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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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