B est ferme dans L∞ , on sait que c'est bon...
Proof: To prove this theorem, we proceed in several steps:
=
ργ
1
ργC XTA + XTB − ξ ∗ .
γC
ait toujours une solution. Si
1:
Step
Let us rst observe that
m
RAB
XTA + XTB = ργC XTA + XTB − ξ ∗ = inf ργA XTA + XTB − Fe − ξ ∗ + ργB Fe ,
e∈X
F
Fe = F + XTB − ξB . Given Proposition 3.5, we obtain directly an expression for the optimal structure
B
Fe∗ as: Fe∗ = γAγ+γ
XTA + XTB − ξ ∗ = γγB
XTA + XTB − ξ ∗ . Moreover, ργB (Fe) = γγB
(XTA + XTB − ξ ∗ ).
B
C
C
where
2:
Step
Rewriting in the reverse order, we naturally set
optimal investment for agent
For
the
sake
of
simplicity
ργB XTB + F − ξB
in
our
∗
∗
ξ ∗ = ηA
+ ηB
is an
notation,
we
GX (ξA , ξB , F ) , ργA XTA − F − ξA +
consider
and
∗
Fe∗ = F ∗ + XTB − ηB
,
we have
∗
∗
= GX (ηA
, ηB
, F ∗)
inf
(A)
F ∈X ,ξA ∈VT
∗
ηB
∗
ηB
B.
=
Then
We then want to prove that
.
Given the optimality of
m
RAB
XTA + XTB
∗
F ∗ = Fe∗ − XTB + ηB
.
is optimal for the problem
(B)
GX (ξA , ξB , F ) ≤
,ξB ∈VT
inf
(B)
ξB ∈VT
ργB (F − ξB ) → inf ξB ∈V (B) .
The optimality of
T
the same arguments.
∗
∗
∗
GX (ηA
, ξB , F ∗ ) ≤ GX (ηA
, ηB
, F ∗) .
∗
ηA
can be proved using
Remark 4.5
(a) We rst assume that both agents have the same access to the nancial market from a
H.
H
cone
Given the fact that the risk measure generated by
is coherent and thus invariant by dilatation,
the market modied risk measures of both agents are generated from the root risk measure
H
ρH
= ργA νγHA = ρν
A = ργA ν
H
γA
= ρH
γA
and
ρν H = ρH
as
H
ρH
B = ργ B .
(b) In a more general framework, when both agents have dierent access to the nancial market, the convex
set
(A+B)
VT
A
ργC ν ν
ν (A+B) = ν A ν B
XTA + XTB .
associated with the risk measure
B
Comments:
XTA
B
+ XT
= ργC ν
(A+B)
plays the same role as the set
H
above, since
Note that when both agents have the same access to the nancial market, it is optimal to
transfer the same proportion of the initial risk as in the problem without market.
This result is very
strong as it does not require any specic assumption either for the non-tradable risk or the nancial market.
Moreover, the optimal structure
F∗
does not depend on the nancial market. The impact of the nancial
market is simply visible through the pricing rule, which depends on the market modied risk measure of
agent B.
34
Standard diversication will also occur in exchange economies as soon as agents have proportional penalty
functions.
The regulator has to impose very dierent rules on agents as to generate risk measures with
non-proportional penalty functions if she wants to increase the diversication in the market. In other words,
diversication occurs when agents are very dierent one from the other. This result supports for instance
the intervention of reinsurance companies on nancial markets in order to increase the diversication on the
reinsurance market.
4.2.4 Optimal Transaction in the General Framework
We now come back to our initial problem of optimal risk transfer between agent A and agent B, when now
they both have access to the nancial market to hedge and diversify their respective portfolio.
General framework Pauline: Malgre cette hypothese de L∞ (P), j'ai tout laisse en mesures
additive pour le moment dans l'ecriture de la CNS.
Je n'ai rien change car cette sous-section est plus delicate par rapport a ce pont.
ρA
As in the dilated framework, we assume that both their risk measures
and
ρB
are dened on
L∞ (P)
and are continuous from above. The investment opportunities of both agents are described by two convex
subsets
VTA
and
VTB
of
L∞ (P)
and the nancial market is assumed to be arbitrage-free.
1. This opportunity to invest optimally in a nancial market reduces the risk of both agents. To assess
their respective risk exposure, they now refer to market modied risk measures
J = A, B
m
as ρJ (Ψ)
, inf ξJ ∈V (J)
T
ρJ (Ψ − ξJ ). As usual, we assume that ρm
J (0)
ρm
A
> −∞
and
ρm
B
dened if
for the individual
hedging programs to make sense. Thanks to Corollary 3.6,
A
ρm
A (Ψ) = ρA ν (Ψ)
and
2. Consequently, the optimization program related to the
A
inf ρm
A XT − F + π
F,π
subject to
B
ρm
B (Ψ) = ρB ν (Ψ).
F -transaction
is simply
B
m
B
ρm
B X T + F − π ≤ ρB X T .
As previously, using the cash translation invariance property and binding the constraint at the optimum, the pricing rule of the
F -structure
is fully determined by the buyer as
B
m
B
π ∗ (F ) = ρm
B X T − ρB X T + F .
It corresponds to an indierence pricing rule from the agent
B 's
(27)
market modied risk measure.
3. Using again the cash translation invariance property, the optimization program simply becomes
A
m
B
B
m
A
B
m
B
inf ρm
− ρm
A X T − F + ρB X T + F
B XT , RAB XT + XT − ρB XT .
F
With the functional
m
RAB
,
m
RAB
XTA + XT
we are in the framework of Theorem 3.1.
B
A
m
B
inf ρm
A X T − F + ρB X T + F
F
A
B
m e
m
A
B
e
= inf ρm
= ρm
A X T + X T − F + ρB F
A ρB XT + XT
e
F
= ρA ν A ρB ν B XTA + XTB .
=
35
(28)
(29)
Pauline:
meme questi
sur
L∞ (P)
The value functional
m
RAB
of this program, resulting from the inf-convolution of four dierent risk
measures, may be interpreted as the residual risk measure after all transactions. This inf-convolution
problem makes sense if the initial condition
m
ρm
A ρB (0) > −∞
is satised.
4. Using the previous Theorem 3.1 on the stability of convex risk measure, provided the initial condition
is satised,
α
A
VT
+α
Comments:
m
RAB
is a convex risk measure with the penalty function
m
m
m
αAB
= αA
+ αB
= αA + αB +
B
VT
.
The general risk transfer problem can be viewed as a game involving four dierent agents if the
access to the nancial market is dierent for agent
A
and agent
B
(or three otherwise). As a consequence,
we end up with an inf-convolution problem involving four dierent risk measures, two per agents.
Optimal design problem
Our problem is to nd an optimal structure
F∗
realizing the minimum of the
Program (28):
m
A
m
B
RAB
XTA + XTB = inf ρm
A X T − F + ρB X T + F
F
Let us rst consider the following simple inf-convolution problem between a convex risk measure
linear function
qA
ρB
as introduced in Subsection 3.1:
qA ρB (X) = inf {EQA [−(X − F )] + ρB (F )}.
(30)
F
Proposition 4.6
and a
F∗
The necessary and sucient condition to have an optimal solution
convolution problem (30) is expressed in terms of the subdierential of
ρB
as
to the linear inf∗
QA ∈ ∂ρB (F ).
This necessary and sucient corresponds to the rst order condition of the optimization problem.
More
generally, the following result is obtained:
Theorem 4.7 (Characterization of the optimal)
Assume that
m
ρm
A ρB (0) > −∞.
The inf-convolution program
m
A
m
B
RAB
XTA + XTB = inf ρm
A XT − F + ρB XT + F
F
is exact at
F∗
if and only if there exists
B
m
A
QX
AB ∈ ∂RAB (XT + XT )
such that
∗
m
A
QX
AB ∈ ∂ρA (XT − F ) ∩
B
∗
∂ρm
B (XT + F ).
F∗
m
In other words, the necessary and sucient condition to have an optimal solution
to the inf-convolution
X
program is that there exists an optimal additive measure QAB for
X
m
A
∗
X
m
optimal for QAB , αB and XT − F is optimal for QAB , αA .
such that
XTA
+
XTB , RAB
XTB + F ∗
is
Both notions of optimality are rather intuitive as they simply translate the fact that the dual representations
of the risk measure on the one hand, and of the penalty function on the other hand, are exact respectively
at a given additive measure and at a given exposure.
A natural interpretation of this theorem is that both agents agree on the measure
their respective residual risk. This agreement enables the transaction.
36
QX
AB
in order to value
Proof:
Let us denote by
XTB ).
QX
AB
the optimal additive measure for
. In this case,
m
A
QX
AB ∈ ∂RAB (XT +
As mentioned in Subsection 1.2, the existence of such an additive measure is guaranteed as soon as
the penalty function is dened by Equation
i)
m
XTA + XTB , RAB
In the proof, we denote by
X,
XTA
X
given additive measure QAB optimal for
−RAB X c
(10).
+ XTB and by
X, RAB
:
Ψc ,
the centered random variable
c
Ψ = Ψ − EQX
[Ψ].
AB
Ψ
with respect to the
So, by denition,
X
= αA QX
AB + αB QAB
= sup − ρA X c − F c + sup − ρB F c
F
F
≥ − inf ρA X c − F c + ρB F c = −RAB X c .
F
In particular, all inequalities are equalities and
sup − ρA X c − F c + sup − ρB F c = sup − ρA X c − F c − ρB F c .
F
Hence,
F
∗
F
F
is optimal for the inf-convolution problem, or equivalently for the program on the right-hand side
of this equality, if and only if
F∗
is optimal for both problems
supF
−ρB F c
and
supF
−ρA X c −F c
.
The second formulation is a straightforward application of Theorem 3.3 ii), considering the problem not at
0
but at
XTA + XTB . In order to obtain an explicit representation of an optimal structure
F ∗,
some technical methods involving
a localization of convex risk measures have to be used. This is the aim of the second part of this chapter,
which is based upon some technical results on BSDEs. Therefore, before localizing convex risk measures and
studying our optimal risk transfer in this new framework, we present in a separate section some quick recalls
on BSDEs, which is essential for a good understanding of the second part on dynamic risk measures.
Part II: Dynamic Risk Measures
We now consider dynamic convex risk measures. Quite recently, many authors have studied dynamic version
of static risk measures, focusing especially on the question of law invariance of these dynamic risk measures:
among many other references, one may quote the papers by Cvitanic and Karatzas [35], Wang [110],Scandolo
[106], Weber [111], Artzner et al. [3], Cheridito, Delbaen and Kupper [29] [30] or [31], Detlefsen and Scandolo
[43], Frittelli and Gianin [58], Frittelli and Scandolo [59], Gianin [61], Riedel [101], Roorda, Schumacher and
Engwerda [105] or the lecture notes of Peng [97]. Very recently, extending the work of El Karoui and Quenez
[49], Klöppel and Schweizer have related dynamic indierence pricing and BSDEs in [77].
In this second part, we extend the axiomatic approach adopted in the static framework and introduce some
additional axioms for the risk measures to be time-consistent. We then relate the dynamic version of convex
risk measures to BSDEs. The associated dynamic risk measure is called
g
g -conditional
risk measure, where
is the BSDE coecient. We will see how the properties of both the risk measure and the coecient
g
are
intimately connected. In particular, one of the key axioms in the characterization of the dynamic convex risk
37
measure will be the translation invariance, as we will see in Section 6, and this will impose the
of the related BSDE to depend only on
g -coecient
z.
In the last two sections, we come back to the essential point of this chapter, the optimal risk transfer problem.
We rst derive some results on the inf-convolution of dynamic convex risk measures and obtain the optimal
structure as a solution to the inf-convolution problem.
The idea behind our approach is to nd a trade-o between static and very abstract risk measures as to
obtain tractable risk measures. Therefore, we are more interested in tractability issues and interpretations
of the dynamic risk measures we obtain rather than the ultimate general results in BSDEs.
5 Some recalls on Backward Stochastic Dierential Equations
In the rest of the chapter, we take into account more information on the risk structure. In particular, we
assume the
σ -eld F
FT -measurable
generated by a d-dimensional Brownian motion between
[0, T ].
Since any bounded
variable is an stochastic integral w.r. to the Brownian motion, the risk measures of interest
have to be robust with respect of this localization principle. To do that, we consider a family of risk measures
described by backward stochastic dierential equations (BSDE).
In this section, we introduce general BSDEs, dening them, recalling some key results on existence and
uniqueness of a solution and presenting the comparison theorem.
Complete proofs and additional useful
results are given in the Chapter dedicated to BSDEs.
5.1 General Framework and Denition
Ω, F, P
Let
TH ),
Ft0
where
be a probability space on which is dened a
TH > 0
is the time horizon of the study.
= σ Ws ; 0 ≤ s ≤ t; t ≥ 0
Denoting by
and
(Ft ; t ≤ TH )
d-dimensional
Brownian motion
W := (Wt ; t ≤
Let us consider the natural Brownian ltration
its completion with the
P-null
sets of
F.
E the expected value with respect to P, we introduce the following spaces which will be important
in the formal setting of BSDEs.
referring to a generic time
Since the time horizon may be sometimes modied, the denitions are
T ≤ TH .
• L2n Ft = {η : Ft − measurable Rn − valued
• Pn (0, T ) = {(φt ; 0 ≤ t ≤ T ) :
random variable s.t.
E(|η|2 ) < ∞}.
progressively measurable process with values in
Rn }
• Sn2 (0, T ) = {(φt ; 0 ≤ t ≤ T ) : φ ∈ Pn s.t. E[supt≤T |Yt |2 ] < ∞} .
RT
• Hn2 (0, T ) = {(φt ; 0 ≤ t ≤ T ) : φ ∈ Pn s.t. E[ 0 |Zs |2 ds] < ∞}.
RT
• Hn1 (0, T ) = {(φt ; 0 ≤ t ≤ T ) : φ ∈ Pn s.t. E[( 0 |Zs |2 ds)1/2 ] < ∞}.
Let us give the denition of the one-dimensional BSDE; the multidimensional case is considered in the
Chapter dedicated to BSDEs.
Verier
Denition 5.1
le statut
Let ξT
∈ L2 (Ω, FT , P) be a R-valued terminal condition and g a coecient P1 ⊗B(R)⊗B(Rd )-
measurable. A solution for the BSDE associated with
(g, ξT )
is a pair of progressively measurable processes
de la
partie
BSDE
theorique
38
(Yt , Zt )t≤T ,
with values in
R × R1×d
such that:
n (Y ) ∈ S 2 (0, T ), (Z ) ∈ H2 (0, T )
t
t
1
1×d
RT
RT
Yt = ξT + t g(s, Ys , Zs )ds − t Zs dWs , 0 ≤ t ≤ T .
(31)
The following dierential form is also useful
−dYt = g(t, Yt , Zt )dt − Zt dWt ,
Conventional notation:
Brownian motion
(1, d)
W
If
ξT
is described as a column vector
and
ZdW
g(t, y, z)
(d, 1)
ξT
are deterministic, then
Zt ≡ 0,
is random, the previous solution is
to introduce the martingale
Rt
0
and the
Z
the
vector is described as a row vector
has to be understood as a matrix product with (1,1)-dimension.
dyt
= −g(t, yt , 0),
dt
If the nal condition
(32)
To simply the writing of the BSDE, we adopt the following notations:
such that the notation
Remark 5.2
YT = ξT .
Zs dWs
and
(Yt )
is the solution of ODE
yT = ξT .
FT -measurable,
and so non adapted. So we need
as a control process to obtain an adapted solution.
5.2 Some Key Results on BSDEs
Before presenting key results of BSDEs, we rst summarize the results concerning the existence and uniqueness of a solution.
The proofs are given in the Chapter dedicated to BSDEs with some complementary
results.
5.2.1 Existence and Uniqueness Results
In the following, we always assume the necessary condition on the terminal condition
1.
ξT ∈ L2 (Ω, FT , P).
(H1): The standard case (uniformly Lipschitz): (g(t, 0, 0); 0 ≤ t ≤ T ) belongs to H2 (0, T ) and
g
uniformly Lipschitz continuous with respect to
(y, z), i.e.
there exists a constant
C≥0
such that
dP × dt − a.s. ∀(y, y 0 , z, z 0 ) |g(ω, t, y, z) − g(ω, t, y 0 , z 0 )| ≤ C(|y − y 0 | + |z − z 0 |).
Under these assumptions, Pardoux and Peng [94] proved in 1990 the existence and uniqueness of a
solution.
2.
(H2) The continuous case with linear growth:
there exists a constant
C≥0
such that
dP × dt − a.s. ∀(y, z) |g(ω, t, y, z)| ≤ k(1 + |y| + |z|).
Moreover we assume that
dP × dt a.s., g(ω, t, ., .)
is continuous in
(y, z).
Then, there exist a maximal
and a minimal solutions (for a precise denition, please refer to the Chapter dedicated to BSDEs), as
proved by Lepeltier and San Martin in 1998 [85].
39
3.
(H3) The continuous case with quadratic growth in z :
In this case, the assumption of square
integrability on the solution is too strong. So we only consider bounded solution and obviously terminal
condition
ξT ∈ L∞ .
k≥0
We also suppose that there exists a constant
such that
dP × dt − a.s. ∀(y, z) |g(ω, t, y, z)| ≤ k(1 + |y| + |z|2 ).
Moreover we assume that
dP × dt − a.s., g(ω, t, ., .)
is continuous in
(y, z).
Then there exist a maximal and a minimal bounded solutions as rst proved by Kobylansky [78] in
2000 and extended by Lepeltier and San Martin [85] in 1998.
The uniqueness of the solution was
proved by Kobylansky [78] under the additional conditions that the coecient
(y, z)
on a compact interval
[−K, K] × R
d
and that there exists
∂g
≤ c1 (1 + |z|),
∂z
g
c1 > 0 and c2 > 0
is dierentiable in
such that:
∂g
≤ c2 (1 + |z|2 )
∂y
Pauline:
(33)
ineq,
5.2.2 Comparison Theorem
est-ce
We rst present an important tool in the study of one-dimensional BSDEs: the so-called comparison theorem.
It is the equivalent of the maximum principle when working with PDEs.
Theorem 5.3 (Comparison Theorem)
Let
(ξT1 , g 1 )
and
(ξT2 , g 2 )
be two pairs (terminal condition, coef-
cient) satisfying one of the above conditions (H1,H2,H3) (but the same for both pairs). Let
(Y 2 , Z 2 )
(i)
(Y 1 , Z 1 )
and
be the maximal associated solutions.
We assume that
ξT1 ≤ ξT2 , P − a.s.
and that
dP × dt − a.s.
∀(y, z)
g 1 (ω, t, y, z) ≤ g 2 (ω, t, y, z).
Then we have
Yt1 ≤ Yt2
(ii)
a.s.
∀ t ∈ [0, T ]
Strict inequality Moreover, under (H1), if in addition Yt1 = Yt2
on
B ∈ Ft ,
then a strict version of
this result holds as
ξT1 = ξT2 , ∀s ≥ t, Ys1 = Ys2
a.s. on B
and
g 1 (s, Ys1 , Zs1 ) = g 2 (s, Ys2 , Zs2 )
dP × ds − a.s. on B × [t, T ]
6 Axiomatic Approach and g-Conditional Risk Measures
In this section, we give a general axiomatic approach for dynamic convex risk measures and see how they are
connected to the existing notions of consistent convex price systems and non-linear expectations, respectively
introduced by El Karoui and Quenez [50] and Peng [95]. Then, we relate the dynamic risk measures with
BSDEs and focus on the properties of the solution of some particular BSDEs associated with a convex
coecient
g,
dans la 2eme
called g-conditional risk measures.
6.1 Axiomatic Approach
Following the study of static risk measures by Föllmer and Schied [53] and [54], we now propose a common
axiomatic approach to dynamic convex risk measures, non-linear expectations and convex price systems and
non-linear.
40
|y|2 ?
Denition 6.1
operator)
Y
Ω, F, P, (Ft ; t ≥ 0)
respect to Ft ; t ≥ 0 is
Let
with
bounded stopping time
T,
a
L2 (FT )
(resp.
be a ltered probability space.
L2 -operator (L∞ -
A dynamic
a family of continuous semi-martingales which maps, for any
L∞ (FT ))
-variable
ξT
onto a process
Yt (ξT ); t ∈ [0, T ] .
Such
an operator is said to be
1.
(P1) Convex:
For any stopping times
S ≤ T,
for any
(ξT1 , ξT2 ),
for any
YS (λξT1 + (1 − λ)ξT2 ) ≤ λYS (ξT1 ) + (1 − λ)YS (ξT2 )
2.
(P2) Monotonic:
(P2+):
(P2-):
3.
the operator is decreasing
For
YS ξ U = Y S Y T
(P5) Arbitrage-free:
for any
YS (ξT1 ) ≥ YS (ξT2 )
S≤T ≤U
ξU
a.s.,
(P6) Conditionally invariant:
(P7) Positive homogeneous:
any
and any
ηS ∈ FS ,
for any
YS (ξT + ηS ) = YS (ξT ) − ηS
(P4-)
YS ξ U = YS − YT ξ U
S ≤ T,
and for any
(ξT1 , ξT2 )
AS = {S < T } =⇒ ξT1 = ξT2
For any stopping times
YS (1B ξT ) = 1B YS (ξT )
7.
ξT1 ≥ ξT2 a.s.,
ξU
ξT ,
a.s.
three bounded stopping times, for any
For any stopping times
on
such that
a.s.
S≤T
(P3-)
a.s.,
(ξT1 , ξT2 )
P − a.s.
a.s.
For any stopping times
YS (ξT1 ) = YS (ξT2 )
6.
S ≤ T,
YS (ξT1 ) ≤ YS (ξT2 )
YS (ξT + ηS ) = YS (ξT ) − ηS
(P4) Time-consistent:
(P4+)
5.
the operator is increasing if
(P3) Translation invariant:
(P3+)
4.
For any stopping times
0 ≤ λ ≤ 1,
For any stopping times
S≤T
a.s.
such that
ξT1 ≥ ξT2 ,
a.s. on AS .
and any
B ∈ FS ,
for any
ξT ,
a.s.
S ≤ T,
for any
λS ≥ 0 (λS ∈ FS )
and for
ξT ,
YS (λS ξT ) = λS YS (ξT )
a.s.
First, note that the property (P5) of no-arbitrage implies that the monotonicity property (P2) is strict.
Most axioms have two dierent versions, depending on the sign involved.
Making such a distinction is
completely coherent with the previous observations in the static part of this chapter about the relationship
between price and risk measure: since the opposite of a risk measure is a price, the axioms with a "+" sign
are related to the characterization of a price system, while the axioms with a "−" sign are related to that of
a dynamic risk measure.
In [50], when studying pricing problems under constraints, El Karoui and Quenez dened a consistent convex
(forward) price system as a convex (P1), increasing (P2+), time-consistency (P4+) dynamic operator
Pt ,
without arbitrage (P5). Time-consistency (P4+) may be view as a dynamic programming principle.
At the same period, Peng introduced the notion of non-linear expectation as a
41
translation invariance
(P3+) convex price system, satisfying the conditional invariance property (P6) which is very intuitive in this
framework, (see for instance, Peng [95]). Note that (P6) of conditional invariance implies some additional
assumptions on the operator
expectation by
Y:
in particular for any t,
Yt (0) = 0.
In the following, we denote the non-linear
E.
Now on, we focus on dynamic convex risk measures, where now only the properties with the "−" sign hold.
Denition 6.2
A dynamic operator satisfying the axioms of convexity (P1), decreasing monotonicity (P2-),
translation invariance (P3-), time-consistency (P4-) and arbitrage-free (P5) is said to be a dynamic convex
risk measure. It will be denoted by
If
R
R
in the following.
also satises the positive homogeneity property (P7), then it is called a dynamic coherent risk measure.
Note that a non-linear expectation denes a dynamic risk measure conditionally invariant and centered.
Remark 6.3
bounded
It is not obvious to nd a negligible set
ξT , ∀ ω ∈
/ N , ξT →
the variable
ξT
N
such that for any bounded stopping time
A typical example is the dynamic entropic risk measure , obtained
by conditioning the static entropic risk measure. For any
1
eγ (ξT ) = γ ln E exp(− ξT )
γ
ξT
and any
itself.
Dynamic Entropic Risk Measure
Since,
S
RgS (ω, ξT ) is a static convex risk measure. The negligible sets may depend on
is bounded,
bounded:
1
eγ,t (ξT ) = γ ln E exp(− ξT )|Ft .
γ
⇒
eγ,t (ξT ) is bounded for any t.
ξT
Therefore, this dynamic operator dened on
L∞
satises
the properties of dynamic convex risk measures. Convexity, decreasing monotonicity, translation invariance,
no-arbitrage are obvious; the time-consistency property
(P 4−)
results from the transitivity of conditional
expectation:
∀t ≥ 0 , ∀h > 0,
eγ,t ξT = eγ,t − eγ,t+h ξT
We give the easy proof of this identity to help the reader to understand the
Proof:
Yt+h
eγ,t − eγ,t+h ξT
a.s.
(−)
sign in the formula.
≡ eγ,t+h ξT = γ ln E exp(− γ1 ξT )|Ft+h
= γ ln E exp(− γ1 (−Yt+h ))|Ft
i
h
= γ ln E exp( γ1 γ ln E exp(− γ1 ξT )|Ft+h )|Ft
h i
= γ ln E E exp(− γ1 ξT )|Ft .
Moreover, it is possible to relate the dynamic entropic risk measure
eγ,t
with the solution of a BSDE, as
follows:
Proposition 6.4
The dynamic entropic measure eγ,t (ξT ); t ∈ [0, T ] is solution of the following BSDE with
2
1
the quadratic coecient g t, z = 2γ z and terminal bounded condition ξT .
1 Zt 2 dt − Zt dWt
−deγ,t ξT =
2γ
42
eγ,T ξT = −ξT .
(34)
Proof:
h
Mt (ξT ) = E exp −
Let us denote by
1
γ ξT
|Ft
i
.
martingale, one can use the multiplicative decomposition to get
1×d
M
As
is a positive and bounded continuous
dMt = γ1 Mt Zt dWt
dimensional square-integrable process. By Itô's formula applied to the function
where
Zt ; t ≥ 0
γ ln(x),
we obtain the
is a
Equation (34).
RT
E t |Zs |2 ds|Ft ] = E eγ,t (ξT ) − ξT |Ft is
RT
(Y,Z) such that YT and E
|Zs |2 ds|Ft ] are
t
Note that the conditional expectation of the quadratic variation
bounded and conversely if the Equation (34) has the solution
bounded, then
Y
is bounded. This point will be detailed in Theorem 7.4.
This relationship between the dynamic entropic risk measure and BSDE can be extended to general dynamic
convex risk measures as we will see in the rest of this section.
6.2 Dynamic Convex Risk Measures and BSDEs
This section is about the relationship between dynamic convex risk measures and BSDEs. More precisely,
we are interested in the correspondence between the properties of the "BSDE" operator and that of the
coecient.
We consider the dynamic operator generated by the maximal solution of a BSDE:
Denition 6.5
Let
g
be a standard coecient. The
g -dynamic
operator, denoted by
Yg,
is such that
Ytg (ξT )
is the maximal solution of the BSDE(g, ξT ).
As a consequence, the adopted point of view is dierent from that of the section dedicated to recalls on
BSDEs where the terminal condition of the BSDE was xed.
It is easy to deduce properties of the
g -dynamic
operator from those of the coecient
is more complex and this study has been initiated by Peng when considering
g.
The converse
g -expectations
(REF). Our
characterization is based upon the following lemma:
Lemma 6.6 (Coecient Uniqueness)
solution for the BSDE(g
1
)
holds. Let
Yg
i
∀(T, ξT ),
a) If the coecients
g1
and
g2
Let
g1
and
g2
g i -dynamic
be
be two regular coecients, such that uniqueness of
operator(i
1
t
and
Assume that
2
Ytg (ξT ) = Ytg (ξT ).
dP × dt − a.s.
simply depend on
= 1, 2).
z,
then
dP × dt − a.s.
∀z
g 1 (t, z) = g 2 (t, z).
b) In the general case, the same identity holds provided the coecients are continuous w.r. to t.
dP × dt − a.s.
∀(y, z)
g 1 (t, y, z) = g 2 (t, y, z).
Proof: Suppose that both coecients g1 and g2 generate the same solution Y (but a priori dierent processes
Z 1 and Z 2 ) for the BSDEs (g 1 , ξT ) and (g 2 , ξT ), for any ξT in the appropriate space (L2 or L∞ ). Given
the uniqueness of the decomposition of the semimartingale
processes of the both decompositions of
Rt
0
Zs dWs , a.s.
and
Rt
0
g 1 (s, Ys , Zs1 )ds =
Y
Rt
0
Y,
the martingale parts and the nite variation
Rt
Zs1 dWs = 0 Zs2 dWs =
Rt 1
Rt
g (s, Ys , Zs )ds = 0 g 2 (s, Ys , Zs )ds.
0
are indistinguishable. In particular,
g 2 (s, Ys , Zs2 )ds.
A priori, these equalities only hold for processes
(Y, Z)
43
Therefore,
obtained through BSDEs.
Rt
0
a)
g1
Assume that
and
g2
do not depend on
following locally bounded semimartingale
of the BSDE(g
R t∧τ
0
1
, UT ∧τ )
2
g (s, Zs )ds.
τ
Given a bounded adapted process
g
1
and
g
2
w.r. to
probability one, for any
z,
we consider the
dUt = g (t, Zt )dt − Zt dWt ; U0 = u0 . (U, Z) is the solution
R t∧τ 1
time s.t. UT ∧τ is bounded. By uniqueness,
g (s, Zs )ds =
0
as
is a stopping
Z,
1
As shown in b) below, this equality implies that
to the continuity of
dt × dP
where
U
y.
g 1 (s, Zs ) = g 2 (s, Zs ), a.s. ds × dP.
we can only consider denumerable rational
1
z
Thanks
to show that, with
2
z , g (s, z) = g (s, z).
b1) In the general case, given a bounded process Z , we consider a solution of the following forward stochastic
Y
dierential equation
rst time, when
|Y |
dYt = g 1 (t, Yt , Zt )dt − Zt dWt ; Y0 = y0
and the stopping time
(YτN ∧t , Zt 1]0,τN ] (t))
the BSDE(g
]0, τN ∧ T ].
, YτN ∧T ).
Since
τN
dened as the
is solution of the BSDE with bounded terminal condition
Thanks to the previous observation, the pair of processes
2
τN
N.
crosses the level
The pair of processes
YτN ∧T .
as
Hence, both processes
goes to innity with
N,
Rt
0
1
g (s, Ys , Zs )ds =
Rt
0
(YτN ∧t , Zt 1]0,τN ] (t))
2
g (s, Ys , Zs )ds
is also solution of
are indistinguishable on
the equality holds at any time, for any bounded process
b2) Assume g1 (s, y, z) and g2 (s, y, z) continuous w.r.
to
s.
Let
z
ξT =
be a given vector. Let
z
Yt+h
Z.
be a forward
Y , at the level z between t and t + h,
Z u
Z u
Yuz = Yt +
g 1 (s, Ysz , z)ds −
zdWs
∀u ∈ [t, t + h].
perturbation of a general solution
t
By assumption,
Rt
t
g 2 (s, Ysz , z)ds.
Then
(Yuz , z)
t
Ru
2
z
, Yt+h
), for u ∈ [t, t + h], and t g 1 (s, Ysz , z)ds =
h
i
z,i
1
1
i
h E Yt+h − Yt |Ft goes in L to g (t, Yt , z) (i = 1, 2) with h → 0.
is also solution of the BSDE(g
Hence, by continuity,
1
g (t, Yt , z) = g 2 (t, Yt , z)
for any solution
Yt
of the BSDE, i.e. for any v.a
Ft -measurable. Citer
Comments
i) Peng ([95] and [97]) and Briand et al. [21] have been among the rst to look at the dynamic
g
operators to deduce local properties through the coecient
of the associated BSDE, when considering
non-linear expectations.
ii) In [21], Briand et al. proved a more accurate result for
coecient such that
g
Lipschitz. More precisely, let
P-a.s., t 7−→ g(t, y, z) is continuous and g(t, 0, 0) ∈ S
2
. Let us x
g
be a standard
(t, y, z) ∈ [0, T ]×R×Rd
n ∈ N∗ , {(Ysn , Zsn ); s ∈ [t, tn = t + n1 ]} solution of the BSDE (g, Xn ) where the terminal
d
given by Xtn = y + z Wtn − Wt . Then for each (t, y, z) ∈ [0, T ] × R × R , we have
and consider for each
condition
Xtn
is
L2 − lim n Ytn − y = g(t, y, z).
n→∞
Some properties automatically hold for the dynamic operator
Yg
simply because it is the maximal solution
of a BSDE. Some others can be obtained by imposing conditions on the coecient
Theorem 6.7
a) Then,
Yg
Let
Yg
be the
g -dynamic
g:
operator.
is increasing monotonic (P2+), time-consistent (P4+) and arbitrage-free (P5).
b) Moreover, under the assumptions of Lemma 6.6,
1.
Yg
is conditionally invariant (P6) if and only if for any
2.
Yg
is translation invariant (P3+) if and only if
g
44
t ∈ [0, T ], z ∈ Rn , g(t, 0, 0) = 0.
does not depend on
y.
la these
du chinois
3.
Yg
is homogeneous if and only if
g
is homogeneous;
c) For properties related to the order, the following implications simply hold:
1. If
g
2. If
g1 ≤ g2
is convex, then
g
Therefore, if
, then
Yg
is convex (P1).
1
Yg ≤ Yg
2
.
is a convex coecient depending only on
z , Rg (ξT ) ≡ Y g (−ξT )
is a dynamic convex risk
measure, called g-conditional risk measure.
Yg
Note that
is a consistent convex price system and moreover, if for any
t ∈ [0, T ], g(t, 0) = 0,
then
Yg
is
a non-linear expectation, called g-expectation.
Proof :
•
a)
The strict version of the comparison Theorem 5.3 leads immediately to both properties (P2+)
and (P5).
•
Up to now, we have dened and considered BSDEs with a terminal condition at a xed given time
always possible to consider it as a BSDE with a time horizon
Obviously, the coecient
the solution
Yt
g
has to be extended as
is constant on
g1[0,T ]
TH ≥ T , even if TH
T.
It is
is a bounded stopping time.
and the terminal condition
ξTH = ξT .
Therefore
[T, TH ].
To obtain the time-consistency property (P4), also called the ow property, we consider three bounded
stopping times
S ≤ T ≤ U
and write the solution of the BSDEs as function of the terminal date.
obvious notations, we want to prove that
With
YS (T, YT (U, ξU )) = YS (U, ξU ) a.s..
By simply noticing that:
RT
YS (T, YT (U, ξU )) = YT (U, ξU )+
RU
RU
= ξU + T g(t, Zt )dt − T Zt dWt
the process which is dened as
of the BSDE
(g, ξU , U ).
Yt (T, YT (U, ξU ))
on
[0, T ]
S
+
g(t, Zt )dt −
RT
S
and by
RT
g(t, Zt )dt −
Yt (U, ξU )
Zt dWt
S
RT
S
on
Zt dWt ,
]T, U ]
is the maximal solution
Uniqueness of the maximal solution implies (P4).
b) The three properties b1), b2) and b3) involve the same type of arguments to be proved, so we simply
present the proof for b2).
gm (t, y, z) = g(t, y + m, z).
Let
the BSDE
processes
Y
(gm , ξT ).
and
Y
m
We simply note that
Y.m = Y. (ξT + m) − m
The translation invariance property is equivalent to the indistinguishability of both
; by the uniqueness Lemma 6.6, this property is equivalent to the identity
g(t, y, z) = gm (t, y, z) = g(t, y + m, z)
This implies that
c)
•
g
does not depend on
and
λ ∈ [0, 1].
(Yt2 , Zt2 )
a.s.
y.
c1) For the convexity property, we consider dierent BSDEs:
(g, ξT1 )
is the maximal solution of
is the (maximal) solution of
(g, ξT2 ).
(Yt1 , Zt1 )
Then, we look at
is the (maximal) solution of
Yet = λYt1 + (1 − λ)Yt2 ,
with
We have:
−dYet = (λg(t, Yt1 , Zt1 ) + (1 − λ)g(t, Yt2 , Zt2 ))dt − (λZt1 + (1 − λ)Zt2 )dWt
45
;
YeT = λξT1 + (1 − λ)ξT2 .
Since
g
is convex, we can rewrite this BSDE as:
et ) + α(t, Y 1 , Y 2 , Z 1 , Z 2 , λ))dt − Z
et dWt
−dYet = (g(t, Yet , Z
t
t
t
t
where
α
is a a.s. non-negative process. Hence, using the comparison theorem, the solution
is for any
t ∈ [0, T ]
1
a.s. greater than the solution Yt of the BSDE (g, λξT
Yet
of this BSDE
+ (1 − λ)ξT2 ). It is a super-solution
in the sense of Denition 2.1 of El Karoui, Peng and Quenez [47].
•
c2) is a direct consequence of the comparison Theorem 5.3.
Some additional comments on the relationship between BSDE and dynamic operators
Since
1995, Peng has focused on nding conditions on dynamic operators so that they are linear growth gexpectations. This dicult problem is solved in particular for dynamic operators satisfying a domination
assumption introduced by Peng [95] in 1997 where
bk (z) = k|z|.
For more details, please refer to his lecture
notes on BSDEs and dynamic operators [97].
Theorem 6.8
Let
(Et ; 0 ≤ t ≤ T )
|λ| ∈ H2
There exists
be a non-linear expectation such that:
and a suciently large real number
t ∈ [0, T ]
and any
Then, there exists a function
g(t, y, z)
k > 0
such that for any
2
ξT ∈ L (FT ):
−bk +|λ|
Et
and for any
b +|λ|
(ξT ) ≤ Et (ξT ) ≤ Et k
(ξT )
(ξT1 , ξT2 ) ∈ L2 (FT ): Et (ξT1 ) − Et (ξT2 ) ≤ Etbk (ξT1 − ξT2 )
satisfying assumption (H1) such that for any
a.s.
t ∈ [0, T ],
∀ ξT ∈ L2 (FT ),
a.s.,
∀t,
Et (ξT ) = Etg (ξT )
For a proof of this theorem, please refer to Peng [96].
Innitesimal Risk Management
The coecient of any
g -conditional
interpreted as the innitesimal risk measure over a time interval
[t, t + dt]
risk measure
Rg
can be naturally
as:
EP [dRgt |Ft ] = −g(t, Zt )dt,
where
Zt
is the local volatility of the
g conditional
Therefore, choosing carefully the coecient
g
risk measure.
enables to generate
g -conditional risk measures that are locally
compatible with the views and practice of the dierent agents in the market.
In other words, knowing
the innitesimal measure of risk used by the agents is enough to generate a dynamic risk measure, locally
compatible. In this sense, the
g -conditional risk measure may appear more tractable than static risk measures.
The following example gives a good intuition of this idea: the
the mean-variance paradigm has a
g -coecient
of the type
g -conditional
risk measure corresponding to
g(t, z) = −λt z + 12 z 2 .
The process
λt
can be
interpreted as the correlation with the market numéraire.
Therefore,
g -Conditional
risk measures are a way to construct a wide family of convex risk measures on a
probability space with Brownian ltration, taking into account the ability to decompose the risk through
46
inter-temporal local risk measures
g -conditional
In the following, to study
framework.
g(t, Zt ).
risk measures, we adopt the same methodology as in the static
In particular, we start by developing a dual representation for these dynamic risk measures,
in terms of the "dual function" of their coecient. This study requires some general properties of convex
functions on
Rn .
7 Dual Representation of g-Conditional Risk Measures
Following the approach adopted in the rst part of this chapter when studying static risk measures, we
now focus on a dual representation for
transform
G
of the coecient
g,
g -conditional
risk measures. The main tool is the Legendre-Fenchel
dened by:
G(t, µ) =
{hµ, −zi − g(t, z)} .
sup
(35)
z∈Qn
rational
The convex function
G
is also called the polar function or the conjugate of
g(t, z) =
g.
Provided that
g
is continuous,
hµ, −zi − G(t, µ) .
sup
(36)
µQn
rational
More precisely,
Denition 7.1
A
A g-conditional risk
Rg
measure is said to have a dual representation if there exists a set
of admissible controls such that for any bounded stopping time
h
RgS (ξT ) = ess supµ∈A EQµ − ξT −
where
Qµ
Z
S≤T
T
µ̄
if the
ess sup
in the appropriate space
i
G(t, µt )dtFS
(37)
S
is a probability measure absolutely continuous with respect to
The dual representation is said to be exact at
ξT
and any
P.
is reached for
µ̄.
In order to obtain this representation, several intermediate steps are needed:
1. Rene results on Girsanov theorem and the integrability properties of martingales with respect to
change of probability measures.
2. Rene results from convex analysis on the Legendre-Fenchel transform and the existence of an optimal
control in both Formulae (35) and (36), including measurability properties,
The next paragraph gives a summary of the main results that are needed.
7.1 Girsanov Theorem and BMO-Martingales
Our main reference on Girsanov theorem and BMO-martingales is the book by Kazamaki [76].
exponential martingale associated with the
exp
R
t
0
µs dWs −
d-dimensional
Brownian motion
The
Rt
W , E( 0 µs dWs ) = Γµt =
1
2
2 |µs | ds , solution of the forward stochastic equation
dΓµt = Γµt µ∗t dWt ,
47
Γµ0 = 1
(38)
is a positive local martingale, if
When
Γµ
µ
is an adapted process such that
is a uniformly integrable (u.i.)
Qµ .
measure denoted by
Moreover, if
martingale,
ΓµT
RT
0
|µs |2 ds < ∞.
is the density (w.r.
to
µ
is a P-Brownian motion, then Wt
W
P)
of a new probability
= Wt −
Rt
0
µs ds
is a
Qµ -
Brownian motion.
Questions around Girsanov theorem are of two main types. They mainly consist of:
Γµ
•
rst, nding conditions on
•
second, giving so;e precision on the integrability properties that are preserved under the new probability
µ
so that
is a u.i. martingale.
measure.
The bounded case, that is recalled below, is well-known. The BMO case is less standard, so we give more
details.
7.1.1 Change of Probability Measures with Bounded Coecient
When
µ
is bounded, it is well-known that the exponential martingale belongs to all
Moreover, if a process is in
then
ctZ =
M
Rt
0
Zs dWsµ
H2 (P), it is in H1+ (Qµ ).
is a u.i. martingale under
In particular, if
Qµ ,
with null
MtZ =
Rt
0
Hp -spaces.
Zs dWs
is a
H2 (P)-martingale,
Qµ -expectation.
7.1.2 Change of Probability Measures with BMO-Martingale
The right extension of the space of bounded processes is the space of BMO processes dened as:
BMO(P) = {ϕ ∈ H2
s.t
∃C
hZ
∀t E
T
i
|ϕs |2 ds|Ft ≤ C a.s.}
t
C
The smallest constant
such that the previous inequality holds is denoted by
In terms of martingale, the stochastic integral
the process
ϕ
Rt
0
ϕs dWs
is said to be a BMO(P)-martingale if and only if
belongs to BMO(P). The following deep result is proved in Kazamaki [76] (Section 3.3).
Theorem 7.2
Let the adapted process
µ
be in BMO(P). Then
Γµ is a u.i. martingale and denes
R
t
Wtµ = Wt − 0 µs ds is a Qµ -Brownian motion.
1. The exponential martingale
µ
Q
2.
. Moreover,
Mtµ =
C ∗ = ||ϕ||2BMO .
Rt
0
µ∗s dWs ,
and more generally any BMO(P)-martingale
ctµ
continuous processes M
=
Rt
0
µ∗s dWsµ
3. The BMO-norms with respect to
P
and
and
Qµ
ctZ =
M
Rt
0
Zs dWsµ
a new equivalent probability measure
MtZ =
Rt
0
Zs dWs ,
µ
that are BMO(Q )-martingales.
are equivalent:
k||Z||BMO(Qµ ) ≤ ||Z||BMO(P) ≤ K||Z||BMO(Qµ ) .
The constants
k
and
K
only depend on the BMO-norm of
48
µ.
are transformed into
Hu, Imkeller and Müller [73] were amongst the rst to use the property that the martingale
dMtZ = Zt dWt
which naturally appears in BSDEs associated with exponential hedging problems, is BMO. Since then, such
a property has been used in dierent papers, mostly dealing with the question of dynamic hedging in an
exponential utility framework (see for instance the recent papers by Mania, Santacroce and Tevzadze [87]
and Mania and Schweizer [88]).
In the proposition below, we extend their results to general quadratic BSDEs.
Proposition 7.3
M
Z
=
R.
0
Zs dWs
Let (Y,Z) be the maximal solution of the quadratic (H3) BSDE with coecient
dYt = g(t, Zt )dt − dMtZ ,
Proof:
Let
k
Y
is bounded, and
be the constant such that
Z
=
YT = ξT .
|g(t, 0)|1/2 ∈ BMO(P), M Z
is a
BMO(P)-martingale
|g(t, z)| ≤ |g(t, 0)| + k|z|2 .
Thanks to Itô's formula applied to the solution
exp(β Yt )
and
Z.W
the stochastic integral
Given that by assumption
g,
T
exp(β YT ) + β
t
(Y, Z)
and to the exponential function:
β2
exp(β Ys )g(s, Zs )ds −
2
Z
T
exp(β Ys )|Zs |2 ds
t
T
Z
− β
exp(β Ys )Zs dWs
t
Z
=
exp(β YT ) + β
t
Given that
β
2
2 |Zs |
− g(s, Zs ) ≥
( β2
T
Z T
β
exp(β Ys ) g(s, Zs ) − |Zs |2 ds − β
exp(β Ys )Zs dWs .
2
t
− k)|Zs |2 − |g(s, 0)| ≥ ε|Zs |2 − |g(s, 0)|
for
β ≥ (k + ε)
and taking the
conditional expected value, we obtain:
hZ
βεE
T
i
hZ
exp(β Ys )|Zs |2 ds|Ft ≤ C + βE
t
where
C
T
i
exp(β Ys )|g(s, 0)|ds|Ft ≤ C
t
is a universal constant that may change from place to place. Since
below and from above, the property holds.
exp(β Ys )
is bounded both from
7.2 Some Results in Convex Analysis
Some key results in convex analysis are needed to obtain the dual representation of
g -conditional
risk mea-
sures. They are presented in the Appendix 10 to preserve the continuity of the arguments in this part. More
details or proofs may be found in Aubin [4], Hiriart-Urruty and Lemaréchal [70] or Rockafellar [102].
7.3 Dual Representation of Risk Measures
We now study the dual representation of
g -conditional
depends on the assumption imposed on the coecient
g.
risk measures.
The space of admissible controls
We consider successively both situations (H1) and
(H3). There is no need to look separately at (H2), as, under our assumptions, the condition (H2) implies
the condition (H1) (for more details, please refer to the Appendix 10.2.1). The (H1) case has been solved in
[47] but the (H3) case is new.
49
Theorem 7.4
g
Let
be a convex coecient satisfying (H1) or (H3) and
G
be the associated polar process,
G(t, µ) = supz∈Qnrational {hµ, −zi − g(t, z)}.
i) For almost all (ω, t), the program g(ω, t, z) = supµ∈Qnrational [hµ, −zi−G(ω, t, µ) ] has an optimal progressively
measurable solution
ii)
Then
R
g
µ̄(ω, t)
in the subdierential of
g
at
z , ∂g(ω, t, z).
has the following dual representation, exact at
T
Z
h
Rgt (ξT ) = esssupµ∈A EQµ − ξT −
µ̄,
T
Z
h
i
G(s, µs )dsFt = EQµ̄ − ξT −
t
i
G(s, µ̄s )dsFt
t
where:
1. Under (H1) (|g(t, z)|
≤ |g(t, 0)|| + k|z|), A
µ
is the space of adapted processes
µ
the associated equivalent probability measure with density ΓT where
µ
Γ
bounded by
k,
and
Qµ
is
is the exponential martingale
dened in (38).
2. Under (H3), (|g(t, z)|
≤ |g(t, 0)| + k|z|2 ), A
µ
is the space of BMO(P)-processes
and
Qµ
is dened as
above.
Proof:1)
Since
g
is a proper function, the dual representation of
g
G
with its polar function
is exact at
mu
¯ ∈ ∂g(z):
g(t, z) =
[hµ, −zi − G(t, µ) ] = hµ̄, −zi − G(t, µ̄),
sup
µ∈Qn
rational
using classical results of convex analysis, recalled in the Appendix 10.
The measurability of
µ̄
is separately studied in Lemma 7.5 just after this proof.
2a) Let us rst consider a coecient
g
with linear growth (H1); so,
g(t, 0)
is in
H2 .
g
is dominated from above by the square integrable process g(t, 0). Then, let Rt (ξT )
the BSDE
By denition,
:= Yt
−G(t, µt )
be the solution of
(g, −ξT ),
−dYt = g(t, Zt )dt − Zt dWt = (g(t, Zt ) − hµt , −Zt i)dt − Zt dWtµ ,
By Girsanov Theorem (Theorem 7.2), for
Qµ
µ ∈ A
YT = −ξT .
the exponential martingale
Wµ = W −
Γµ
is u.i.
(39)
and denes a
R.
µ ds is a Qµ -Brownian motion. Moreover,
0 s
Z
cZ = Zs dWsµ is a u.i. Qµ -martingale. Moreover, since µ is bounded
since M
= 0 Zs dWs is in H2 (P), M
0
2
1+
and g uniformly Lipschitz, the process (g(t, Zt ) − Zt µt ) belongs to H (P) but also to H
(Qµ ). So we can
probability measure
on
FT
such that the process
R.
R.
use an integral representation of the BSDE (39) in terms of
Z
h
Yt = EQµ − ξT +
T
h
i
(g(s, Zs ) − hµs , −Zs i)dsFt ≥ EQµ − ξT −
Z
T
We do not need to prove that the last term is nite. It is enough to recall that
2b) Let
µ̄
dQ × ds
integrable process
be an optimal control, bounded by
result,
Yt = EQµ̄ − ξT −
RT
t
(−G(s, µs ))+
is dominated
(g(s, 0))+ .
k,
such that
g(t, Zt ) = hµ̄t , −Zt i − G(t, µ̄t )
(see Lemma 7.5 for
−G(t, µ̄t ) belongs to H2 (P) and so to H1+ (Qµ̄ ). By the previous
i
G(s, µ̄s )dsFt . So the process Y is the value function of the maximization dual
measurability results). Then the process
h
(40)
t
t
from above by the
i
G(s, µs )dsFt .
50
h
i
RT
Yt = esssupµ∈A EQµ − ξT − t G(s, µs )dsFt .
problem
g
3a) We now consider a coecient
with quadratic growth (H3) and bounded solution
notation, we know by Girsanov Theorem 7.2 that if
measure
Qµ
µ
µ ∈ BMO(P), Γ
Yt .
Using the same
is a u.i. martingale and the probability
is well-dened. The proof of the dual representation is very similar to that of the previous case,
after solving some integrability questions. It is enough to notice that
So
1
•
by assumption,
•
by Proposition 7.3,
•
by Girsanov Theorem 7.2, for any
1
|g(t, Zt )| 2
above by
3b) Let
µ̄
Z
1
|µt Zt | 2
and
|g(t, 0)|
|g(., 0)| 2
is a
is
is
BMO(P),
BMO(P),
are in
µ ∈ BMO(P),
BMO(Qµ ).
µ
Q × dt-integrable
G(t, .)
Moreover, the process
µ, Z
and
1
|g(., 0)| 2
(−G(t, µ))+
BMO(Qµ ).
are in
which is dominated from
process. Then the inequality (40) holds.
be an optimal control, such that
growth, the polar function
the processes
g(t, Zt ) = hµ̄t , −Zt i − G(t, µ̄t ).
satises the following inequality,
Given that
g(t, .)
G(t, µ̄t ) ≥ −|g(t, 0)| +
1
4k ,
1
for small ε <
( 4k
− ε)|µ̄t |2 ≤ G(t, µ̄t ) + |g(t, 0)| − ε|µ̄t |2 ≤ |g(t, 0)|
1
|g(t, 0)| − g(t, Zt ) + 4ε
|Zt |2 . Since both processes |g(t, Zt )|1/2 and Z are
has quadratic
1
2
4k |µ̄t | .
− g(t, Zt ) + hµ̄t , −Zt i − ε|µ̄t |2 ≤
BMO(P), µ̄
is also
BMO(P),
the other processes hold nice integrability properties with respect to both probability measures
and the integral representation follows.
Let
g
P
and
and
Qµ
The question of the measurability of the optimal solution(s)
Lemma 7.5
Then,
µ̄
is considered in the following lemma.
be a convex coecient satisfying (H1) or (H3) and
There exists an progressively measurable optimal solution
G
be the associated polar function.
µ̄ such that g(t, Zt ) = hµ̄t , −Zt i−G(t, µ̄t )
a.s.dP×
dt.
Proof:
For each
(ω, t) ∈ Ω×[0, T ], the sets given by: {µ ∈ Rn : g(ω, t, Zt ) = Zt µ−G(ω, t, µ)} are nonempty.
Hence, by a measurable selection theorem (see for instance Dellacherie and Meyer [41] or Benes [14]), there
exists a
Rn -valued
dt − a.s..
progressively measurable process
µ̄
such that:
g(ω, t, Zt ) = hµ̄t , −Zt i − G(ω, t, µ̄t ) dP ×
Penalty Functional and Duality Pauline le 14 juin: j'ai repris cette partie en suivant vos
remarques. Je ne suis pas sure totalement de ec que j'ai ecrit... notamment pour la proposition.
Pouvez-vous y jeter un coup d'oeil?
As in the static framework, the converse dual representation holds:
EQµ
hR
T
S
i
G(t, µt )dtFS
of the
g -conditional
risk measure
Rg
the penalty functional
αS,T Qµ =
can be written as the following supremum:
αS,T Qµ = esssupξT EQµ ξT − Rg (ξT )FS .
However, some additional assumptions need to be imposed on the coecient
Indeed, the dual representation (36) of the polar function
to the interior of
Dom(G).
G is valid in Rn
g
of the associated BSDE.
under the condition that
µ belongs
This is an essential dierence with the direct case (in order to obtain the dual
51
representation (35) for the convex function
g ).
A second major dierence comes from the dierent properties
of both Fenchel-Legendre transforms at the innity under the previous assumptions. However, if
G has linear
or quadratic growth, the previous duality result can be applied. That kind of growth property of the polar
function
G
requires "more" convexity for the primal function
Proposition 7.6
Let
g(t, .)
be a strongly convex function (i.e.
G(t, µ)
the Fenchel-Legendre transform
holds true:
hZ
T
EQµ
S
Proof:
Let
g
in the following sense:
g(t, z) − 12 C|z|2
has a quadratic growth in
µ
is a convex function). Then
and the following dual representation
i
G(t, µt )dtFS = esssupξT EQµ ξT − Rg (ξT )FS .
h(t, z) = g(t, z) − 21 C|z|2
g.
be the convex function associated with
Since
g
is the sum of
1
2
two convex functions h and C|.| , its Fenchel-Legendre transform G is the inf-convolution of the Fenchel2
1
2
Legendre transforms of both h and
2 C|.| . But the Fenchel-Legendre transform of the quadratic function
1
1
2
2
2 C|.| is still a quadratic function, 2C |µ| and G has a quadratic growth (as the inf-convolution of a convex
function
7.4
H
with a quadratic function). We can apply the previous result and conclude.
g -Conditional γ -Tolerant
Risk Measures and Asymptotics
In this subsection, we pursue our presentation and study of g-conditional risk measures using an approach
similar to that we have adopted in the static framework.
7.4.1
g -Conditional γ -Tolerant
Risk Measures
As in the static framework, we can dene dynamic versions for both coherent and
γ -tolerant
risk measures
based on the properties of their coecients using the uniqueness Lemma 6.6.
More precisely, let
γ > 0
be a risk-tolerance coecient.
of any static convex risk measure
g
measure, Rγ ,
of
γ -tolerant
of
R
g
ρ
is dened by
As in the static framework, where the
ργ (ξT ) = γρ
1
γ ξT
γ -dilated
we can dene the g-conditional risk
, as the risk measure associated with the coecient
gγ ,
which is the
γ -dilated
1
g : gγ (t, z) = γg( t,γ
z).
Note that if
g
is Lipschitz continuous (H1),
growth (H3) with parameter
function of
gγ , Gγ ,
k,
then
g
gγ
also satises (H1), and if
can be expressed in terms of
G,
the dual function of
g -conditional γ -tolerant risk measure is certainly
eγ,t (ξT ) = γ ln E exp(− γ1 ξT )|Ft , which is the γ -tolerant of e1,t .
Asymptotic behaviour of entropic risk reasure
γ
go to
+∞,
the BSDE-coecient
natural extension of the static case,
is continuous with quadratic
also satises (H3), but with parameter
A standard example of
measure. Letting
g
g
as
k
γ . Note also that the dual
Gγ (µ) = γG(µ).
the dynamic entropic risk measure
Let us look more closely at the dynamic entropic risk
qγ (z) =
1
2
2γ |z| tends to
0
and we directly obtain the
e∞,t (ξT ) = EP [−ξT |Ft ].
γ tend to 0, the BSDE coecient explodes if |z| =
6 0 and intuitively the martingale of this BSDE has to
γ
1
be equal to 0. More precisely, since by denition exp(eγ,t (ξT )) = E exp(− ξT )|Ft , limγ→0 exp(eγ,t (ξT )) =
γ
Letting
|| exp(−ξT )||∞
t = inf {Y ∈ Ft : Yt ≥ exp(−ξT )}.
So we have
52
e0+ ,t (ξT ) = || − ξT ||∞
t .
This conditional risk
measure is a
that
g -conditional
e0+ ,t (ξT )
risk measure associated with the indicator function of
{0}.
Let us also observe
is an adapted non-increasing process without martingale part.
7.4.2 Marginal Risk Measure
Pauline: cette partie est a reprendre. Il nous manque la representation duale de α∞ pour
pouvoir proceder comme dans le cas statique
NEK le 14 juin: j'y reechis.
On n'a qu'a considerer le cas strongly convex si on ne sait pas quoi faire d'autre, puisque l'on
a la dualite dans ce cas-la
Pauline le 14 juin: j'attends de vos nouvelles sur ce point
In the general
g(t, 0) = 0
equivalently
behavior of the
of
g
γ -tolerant
case, assuming that the
G(t, .) ≥ 0),
γ -dilated
g -conditional
the same type of results can be obtained concerning the asymptotic
coecient and the duality. Then, the limit of
at the origin in the direction of
G
has a unique
0,
i.e.
gγ
when
γ → +∞
is the derivative
z.
Rg∞ is the non-increasing limit of Rgγ dened by its
ξT Ft G(u, µu ) = 0, ∀u ≥ t, −a.s.} ; in some cases
function
risk measures are centered (equivalently
G(u, 0) = 0
dual representation
Rg∞,t (ξT ) = ess supµ∈A {EQµ −
(in particular, in the quadratic case when the polar
is unique),
−Rg∞
is a linear pricing rule and can be seen as an
extension of the Davis price (see Davis [37]).
7.4.3 Conservative Risk Measures and Super Pricing
We now focus on the properties of the
g -conditional γ -tolerant
risk measures when the risk tolerance co-
ecient goes to zero. To do so, we need some results in convex analysis regarding the so-called recession
function, dened for any
z ∈ Dome(g)
g0+ (z) := limγ↓0 γg
by
1
γz
= limγ↓0 γ g(y + γ1 z) − g(y) .
The key
properties of this function are recalled in the Appendix 10.2.1.
Conservative Risk Measure •
the polar function
g0+ (t, .)
G is non negative.
Under assumption
Since
(H1),
we may assume that
g(t, 0) = 0.
g(t, .) has a linear growth with constant k ,
the recession function
is nite everywhere with linear growth, and the domain of the dual function
The BSDE(g0+ , ξT ) has a unique solution
polar functions ldom(G) and
Yt0 (ξT ) ≥
G
is bounded by
Rgγ,t (ξT )
RT
= ess supµ∈Ak EQµ − ξT − t ldom(G) (u, µu )duFt ,
RT
= ess supµ∈Ak EQµ − ξT − γ t G(u, µu )duFt .
we can take the non decreasing limit in the second line and show that
Rg0+ ,t (ξT )
k.
g
Rt γ (ξT ). Using their dual representation through their
γG,
Yt0 (ξT )
Therefore,
= limγ↓0 Rgγ,t (ξT ) = Yt0 (ξT )
= ess supµ∈Ak EQµ − ξT Ft G(u, µu ) < ∞ ∀u ≥ t, du − a.s..
= ess supµ∈Ak ∩dom(G) EQµ − ξT Ft .
53
•
When the coecient
g
has a quadratic growth
(H3),
the recession function may be innite on a set with
positive measure and the BSDE(g0+ , ξT ) is not well-dened. However, we can still take the limit in the dual
representation of
Rgγ,t , obtain the same characterization of Rg0+ ,t , and consider Rg0+
of BSDE whose the coecient
g0+ = l{0} , G
said to be in
g0+
may be take innite values.
as a generalized solution
In particular if, as in the entropic case,
is nite everywhere and any equivalent probability measure associated with BMO coecient,
Q(BMO),
is admissible. Then,
l
∞
R0{0}
= e0+ ,t (ξT ).
+ ,t (ξT ) = ess supQ∈Q(BMO) EQ − ξT Ft = || − ξT ||t
Super Price System
ξT Ft
Note that the conservative risk-measure
−ξT
is the equivalent of the super-pricing rule of
(this notion was rst introduced by El Karoui and
Quenez [49] under the name "upper hedging price"). When the
the recession function
Rg0+ ,t (ξT ) = ess supµ∈A∩Dom(G) EQµ −
λt -translated
of
Dom(G)t
is a vector space,
g0+ (t, z) is the indicator function of the orthogonal vector space Dom(G)>
t
plus a linear
g
function hz, −λt i. Then, R + (−ξT ) is exactly the upper-hedging price associated with hedging portfolios
0 ,t
>
constrained to live in Dom(G) .
The conservative measure is the smallest of coherent risk measure such that
Rcoh
t (−ξT + ηT )
for any
(ξT , ηT )
in the appropriate space.
Volume Perspective Risk Measure
any convex risk measure
R
g
It is also possible to associate a coherent risk measure
, using the perspective function
normalized for the sake of simplicity (g(t, 0)
g̃(t, γ, z) =
measure
R
g̃
g̃
= 0).
g̃
of the coecient
The perspective function



More details about
Rgt (−ξT ) − Rgt (−ηT ) ≤
γg(t, γz )
z
γ)
limγ→0 γg(t,
g̃
g,
Rg̃
with
which is assumed to be
is dened as:
if γ > 0
= g0+ (t, z)
if γ = 0
can be found in the Appendix 10.2.2. As a direct consequence, the
g̃ -conditional
risk
is a coherent risk measure.
8 Inf-Convolution of g-Conditional Risk Measures
In this section, we come back to inf-convolution of risk measures, when they are g-conditional risk measures.
This study is based upon the inf-convolution of their respective coecients.
More precisely, we will study for any
t
the inf-convolution of the g-conditional risk measures
RA
t
and
RB
t
dened as
RA RB
where both
ξT
and
FT
t
B
ξT = ess inf FT RA
t ξT − FT + Rt FT
(41)
are taken in the appropriate space and show that this new dynamic risk measure is
under mild assumptions the (maximal solution)
ess inf z (g(., t, x − z) + g(., t, z)).
RA,B
of the BSDE
(g A g B , −ξT )
where
(g A g B )(., t, z) =
Then, the next step is to characterize the optimal transfer of risk between
both agents A and B, agent A being exposed to
ξT
at time
convex functions are recalled in the Appendix 10.3.
54
T.
Some key results on the inf-convolution of
8.1 Exact Inf-Convolution of g-Conditional Risk Measures
Pauline: Le plan que l'on avait pense faire est le suivant, apres une remarque generale sur les
mesures dilatees: Il doit y avoir un theoreme dont le label est "theorem inf-convolution
atteint"
Theorem 8.1
8.2 Characterization of Optima
We now focus on our main problem of inf-convolution of
g -conditional risk measures as expressed in Equation
(41). Before presenting the main result in a very general framework, we rst consider the case of dilated
g -conditional
risk measures.
The following theorem gives us an explicit characterization of an optimum for the inf-convolution problem:
Theorem 8.2
Let
gA
and
gB
be two convex coecients depending only on
of the Theorem 8.1. For a given
be the solution of the BSDE
Let
btB
Z
ξT
in the appropriate space (either
2
L (FT )
z
and satisfying the assumptions
or
L∞ (FT )),
let
(RA,B
(ξT ), Zt )
t
(g A g B , −ξT ).
n
o
btB = arg minx g A t, Zt − x + g B t, x
Z
be a measurable process such that
dt × dP − a.s..
Then, the following results hold:
(1)
For any
t ∈ [0, T ]
and for any
FT
RA
t (ξT − FT )
such that both
B
RA,B
ξ T ≤ RA
t
t (ξT − FT ) + Rt (FT )
(2)
If the process
bB
Z
is admissible, then for any
FT∗
RB
t (FT )
are well dened:
P − a.s.
t ∈ [0, T ]
RA,B
ξT = (RA RB t ξT
t
and the structure
and
P − a.s.
dened by the forward equation
FT∗
ZT
=
g
B
btB dt −
t, Z
0
ZT
btB dWt
Z
0
is an optimal solution for the inf-convolution problem (41) of the g-conditional risk measures.
Proof:
(1)
First, note that the existence of such a measurable process
In the following, we consider any
Let us now focus on
FT
such that both
B
RA
t (ξT − FT ) + Rt (FT ).
B
−d RA
t (ξT − FT ) + Rt (FT )
Therefore,
RA
t (ξT − FT )
is guaranteed by Theorem 8.1.
and
RB
t (FT )
are well dened.
It satises
=
=
and at time
btB
Z
g A (t, ZtA ) + g B (t, ZtB ) dt − ZtA + ZtB dWt
g A (t, Zt − ZtB ) + g B (t, ZtB ) dt − Zt dWt ,
B
T , RA
T (ξT − FT ) + RT (FT ) = −ξT .
B
(RA
t (ξT − FT ) + Rt (FT ), Zt )
is solution of the BSDE with terminal condition
55
−ξT ,
which is also
the terminal condition of the BSDE
of the BSDE
(g B , FT )
as:
(g A g B , −ξT ),
g(t, z) = g A (t, z − ZtB ) + g B (t, ZtB ).
this coecient is then always greater than
solution of the BSDEs
A
g A g B .
g
and a coecient
written in terms of the solution
ZtB
Using the denition of the inf-convolution,
Thus, we can compare
B
RA
t (ξT − FT ) + Rt (FT )
with the
B
(g g , −ξT ) using the comparison Theorem (5.3) and obtain the desired inequality.
***************************
Admissibility de ZbtB
***************************************
(2)
Let now assume that the process
btB
Z
is admissible, using dierent notion of admissibility when either
(H1) or (H3) (square integrability or BMO).
Thanks to Theorem 8.1, we can show that both dynamic risk measures coincide.
We now introduce the structure
FT∗
dened by the forward equation
Rt
Ft∗ =
Rt B
bsB ds − Z
bs dWs .
g B s, Z
0
Note rst that thanks to the admissibility of the process
the appropriate space (either
Let us also observe that
−Ft∗
L2 (FT )
or
btB ,
Z
0
such a structure is well-dened and belongs to
L∞ (FT )).
is also solution of the BSDE
(g B , −FT∗ )
since
−Ft∗ = −FT∗ +
RT
buB dt −
g B u, Z
t
RT
buB dWu .
Z
By uniqueness, this process is
∗
RB
t (FT ).
t
Since
∗
B
∗
RA
t (ξT − FT ) + Rt (FT ) is
and terminal condition
we also have
g A g B t, Zt = g A
ξT = RA RB t ξT
P a.s.
−ξT
∀t ≥ 0, RA,B
t
btB )
btB ) + g B (t, Z
g A (t, Zt − Z
btB + g B t, Z
btB , by uniqueness,
t, Zt − Z
solution of the BSDE with coecient written as
and given that
The proof also gives the optimality for the Problem (41) of the structure
FT∗ =
RT
RT B
bB dt − Z
bt dWt .
g B t, Z
t
0
0
Remark 8.3 (On uniqueness on the optimum)
Note that the optimal structure
FT∗
is determined to
within a constant because of the translation invariance property (P3-) satised by both risk measures
RA
t
and
RB
t since:
B
ess inf FT RA
t ξT − (FT + m) + Rt FT + m
=
=
B
ess inf FT RA
t ξ T − FT + m + R t FT − m
RA RB t ξT .
Comments:
(i)
Just as in the static framework, we obtain the same result when considering risk measures of the same
family.
The Borch theorem is therefore extremely robust since the quota sharing of the initial exposure
remains an optimal way of transferring the risk between dierent agents.
(ii)
Under some particular assumptions, the underlying logic of the transaction is non-speculative since
there is no interest for the rst agent to transfer some risk or equivalently to issue a structure if she is not
initially exposed. This result is completely consistent with the result we have already obtained in the static
framework.
Nicole le 14 juin: pour la partie suivante, il faut absolument reprende les notations de la
premiere partie pour aller plus vite. Il faut centrer le message sur le fait que l'inf-convolution
56
est avec une fonctionnelle quadratique
Pauline le 14 juin: je reprends les notations. Il me semble que la redaction nale viendra apres
celle du gros theoreme d'au-dessus
9 Optimal Hedging and Risk Transfer in a g-Conditional Risk Measure Framework
In this section, we come back to the questions of optimal hedging and risk transfer in this dynamic framework
and study how it can be expressed in terms of an inf-convolution of g-conditional risk measures.
As in subsection 4.2.2, we consider a single agent, denoted by agent A. She wants to hedge her terminal
wealth
XTA
by optimally investing on nancial market. She now assesses her risk using a general g-conditional
risk measure
RgA .
9.1 Framework
We consider the same framework as that introduced in Subsection 1.5.3 when looking at the question of
dynamic hedging in the static part.
More precisely, we assume that
market. Their forward (non-negative) vector price process
S
d
basic securities are traded on the
follows an Itô semi-martingale with a uniformly
bounded drift coecient and an invertible and bounded volatility matrix
BM O(P),
QT -local martingale.
of
QT ,
(or diusion coecients in
depending on the framework we consider).
To avoid arbitrage, we assume
a
σt
(AAO): there exists a probability measure QT , equivalent to P, such that S is
From the completeness of this basic arbitrage-free market, we deduce the uniqueness
which is usually called the forward risk-neutral probability measure. Its Radon-Nikodym density is
given as
Γ−λ
0,T
where the bounded (or in
under
T
Z
−λ∗u dWu
= exp
0
1
−
2
2
|λu | du
0
BM O(P)) parameter −λt can be interpreted as the risk premium vector.
P,
dSt
= σt (dWt + λt dt)
St
;
S0
As in Subsection 1.5.3, the agent can invest in dynamic strategies
(θ.S)t )
!
T
Z
given.
Therefore,
(42)
θ, i.e. d-predictable processes and (Gt (θ) =
denotes the associated gain process.
We assume that not all strategies are admissible and that, for instance, the agent has some restriction
imposed on the transaction size. These constraints create some market incompleteness in the framework we
consider.
K
ΘST = {GT (θ) | θ.S is bounded by below , θ ∈ K}
is a convex subset of
H
2
(or of
BM O(P))
is the set of admissible hedging gain processes.
such that any admissible strategies
θ
is in
K
(equivalently,
∀ t, θt ∈ Kt ).
Because of the boudedness of the dierent parameters (or because they all belong to
value of the gain process
GT (θ)
is bounded in
L2 (P)
(or in
57
L∞ (P)).
BM O(P)), the terminal
Note that it is possible to extend this framework to other possible hedging instruments, including for instance
derivatives, the price of which is known at any time and agreed by all agents in the market. In other words,
these derivatives are very liquid and used as reference instruments for calibration purposes.
Pauline: La suite est a reprendre en fonction de ce qui a ete fait precedemment, surtout depuis
que l'on a enleve les mesures innies...
9.2 Hedging Problem
The hedging problem of agent A can be expressed as the determination of an optimal admissible strategy
Zφ
as to minimize at time
0
the initial g-conditional risk measure of her terminal wealth
φ
A
inf RA
0 XT − ξT .
(43)
φ∈AK
To solve this problem, we use an inf-convolution argument between the initial dynamic risk measure of agent
A, RA ,
and the dynamic risk measure associated with nancial investments denoted as previously by
LK :
φ
A
A
K
(XTA )
inf RA
(XTA ) = RA,m
0 XT − ξT = R L
0
0
φ∈AK
where
RA,m
is the dynamic market modied risk measure of agent A.
Solving the hedging problem at time
0
leads to the characterization of a particular probability measure,
which can be called calibration probability measure as the prices of any hedging instruments made with
respect to this measure coincide with the observed market prices on which all agents agree.
Solving the hedging problem at any time
t is equivalent to solving the same problem at time 0 as soon as the
prices of these hedging instruments at this time
t
are given as the expected value of their discounted future
cash ows under the optimal calibration probability measure determined at time
0.
This optimal probability
measure is in this sense very robust as it remain the pricing measure for hedging instruments between
0
and
T.
Therefore, we can introduce the same problem at any time
t,
that is
φ
A
A
K
ess inf RA
(XTA ) = RA,m
(XTA ).
t
t XT − ξT = R L
t
(44)
φ∈AK
Then BSDEs time-consistency and uniqueness are key arguments to show that if
at time
0,
then
φ
is optimal for the optimization program at any time
φ is optimal for the problem
t.
The residual risk measure of agent A, after her optimal nancial investment, also called market modied
risk measure and denoted by
RA,m
,
t
is again a
g -conditional risk measure,
using Theorem 8.2. It is explicitly
given as solution of the following BSDE:
−dRA,m
(X) = g A,m (t, Zt )dt − ZT dWt
t
RA,m
(X) = −X
T
g A,m (t, Zt ) = inf φ∈AK g A (Zt −φσt ). This corresponds to the inf-convolution of g A
ct = σt∗ Kt .
indicator function of the convex set K
where
the
;
Obviously, the characteristics of the space
K
with lσ ∗ (Kt )
t
= lK
ct ,
are essential to the existence of a solution for this inf-convolution
58
problem. More precisely, under (H1) or (H3) for the coecient
gA
notations), the coecient of the market modied risk measure,
the space
K,
g
(i.e., if
A,m
gA
is either
bk
or
qk
using previous
, is obtained through a projection over
which should be rich enough so that the projection satises the desired properties as already
mentioned in Subsection 8. Note that
where
d(z, K)
under (H1)
c
g A,m (t, z) = bk lK
ct (t, z) = dk (z, Kt ),
under (H3)
g A,m (t, z) = qk lK
ct (t, z) =
is the distance function to
k
ct )2
dk (z, K
2
K.
Note that in both cases ((H1) and (H3)), the optimal nancial strategy is obtained as the projection of
solution of the BSDE
(g
A,m
, −XTA ) over the space
ct .
K
Zt ,
The optimal hedge is identical even if two dierent
criteria are used.
*****************
Preciser ici les proprietes que l'on va donner a K pour s'assurer de l'existence d'une "bonne"
solution....
*********************
We now consider the particular hedging problem when Agent A has a dynamic entropic risk measure.
Hedging and Pricing in the Dynamic Entropic Framework
dynamic entropic risk measure: in other words, for any bounded
Ψ,
Agent A assesses her risk using a
her risk measure is expressed as:
1
eγA ,t (Ψ) = γA ln EP exp(− Ψ)|Ft .
γA
We already know from equation (34) that
coecient
qk (t, z) =
eγA ,t
is a g-conditional risk measure associated with the quadratic
1
2
2γA |z| .
The hedging problem of agent A, described in equation (44) in the general framework, now becomes:
ess inf eγA ,t XTA − ξTφ = eγA LK t (XTA ).
φ∈AK
(45)
This entropic framework has been intensively studied in the literature. But only a few papers are using a
BSDEs framework. After the seminal paper by El Karoui and Rouge [51], dierent authors have used BSDEs
to solve this problem under various assumptions (see in particular Sekine [109], Mania et al. [87] and more
recently Hu, Imkeller and Müller [73] or Mania and Schweizer [88]).
Two main approaches can be used to solve this problem. On the one hand, as in most papers of the literature,
it is solved using the dual representation for the dynamic entropic risk measure as given by Theorem 7.4:
Z
eγA ,t (Ψ) = sup EQµ − Ψ − γA
µ∈Aq
t
T
|µs |2
ds|Ft .
2
Therefore, the hedging problem (45) can be rewritten as:
n
o
inf ess supµ∈Aq EQµ (−XTA + ξTφ |Ft ) − γA h(Qµ |P)
φ∈AK
59
and it may be solved by using dynamic programming arguments.
On the other hand, the inf-convolution approach gives a direct and quicker way to solve this problem.
As previously mentioned in the general framework, the hedging problem (45) can be rewritten as an infconvolution:
A
ess inf eγA ,t XTA − ξTφ = eγA LK t (XTA ) = em
γA ,t XT .
φ∈AK
em
γA ,t is the market modied risk measure of agent A. It is solution of the following BSDE:
A,m
−dem
(t, Zt )dt − ZT dWt
γA ,t (X) = g
;
em
γA ,T (X) = −X
g A,m (t, Zt ) = inf φ∈AK g A (Zt − φσt ). This corresponds to the inf-convolution of g A = q γ1
A
ct . Using the previous result when (H3) holds, we have:
indicator function of the convex set K
where
g A,m (t, z) =
The optimal investment strategy
φ?
1
ct )2 .
d 1 (z, K
2γA γA
is obtained by solving the inf-convolution problem of coecients. In
this quadratic case, we know from Subsection 8 that it is given as the projection of
(g
A,m
, −XTA ) over the space
with the
Zt ,
solution of the BSDE
K.
?
The terminal value of the investment portfolio associated with the optimal strategy
equation (
??) as:
∗
ξTφ = x +
Z
T
(φ?t )∗ σt λt dt +
0
Z
φ? , ξTφ
, is given using
T
(φ?t )∗ σt dWt .
0
9.3 General Optimal Hedging Problem and Optimal Risk Transfer
As previously noticed in the static framework (Subsection 4.2.2), the problem of optimal hedging can be
seen as a particular optimal risk transfer. Following this logic, we now extend the previous optimal hedging
problem and consider the question of optimal risk transfer between two agents. More precisely, in this new
framework, two agents A and B assess their risk using a general g-conditional risk measure, respectively
denoted by
RA
t
and
RB
t .
Agent A has a particular risk exposure
XTA
at time
T
and wants to reduce it by
transferring a part of it to agent B. Both agents can also invest on the nancial market. The set of admissible
strategies we consider may be dierent for both agents since their access to the nancial market just as the
constraint imposed on their investment may be dierent. Therefore, we denote these sets by
AA
and
AB
respectively.
At time
0,
agent A's problem is to choose both the optimal structure
(FT , π0 )
to issue and her nancial
investment as to minimize her risk measure under the constraint that agent B has an interest to buy the
structure.
Note that using the dynamic representation of both risk measures together with that of the respective
hedging portfolios of both agents, we can introduce the same problem at any time
the pricing rule at any time
time
t
to buy the structure
t, πt (FT ),
FT
t ∈ [0, T ].
In particular,
corresponds to the price agent B is willing to pay at that particular
given the observation of what happened between
60
0
and
t.
Coming back to time
inf
0,
the optimization problem can thus be written as:
φ
A
inf RA
0 X T − F T − ξ T + π0
π0 ,FT φ∈AA
s.t.
φ
φ
B
inf RB
0 FT − π0 − ξT ≤ inf R0 − ξT .
φ∈AB
φ∈AB
The optimal pricing rule at time 0 is obtained by binding the constraint at the optimum and using the cash
translation invariance property (P3-) of the g-conditional risk measure
RB
as:
φ
φ
B
π0∗ (FT ) = inf RB
0 − ξT − inf R0 FT − ξT .
φ∈AB
φ∈AB
Therefore, using the cash translation invariance property (P3-) of the g-conditional risk measure
with the optimal pricing rule, the program can be rewritten, to within the constant
inf
FT
n
φ
A
B
inf RA
0 XT − FT − ξT + inf R0
φ∈AA
φ∈AB
inf φ∈AB
RB
0
LB ,
where
LA
and
LB
together
− ξTφ
o
FT − ξTφ .
as:
(46)
This program corresponds to the inf-convolution of four dierent g-conditional risk measures:
and
RA
RA , LA , RB
are the market risk measures of both agents. Solving it combines the solving of
two inf-convolution problems, involving rst the inf-convolution between each individual risk measure and
RA
the associated market risk measure (in other words, the inf-convolution between
R
B
and
B
L
LA
and
and between
) solving there both individual hedging problems and then involving the global inf-convolution
problem using Theorem 8.2 to obtain an explicit characterization of the optimal risk transfer.
Condition for the transaction feasibility:
the feasibility of a transaction at any time
R
A
and
R
B
t
As previously mentioned in Section 8, the condition ensuring
between both agents
a
and
B,
having respective risk measures
, is expressed in terms of the associated recession risk measures as
B
RA
0,t (ΨT ) + R0,t (−ΨT ) ≥ 0.
Such a condition can be translated in terms of a condition on the super-prices of both agents: more precisely,
−RA
0,t (ΨT )
corresponds to the conservative seller's price of
corresponds to the conservative buyer's price of
ΨT
ΨT
made by agent
made by agent
B
at time
A
t.
at time
t
while
RB
0,t (−ΨT )
Therefore, it is equivalent
to impose a condition on the spread: a transaction can take place between both agents if the conservative
seller's price of agent
A
is less than the conservative buyer's price of agent
B
for any possible derivative
ΨT .
Such a condition appears very intuitive.
Here, in our problem, three inf-convolution problems are considered. The individual hedging problem will
have a solution under the condition:
Ri0,t (−ξT ) + Li0,t (ξT ) ≥ 0.
Such a condition can also be translated in terms of conservative prices: more precisely,
to the conservative buyer's price of
price of
ξT
ξT
made by agent
Ri0,t (−ξT ) corresponds
i while −Li0,t (ξT ) corresponds to the conservative seller's
made by the marker. A transaction with the market can take place as soon as the conservative
seller's price of the market is less than the conservative buyer's price of agent
i
for any possible hedging
instrument.
Finally, the nal inf-convolution problem involving both market modied risk measures makes sense if the
conservative seller's price of agent
contract
A
is less than the conservative buyer's price of agent
FT .
61
B
for any possible
10 Appendix: Some Results in Convex Analysis
We now present some key results in convex analysis that will be useful to obtain the dual representation of
g-
conditional risk measures. More details or proofs may be found in Aubin [4], Hiriart-Urruty and Lemaréchal
[70] or Rockafellar [102].
All the notations and denitions we introduce are consistent with the notations of risk measures. They may
dier from the standard framework of convex analysis (especially regaring the sign).
Even if the coecient of the BSDE is nite, we are also interested in convex functions taking innite values.
The main motivation is the denition of its convex polar function
G.
In that follows, as in [70], we always
+∞ and are bounded from below by a ane function
assume that the considered functions are not identically
(note that this assumption is rather general and does not necessarily require that the functions are convex).
The domain of a function
g
is dened as the nonempty set
convex function is the subset of
semicontinuous (lsc),
epig
n
R ×R
as:
dom(g) = {z : g(z) < +∞}.
epi g = {(x, λ) | g(x) ≤ λ}.
The epigraph of
When the convex functions are lower
is closed, and they are said to be closed.
10.1 Duality
10.1.1 Legendre-Fenchel Transformation
Let
g
be a convex function. The polar function
G
G(µ) = sup(hµ, zi − g(z)) =
z
The function
G
Rn
is dened on
by
sup (hµ, −zi − g(z)).
(47)
z∈dom(g)
is a closed convex function, which can take innite values. The conjugacy operation induces
a symmetric one-to-one correspondence in the class of all closed convex functions on
g(z) = sup(hµ, −zi − G(µ)),
analysis terminology) of S, lS
z
Given a nonempty subset
n
+
: R → R ∪ {+∞},
lS (z) = 0
lS
:
G(µ) = sup(hµ, −zi − g(z)).
µ
Convex set and duality
Rn
if
The polar function of lS is the support function of
the indicator function (in the convex
is dened by:
z∈S
is convex (closed), i S is convex (closed) since
S ⊂ Rn ,
and
+ ∞ if not.
epi lS = S × R+ .
−S :
σS (z) := suphs, −zi = sup{hs, −zi − lS (s)}.
s
s∈S
The support function is closed, convex, homgeneous function:
and its domain are convex cones.
62
σS (λz) = λσS (z)
for all
λ > 0.
Its epigraph
10.1.2 Subdierential and Optimization
The sub-dierential of the convex function
set
∂g(z)
g
in
z,
whose the elements are called subgradient of
z ∈
/ dom(g), ∂g(z) = ∅.
z
But if
ridom(g)
is in the interior of
dom(g),
z.
When
Note that when the function
is the
∂g(z)
g
is non-empty (see
belongs to the relative interior of
is the indicator function of the convex set
C
+
at z , NC (z)
Subgradients are solutions of minimization programs as
inf µ (G(µ) − hµ, −zi) (= −g(z)).
convex function and
G
(48)
g
dom(g),
is nite, then
∂g(z) is reduced to a single point, the function is said to be dierentiable
+
is the positive normal cone NC (z) to
z∈C
z
the subgradient
is dened in Section A in [70] and in Chapter 6 in [102]. In particular, if
∂g(z) is nonempty for any z .
in
z,
∀x} = { µ |g(z) − hµ, −zi ≥ G(µ)}.
Section E in [70] or Chapter 23 in [102]); in fact, it is enough that
where
at
dened as:
∂g(z) = { µ | g(x) ≥ g(z) − hµ, x − zi,
If
g
= {s ∈ Rn | ∀y ∈ C
C,
the sub-dierential of
g
in
− hs, y − zi} ≤ 0}.
inf z (g(z) − hµ, −zi) (= −G(µ)), or its dual program,
The precise result is the following (see Section E in [70]): Let
g
be a closed
its polar function.
• µ
b ∈ ∂g(b
z ) ⇐⇒ µ
b
is the optimal for the following minimization program,
that is
−g(z) =
inf µ (G(µ) − hµ, −zi) = G(b
µ) − hb
µ, −zi.
• zb ∈ ∂G(b
µ) ⇐⇒ zb
is optimal for the following minimization program,
that is in
−G(µ) =
inf z (g(z) − hµ, −zi) = g(b
z ) − hµ, −b
z i.
In the following, when working with BSDEs, we will denote by
z
and the column vector
zµ the scalar product between the line vector
µ.
10.2 Recession function
10.2.1 Recession Function
g is the homogeneous convex function dened
g0+ (z) := limγ↓0 γg γ1 z = limγ↓0 γ g(y + γ1 z) − g(y) . This function g0+ is the smallest
The recession function associated with a closed convex function
for
z ∈ Dom(g)
by
homogeneous function
g0+ (z) ≤ k|z|
coecient
k
h
such that for any
z, y ∈ Dom(g), g(z) − g(y) ≤ h(z − y).
is a nite convex function, and the function
since
g
When
g(z) ≤ c + k|z|,
is Lipschitz-continuous function with Lipschitz
g(z) − g(y) ≤ k|z − y|.
This property explains why any convex coecient of BSDE satisfying the assumption
(H2)
in fact satises
(H1).
Let
G
be the polar function of
limγ↓0 (γG(µ)) = 0.
g0+
So,
g
Using obvious notations, for any
polar g0+ = ldom G .
is the support function of
Lipschitz, or nally i
g.
dom(G); so g0+
µ ∈ Dom(g), polar g0+ (µ) =
By the conjugacy relationship applied to closed functions,
is nite everywhere i
has linear growth.
63
dom(G) is bounded, or i g
is uniformly
The recession function of the quadratic function
qk (z) = c + k|z|2
is innite except in
lar function is the null function. More generally, convex functions such that
function
G
g0+ = l{0}
z = 0,
and its po-
admit nite polar
and this condition is sucient.
10.2.2 Perspective Function
g
Let us consider a closed convex function
g̃
the function
dened on
+
R ×R
n


γg( γz )
limγ→0 γg( γz )

g
Note rst that the perspective function of
z
and
γ,
when
γ > 0.
g
The perspective function associated with
is
as:
g̃(γ, z) =
variables
g(0) = 0.
such that
if γ > 0
= g0+ (z)
corresponds to the
It is prorogated for
γ=0
tolerance coecient is considered as a risk factor itself.
if γ = 0
γ -dilated
of
g,
seen as a function of both
by the recession function
g̃
g0+ .
Note that the risk
is a positive homogeneous convex function (for
more details, please refer to Part B [70]).
The dual function of
g̃ ,
dened on
R × Rn ,
if G(µ) ≤ −θ ,
G̃(θ, µ) = 0
If
g(0) < ∞,
note that
G(µ)
is given by:
+ ∞ otherwise.
and
is bounded.
10.3 Inmal Convolution of Convex Functions and Minimization Programs
Addition and inf-convolution of closed convex functions are two dual operations with respect to the conjugacy
relation.
Let
gA
and
gA
and
gB
gB
be two closed convex functions from
is the function
g A g B
g A g B (z) =
If
g
g A g B 6≡ ∞,
B
Rn ∪ {+∞}.
By denition, the inmal convolution of
dened as:
inf
y A +y B =z
(g A (y A ) + g B (y B )) = inf (g A (z − y) + g B (y)).
(49)
y
then its polar function, denoted by
GAB ,
is simply the sum of the polar functions of
gA
and
:
GAB (µ) = GA (µ) + GB (µ)
10.3.1 Inf-Convolution as a Proper Convex Function
The function
g A g B
may take the value
−∞, which is contrary to the assumption made in Subsection 7.2.
avoid this diculty, we assume that both functions
gA
and
gB
have a common ane minorant
hs, .i − b.
To
This
assumption may be expressed in terms of their recession functions, both of them being also bounded from
below by
hs, .i.
Therefore,
g0A+ (z) + g0B+ (−z) ≥ 0
for any
z
this condition can also be expressed in terms of the polar functions of
64
g0A+ g0B+ (0) ≥ 0.
and consequently
g
A
and
g
B
as
A
Note that
dom(G )∩dom(GB ) 6= ∅.
10.3.2 Existence of Exact Inf-Convolution
We are interested in the existence of a solution to the inf-convolution problem (49). When a solution exists,
the inmal convolution is said to be exact.
The previous conditions are almost sucient, as proved in Rockafellar [102] since, if we assume
g0A+ (z) + g0B+ (−z) > 0,
then
g A g B
∀z 6= 0
(50)
is a closed convex function, and for any z, the inmum is attained by some
x∗ :
g A g B (z) = inf {g A (z − x) + g B (x)} = g A (z − x∗ ) + g B (x∗ ).
x
The condition (50) is satised if
intdom(GA ) ∩ intdom(GB ) 6= ∅
(in fact, the true interior corresponds to the
relative interior dened in Section A by Hiriart-Urruty and Lemaréchal [70]).
Examples of exact inf-convolution:
We now mention dierent cases where the inf-convolution has a
solution.
•
First, when both convex functions
gA
gB
and
are dilated, then their inf-convolution is exact without
having to impose any particular assumption, as we have already noticed when working with static risk
measures in the rst part (see Proposition 3.5). More precisely, assume that
from a given convex function
z,
•
an optimal solution
More generally, if
that
gA
∗
x
g
such that
A
g = gγA
and
g
B
= gγB ,
then
to the inf-convolution problem is given by
is bounded from below and if
B
inf z g (z) is reached for some z
gB
(in other words,
g
B
A
g g
∗
x =
gA
B
gB
are dilated
= gγA +γB
and for any
and
γB
γA +γB .
satises the qualication constraint ensuring
has a strictly positive recession function
then the condition (50) is satised and the inf-convolution
g A g B
g0B+ ),
has a non-empty compact set of
solutions.
10.3.3 Characterization of Optima
We are now interested on the characterization of optima in the case of exact inf-convolution.
be done in terms of the subdierentials of the dierent convex functions involved.
consider
zA
and
zB
respectively in
dom(g A )
and in
dom(g B )
and
z = zA + zB
in
This can
More precisely, let us
dom(g A g B ).
Then,
∂g A (z A ) ∩ ∂g B (z B ) ⊂ ∂(g A g B )(z).
Moreover, if
∂g A (z A ) ∩ ∂g B (z B ) 6= ∅, then the inf-convolution g A g B
∂g B (z B ) = ∂(g A g B )(z).
In particular, as
∂g A (0) ∩ ∂g B (0)
0
z = zA + zB
and
∂g A (z A ) ∩
(For more details, please refer to [70]).
belongs to the domain of
gA
and
and the inf-convolution is exact at
Moreover, if both functions are centered, i.e.
as
is exact at
gB ,
if
∂g A (0) ∩ ∂g B (0) 6= ∅,
then
∂(g A g B )(0) =
0.
A
g (0) = g B (0) = 0,
(g A g B )(0) = g A (0) + g B (0) = 0.
65
then the inf-convolution is also centered
10.3.4 Regularization by Inf-Convolution
As convolution, the inmal convolution is used in regularization procedures. The most famous regularizations
are certainly, on the one hand, the Lipschitz regularization
g(k)
of
g
using the inf-convolution with the kernel
bk (z) = k|z| and on the other hand, the dierentiable regularization, also called Moreau-Yosida regularization,
g[k]
of
g
using the inf-convolution with the kernel
qk (z) = k2 |z|2 .
Both regularizations do not have however
the same "eciency".
Lipschitz Regularization
We rst consider the inf-convolution
g(k)
of
g
using the kernel
bk (z) = k|z|
or more generally using functions whose polar's domain is bounded (or equivalently with a nite recession
function).
The function
g(k)
is nite, convex, non decreasing w.r. to
continuous, with Lipschitz constant
k.
k.
Moreover, its inf-convolution
g(k)
is Lipschitz-
More generally, the inf-convolution of two convex functions, one of
them satisfying (H1), also satises (H1) without any condition on the other function.
If
z0 ∈ int domg ,
convex set
then
g(k) (z0 ) = g(z0 )
for
k
large enough. When
g = lC
is the indicator function of a closed
C , gk = kdist(., C).
This regularization is used in the
Appendix or BSDE chapter??
to show the existence of BSDE with
continuous coecient.
Moreau-Yosida Regularization
k
2
2 |z| .
The function
g[k]
We now consider the inf-convolution
is nite, convex, non decreasing w.r.
its gradient is Lipschitz-continuous with Lipschitz constant
strongly convex with module
k,
G[k] (.) −
equivalently
k.
to
k.
g[k]
of
g
Moreover,
using the kernel
g[k]
qk (z) =
is dierentiable and
In other words, the polar function of
g[k]
is
k 2
2 |.| is still a convex function (for more details, please
refer to Cohen [32]).
There exists a point
Jk (z)
that attains the minimum in the inf-convolution problem with
z → Jk (z) are Lipschitz continuous with a constant 1, independent of k
∗
2
(Jk (z) − Jk (y)) (z − y) ≥ ||Jk (z) − Jk (y)||
. Moreover,∇g[k]
qk .
The maps
and monotonic in the following sense
= k(z − Jk (z)).
More generally, the inf-convolution of two convex functions, one of them being strongly convex, satises (H3)
without any condition on the other function.
Pauline: il faut rajouter les fortement convexes....
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203-228 (1999).
66
[3] Artzner, P., Delbaen, F., Eber, J.M., Heath,D., Ku, H.: Coherent Multiperiod risk adjusted values and
Bellman's principle, Working Paper (2004) (http://www.math.ethz.ch/ delbaen/ftp/preprints/adehkor.pdf|).
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