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Optimal Claims with Fixed Payoff Structure
C. Bernard∗
,L.R¨uschendorf†
, S. Vanduffel‡
January 31, 2014
Abstract
Dybvig (1988) introduced the interesting problem of how to construct in
the cheapest possible way a terminal wealth with desired distribution. This
idea has induced a series of papers concerning generality, consequences and
applications. As the optimized claims typically follow the trend in the market,
they are not useful for investors who wish to use them to protect an existing
portfolio. For this reason, Bernard et al. (2013a) impose additional state-
dependent constraints as a way of controlling the payoff structure. The present
paper extends this work in various ways.
In order to get optimal claims in general models we allow in this paper
for extended contracts. We deal with general multivariate price processes and
dismiss with several of the regularity assumptions in the previous work (in par-
ticular, we omit any continuity assumption). State-dependence is modeled by
requiringthatterminalwealth hasafixedcopulawithabenchmarkwealth. In
this setting, we are able to characterize optimal claims. We apply the theo-
retical results to deal with several hedging and expected utility maximization
problems of interest.
Key-words: cost-efficient payoffs, optimal portfolio, state-dependent utilities
AMS classification: 91G10, 91B16
1 Introduction
We consider optimal investment problems in a financial market given by a market
model S=(St)0≤t≤Tin a filtered probability space (Ω,A,(At)0≤t≤T,P). Smay
∗Carole Bernard, University of Waterloo, 200 University Avenue West, Waterloo, Ontario,
N2L3G1, Canada. (email: c3bernar@uwaterloo.ca). Carole Bernard acknowledges support from
NSERC.
†Ludger R¨uschendorf, University of Freiburg, Eckerstraße 1, 79104 Freiburg, Germany. (email:
ruschen@stochastik.uni-freiburg.de).
‡Steven Vanduffel, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Bruxelles, Belgium. (email:
steven.vanduffel@vub.ac.be). Steven Vanduffel acknowledges support from BNP Paribas Fortis.
1
consist of several stocks and also bank accounts. Our basic assumption is that state
prices at time tare determined by a pricing process ξ=(ξt)0≤t≤Tthat is adapted
to the filtration. Typically, ξwill be the (discounted) pricing density process of a
martingale pricing rule. Typical examples include exponential L´evy models in which
one uses an Esscher pricing measure that has pricing process of the form ξt=gt(St)
and note that the case of the multidimensional Black–Scholes market is also covered
herein. But one may also think of a stochastic volatility model in which ξtis a function
of the process (Su)0≤u≤tand some additional volatility process (σu)0≤u≤t.
Let XTbe a payoff at time T(i.e., XTis AT-measurable) with payoff distribution
Fand cost c(XT):=EξTXT.The aim of an investor with law invariant (state-
independent)preferences1is to construct a payoff X∗
Twith the same payoff distribution
Fat lowest possible cost, i.e.
c(X∗
T) = inf{EξTYT;YT∼F},(1)
where YT∼Fmeans that YThas the same payoff distribution Fas XT.Notethat
problem (1) depends only on the distribution Fand not on the specific form of the
payoff XT. Therefore, XTdenotes in what follows any generic payoff with distribution
function F.
The optimization problem in (1) represents the static version of the optimal port-
folio problem (He and Pearson (1991a,b)). The optimal payoff can (in a second
step) always be attained by hedging strategies in complete markets. Characteriza-
tions and sufficient conditions for representation of the optimal claims in incomplete
markets by continuous time trading strategies have been established in the literature
and are related to the optional decomposition theorem (see Jacka (1992), Ansel and
Stricker (1994), Delbaen and Schachermayer (1995), Goll and R¨uschendorf (2001) and
Rheinl¨ander and Sexton (2011)).
The cost minimization problem in (1) has been stated and solved in various gen-
erality in Dybvig (1988), Bernard, Boyle, and Vanduffel (2014), Carlier and Dana
(2011), R¨uschendorf (2012), and others under various assumptions on the distribu-
tions. Several explicit calculations of optimal claims2have been given in the frame-
work of the Black–Scholes Model (see Bernard et al. (2011,2014) ) and in exponential
L´evy models (see Hammerstein et al. (2013)). In Section 2, we introduce the class of
extended payoffs, which are based on the market information ATup to time Tbut
also allow for external randomization. We refer to them as randomized payoffs. The
use of randomization allows us to construct optimal claims explicitly without posing
the regularity conditions as in Bernard et al. (2013a). Indeed, we provide a simple
proof showing that optimal claims are only dependent on ξTand possibly on some in-
dependent randomization V. In the particular case of L´evy models this result implies
path-independence of optimal claims, i.e. optimal claims are of the form X∗
T=f(ST)
resp. f(ST,V).
1Examples include lots of classical behavioral theories including mean-variance optimization
(Markovitz (1952)), expected utility theory (von Neumann and Morgenstern (1947)), dual theory
(Yaari (1987)), rank dependent utility theory (Quiggin (1993)), cumulative prospect theory (Tversky
and Kahneman (1992)), and sp/a theory (Shefrin and Statman (1990)).
2We use in this paper the notion of payoffs and claims synonymously.
2
Bernard et al. (2014), (2013a) point out that solutions to the cost minimization
problem (1) are not suitable for investors who are exposed to some external risk
that they want to protect against. These investors are prepared to pay more to
obtain a certain distribution, simply because they want the optimal payoff to pay
our more in some desired states. For example, a put option gives its best outcomes
in the worst states of the market and thus allows investors to protect3the value of
an existing investment portfolio that is long with the market. In other words, two
payoffs with the same distribution do not necessarily present the same “value” for
an investor; see also the discussion in Vanduffel et al. (2012). Therefore, in Section
3, we introduce and discuss, following the development in Bernard et al. (2013a),
additional restrictions on the form of the payoffs. These restrictions are determined
by fixing the desired copula of the claim with a random benchmark AT.Thistype
of constraint allows to control for the states of the economy in which the investor
wants to receive payments. Note that in the case where ATis deterministic there
is no imposed restriction and we obtain again the optimal payoffs in the classical
context without constraint. As a main result, we determine payoffs with minimal
price and given payoff distribution Funder state-dependent constraints in general
markets. In comparison to the results in Bernard et al. (2013a) we obtain with the
extended notion of (randomized) claims optimal solutions that are functions of ξT,
ATand some independent randomization. This characterization extends the concept
of ‘twins’ as optimal solutions as in Bernard et al. (2013a).
We use this characterization result to deal with several hedging and investment
problems of interest. In Section 4, we provide the optimal claim for an expected utility
maximizer with state-dependent constraints. In Section 5 we solve some optimal
hedging problems and also determine the optimal contract for an expected return
maximizer with constraints on the minimum and maximum desired return.
2 Randomized claims and cost-efficient payoffs
Denote by L(AT)theclassofallAT-measurable claims (payoffs) at time T.For
the construction of optimal claims it will be useful to extend the notion of claims
(payoffs) to randomized claims (randomized payoffs). We generally assume that the
underlying probability space (Ω,AT,P) is rich enough to allow to construct for each
element YTa random variable Vthat is independent of YTand uniformly distributed
on (0,1).
A ‘randomized claim’ is a claim of the form f(YT,V) involving a randomization
Vthat is independent of YT. The use of randomized claims is an essential point in
this paper, which allows to solve portfolio optimization problems in general market
models. Under continuity assumptions as used in Bernard et al. (2013a) one can avoid
this additional randomization. At first glance, it may seem strange to an investor to
3The same observation is also at the core of insurance business. People buy a fire insurance
contract and not a cheaper financial contract with identical distribution (“digital option”) because
the insurance contract provides wealth when it is actually needed; see also Bernard and Vanduffel
(2014a).
3
use an independent randomization for the construction of an investment. A similar
objection also concerned the use of randomized tests in classical testing theory. As
in testing theory, where one obtains existence of optimal tests only in the class of
randomized tests, one can expect existence of optimal claims only within the more
general class of randomized claims. In some market models it may be possible to use
the market model to construct this independent randomization. This is underlying
the concept of twins in Bernard et al. (2013a), but in general the investor should
be prepared to throw the dice in order to be better off. In what follows we use
randomized claims without further ado.
For a given payoff distribution Fa claim X∗
T∈L(AT) with payoff distribution F
is called ‘cost-efficient’ if it minimizes the cost c(YT)overallclaimsYTwith payoff
distribution F, i.e. if X∗
Tsolves (1) (see Bernard et al. (2014)). For the construction
of cost-efficient payoffs we shall make use of the following two classical results:
Hoeffding–Fr´echet bounds (Hoeffding (1940) and Fr´echet (1940), (1951)): Let
X,Ybe random variables with distribution functions F,Gand let U∼U(0,1) be
uniformly distributed on (0,1). Then
EF−1(U)G−1(1 −U)≤EXY ≤EF−1(U)G−1(U).(2)
The upper bond is attained only if (X, Y )∼(F−1(U),G
−1(U)) (where ∼refers to
equality in distribution), i.e. X,Yare comonotonic. The lower bound is attained
only if (X, Y )∼(F−1(U),G
−1(1 −U)), i.e. X,Yare anti-monotonic.
Distributional transform (R¨uschendorf (1981, 2009)): For a random variable X∼
Fand a random variable V∼U(0,1) that is independent of X, the ‘distributional
transform’τXis defined by
τX=F(X, V ),(3)
where (with slight abuse of notation), F(x, λ):=P(X<x)+λP (X=x) and note
that F(x, λ)=F(x)whenFis continuous. Then
τX∼U(0,1) and X=F−1(τX)a.s. (4)
The variable τXcan thus be seen as a uniformly distributed variable that is associated
to (or, transformed from) X.
For a payoff distribution function F, we denote by K(F) the class of all claims
that have payoff distribution F:
K(F)={YT∈L(AT); YT∼F}.
Combining the Hoeffding–Fr´echet bounds in (2) and the distributional transform in
(3) allows us to obtain in a straightforward way the following general form of the
cost-efficient claim.
Theorem 2.1 (Cost-efficient claim).For a given payoff distribution Fthe claim
X∗
T=F−1(1 −τξT)(5)
is cost-efficient, i.e.
c(X∗
T) = inf
YT∈K(F)c(YT)(6)
4
Proof. The distributional transform τξT=F(ξT,V) is by (4) uniformly distributed
on (0,1) and ξT=F−1(τξT) a.s. This implies that the pair (ξT,X∗
T) is anti-monotonic
and thus (6) is a consequence of the Hoeffding–Fr´echet lower bound in (2).
Remark 2.2.
1. When FξTis continuous, the additional randomization Vcan be omitted and
(5) coincides with the classical result on cost-efficient claims (see Dybvig (1988),
Bernard et al. (2014)).
InthecasethatξT=gT(ST) for an appropriate function gTone obtains that
X∗
T=h(ST)(7)
for some function h. Thus any path-dependent option can be improved by a
path-independent option. For this observation, see Bernard et al. (2014).
2. Several explicit results on lookback options, Asian options and related path-
dependent options have been given in the context of Black–Scholes models and
L´evy models in Bernard et al. (2011,2014) and Hammerstein et al. (2013).
3. It is not difficult to see (cf. the proof of Theorem 4.1) that optimal claims
that follow from optimizing a law-invariant objective (e.g. expected utility) at
a given horizon Tmust be cost-efficient.
3 Payoffs with fixed payoff structure
If ξTis a decreasing function of ST(a property that is predicted by economic theory
and confirmed by many popular pricing models including increasing exponential L´evy
type models) then an optimized payoff X∗
Tis increasing in ST.The optimal payoff can
thus be quite different from the initial payoff XTand performs poorly when the market
asset STreaches low levels. These qualitative features do not demonstrate a defect
of the solution, but rather show that portfolio optimization which only considers
distributional properties of terminal wealth is not suitable in all situations. For
example, some investors buy put options to protect their existing portfolio (as a source
of benchmark risk) and they are not interested in the cost-efficient alternatives as
these are long with the market and do no longer offer protection. These observations
let Bernard et al. (2013a) to include constraints in the optimization problem that
allow controlling for the states in which payments are received. In this paper, we
build further on this development. We restrict the class of admissible options in the
portfolio optimization problem by requiring that the admissible claims pay out more
in some desired states (e.g. when STis low) and less in other states (state-dependence
constraints).
To model the state-dependence constraints we use a random benchmark ATand
we couple the admissible claims YTto the behavior of AT. More precisely, let ATbe
some random benchmark such as e.g. AT=STor AT=(ST−K)+or some other
available claim in the market and let Cdenote a copula which describes the wished
5
payoff structure of admissible claims. The copula Cis not necessarily the copula of a
given initial claim with the benchmark AT, but it is a tool to describe in which states
of the benchmark the investor wants to receive income (or protection). We consider
a claim YTto be admissible if the copula of the pair (YT,A
T)isC, i.e.
C(YT,AT)=C. (8)
The copula Cdetermines how the payoff structure of YTis coupled to the benchmark
AT. in this way we are able to prescribe that payoffs are (approximately) increasing
or decreasing in ATor take place for ATeither big or small, as described by the
following examples of copulas.
0000
11
111
1
1
1
Figure 1: Various dependence prescriptions
The portfolio optimization problem in (1) is now modified to include a fixed payoff
structure, i.e. we determine X∗
T∼Fwith copula C(X∗
T,AT)=Csuch that
c(X∗
T) = inf{c(YT); YT∼F, C(YT,AT)=C}.(9)
Since the joint distribution function Gof (YT,A
T)isgivenby
G=C(F, FAT),(10)
problem (9) is equivalent to the cost minimization problem when fixing the joint
distribution of (YT,A
T)tobeequaltoG, i.e.
c(X∗
T) = inf{c(YT); (YT,A
T)∼G}.(11)
For the construction of the solution of the portfolio optimization problem in (9)
resp. (11) we will use the concept of the conditional distributional transform.
Conditional distributional transform: The conditional distributional transform
of Xgiven Yis defined as
τX|Y=FX|Y(X, V ) (12)
where for all y,Vis independent of (X|Y=y).
It is clear that by property (4) of the distributional transform,
τX|Y∼U(0,1) and τX|Yis stochastically independent of Y. (13)
In the following theorem we determine the optimal solution of the portfolio opti-
mization problem in (9), (11). Based on the concept of randomized claims it gives
6
an extension of Theorem 3.3 in Bernard et al. (2013a) to the case of general mar-
ket models (avoiding the regularity conditions imposed in that paper). Let XTbe
a payoff with distribution Fandsuchthat(XT,A
T) has copula C, or, equivalently,
(XT,A
T)∼G.
Theorem 3.1 (Cost-efficient claim with fixed payoff structure).Let
XTbe a claim with (XT,A
T)∼G,then
X∗
T:= F−1
XT|AT(1 −τξT|AT) (14)
is a cost-efficient claim with fixed dependence structure, i.e. X∗
Tis a solution of the
portfolio optimization problem with fixed payoff structure
c(X∗
T) = inf{c(YT); (YT,A
T)∼G}.
Proof. Let us denote U=τξT|AT.ForX∗
T=F−1
XT|AT(1 −U)holds
(X∗
T|AT=a)=(F−1
XT|AT=a(1 −U)|AT=a)=F−1
XT|AT=a(1 −U)
since U,ATare independent (see (13)). Consequently, we obtain (X∗
T,A
T)∼(XT,A
T)
∼Gand thus X∗
Tis admissible. Furthermore, since conditionally on AT=a,
((X∗
T,ξ
T)|AT=a)∼((F−1
XT|AT=a(1 −FξT|AT=a(ξT)),ξ
T)|AT=a),
we obtain that X∗
T,ξTare anti-monotonic conditionally on AT=a. Thisimpliesby
the Hoeffding–Fr´echet bounds in (2)
EX∗
TξT=EE(X∗
TξT|AT)
≤EE(XTξT|AT)=EXTξT,
i.e. X∗
Tis cost-efficient in the class of portfolios with fixed dependence structure.
Remark 3.2.
1. The proof shows that the cost-efficient claim with fixed dependence structure
is characterized by the property that conditionally on ATit is anti-monotonic
with the state-price ξT. Note that Theorem 3.1 holds true in the case that C
is any copula (not necessarily the copula of a given initial claim XTwith AT).
The construction of X∗
Tdepends only on F,ATand on the copula C, i.e. the
aimed payoff structure.
2. When the state-price ξT=gT(ST) is a decreasing function of the stock ST,
as in increasing exponential L´evy models, we obtain that a cost-efficient claim
X∗
Tis characterized by the property that conditionally on AT,X∗
Tand STare
comonotonic.
3. In the case that the independent randomization Vcan be generated from the
market pair (St,S
T) by a transformation we obtain a cost-efficient claim of the
form f(St,S
T)ifAT=ST,resp. f(St,S
T,A
T) in the general case. Claims
of this form are called ‘twins’ in Bernard et al. (2013a). It is shown in that
paper that under some conditions cost-efficient payoffs are given by twins. With
the notion of extended payoffs in this paper we obtain that generally optimal
payoffs are of the form f(ST,V)resp. f(ST,A
T,V) with some independent
randomization V.
7
4 Utility optimal payoffs with fixed payoff struc-
ture
The basic optimization problem of maximizing the expected utility of final wealth XT
at a given horizon Twith an initial budget w, i.e.
max
c(XT)=wEu(XT) (15)
was solved in various generality in classical papers of Merton (1971), Cox and Huang
(1989) and He and Pearson (1991a,b). The optimal solution for differentiable increas-
ing concave utility functions uon (a, b)isoftheform
X∗
T=(u)−1(λξT),(16)
where λis such that c(X∗
T)=w. For the existence of λsuch that c(X∗
T)=wit is
assumed that uis strictly decreasing and u(a+)=∞,u(b−)=0.
An extension of the utility optimization problem to the case with a fixed payoff
structure was introduced in Bernard et al. (2013a) as
max
c(XT)=w
C(XT,AT)=C
Eu(XT) (17)
To deal with problem (17), we define
ZT=C−1
1|AT(1 −τξT|AT),(18)
where C1|AT=C1|τATis the conditional distribution function (w.r.t. C) of the first
component given that the second component is the distributional transform τAT.
Then ZT∼U(0,1), ZThas copula Cwith ATand the pair (ZT,ξ
T) is anti-monotonic
conditionally on AT(see also (13)). Next, we introduce the following condition
(D) HT=E(ξT|ZT)=ϕ(ZT) is a decreasing function of ZT.
Condition (D) does not always hold but is natural since ZT,ξ
Tare anti-monotonic
conditionally on AT. In the strict sense it needs however some regularity condition
to be fulfilled.
The following theorem describes the utility optimal payoff with fixed payoff struc-
ture and given budget wunder condition (D).
Theorem 4.1 (Utility optimal payoff with given payoff structure).Under
condition (D) the solution of the restricted portfolio optimization problem (17) is
given by
X∗
T=(u)−1(λHT) (19)
with λsuch that c(X∗
T)=w.
Proof. The utility optimal payoff must be a cost-efficient claim with fixed payoff
structure (with cost w) as in Theorem 3.1. Otherwise, it is possible to construct a
8
strictly cheaper solution which yields the same utility while respecting the dependence
constraint. Consequently the solution XT(when it exists) is characterized by the
property that conditionally on ATit is anti-monotonic with the state-price ξTand,
therefore,
XT=F−1
XT|AT(1 −τξT|AT).(20)
The payoff F−1
XT(ZT) has distribution function FXT, has copula Cwith ATand is con-
ditionally on ATanti-monotonic with the state-price ξT. By the uniqueness property
of cost-efficient claims this implies that
XT=F−1
XT(ZT) a.s.
In particular, the optimal solution is increasing in ZTand the constraint on its
cost can be written as
c(XT)=EξTF−1
XT(ZT)=EHTF−1
XT(ZT),(21)
where HT=E(ξT|ZT)=ϕ(ZT) is decreasing in ZTby assumption (D).
The utility optimization problem of interest can thus be rewritten as
max
EXTHT=w
XT=k(ZT)
kincreasing
Eu(XT).(22)
Considering the relaxed problem
max
EXTHT=wEu(XT) (23)
we obtain an utility optimization problem in standard form with price density HT
instead of ξT. By (16) its solution is given by
X∗
T=(u)−1(λHT)=(u)−1(λϕ(ZT)) (24)
where λ>0 is chosen such that EHTX∗
T=w.Sinceϕis decreasing by assumption
(D) it follows that X∗
Tis increasing in ZTand thus it also solves the restricted portfolio
optimization problem (19).
Bernard and Vanduffel (2014b) derive optimal mean-variance efficient portfolios
in presence of a stochastic benchmark (Propositions 5.1 and 5.2). Their results also
follow from Theorem 4.1. An application of Theorem 4.1 in the univariate Black–
Scholes model can be found in Bernard et al. (2013a). These authors use a Gaussian
copula to fix the portfolio structure and verify that condition (D) is satisfied. Note
that this example can be extended to the multivariate Black–Scholes model.
Interestingly, Theorem 4.1 can be extended to the general case without assuming
condition (D). As a result the optimal claim will be slightly more complex. For the
extension we need to project the function ϕfrom the representation of HTto the
convex cone of decreasing L2-functions M↓on [0,1]
M↓={f∈L2[0,1]; fnon-increasing}.
Let ϕ∈L2[0,1], supplied with the Lebesgue-measure and the Euclidean norm, and
ϕ=πM↓(ϕ) denotes the projection of ϕon M↓. Then we obtain
9
Theorem 4.2 (Utility optimal payoff).Assume that HT=E(ξT|ZT)=ϕ(ZT)with
ϕ∈L2[0,1]. Then the solution to the restricted utility optimization problem (17) is
given by
X∗
T=(u)−1(λ
HT),(25)
where
HT=ϕ(ZT)and λis such that c(X∗
T)=w.
Proof. The proof is analogous to the proof of Theorem 5.2 in Bernard et al. (2013a).
It is based on properties of the projection on convex cones which can be found in
Barlow et al. (1972).
Remark 4.3.
1. The projection ϕof ϕon M↓is given as the slope of the smallest concave
majorant SCM(ϕ)ofϕ, i.e. ϕ=(SCM(ϕ)). Fast algorithms are known to
determine ϕ.
2. The condition ϕ∈L2[0,1] is implied by the condition ξT∈L2(P).
5 Optimal hedging and quantile hedging
In this section we use the results of Sections 2–4 to solve various forms of static partial
hedging problems. Let LTbe a financial derivative (liability) and let wL=c(LT)
denote the price of LTw.r.t. the underlying pricing measure. If the available budget
wis smaller than wLthen it is of interest to find a best possible partial hedge (cover)
of LTwith cost wunder various optimality criteria. This leads to the following basic
static partial hedging problems.
The quantile (super-)hedging problem is defined as
max
c(XT)=wP(XTLT).(26)
The utility optimal hedging problem is a natural variant of (26) and is stated as
max
c(XT)=wEu(XT−LT),(27)
where uis a given concave utility function defined on Rthat satisfies the same regu-
larity conditions as in Section 4.
A more general version of the hedging problem in (27) is obtained by replacing
expected utility by some law invariant, convex risk measure Ψ (Ψ monotonic in the
natural order), i.e.
max
c(XT)=wΨ(XT−LT).(28)
We also consider (state-dependent) variants of the partial hedging problems (26)–(28)
in which the excess XT−LTsatisfies additional restrictions, allowing us to control
its excess structure. For example, we may want that XT−LThas a certain copula
10
Cwith a benchmark AT,i.e. C(XT−LT,AT)=C, or we may impose certain additional
boundedness conditions on XT.
We start with the (unconstrained) optimal hedging problem (27). Its solution is
given in the following proposition.
Proposition 5.1 (Utility optimal hedge).Let LTbe a financial claim with price
c(LT)=wLand let w<w
Lbe the budget available for hedging. Then the optimal
hedge for the utility optimal hedging problem (27) is given by
X∗
T=LT+(u)−1(λξT),(29)
where λ0is such that
c((u)−1(λξT)) = w−wL.
Proof. By the classical portfolio optimization result (see (16)) we obtain that the
optimal solution of the utility optimization problem
max
c(XT)=w−wL
Eu(XT) (30)
is given by
XT=(u)−1(λξT) (31)
with λchosen in such a way that c(
XT)=EξT
XT=w−wL.TheclaimX∗
T:=
XT+LT
therefore has price w,c(X∗
T)=w. By definition X∗
Tsolves the utility optimal hedging
problem in (27).
In the following two extensions we fix the joint dependence structure of the excess
XT−LTwith a given benchmark AT.
Proposition 5.2 (Utility optimal hedge with dependence restriction on the excess).
Let LTbe a financial claim with price c(LT)=wL,letw<w
Lbe the budget available
for hedging and let ATbe a given benchmark. Then the restricted utility optimal
hedging problem
max
c(XT)=w
C(XT−LT,AT)=C
Eu(XT−LT)
has the solution
X∗
T=LT+(u)−1(λ
HT),(32)
where
HT=ϕ(ZT)and λ0is such that
c((u)−1(λ
HT)) = w−wL.
Proof. The optimality of X∗
Tis a consequence of Theorem 4.2 and is based on a simple
replacement strategy as in the proof of Proposition 5.1.
In the following variant of the hedging problem it is our aim to avoid super-hedging
of LT.
11
Proposition 5.3 (Utility optimal hedge with negative excess).Let LTbe a financial
claim with price c(LT)=wLand let w<w
Lbe the budget available for hedging. The
optimal hedge with lower bound constraint, i.e. the solution of
max
XT≤LT
c(XT)=w
Eu(LT−XT) (33)
is given by
X∗
T= min(LT,L
T+(u)−1(λξT)),(34)
where λ0is such that
c((u)−1(λξT)) = w−wL.
Proof. With YT=LT−XTthe hedging problem in (33) is reduced to the classical
utility optimization problem in (15) with the additional constraints YT≥0. The
solution of this problem is easily shown to be the classical solution Y∗
Trestricted to
this boundary. As consequence we obtain X∗
T= min(LT,Y∗
T+LT) as in (34).
When there are enough financial resources w>w
Lavailable it might be of interest
to obtain the best super-hedge XT≥LT.We omit the details of the proof for this
case.
Proposition 5.4 (Utility optimal hedge with boundedness restriction on the excess).
Let LTbe a financial claim with price c(LT)=wLand let w>w
Lbe the budget
available. The optimal super-hedge, i.e. the solution of
max
LT≤XT
c(XT)=w
Eu(XT−LT) (35)
is given by
X∗
T=max(LT,L
T+(u)−1(λξT)),(36)
where λ0is such that
c((u)−1(λξT)) = w−w0.
The quantile super-hedging problem (26) was introduced in Browne (1999) for
a deterministic target LTin a Black–Scholes model. This result was extended in
Bernard et al. (2013a, Theorem 5.6) to random targets LT0 under regularity con-
ditions. The following proposition solves this problem without posing any regularity
conditions.
Proposition 5.5 (Quantile super-hedging).Let LT0be a financial claim with
price c(LT)=wLand let w≤wLbe the budget available. Then the solution to the
quantile super-hedging problem in (26), i.e.
max
0≤XT,
c(XT)=w
P(XTLT)
is given by
X∗
T=LT1{τLTξT<λ},(37)
where λis such that c(X∗
T)=w.
12
Proof. The optimal solution X∗
Tof (26) has a joint distribution Gwith the ‘bench-
mark’ LT. Thus from Theorem 3.1 we obtain that X∗
Tis an optimal claim with
fixed payoff structure. Therefore, conditionally on LT,X∗
Tis anti-monotonic with the
state-price ξTand is of the form X∗
T=f(ξT,L
T,V), where Vis some independent
randomization.
We define the sets A0={f(ξT,L
T,V)=0}and A1={f(ξT,L
T,V)=LT};then
P(A0∪A1) = 1, since in the other case it would be possible to construct an improved
solution. In consequence we get that fcan be represented in the form
f(ξT,L
T,V)=LT1{h(ξT,LT,V )∈A}
for some function hand measurable set A. Define λ>0 such that
P({h(ξT,L
T,V)∈A})=P({τLTξT<λ}).
Then 1{h(ξT,LT,V )∈A}and 1{τLTξT<λ}have the same distribution and further LTξTand
1{τLTξT<λ}are anti-monotonic. Thus by the Hoeffding–Fr´echet lower bound in (2) we
obtain
c(LT1{τLTξT<λ})=ELTξT1{τLTξT<λ}
≤ELTξT1{h(ξT,LT,V )∈A}
and thus LT1{τLTξT<λ}is optimal.
Remark 5.6.It was pointed out to the authors by a reviewer that the optimization
results in Proposition 5.5 and Proposition 5.8 can be cast in an unconstrained opti-
mization problem in Lagrangian form. For Proposition 5.5 this takes the forms
sup
XT≥0
E(1{XT≥LT}−λξTXT)+λw. (38)
Under a continuity assumption a solution of (38) is achieved by XT=LT1{λξLT<1}
with λsuitably chosen. The Lagrangian form in case of Proposition 5.8 is similar.
As a last application on hedging problems, we extend Proposition 5.5 by con-
sidering the combined case of a random claim LTthat needs to be hedged and the
requirement that the hedging portfolio has some copula Cwith a random benchmark
AT.
Proposition 5.7 (Quantile hedging with fixed payoff structure).For a
random claim LT0, a benchmark ATand a given copula C, the solution to the
restricted hedging problem
max
0≤XT
c(XT)=w
C(XT,AT)=C
P(XTLT) (39)
is given by
X∗
T=LT1{ZTλ},(40)
where ZT=C−1
1|AT(1 −τLTξT|AT)and where λis such that c(X∗
T)=w.
13
Proof. Let Gbe the joint distribution of the optimal claim X∗
Twith AT, then by the
randomization technique in Section 2 there exists a claim of the form f(ξT,A
T,V)
with a randomization Vindependent of (ξT,A
T), such that
(f(ξT,A
T,V),A
T)∼(X∗
T,A
T)∼Gand c(f(ξT,A
T,V),A
T)=c(X∗
T)=w.
Thus also f(ξT,A
T,V) is an optimal claim.
Defining as in the proof of Proposition 5.5 A0={f(ξT,A
T,V)=0},A1=
{f(ξT,A
T,V)=LT}we get P(Ao∪A1) = 1 and thus there exist a measurable
set Aand a function hsuch that
f(ξT,A
T,V)=LT1{h(ξT,AT,V )∈A}.
Defining λ>0 by the equation
P(h(ST,A
T,V)∈A)=P(ZTλ)
we obtain that 1{h(ξT,AT,V )∈A}and 1{ZTλ}have the same distribution. This implies
by the Hoeffding–Fr´echet inequalities in (2) that
c(LT1{ZTλ})=EξTLT1{ZTλ}
≤c(LT1{h(ξT,AT)∈A})
since conditionally on AT,ZTis anti-monotonic with ξTLTand thus 1{ZTλ}is anti-
monotonic with ξTLT. This implies optimality of X∗
T.
In the final application we consider the related problem of maximizing expected
return with given cost and target bounds. For given bounds a<bwe assume existence
of a claim XTsuch that a≤XT≤band c(XT)=w.
Proposition 5.8 (Maximizing expected return with given target bounds).
The solution of the expected returns maximization problem
max
a≤XT≤b
c(XT)=w
EXT
is given by the payoff
X∗
T=a1{τξT>λ}+b1{τξT≤λ}(41)
with λsuch that c(X∗
T)=w.
Proof. Assume that X∗
Tis not an optimal payoff. Then there exists an admissible
payoff YTsuch that EYT>EX
∗
T. We can then also find a strategy Y∗
Tof the form
Y∗
T=a1{τξT>d}+b1{τξT≤d}
with dchosen such that EY ∗
T=EYT.SinceEY ∗
T>EX
∗
Tit follows that d>λand,
therefore,
c(Y∗
T)=EξTY∗
T>c(X∗
T)=w.
14
On the other hand, YTis smaller than Y∗
Tin convex order ≤cx, i.e.
YT≤cx Y∗
T,
since Y∗
Thas the same expectation and shifts all mass to the boundaries. Let
YT=
F−1
YT(1 −τξT) be the random variable with
YT∼YTand such that
YT,ξTare anti-
monotonic. Then from the Hoeffding inequality and the Lorentz ordering theorem
(see R¨uschendorf (2013)) we obtain
c(YT)=EξTYTEξT
YTEξTY∗
T=c(Y∗),(42)
using that
YT∼YTand thus
YT≤cx Y∗
T. This leads to c(YT)>wand thus YTis not
admissible. This contradiction implies the result.
Illustration of Proposition 5.8 in a Black–Scholes Market: In the n-dimensional
Black–Scholes market there is a (risk-free) bond with price process (S0
t)=(S0
0ert)and
nrisky assets S1,S2,...,Snwith price processes,
dSi
t
Si
t
=μidt+σidBi
t,i=1,2,...,n,
where the (Bi
t) are (correlated) standard Brownian motions, with constant corre-
lation coefficients ρij := Corr Bi
t,B
j
t+s(t, s 0; i, j =1,2,...,n).Let μT=
(μ1,μ
2,...,μ
n),(Σ)ij =ρij σiσjand assume there exists isuch that μi=r. Let
Σ be positive definite. Then the state-price takes the form (see e.g. Bernard et al.
(2011)),
ξt=aSt
S0−b
,(43)
where a=exp
bμ−σ2
2t−r+(μ−r)2
2σ2tand b=μ−r
σ2.Here, the process (St)
satisfies the SDE dSt
St
=mdt+σdBt,(44)
where (Bt) is the standard Brownian motion defined by Bt=n
i=1 πiσiBi
t
√πT·Σ·π,m=
r+πT·(μ−r1),σ
2=πT·Σ·πand π=Σ−1·(μ−r1)
1T·Σ−1·(μ−r1). The process (St) is the price
process that corresponds to a so-called constant-mix trading strategy (at each time
t>0 a fixed proportion πiis invested in the i-th risky asset).
We make the (economic appealing) assumption that μ>r.From Proposition 5.8,
since τξTis decreasing in ST,the optimal payoff is of the form
X∗
T=a1{ST<α}+b1{STα},
with αsuch that EQ1(STα)=werT−a
b−awhere dQ
dP =erT ξT.It follows that αis given as
α=expr−σ2
2T−σ√TΦ−1werT −a
b−a.
15
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