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Optimal Payoﬀs under State-dependent Preferences

C. Bernard∗

, F. Moraux†

, L. R¨uschendorf‡and S. Vanduﬀel§

October 23, 2014

Abstract

Most decision theories, including expected utility theory, rank dependent

utility theory and cumulative prospect theory, assume that investors are only

interested in the distribution of returns and not in the states of the economy in

which income is received. Optimal payoﬀs have their lowest outcomes when the

economy is in a downturn, and this feature is often at odds with the needs of

many investors. We introduce a framework for portfolio selection within which

state-dependent preferences can be accommodated. Speciﬁcally, we assume

that investors care about the distribution of ﬁnal wealth and its interaction

with some benchmark. In this context, we are able to characterize optimal

payoﬀs in explicit form. Furthermore, we extend the classical expected util-

ity optimization problem of Merton to the state-dependent situation. Some

applications in security design are discussed in detail and we also solve some

stochastic extensions of the target probability optimization problem.

Key-words: Optimal portfolio selection, state-dependent preferences, condi-

tional distribution, hedging, state-dependent constraints.

∗Corresponding author: Carole Bernard, University of Waterloo, 200 University Avenue West,

Waterloo, Ontario, N2L3G1, Canada. (email: c3bernar@uwaterloo.ca). Carole Bernard acknowl-

edges support from NSERC and from the Humboldt foundation.

†Franck Moraux, Univ. Rennes 1, 11 rue Jean Mac´e, 35000 Rennes, France. (email:

franck.moraux@univ-rennes1.fr). Franck Moraux acknowledges ﬁnancial supports from CREM

(CNRS research center) and IAE of Rennes.

‡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.

1

Introduction

Studies of optimal investment strategies are usually based on the optimization of

an expected utility, a target probability or some other (increasing) law-invariant

measure. Assuming that investors have law-invariant preferences is equivalent to

supposing that they care only about the distribution of returns and not about the

states of the economy in which the returns are received. This is, for example, the

case under expected utility theory, Yaari’s dual theory, rank-dependent utility the-

ory, mean-variance optimization and cumulative prospect theory. Clearly, an optimal

strategy has some distribution of terminal wealth and must be the cheapest possible

strategy that attains this distribution. Otherwise, it is possible to strictly improve

the objective and to contradict its optimality. Dybvig (1988) was the ﬁrst to study

strategies that reach a given return distribution at lowest possible cost. Bernard and

Boyle (2010) call these strategies cost-eﬃcient and their properties have been exam-

ined further in Bernard, Boyle and Vanduﬀel (2014a). In a fairly general market

setting these authors show that the cheapest way to generate a given distribution

is obtained by a contract whose payoﬀ is decreasing in the pricing kernel (see also

Carlier and Dana (2011)). The basic intuition is that investors consume less in states

of economic recession because it is more expensive to insure returns under these con-

ditions. This feature is also explicit in a Black-Scholes framework, in which optimal

payoﬀs at time horizon Tare shown to be an increasing function of the price of the

risky asset (as a representation of the economy) at time T. In particular, such payoﬀs

are path-independent.

An important issue with respect to the optimization criteria and the resulting

payoﬀs under most standard frameworks, is that their worst outcomes are obtained

when the market declines. Arguably, this property of optimal payoﬀs does not ﬁt

with the aspirations of investors, who may seek protection against declining markets

or, more generally, may consider a benchmark when making investment decisions. In

other words, two payoﬀs with the same distribution do not necessarily present the

same “value” for a given investor. Bernard and Vanduﬀel (2014b) show that insurance

contracts can usually be substituted by ﬁnancial contracts that have the same pay-

oﬀ distribution but are cheaper. The existence of insurance contracts that provide

protection against speciﬁc events shows that these instruments must present more

value for an investor than ﬁnancial payoﬀs that lack this feature. This observation

supports the general observation that investors are more inclined to receive income

in a “crisis” (for example when their property burns down or when the economy is in

recession) than under “normal” conditions.

This paper makes several theoretical contributions to the study of optimal invest-

ment strategies and highlights valuable applications of its ﬁndings in the areas of

portfolio management and security design. First, we clarify the setting under which

optimal investment strategies necessarily exhibit path-independence. These ﬁndings

complement Cox and Leland (1982, 2000) and Dybvig’s (1988) seminal results and

underscore the important role of path-independence in traditional optimal portfolio

selection. Thereafter, as our main contribution, we introduce a framework for portfo-

lio selection that makes it possible to consider the states in which income is received.

2

More precisely, it is assumed that investors target some distribution for their termi-

nal wealth and additionally aim for a certain (desired) interaction with a random

benchmark.1For example, the investor may want his strategy to be unrelated to the

benchmark when it decreases but to follow this benchmark when it performs well.

Using our framework, we can characterize optimal payoﬀs explicitly (Theorems 3.2

and 3.4) in this setting. Such explicit characterizations are derived independently in

Theorems 3.1 and 3.3 of Takahashi and Yamamoto (2013) but proved only for cases

in which there is a countable number of states2. Furthermore, we show that optimal

strategies in this setting become conditionally increasing functions of the terminal

value of the underlying risky asset.

A further main contribution in this part of the paper is the extension of the clas-

sical result of portfolio optimization under expected utility (Cox and Huang (1989)).

Speciﬁcally, we determine the optimal payoﬀ for an expected utility maximizer under

a dependence constraint, reﬂecting a desired interaction with the benchmark (Theo-

rem 5.2). The proof builds on isotonic approximations and their properties (Barlow

et al. (1972)). We also solve two stochastic generalizations of Browne (1999) and

Cvitani`c and Spivak’s (1999) classical target optimization problem in the given state-

dependent context.

Finally, we show how these theoretical results are useful in security design and

can help to simplify (and improve) payoﬀs commonly oﬀered in the ﬁnancial markets.

We show how to substitute highly path-dependent products by payoﬀs that depend

only on two underlying assets, which we refer to as “twins”. This result is illustrated

with an extensive discussion of the optimality of Asian options. We also construct

alternative payoﬀs with appealing properties.

The paper is organized as follows. Section 1 outlines the setting of the investment

problem under study. In Section 2, we restate basic optimality results for path-

independent payoﬀs for investors with law-invariant preferences. We also discuss in

detail the suﬃciency of path-independent payoﬀs when allocating wealth. In Section

3, we point out drawbacks of optimal path-independent payoﬀs and introduce the

concept of state-dependence used in the following sections. We show that “twins”,

deﬁned as payoﬀs that depend only on two underlying asset values, are optimal for

state-dependent preferences. In Section 4, we discuss applications to improve security

designs. In particular, we propose several improvements in the design of geometric

Asian options. In Section 5, we solve the standard Merton problem of maximization

of expected utility of ﬁnal wealth when the investor constrains the interaction of the

1The paper draws its inspiration from the last section in Bernard, Boyle and Vanduﬀel (2014a),

in which a constrained cost-eﬃciency problem is solved when the joint distribution between the

wealth and some benchmark is determined in some speciﬁc area (local dependence constraint).

2The results of Takahashi and Yamamoto (2013) are stated in a general market, but the proof of

their basic Theorem 3.1 in Appendix A.1 only holds when the number of states is countable. The

proof of their main theorem, Theorem 3.3 (Appendix A.3.), is based on the same idea as in Theorem

3.1 (see statement A.3 on page 1571) and is thus also valid in the case of countable states. Their set

up also diﬀers from ours in that these authors assume that stock prices follow diﬀusion processes,

and they derive the speciﬁc form of the state price density process in this setting (page 1561). In

this paper, we do not assume that the underlying stock prices are diﬀusion processes and hence the

state price process does not need to be of a speciﬁc form (see also our ﬁnal remarks).

3

ﬁnal wealth with a given benchmark. In this context, we also generalize the results

of Browne (1999) and Cvitani`c and Spivak (1999) with regard to target probability

maximization. Final remarks are presented in Section 6. Most of the proofs are

provided in the Appendix.

1 Framework and notation

Consider investors with a given ﬁnite investment horizon Tand no intermediate

consumption. We model the ﬁnancial market on a ﬁltered probability space (Ω,F,P),

in which Pis the real-world probability measure. The market consists of a bank

account Bpaying a constant risk-free rate r > 0, so that B0invested in a bank

account at time 0 yields Bt=B0ert at time t. Furthermore, there is a risky asset (say,

an investment in stock) whose price process is denoted by S= (St)06t6T.We assume

that St(0 <t<T) has a continuous distribution FSt. The no-arbitrage price3at

time 0 of a payoﬀ XTpaid at time T > 0 is given by

c0(XT) = E[ξTXT],(1)

where (ξt)tis the state-price density process4ensuring that (ξtSt)tis a martingale.

Moreover, based on standard economic theory, we assume throughout this paper that

state prices are decreasing with asset prices,5i.e.,

ξt=gt(St), t >0,(2)

where gtis decreasing (in markets where E[ST]> S0erT ). There is empirical evidence

that this relationship may not hold in practice, which is called the pricing kernel puzzle

(Brown and Jackwerth (2004), Grith et al. (2013)). Many explanations have been

provided in the literature (Brown and Jackwerth (2004), Hens and Reichlin (2013)),

including state-dependence of preferences (Chabi-Yo et al. (2008)). Therefore, (2) is

not consistent with a market populated by investors with state dependent preferences.

However, we do not tackle the problem of equilibrium and instead study the situation

of a small investor whose state-dependent preferences do not inﬂuence the pricing

kernel that is exogenously given in the market. This is a commonly studied situation

since the work of Karatzas et al. (1987).

The functional form (2) for (ξt)tallows us to present our results regarding optimal

portfolios using (St)tas a reference, which is practical. We will explain in Section

3The payoﬀs we consider are all tacitly assumed to be square integrable, to ensure that all

expectations mentioned in the paper exist. In particular, c0(XT)<+∞for any payoﬀ XTconsidered

throughout this paper.

4The process is commonly so designated. However, strictly speaking, it is not a density that

is at issue, but rather the product of a discount factor (generally strictly less than 1) and the

Radon-Nikodym derivative between the physical measure and the risk-neutral measure.

5See e.g., Cox, Ingersoll and Ross (1985) and Bondarenko (2003), who shows that property (2)

must hold if the market does not allow for statistical arbitrage opportunities, where a statistical

arbitrage opportunity is deﬁned as a zero-cost trading strategy delivering at T, a positive expected

payoﬀ unconditionally, and non-negative expected payoﬀs conditionally on ξT.

4

6 how the results and characterizations of the optimality of a payoﬀ XTare tied

to its (conditional) anti-monotonicity with ξTand do not depend on the functional

form (2) per se. Note that assumption (2) is satisﬁed by many popular pricing

models, including the CAPM, the consumption-based models and by exponential

L´evy markets in which the market participants use Esscher pricing (Vanduﬀel et al.

(2008), Von Hammerstein et al. (2014)). It is also possible to use a market model in

which prices are obtained using the Growth Optimal portfolio (GOP) as num´eraire

(Platen and Heath (2006)), as is discussed further in Section 6.

The Black–Scholes model can be seen as a special case of this latter setting. Since

we will use it to illustrate our theoretical results, we recall here its main properties.

In the Black–Scholes market, under the real probability P, the price process (St)t

satisﬁes dSt

St

=µdt +σdZt,

with solution St=S0exp µ−σ2

2t+σZt. Here, (Zt)tis a standard Brownian

motion, µ(> r) the drift and σ > 0 the volatility. The distribution (cdf) of STis

given as

FST(x) = P(ST6x) = Φ

ln x

S0−(µ−σ2

2)T

σ√T

,(3)

where Φ is the cdf of a standard normal random variable. In the Black–Scholes

market, the state-price density process (ξt)tis unique and ξt=e−rte−θZt−θ2t

2where

θ=µ−r

σ. Consequently, ξtcan also be expressed as a decreasing function of the stock

price St,

ξt=αtSt

S0−β

,(4)

where αt= exp θ

σµ−σ2

2t−r+θ2

2t, β =θ

σ>0 (because we assume that

E[ST] = S0eµT > S0erT ).

2 Law-invariant preferences and optimality of path-

independent payoﬀs

In this section, it is understood that investors have law-invariant (state-independent)

preferences. This means that they are indiﬀerent between two payoﬀs having the

same payoﬀ distribution (under P). In this case, any random payoﬀ XT(that pos-

sibly depends on the path of the underlying asset price) admits a path-independent

alternative with the same price, which is at least as good for (i.e., desirable in the eyes

of) these investors. Recall that a payoﬀ is path-independent if there exists some func-

tion fsuch that XT=f(ST) holds almost surely. Hence, investors with law-invariant

preferences only need to consider path-independent payoﬀs when making investment

decisions. Under the additional (typical) assumption that preferences are increasing,

5

any path-dependent payoﬀ can be strictly dominated by a path-independent one that

is increasing in the risky asset.6

Note that results in this section are related closely to the original work of Cox

and Leland (1982, 2000), Dybvig (1988), Bernard, Boyle and Vanduﬀel (2014a) and

Carlier and Dana (2011). These overview results are recalled here to facilitate the

exposition of the extensions that are developed in the following sections.

2.1 Suﬃciency of path-independent Payoﬀs

Proposition 2.1 shows that for any given payoﬀ there exists a path-independent al-

ternative with the same price that is at least as good for investors with law-invariant

preferences. Thus, such an investor needs only to consider path-independent pay-

oﬀs. All other payoﬀs are indeed redundant in the sense that they are not needed to

optimize the investor’s objective. The proof of Proposition 2.1 provides an explicit

construction of an equivalent path-independent payoﬀ.

Proposition 2.1 (Suﬃciency of path-independent payoﬀs).Let XTbe a payoﬀ with

price cand having a cdf F. Then, there exists at least one path-independent payoﬀ

f(ST)with price c:= c0(f(ST)) and cdf F.

The proof of Proposition 2.1 is provided in Appendix A.1.

Proposition 2.1, however, does not conclude that a given path-dependent payoﬀ

can be strictly dominated by a path-independent one. The following section shows

that the dominance becomes strict as soon as preferences are increasing.

2.2 Optimality of path-independent payoﬀs

Let Fbe a payoﬀ distribution with (left-continuous) inverse deﬁned as

F−1(p) = inf {x|F(x)>p}.(5)

The basic result provided here was originally derived by Dybvig (1988) and was

presented more generally in Bernard, Boyle and Vanduﬀel (2014a). It shows how to

construct a payoﬀ that generates the distribution Fat minimal price. Such payoﬀ is

referred to as cost-eﬃcient by Bernard and Boyle (2010).

Theorem 2.2 (Cost optimality of path-independent payoﬀs).Let Fbe a cdf. The

optimization problem

min

XT∼Fc0(XT) (6)

has an almost surely unique solution X∗

Tthat is path-independent, almost surely in-

creasing in STand given by

X∗

T=F−1(FST(ST)) (7)

6This dominance can easily be implemented in practice, as all path-independent payoﬀs can be

replicated statistically with European call and put options as shown e.g., by Carr and Chou (1997)

and by Breeden and Litzenberger (1978).

6

This theorem can be seen as an application of the Hoeﬀding–Fr´echet bounds recalled

in Lemma A.1, which is presented in the Appendix. This result implies that investors

with increasing law-invariant preferences may restrict their optimization strictly to

the set of path-independent payoﬀs when making investment decisions.7The payoﬀ

(7) is obviously increasing in ST. In fact, this property characterizes cost-eﬃciency

because of the a.s. uniqueness of the cost-eﬃcient payoﬀ established in Theorem 2.2.

Consequently, this implies the following corollary.

Corollary 2.3 (Cost-eﬃcient payoﬀs).A payoﬀ is cost-eﬃcient if and only if it is

almost surely increasing in ST.

Theorem 2.2 also implies that investors with increasing law-invariant preferences

only invest in path-independent payoﬀs that are increasing in ST. This is consistent

with the literature on optimal investment problems in which optimal payoﬀs derived

using various techniques always turn out to exhibit this property.

Corollary 2.4 (Optimal payoﬀs for increasing law-invariant preferences).For any

payoﬀ YTat price cthat is not almost surely increasing in STthere exists a path-

independent payoﬀ Y∗

Tat price cthat is a strict improvement for any investor with

increasing and law-invariant preferences.

A possible choice for Y∗

Tis given by Y∗

T:= F−1(FST(ST)) + (c−c∗

0)erT ,in which

c∗

0denotes the price of 7. Note that the payoﬀ Y∗

Thas price cand is almost surely

increasing in ST. It consists in investing an amount c∗

0< c in the cost-eﬃcient payoﬀ

(also distributed with F) and leaving the remaining funds c−c∗

0>0 in the bank

account, so that it is a strict improvement of the payoﬀ YT.

3 Optimal payoﬀs under state-dependent prefer-

ences.

Many of the contracts chosen by law-invariant investors do not oﬀer protection in

times of economic hardship. In fact, due to the observed monotonicity property with

ST,the lowest outcomes for an optimal (thus, cost-eﬃcient) payoﬀ occur when the

stock price STreaches its lowest levels. More speciﬁcally, denote by f(ST) a cost-

eﬃcient payoﬀ (with an increasing function f) and by XTanother payoﬀ such that

both are distributed with Fat maturity. Then, f(ST) delivers low outcomes when

STis low and it holds8for all a>0 that

E[f(ST)|ST< a]6E[XT|ST< a].(8)

7Similar optimality results to those in Theorem 2.2 have been given in the class of admissible

claims XTthat are smaller than Fin convex order in Dana and Jeanblanc (2005) and in Burgert

and R¨uschendorf (2006).

8We provide here a short proof of (8). It is clear that the couple (f(ST),1ST<a ) has the same

marginal distributions as (XT,1ST<a),but E[f(ST)1ST<a ]6E[XT1ST<a] because f(ST) and 1ST<a

are anti-monotonic (from Lemma A.1).

7

Let Fbe the distribution of a put option with payoﬀ XT:= (K−ST)+= max(K−

ST,0). Bernard, Boyle and Vanduﬀel (2014a) show that the payoﬀ of the cheapest

strategy with cdf Fcan be computed as in (7). It is given by X∗

T= (K−a S−1

T)+with

a:= S2

0exp(2 (µ−σ2/2) T) and is a power put option (with power -1). X∗

Tis the

cheapest way to achieve the distribution F, whereas the ﬁrst “ordinary” put strategy

(with payoﬀ XT) is actually the most expensive way to do so. These payoﬀs interact

with STin fundamentally diﬀerent ways, as one payoﬀ is increasing in STwhile the

other is decreasing in it. A put option protects the investor against a declining market,

in which consumption is more expensive than is otherwise typical, whereas the cost-

eﬃcient counterpart X∗

Tprovides no protection but rather emphasizes the eﬀect of a

market deterioration on the wealth received.

As mentioned in the introduction, the use of put options and the demand for in-

surance (Bernard and Vanduﬀel (2014b)) are signals that many investors care about

states of the economy in which income derived from investment strategies is received.

In particular, they may seek strategies that provide protection against declining mar-

kets or, more generally, that exhibit a desired dependence with some benchmark.

Hence, in the remainder of this paper, we consider investors who exhibit state-

dependent preferences in the sense that they seek a payoﬀ XTwith a desired distribu-

tion and a desired dependence with a benchmark asset AT. In other words, they ﬁx

the joint distribution Gof the random couple (XT, AT).The optimal state-dependent

strategy is the one that solves for

min

(XT,AT)∼Gc0(XT).(9)

Note that the setting also includes law-invariant preferences as a special (limiting)

case when ATis deterministic. In this case, we eﬀectively revert to the framework

of state-independent preferences that we discussed in the previous section. In what

follows, we consider as benchmark the underlying risky asset or any other asset in

the market, considered at ﬁnal or intermediate time(s). Moreover, to ensure that the

impact of state-dependent preferences on the structure of optimal payoﬀs is clear, we

have organized the rest of the present section along similar lines to those of Section

2.

Remark 3.1.One can use a copula as a device to model the interaction between payoﬀs

and benchmarks. The joint distribution Gof the couple (XT, AT) can be written using

a copula C. From Sklar’s theorem, G(x, a) = C(FXT(x), FAT(a)), where Cis a copula

(this representation is unique for continuously distributed random variables). It is

then clear that the determination of optimal strategies in (9) can also be formulated

as

min

XT∼F,

C(XT,AT)=C

c0(XT),(10)

where “C(XT,AT)=C” means that the copula between the payoﬀ XTand the bench-

mark ATis C. In particular, (10) shows that knowledge of the distribution of ATis

not necessary in order to determine optimal state-dependent strategies.

8

3.1 Suﬃciency of twins

In this paper, any payoﬀ that writes as f(ST, AT) or f(ST, St) is called a twin. We

show ﬁrst that, in our state-dependent setting, for any payoﬀ there exists a twin

that is at least as good. When also assuming that preferences are increasing, we ﬁnd

that optimal payoﬀs write as twins, and we are able to characterize them explicitly.

Conditionally on AT, optimal twins are increasing in the terminal value of the risky

asset ST.

The following theorems show that for any given payoﬀ there is a twin that is at

least as good for investors with state-dependent preferences.

Theorem 3.2. (Twins as payoﬀs with a given joint distribution with a benchmark

ATand price c).Let XTbe a payoﬀ with price chaving joint distribution Gwith

some benchmark AT, where (ST, AT)is assumed to have a joint density with respect

to the Lebesgue measure. Then, there exists at least one twin f(ST, AT)with price

c=c0(f(ST, AT)) having the same joint distribution Gwith AT.

Theorem 3.2 does not cover the case in which STplays the role of the benchmark

(because (ST, ST) has no density). This interesting case is considered in the following

theorem (Theorem 3.3).

Theorem 3.3 (Twins as payoﬀs with a given joint distribution with STand price

c).Let XTbe a payoﬀ with price chaving joint distribution Gwith the benchmark

ST. Assume that (ST, St)for some 0<t<Thas a joint density with respect

to the Lebesgue measure. Then, there exists at least one twin f(St, ST)with price

c=c0(f(St, ST)) having a joint distribution Gwith ST. An example is given by

f(St, ST) := F−1

XT|ST(FSt|ST(St)).(11)

The proofs for Theorems 3.2 and 3.3 are in Appendix A.3 and A.4.

Theorems 3.2 and 3.3 imply that investors who care about the joint distribution of

terminal wealth with some benchmark ATneed only consider the twins in both cases,

i.e., when (AT, ST) is continuously distributed, as in Theorem 3.2, or when ATis

equal to ST,as in Theorem 3.3. These results extend Proposition 2.1 to the presence

of a benchmark and state-dependent preferences. All other payoﬀs are useless in the

sense that they are not needed for these investors per se.9

Note that in Theorem 3.3, tcan be chosen freely in (0, T ) and the dependence

with respect to Stis not ﬁxed. So, for instance, replacing FSt(St) with 1 −FSt(St) in

(11) would also lead to the appropriate properties. Hence, there is an inﬁnite number

of twins f(St, ST) having the joint distribution Gwith ST.All of them have the same

price.10 The question then arises: how does one select one among them. A natural

9This ﬁnding is consistent with the result obtained by Takahashi and Yamamoto (2013), who

apply it to replicate a joint distribution in the hedge fund industry.

10To see this, recall that the joint distribution between the twin f(St, ST) and STis ﬁxed and

thus also the joint distribution between the twin and ξT(as ξTis a decreasing function of STdue

to (2)). All twins f(St, ST) with such a property have the same price E[ξTf(St, ST)].

9

possibility is to determine the optimal twin XT=f(St, ST) by imposing an additional

criterion. For example, one could deﬁne the best twin XTas the one that minimizes

E(XT−HT)2,(12)

where HTis another payoﬀ that is not a function of ST. This approach appears

natural in the context of simplifying the design of contracts. For instance, start with

a geometric Asian option and compute its joint distribution Gwith ST. Then, all

twins as in (11) have the same price but one of them may be closer to the original

Asian derivative (in the sense of minimizing the distance, as in (12)). Note that since

all marginal distributions are ﬁxed, the criterion (12) is equivalent to maximizing the

correlation between XTand HT. We use this criterion in one of our applications (see

Section 4.1).

3.2 Optimality of twins

Next, we investigate the cost optimality of twins. As discussed above, if the bench-

mark ATcoincides with ST, then all twins that satisfy (XT, AT)∼Ghave the same

cost and the problem of searching for the cheapest one is not meaningful. However,

this observation is no longer true when the benchmark AThas a density with ST. In

this case, the cheapest twin is determined by Theorem 3.4 that extends Theorem 2.2

to the state-dependent case. Theorem 2.2 ﬁnds that among the inﬁnite number of

payoﬀs with a given distribution F, the cheapest one is increasing in ST. In the state-

dependent setting one has that optimal payoﬀs are increasing in ST,conditionally on

AT.

Theorem 3.4 (Cost optimality of twins).Assume that (ST, AT)has joint density

with respect to the Lebesgue measure. Let Gbe a bivariate cumulative distribution

function. The optimal state-dependent strategy determined by

min

(XT,AT)∼Gc0(XT) (13)

has an almost surely unique solution X∗

Twhich is a twin of the form f(ST, AT).X∗

T

is almost surely increasing in ST, conditionally on AT,and given by

X∗

T:= F−1

XT|AT(FST|AT(ST)).(14)

The proof of Theorem 3.4 is provided in Appendix A.5.

Recall from Section 2 that when preferences are law-invariant, optimal payoﬀs

are path-independent and increasing in ST.When preferences are state-dependent,

we observe from expression (14) that optimal state-dependent payoﬀs may become

path-dependent, and are increasing in ST,conditionally on AT. We end this section

with a corollary derived from Theorem 3.4. The result echoes the one established for

investors with law-invariant preferences in the previous section (Corollary 2.4)

10

Corollary 3.5 (Cheapest twin).Assume that (ST, AT)has joint density with respect

to the Lebesgue measure. Let Gbe a bivariate cumulative distribution function. Let

XTbe a payoﬀ such that (XT, AT)∼G. Then, XTis the cheapest payoﬀ if and only

if, conditionally on AT,XTis (almost surely) increasing in ST.

The proof of Corollary 3.5 is provided in Appendix A.6.

4 Improving security design

In this section, we show that the results above are useful in designing balanced and

transparent investment policies for retail investors as well as ﬁnancial institutions:

1. If the investor who buys the ﬁnancial contract has law-invariant preferences

and if the contract is not increasing in ST,then there exists a strictly cheaper

derivative (cost-eﬃcient contract) that is strictly better for this investor. We

ﬁnd its design by applying Theorem 2.2.

2. If the investor buys the contract because of the interaction with the market asset

ST, and the contract depends on another asset, then we can apply Theorem 3.3

to simplify its design while keeping it “at least as good.” The contract then

depends, for example, on STand Stfor some t∈(0, T ).

3. If the investor buys the contract because he likes the dependence with a bench-

mark AT,which is not ST, and if the contract does not only depend on AT

and ST, then we use Theorem 3.2 to construct a simpler one that is “at least

as good”and that writes as a function of STand AT. Finally, if the obtained

contract is not increasing in STconditionally on AT,then it is also possible to

construct a strictly cheaper alternative using Theorem 3.4 and Corollary 3.5.

We now use the Black–Scholes market to illustrate these three situations. We

begin with the example of an Asian option with ﬁxed strike, followed by the example

of one with ﬂoating strike.

4.1 The geometric Asian twin with ﬁxed strike

Consider a ﬁxed strike (continuously monitored) geometric Asian call with payoﬀ

given by

YT:= (GT−K)+.(15)

Here, Kdenotes the ﬁxed strike and GTis the geometric average of stock prices from

0 to T, deﬁned as

ln(GT) := 1

TZT

0

ln (Ss)ds. (16)

We can now apply the results derived above to design products that improve upon

YT.

11

Use of cost-eﬃciency payoﬀ for investors with increasing law-invariant

preferences. By applying Theorem 2.2 to the payoﬀ YT(15), one ﬁnds that the

cost-eﬃcient payoﬀ associated with a ﬁxed strike (continuously monitored) geometric

Asian call is

Y∗

T=dS1/√3

T−K

d+

,(17)

where d=S1−1

√3

0e1

2−√1

3µ−σ2

2T. This is also the payoﬀ of a power call option, with

well-known price

c0(Y∗

T) = S0e(1

√3−1)rT +( 1

2−1

√3)µT −σ2T

12 Φ(h1)−Ke−rT Φ(h2) (18)

where

h1=ln S0

K+ (1

2−1

√3)µT +r

√3T+1

12 σ2T

σqT

3

, h2=h1−σrT

3.

While the above results can also be found in Bernard, Boyle and Vanduﬀel (2014a),

they are worth considering here for the purpose of comparison with what follows.

Note that letting Kgo to zero provides a cost-eﬃcient payoﬀ that is equivalent to

the geometric average GT.

A twin that is useful for investors who care about the dependence with ST.

By applying Theorem 3.3 to the payoﬀ GT, we can ﬁnd a twin payoﬀ RT(t) = f(St, ST)

such that

(ST, RT(t)) ∼(ST, GT).(19)

By deﬁnition, this twin preserves existing dependence between GTand ST. However,

compared to the original contract it is simpler and “less” path-dependent, as it de-

pends only on two values of the path of the stock price. Interestingly, the call option

written on RT(t) and the call option written on GThave the same joint distribution

with ST. Consequently,

ST,(RT(t)−K)+∼ST,(GT−K)+.(20)

(RT(t)−K)+is therefore a twin equivalent to the ﬁxed strike geometric Asian call

(as in Theorem 3.3). We can compute RT(t) by applying Theorem 3.3, and we ﬁnd

that

RT(t) = S

1

2−1

2√3√T−t

t

0S

T

t

1

2√3√t

T−t

tS

1

2−1

2√3√t

T−t

T,(21)

where tis freely chosen in (0, T ). Details on how (11) becomes (21) are provided in

Appendix B.1.11 The equality of joint distributions exposed in (20) implies that the

11Formula (21) is based on the expression (11) for a twin dependent on Stand ST. Note that

there is no uniqueness. For example, 1 −FSt|ST(St) is also independent of ST,and we can thus

also consider HT(t) := F−1

XT|ST(1 −FSt|ST(St)) as a suitable twin (0 <t<T) satisfying the joint

distribution, as in (19). In this case, one obtains HT(t) = S

1

2+1

2√3√T−t

t

0S−T

t

1

2√3√t

T−t

tS

1

2+1

2√3√t

T−t

T.

12

call option written on RT(t) has the same price as the original ﬁxed strike (contin-

uously monitored) geometric Asian call (15). The time−0 price of both contracts is

therefore

c0((RT(t)−K)+) = S0e−rT

2−σ2T

12 Φ( ˜

d1)−Ke−rT Φ( ˜

d2),(22)

where ˜

d1=ln(S0/K)+rT/2+σ2T/12

σ√T/3and ˜

d2=˜

d1−σpT/3 (see Kemna and Vorst (1990)).

Choosing among twins. The construction in Theorem 3.3 depends on t. Maxi-

mizing the correlation between ln (RT(t)) and ln (GT) is nevertheless a possible way

to select a speciﬁc t. The covariance between ln(RT(t)) and ln(GT) is provided by

cov (ln (RT(t)) ,ln (GT)) = σ2

2T

2+√t√T−t

2√3

and, by construction of RT(t), the standard deviations of ln (RT(t)) and ln (GT) are

both equal to σqT

3. Maximizing the correlation coeﬃcient is therefore equivalent to

maximizing the covariance, and thus of f(t) = (T−t)t. This maximum is obtained

for t∗=T

2, and the maximal correlation ρmax between ln(RT(t)) and ln(GT) is

ρmax =3

4+√3p(T−t∗)t∗

4T=3

4+√3

8≈0.9665,

which shows that the optimal twin is highly correlated to the initial Asian, while being

considerably simpler. Note that both the maximum correlation and the optimum

RT(T

2) are robust to changes in market parameters.

4.2 The geometric Asian twin with ﬂoating strike

Consider now a ﬂoating strike (continuously monitored) Asian put option deﬁned by

YT= (GT−ST)+.(23)

For increasing law-invariant preferences, Corollary 2.4 may be used to ﬁnd a

cheaper contract that depends on STonly. The cheapest contract with cdf FYTis

known to be F−1

YTΦlnST

S0−(µ−σ2

2)T

σ√T. Notice that F−1

YTcan only be numerically

approximated because the distribution of the diﬀerence between two lognormal dis-

tributions is unknown.

If investors care about the dependence with ST, by applying Theorem 3.3, one

can ﬁnd twins F−1

YT|ST(FSt|ST(St)) as functions of Stand ST,which are explicitly given

as S

1

2−1

2√3√T−t

t

0S

T

t

1

2√3√t

T−t

tS

1

2−1

2√3√t

T−t

T−ST+

.(24)

Details can be found in Appendix B.2.

13

Finally, if investors care about the dependence with GT, then it is possible to

construct a cheaper twin because the payoﬀ (23) is not conditionally increasing in

ST.Therefore, it can be strictly improved using Theorem 3.4. The reason is that

we can improve the payoﬀ (23) by making it cheaper while maintaining dependence

with GT.Hence, we invoke Theorem 3.4 (expression 14) to exhibit another payoﬀ

XT=F−1

YT|GTFST|GT(ST)such that

(YT, GT)∼(XT, GT),

but so that XTis strictly cheaper. After some calculations, we ﬁnd that XTwrites

as

XT=GT−aG3

T

ST+

,(25)

where a=eµ−σ2

2T

2

S0.Details can be found in Appendix B.3.

Finally, one can easily assess the extent to which the twin (25) is cheaper than

the initial payoﬀ YT. To do so, we recall the price of a geometric Asian option with

ﬂoating strike (the no-arbitrage price of YT):

c0(YT) = e−rT EQ(GT−ST)+=S0e−rT

2 Φ (f)e−σ2T

12 −erT

2Φ f−σrT

3!!,(26)

where f=σ2

12 T−rT

2

σ√T

3

. Similarly, one ﬁnds that

c0(XT) = e−rT EQGT−aG3

T

ST+

=S0e−rT

2 Φ (d)e−σ2T

12 −eµT

2Φ d−σrT

3!!

(27)

where d=σ2T

12 −µT

2

σ√T

3

,which we need to compare numerically to (26). For example, when

µ= 0.06, r = 0.02, σ = 0.3 and T= 1,one has c0(YT)=6.74 and c0(XT)=5.86,

indicating that cost savings can be substantial. Also note the close correspondence

between formulas (26) and (27). The proofs for these formulas are provided in Ap-

pendix B.4.

5 Portfolio management

This section provides several contributions to the ﬁeld of portfolio management. We

ﬁrst derive the optimal investment for an expected utility maximizer who has a con-

straint on the dependence with a given benchmark. Next, we revisit optimal strate-

gies for target probability maximizers (see Browne (1999) and Cvitani`c and Spivak

(1999)), and we extend this problem in two directions by adding dependence con-

straints and by considering a random target. In both cases, we derive analytical

solutions that are given by twins. From now on, we denote by W0the initial wealth.

14

5.1 Expected utility maximization with dependence constraints

The most prominent decision theory used in various ﬁelds of economics is the expected

utility theory (EUT) of von Neumann & Morgenstern (1947). In the expected utility

framework investors assign a utility u(x) to each possible level of wealth x. Increasing

preferences are equivalent to an increasing utility function u(·). Assuming that u(·) is

concave is equivalent to assuming that investors are risk averse in the sense that for

a given budget they prefer a sure income over a random one with the same mean. In

their seminal paper on optimal portfolio selection, Cox and Huang (1989) showed how

to obtain the optimal strategy for a risk averse expected utility maximizer; see also

Merton (1971) and He and Pearson (1991a),(1991b). We recall this classical result in

the following theorem.

Theorem 5.1 (Optimal payoﬀ in EUT).Consider a utility function u(·)deﬁned

on (a, b)such that u(·)is continuously diﬀerentiable and strictly increasing, u0(·)is

strictly decreasing, limx&au0(x) = +∞and limx%bu0(x) = 0.Consider the following

portfolio optimization problem:

max

E[ξTXT]=W0

E[u(XT)].(28)

The optimal solution to this problem is given by

X∗

T= (u0)−1(λξT),(29)

where λis such that EξT(u0)−1λξT=W0.

Note that the optimal EUT payoﬀ X∗

Tis decreasing in ξTand thus increasing in ST

(illustration of the results derived in Section 2), which highlights the lack of protection

of optimal portfolios when markets decline. To account for this, we give the investor

the opportunity to maintain a desired dependence with a benchmark portfolio (e.g.,

representing the ﬁnancial market). This extends earlier results on expected utility

maximization with constraints, such as those of Brennan and Solanki (1981), Brennan

and Schwartz (1989), He and Pearson (1991a),(1991b), Basak (1995), Grossman and

Zhou (1996), Sorensen (1999) and Jensen and Sorensen (2001). These studies were

for the most part concerned with the expected utility maximization problem when

investors want a lower bound on their optimal wealth either at maturity or throughout

some time interval. When this bound is deterministic, this corresponds to classical

portfolio insurance. Boyle and Tian (2007) extend and unify the various results

by allowing the benchmark to be beaten with some conﬁdence. They consider the

following maximization problem over all payoﬀs XT:

max

P(XT>AT)>α,

c0(XT)=W0

E[u(XT)],(30)

where ATis some benchmark (e.g., the portfolio of another manager in the market).

In Theorem 2.1 (page 327) of Boyle and Tian (2007), the optimal contract X∗

Tis

15

derived explicitly (under some regularity conditions ensuring feasibility of the stated

problem), and it is an optimal twin.12

This also follows from our results. Assume that the solution to (30) exists, and

denote it by X∗

T. Then let Gbe the bivariate cdf of (X∗

T, AT). The cheapest way to

preserve this joint bivariate cdf is obtained by a twin f(AT, ST),which is increasing

in STconditionally on AT(see Corollary 3.5). Hence, X∗

Tmust also be of this form,

otherwise one can easily contradict the optimality of X∗

Tto the problem. Thus, the

solution to optimal expected utility maximization with the additional probability

constraint (when it exists) is an optimal twin. By similar reasoning, this result also

holds when there are several probability constraints involving the joint distribution

of terminal portfolio value XTand benchmark AT.

The following theorem extends Theorem 5.1 and the referenced literature above

by considering an expected utility maximization problem in which the investor ﬁxes

the dependence with a benchmark. Doing so amounts to specifying up front the

joint copula of (XT, AT). Hence, let us assume that the copula between XTand ATis

speciﬁed to be C, i.e., C(XT,AT)=C. We formulate the following portfolio optimization

problem

max

c0(XT)=W0

C(XT,AT)=C

E(u(XT)) .(31)

In order to solve the expected utility optimization problem with dependence con-

straints (31), we denote by C1|ATthe conditional distribution of the ﬁrst component,

given AT(or equivalently given FAT(AT)) and deﬁne

UT=FST|AT(ST) and ZT=C−1

1|AT(UT).(32)

Note that when (AT, ST) has a joint density, then UTand ZTare uniformly distributed

on (0,1) and (ZT, AT) has copula C(see also Lemma A.2). Theorem 5.2 makes also

use of the projection on the convex cone

M↓:= {f∈L2[0,1]; fdecreasing},(33)

which is a subset of L2[0,1] equipped with the Lebesgue measure and the standard

|| · ||2norm. For an element ϕ∈L2[0,1], we denote by bϕ=πM↓(ϕ) the projection

of ϕon M↓.bϕcan be interpreted as the best approximation of ϕby a decreasing

function for the || · ||2norm.

Theorem 5.2 (Optimal payoﬀ in EUT with dependence constraint).Consider a

utility function u(·)as in Theorem 5.1 and assume that (AT, ST)has a joint density.

Let HT=E(ξT|ZT) = ϕ(ZT)and b

HT=bϕ(ZT)in which ZTis deﬁned as in (32).

Then, the solution to the optimization problem (31) is given by

b

XT= (u0)−1λb

HT,(34)

where λis such that EhξT(u0)−1λb

HTi=W0.

12The observation that in the given context optimal payoﬀs write as twins is also consistent with

the solutions of the constrained portfolio optimization problems considered in Bernard, Chen and

Vanduﬀel (2014d) and Bernard and Vanduﬀel (2014c).

16

The proof of Theorem 5.2 is provided in Appendix C.1.

Remark 5.3.In the case that HT=E(ξT|ZT) is decreasing in ZT, we obtain, as

solution to (31),

b

XT= (u0)−1(λHT).(35)

In this case, the proof of Theorem 5.2 can be simpliﬁed and reduced to the classical

optimization result in Theorem 5.1 since by Theorem 3.4 an optimal solution XTis

unique and satisﬁes

XT=F−1

XT|AT(FST|AT(ST)).

By Lemma A.2 one can conclude that XT=F−1

XT(ZT), i.e., XTis an increasing

function of ZT. Theorem 5.1 then allows one to ﬁnd the optimal element in this class.

Remark 5.4.The determination of the isotonic approximation bϕof ϕis a well-studied

problem (see Theorem 1.1 in Barlow et al. (1972)). bϕis the slope of the smallest

concave majorant SCM (ϕ) of ϕ, i.e., bϕ= (SCM(ϕ))0. In Barlow et al. (1972) the

projection on M↑is given as the slope of the greatest convex minorant GCM (ϕ) of

ϕ. Fast algorithms are known to determine bϕ.

Remark 5.5.Some special cases of interest concern the study of the optimum when

the copula constraint is the lower or upper Fr´echet bound. If in Theorem 5.2 the

copula Cis the upper Fr´echet bound, then ZT=FAT(AT). When AT=ST, then

HT=E[ξT|AT] = ξTand we ﬁnd that b

XTis equal to the optimal portfolio when

there is no dependence constraint (Theorem 5.1). This result is intuitive because the

dependence constraint that we impose implies that that the optimum is increasing in

ST, which is a feature that arises naturally in the unconstrained problem. If AT=St,

then HT=E[ξT|St] is decreasing in St. Thus, b

HT=HTand the optimum can

be explicitly calculated (see also the example below). Finally, if in Theorem 5.2 the

copula Cis the lower Fr´echet bound , then ZT= 1−FAT(AT). Assume that AT=ST,

then HT=E[ξT|ZT] = ξT, which is increasing in STand therefore decreasing in ZT.

The isotonic approximation is the constant. Hence, the optimal portfolio is also a

constant, i.e., the budget is entirely invested in the risk-free asset.

Example (CRRA investor) Next, we illustrate Theorem 5.2 by a comparison of

the optimal wealth b

XTderived under a dependence constraint (Theorem 5.2) with

the optimal wealth X?

Tderived with no constraints on dependence (Theorem 5.1).

W0stands for the initial wealth and we set the benchmark ATequal to Stfor some

0<t<T.We assume also that the dependence between Stand the ﬁnal wealth

is described by a Gaussian copula Cwith correlation coeﬃcient ρ∈h−q1−t

T,1.

Consider a CRRA utility function with risk aversion η > 0 :

u(x) := x1−η

1−ηwhen η6= 1

ln (x) when η= 1 .

17

The standard Merton problem (28) exposed in Theorem 5.1 involves no dependence

constraint on the ﬁnal wealth. The solution is X?

T= (u0)−1(λξT) where λis found

to meet the initial wealth constraint (E[ξTX?

T] = W0). It is straightforward to verify

that for all η > 0 the optimal wealth is given by

X?

T(η) = λ−1

ηξ−1

η

T=W0erT e−1

η

θ

σµ−σ2

2T+1

η−1

2η2θ2TST

S01

η

θ

σ

(36)

Observe that the dependence between X∗

T(η) and Stis characterized by the Gaussian

copula with correlation parameter

corr (ln(X∗

T(η)) ,ln(St)) = rt

T.(37)

When there is a constraint on the dependence, we show in Appendix C.2 that the

solution to the optimization problem (31) (that is the optimal wealth satisfying the

initial budget and the dependence constraint) is given as

b

XT(η) = W0erT e−1

η

θ

σµ−σ2

2ρ√t+√(1−ρ2)(T−t)2+1

η−1

2η2θ2ρ√t+√(1−ρ2)(T−t)2

×

ST

S0θ

ησ ρ√t+√(1−ρ2)(T−t)√1−ρ2

√T−tSt

S0θ

ησ ρ√t+√(1−ρ2)(T−t)ρ

√t−√1−ρ2

√T−t.(38)

Note that the expressions (36) and (38) coincide when ρ=qt

T.The basis reason for

this feature is that the unconstrained optimum has correlation qt

Twith St.When

η6= 1,the expected utilities of X?

T(η) and b

XT(η) are given by

E[u(X?

T(η))] = 1

1−ηW1−η

0e(1−η)rT +1

2

1−η

ηθ2T

and

Ehub

XT(η)i=1

1−ηW1−η

0e(1−η)rT +1

2

1−η

ηθ2ρ√t+√(1−ρ2)(T−t)2

,

respectively. In the case that η= 1, i.e., the log-utility case u(x) := ln (x), we ﬁnd

that

Ehub

XTi= ln (W0) + rT +1

2θ2ρ√t+p1−ρ2√T−t2

and

E[u(X?

T)] = ln (W0) + rT +1

2θ2T,

respectively.

Assume that t=T/2 for the numerical application so that ST/2is the benchmark.

Using an initial wealth W0= 100 and the same set of parameters as in the previous

section, µ= 0.06, r = 0.02, σ = 0.3 and T= 1.Figure 1 plots the expected utility

as a function of ρfor the constrained payoﬀ ( b

XT) and we have an horizontal line

corresponding to the expected utility of X∗

T. Note that they share exactly one common

point corresponding to the level of correlation found in (37).

18

E@uHX

`TLD

E@uHXT

*LD

-0.5

0.0

0.5

1.0

4.620

4.625

4.630

4.635

4.640

Ρ

E@uHXTLD

E@uHX

`THΗLLD

E@uHXT

*HΗLLD

-0.5

0.0

0.5

1.0

-0.982

-0.981

-0.980

-0.979

-0.978

-0.977

-0.976

-0.975

Ρ

E@uHXTHΗLLD

η= 1 (log utility) η= 2

Figure 1: Expected utility as a function of ρfor a CRRA investor, with and without

dependence constraint.

5.2 Target probability maximization

Target probability maximizers are investors who, for a given budget (initial wealth)

and a given time frame, want to maximize the probability that the ﬁnal wealth

achieves some ﬁxed target b. In a Black–Scholes ﬁnancial market model, Browne

(1999) and Cvitani`c and Spivak (1999) derive the optimal investment strategy for

these investors using stochastic control theory and show that it is optimal to purchase

a digital option written on the risky asset. We show that their results follow from

Theorem 2.2 in a more straightforward way.

Proposition 5.6 (Browne’s original problem).Let W0be the initial wealth and let

b > W0erT be the desired target.13 The solution to the following target probability

maximization problem,

max

XT>0, c0(XT)=W0

P[XT>b],(39)

is given by the payoﬀ

X∗

T=b1{ST>λ},(40)

in which λis given by bEξT1{ST>λ}=W0.

The proof of this proposition is provided in Appendix C.3. In a Black–Scholes market

one easily veriﬁes that λ=S0exp (r−σ2

2)T−σ√TΦ−1W0erT

b.

A target probability maximizing strategy is essentially an all-or-nothing strategy.

Intuitively, investors might not be attracted by the design of the optimal payoﬀ, which

13If b6W0erT ,then the problem is not interesting since an investment in the risk-free asset allows

the investor to reach a 100% probability of beating the target b.

19

maximizes the probability beating a ﬁxed target. The obtained wealth depends solely

on the ultimate value of the underlying risky asset, which makes it highly dependent

on ﬁnal market behavior and thus prone to unexpected and brutal changes. Our

ﬁrst extension concerns the case of a stochastic target, so that preferences become

state-dependent.

Theorem 5.7 (Target probability maximization with a random target).Let W0be

the initial wealth and let Bbe the random target such that (B, ST)has a density. The

solution to the random target probability maximization problem,

max

XT>0, c0(XT)=W0

P[XT>B],(41)

is given by the payoﬀ

X∗

T=B1{BξT<λ},(42)

in which λis implicitly given by EBξT1{BξT<λ}=W0.

The proof of this proposition is provided in Appendix C.4.

Our second extension assumes a ﬁxed dependence with a benchmark in the ﬁnan-

cial market. We now consider the problem of an investor who, for a given budget,

aims to maximize the probability that the ﬁnal wealth will achieve some ﬁxed target

while preserving a certain dependence with a benchmark.

Theorem 5.8 (Target probability maximization with a random benchmark).Let

W0be the initial wealth and let b>W0erT the desired target for ﬁnal wealth. As-

sume that the pair (AT, ST)has a density. Then the solution to the target probability

optimization problem with random benchmark AT,

max

XT>0,c0(XT)=W0,

C(XT,AT)=C

P[XT>b],(43)

is given by

X∗

T=b1{ZT>λ},(44)

in which λis determined by bEξT1{ZT>λ}=W0and ZTis deﬁned as in (32).

The proof of this result is provided in Appendix C.5.

The result derived in Theorem 5.8 holds in particular when AT=St(0 <t<T)

and when Cis a Gaussian copula with correlation coeﬃcient ρ. Then, the optimal

solution is explicit and equal to

X∗

T=b1{Sα

tST>λ},(45)

with α=qT−t

t(1−ρ2)ρ−1,and λ=Sα+1

0exp (r−σ2

2)(αt +T)−σ√kΦ−1W0erT

b

with k= (α+ 1)2t+ (T−t) = T−t

1−ρ2. The proof of (45) is provided in Appendix C.6.

20

Illustration of target probability maximization Let us compare the payoﬀs

that arise from the unconstrained target probability maximization problem in Theo-

rem 5.6 and the constrained maximization problem in Theorem 5.8. We use the same

set of parameters as in Section 5.1, i.e., µ= 0.06, r = 0.02, σ = 0.3 and T= 1.We

also take S0= 100 and b= 106. In Figure 2, we plot for both payoﬀs their expected

value as a function of ρ. The optimum for the unconstrained target optimization

problem in Theorem 5.6 is given by b1{ST>λ1}in which λ1is such that the budget

constraint is satisﬁed. Its expected value is given as

Eb1{ST>λ1}=bΦθ√T+ Φ−1W0erT

b.

By similar reasoning, we ﬁnd for the expected value of the optimum of Theorem 5.8,

Eb1{Sα

tST>λ2}=bΦθαt +T

√k+ Φ−1W0erT

b,

in which α=qT−t

t(1−ρ2)ρ−1, k =T−t

1−ρ2and λ2is such that the budget constraint is

satisﬁed. Note that the expected values are proportional to the probabilities to beat

the target value b. We observe that in the constrained target probability maximization

problem the expected value (and the corresponding success probability) is smaller

than in the unconstrained problem.

E@b.18ST>Λ1<D

E@b.19St

ΑST> Λ2=D

-1.0

-0.5

0.0

0.5

1.0

101.5

102.0

102.5

103.0

Ρ

E@XTD

Figure 2: Expected payoﬀ as a function of ρfor the diﬀerent target probability

maximization strategies considered in Theorem 5.6 and Theorem 5.8.

6 Final remarks

In this paper, we introduce a state-dependent version of the optimal investment prob-

lem. We deal with investors who target a known wealth distribution at maturity (as

21

in the traditional setting) and additionally desire a particular interaction with a ran-

dom benchmark. We show that optimal contracts depend at most on two underlying

assets, or on one asset evaluated at two diﬀerent dates, and we are able to characterize

and determine them explicitly. Our characterization of optimal strategies allows us

to extend the classical expected utility optimization problem of Merton to the state-

dependent situation. Throughout the paper, we have assumed that the state-price

density process ξTis a decreasing functional of the risky asset price STand that there

is a single risky asset. It is possible to relax these assumptions and yet still to provide

explicit representations of optimal payoﬀs. However, the optimality is then no longer

related to path-independence properties.

Throughout the paper, we assumed that ξTis decreasing in ST(in (2)). Moreover,

we use the one-dimensional Black-Scholes model to illustrate our ﬁndings. However,

the case of multidimensional markets described by a price process (S(1)

t, . . . , S(d)

t)tis

essentially included in the results presented in this paper, assuming that the state-

price density process (ξt)tof the risk-neutral measure chosen for pricing is of the

form ξt=gtht(S(1)

t, . . . , S(d)

t)with some real functions gt,ht(as in Bernard, Maj

and Vanduﬀel (2011) who considered the state-independent case). All results in the

paper apply by replacing the one-dimensional stock price process Stby the one-

dimensional process ht(S(1)

t, . . . , S(d)

t). In addition, we have assumed that asset prices

are continuously distributed, which amounts essentially to assuming that the state-

price density process ξtis continuously distributed at any time. An extension to the

case in which ξtmay have atoms is possible but not in the scope of the present paper.

A straightforward extension of the results presented in this paper is to consider the

market model of Platen and Heath (2006) using the Growth Optimal Portfolio (GOP).

Its origins can be traced back to Kelly (1956). It consists of replacing the state-price

density process ξtby 1/S∗

t, where S∗

tdenotes the value of the GOP at time t. In the

Black-Scholes setting, S∗

tis simply the value of one unit investment in a constant-mix

strategy, where a fraction θ

σis invested in the risky asset and the remaining fraction

1−θ

σin the bank account. It is easy to prove that this strategy is optimal for an

expected log-utility maximizer. Using a milder notion of arbitrage, Platen and Heath

(2006) argue that, in general, the price of (non-negative) payoﬀs could be achieved

using the pricing rule (1) where the role of ξTis now played by the inverse of the

GOP. Hence, our results are also valid in their setting, where the GOP is taken as the

reference (see Bernard et al. (2014d) for an example). Other dependence constraints

can be considered, e.g. a constraint on the correlation between the terminal wealth

and a benchmark (developed in the context of mean-variance optimization by Bernard

and Vanduﬀel (2014c)).

22

A Proofs

Throughout the paper and the diﬀerent proofs, we make repeatedly use of the follow-

ing lemmas. The ﬁrst lemma gives a restatement of the classical Hoeﬀding–Fr´echet

bounds going back to the early work of Hoeﬀding (1940) and Fr´echet (1940), (1951).

Lemma A.1 (Hoeﬀding–Fr´echet bounds).Let (X, Y )be a random pair and Uuni-

formly distributed on (0,1). Then

EF−1

X(U)F−1

Y(1 −U)6E[XY ]6EF−1

X(U)F−1

Y(U).(46)

The upper bound for E[XY ]is attained if and only if (X, Y )is comonotonic, i.e.

(X, Y )∼(F−1

X(U), F −1

Y(U)).Similarly, the lower bound for E[XY ]is attained if and

only if (X, Y )is anti-monotonic, i.e. (X, Y )∼(F−1

X(U), F −1

Y(1 −U)).

The following lemma combines special cases of two classical construction results.

The Rosenblatt transformation describes a transform of a random vector to iid uni-

formly distributed random variables (see Rosenblatt (1952)). The second result is a

special form of the standard recursive construction method for a random vector with

given distribution out of iid uniform random variables due to O’Brien (1975), Arjas

and Lehtonen (1978) and R¨uschendorf (1981).

Lemma A.2 (Construction method).Let (X, Y )be a random pair and assume that

FY|X=x(·)is continuous ∀x. Denote V=FY|X(Y).Then Vis uniformly distributed

on (0,1) and independent of X. It is also increasing in Yconditionally on X. Fur-

thermore, for every variable Z,(X, F −1

Z|X(V)) ∼(X, Z).

For the proof of the ﬁrst part note that by the continuity assumption on FY|X=x

we get from the standard transformation

(V|X=x)∼FY|X=x(Y)|X=x∼U(0,1),∀x.

Clearly V∼U(0,1).Furthermore, the conditional distribution FV|X=xdoes not de-

pend on xand thus Vand Xare independent. For the second part one gets by the

usual quantile construction that F−1

Z|X=x(V) has distribution function FZ|X=x. This

implies that (X, F −1

Z|X(V)) ∼(X, Z) since both sides have the same ﬁrst marginal

distribution and the same conditional distribution.

Lemma A.3. Let (X, Y )be jointly normally distributed. Then, conditionally on Y,

Xis normally distributed and,

E(X|Y) = E(X) + cov(X, Y )

var(Y)(Y−E(Y)

var(X|Y) = (1 −ρ2) var(X).

Denote the density of Yby fY(y). One has,

Zc

−∞

ea+byfY(y)dy =ea+bE(Y)+ b2

2var(Y)1

p2πvar(Y)Zc

−∞

e−1

2y−(E(Y)+bvar(Y))

√var(Y)2

dy.

The results in this lemma are well-known and we omit its proof.

23

A.1 Proof of Proposition 2.1

Let U=FST(ST) a uniformly distributed variable on (0,1).Consider a payoﬀ XT.

One has,

c0(XT) = E[XTξT]>EF−1

XT(U)ξT=c0(X∗

T),

where the inequality follows from the fact that F−1

XT(U) and ξTare anti-monotonic

and using the Hoeﬀding–Fr´echet bounds in Lemma A.1. Hence, X∗

T=F−1(FS(ST))

is the cheapest payoﬀ with cdf F. Similarly, the most expensive payoﬀ with cdf F

writes as Z∗

T=F−1(1 −FS(ST)). Since cis the price of a payoﬀ XTwith cdf F, one

has

c∈[c0(X∗

T), c0(Z∗

T)].

If c=c0(X∗

T) then X∗

Tis a solution. Similarly, if c=c0(Z∗

T) then Z∗

Tis a solution.

Next, let c∈(c0(X∗

T), c0(Z∗

T)) and deﬁne the payoﬀ fa(ST) with a∈R,

fa(ST) = F−1[(1 −FST(ST))1ST6a+ (FST(ST)−FST(a))1ST>a].

Then fa(ST) is distributed with cdf F. The price c0(fa(ST)) of this payoﬀ is a

continuous function of the parameter a. Since lima→0+c0(fa(ST)) = c0(X∗

T) and

lima→+∞c0(fa(ST)) = c0(Z∗

T), using the theorem of intermediary values for continuous

functions, there exists a*such that c0(fa*(ST)) = c. This ends the proof.

A.2 Proof of Corollary 2.3

Let XT∼Fbe cost-eﬃcient. Then XTsolves (6) and Theorem 2.2 implies that

XT=F−1(FST(ST)) almost surely. Reciprocally, let XT∼Fbe increasing in ST.

Then, by our continuity assumption, XT=F−1(FST(ST)) almost surely and thus XT

is cost-eﬃcient.

A.3 Proof of Theorem 3.2

The idea of the proof is very similar to the proof of Proposition 2.1. Let Ube given

by U=FST|AT(ST).It is uniformly distributed over (0,1) and independent of AT(see

Lemma A.2). Furthermore, conditionally on AT, U is increasing in ST. Consider next

a payoﬀ XTand note that F−1

XT|AT(U)∼XT.We ﬁnd that

c0(XT) = E[XTξT] = E[E[XTξT|AT]]

>EhEhF−1

XT|AT(U)ξTATii=EhF−1

XT|AT(U)ξTi,(47)

where the inequality follows from the fact that F−1

XT|AT(U) and ξTare conditionally

(on AT) anti-monotonic and using (46) in Lemma A.1 for the conditional expectation

(conditionally on AT). Similarly, one ﬁnds that

c0(XT)6EhF−1

XT|AT(1 −U)ξTi.

24

Next we deﬁne the uniform (0,1) distributed variable,

ga(ST) = (1 −FST(ST))1ST6a+ (FST(ST)−FST(a))1ST>a.

We observe that thanks to Lemma A.2, Fga(ST)|AT(ga(ST)) is independent of ATand

also that fa(ST, AT) given as

fa(ST, AT) = F−1

XT|AT(Fga(ST)|AT(ga(ST)))

is a twin with the desired joint distribution Gwith AT.Denote by X∗

T=F−1

XT|AT(U)

and by Z∗

T=F−1

XT|AT(1−U). Note that X∗

T=f0(ST, AT) and Z∗

T=f1(ST, AT) almost

surely. The same discussion as in the proof of Proposition 2.1 applies here. When

c=c0(X∗

T) then X∗

Tis a twin with the desired properties. Similarly, when c=c0(Z∗

T)

then Z∗

Tis a twin with the desired properties. Otherwise, when c∈(c0(X∗

T), c0(Z∗

T))

then the continuity of c0(fa(ST, AT)) with respect to aensures that there exists a∗

such that c:= c0(fa∗(ST, AT)). Thus, fa∗(ST, AT) is a twin with the desired joint

distribution Gwith ATand with cost c. This ends the proof.

A.4 Proof of Theorem 3.3

Let 0 < t < T. It follows from Lemma A.2 that FSt|ST(St) is uniformly distributed on

(0,1) and independent of ST.Let the twin f(St, ST) be given as

f(St, ST) := F−1

XT|ST(FSt|ST(St)).

Using Lemma A.2 again, one ﬁnds that (f(St, ST), ST)∼(XT, ST)∼G. This also

implies,

c0(f(St, ST)) = E[f(St, ST)ξT] = E[XTξT] = c0(XT),

and this ends the proof.

A.5 Proof of Theorem 3.4

It follows from Lemma A.2 that U=FST|AT(ST) is uniformly distributed on (0,1),

stochastically independent of ATand increasing in STconditionally on AT. Let the

twin X∗

Tbe given as

X∗

T=F−1

XT|AT(U).

Invoking Lemma A.2 again, (X∗

T, AT)∼(XT, AT)∼G. Moreover,

c0(XT) = E[XTξT] = E[E[XTξT|AT]]

>EhEhF−1

XT|AT(U)ξTATii

=EhF−1

XT|AT(U)ξTi=c0(X∗

T)

where the inequality follows from the fact that F−1

XT|AT(U) and STare conditionally

(on AT) comonotonic and using (46) in Lemma A.1 for the conditional expectation

(conditionally on AT).

25

A.6 Proof of Corollary 3.5

Let us ﬁrst assume that XTis a cheapest twin. By Theorem 3.4, XTis (almost

surely) equal to X∗

Tas deﬁned by (14) which is, conditionally on AT, increasing

in ST. Reciprocally, we now assume that XT=f(ST, AT) is conditionally on AT

increasing in ST. Hence XT=F−1

XT|ATFST|AT(ST)almost surely, which means it is

a solution to (13) and thus a cheapest twin.

B Security design

B.1 Twin of the ﬁxed strike (continuously monitored) geo-

metric Asian call option

Expression (11) allows us to ﬁnd twins satisfying the constraint (19) on the depen-

dence with the benchmark ST. Using Lemma A.3 we ﬁnd that

ln(St/S0)|ln(ST/S0)∼ N t

Tln ST

S0, σ2t1−t

T,

and thus

FSt|ST(St)=Φ

ln StS

t

T−1

0

S

t

T

T

σqtT −t2

T

.

Furthermore, the couple (ln (GT),ln (ST)) is bivariate normally distributed with mean

and variance for the marginal distributions that are given as E[ln(GT)] = ln S0+

µ−1

2σ2T

2, var[ln(GT)] = σ2T

3and E[ln(ST)] = ln S0+µ−1

2σ2T, var[ln(ST)] =

σ2T. For the correlation coeﬃcient one has ρ(ln(ST),ln(GT)) = √3

2.Applying Lemma

A.3 again one ﬁnds that,

ln(GT)|ln(ST)∼ N ln S1/2

0S1/2

T,σ2T

12 ,(48)

and thus,

FGT|ST(x)=Φ

ln(x)−ln S1/2

0S1/2

T

σ√T

2√3

.

Therefore,

F−1

GT|ST(y) = exp ln S1/2

0S1/2

T+σ√T

2√3Φ−1(y)!.

The expression of RT(t) given in (21) is then straightforward to derive.

26

For choosing a speciﬁc twin among others, we suggest to maximize ρ(ln RT(t),ln GT).

First, we calculate,

cov ln ST,1

TZT

0

ln (Ss)ds=1

TZT

0

cov (ln ST,ln (Ss)) ds

=σ2

TZT

0

(s∧T)ds =σ2T

2.

Furthermore, by denoting a=1

2−1

2√3qT−t

t,b=T

t

1

2√3qt

T−tand c=1

2−1

2√3qt

T−t,

equation (21) may be rewritten as ln RT(t) = aln S0+bln St+cln ST. The covariance

being bilinear, one then has,

cov (ln RT(t),ln GT) = bcov ln St,1

TZT

0

ln (Ss)ds+ccov ln ST,1

TZT

0

ln (Ss)ds

=σ2

2T

2+√t√T−t

2√3.

Denote by σln RT(t)and by σln GTthe respective standard deviations. For the correla-

tion we ﬁnd that

ρ(ln RT(t),ln GT) = cov (ln RT(t),ln GT)

σln RT(t)σln GT

=3

4+√3p(T−t)t

4T.

Hence ρ(ln RT(t),ln GT) is maximized for t=T

2.

B.2 Twin of the ﬂoating strike (continuously monitored) ge-

ometric Asian put option

We ﬁrst recall from equation (48) that,

ln(GT)|ln(ST)∼ N ln S

1

2

0S

1

2

T,σ2T

12 .

Therefore YT= (GT−ST)+has the following conditional cdf

P(YT6y|ST=s) = Φ

ln(s+y)−ln S1/2

0s1/2

σ√T

2√3

1y>0

Then

F−1

YT|ST(z) = S

1

2

0S

1

2

Teσ

2√T

3Φ−1(z)−ST+

.

Therefore F−1

YT|STFSt|ST(St))can then easily be computed and after some calcula-

tions it simpliﬁes to (24).

27

B.3 Cheapest Twin of the ﬂoating strike (continuously mon-

itored) geometric Asian put option

Applying Lemma A.3 we ﬁnd,

ln(ST)|ln(GT)∼ N ln G3/2

T

S

1

2

0!+1

4µ−σ2

2T, σ2T

4!.

Hence,

FST|GT(ST)) = Φ

ln STS

1

2

0

G

3

2

T−µ−σ2

2T

4

σ√T

2

.(49)

Furthermore, YT= (GT−ST)+has the following conditional cdf,

P(YT6y|GT=g) =

1 if y>g,

Φ

ln

g3/2

S

1

2

0

+1

4µ−σ2

2T−ln(g−y)

σ√T

2

if 0 6y6g,

0 if y < 0.

Then

F−1

YT|GT(z) = GT−G

3

2

T

S

1

2

0

e1

4µ−σ2

2T−σ

2√TΦ−1(z)!+

.

Replacing zby the expression (49) for FST|GT(ST)) derived above, then gives rise to

expression (25).

B.4 Derivation of prices (26) and (27)

Price (26)

Let us observe that,

(GT−ST)+=GT1−ST

GT+=S0eY1−eZ+,

where Z=X−Y, Y = ln GT

S0, X = ln ST

S0. We ﬁnd, with respect to the risk

neutral measure Q,

EQ(GT−ST)+=S0EQEQeY|Z1−eZ+

=S0EQeEQ(Y|Z)+ 1

2varQ(Y|Z)−eEQ(Y|Z)+1

2varQ(Y|Z)+Z+.

28

We now compute (still with respect to Q),

EQ(Y|Z) = EQ(Y) + covQ(Y, Z)

varQ(Z)(Z−EQ(Z)) = r−σ2

2T

4+1

2Z

varQ(Y|Z) = (1 −ρ2) varQ(Y) = 3

4

σ2T

3=σ2T

4.

Hence,

EQ(GT−ST)+=S0EQerT

4+1

2Z−erT

4+3

2Z+

=S0Z0

−∞

erT

4+1

2ZfZ(z)dz −S0Z0

−∞

erT

4+3

2ZfZ(z)dz,

where fZ(z) is now denoting the density of Zunder Q.Here Zis normally distributed

with parameters (r−σ2

2)T

2and variance σ2T

3.Hence, taking into account Lemma A.3,

EQ(GT−ST)+=S0erT

2−σ2T

12 Φ

−r−σ2

2T

2−σ2T

6

qσ2T

3

−S0erT Φ

−r−σ2

2T

2−σ2T

2

qσ2T

3

Choose f=−rT

2+σ2T

12

σ√T

3

to obtain (26).

Price (27)

One has, GT−aG3

T

ST+

=GT1−aG2

T

ST+

=S0eY1−ceZ+

where Z= 2Y−X, Y = ln GT

S0, X = ln ST

S0, c =eµ−σ2

2T

2.Hence, with respect

to the risk neutral measure Q,

EQGT−aG3

T

ST+

=S0EQEQ(eY|Z)1−ceZ+

=S0EQeEQ(Y|Z)+ 1

2varQ(Y|Z)−ceEQ(Y|Z)+1

2varQ(Y|Z)+Z+.

We now compute,

EQ(Y|Z) = r−σ2

2T

2+1

2Zand varQ(Y|Z) = σ2T

4.

Hence,

EQGT−aG3

T

ST+

=S0EQerT

2−σ2T

8+1

2Z−cerT

2−σ2T

8+3

2Z+

=S0Zln(c)

−∞

erT

2−σ2T

8+1

2ZfZ(z)dz −S0cZln(c)

−∞

erT

2−σ2T

8+3

2ZfZ(z)dz,

29

where fZ(z) is the density of Z, under Q.Note that Zis normally distributed with

parameters 0 and variance σ2T

3.Taking into account Lemma A.3,

EQGT−aG3

T

ST+

=S0erT

2 Φ (d)e−σ2T

12 −eµT

2Φ d−σ√T

√3!!

where d=−ln(c)−σ2T

6

σ√T

3

=σ2T

12 −µT

2

σ√T

3

.

C Portfolio Management

C.1 Proof of Theorem 5.2

Let HT=E(ξT|ZT) = ϕ(ZT) and let bϕdenote the projection of ϕon the cone M↓

deﬁned as in (33) with respect to L2(λ[0,1]). Then we deﬁne b

XTand k(·) by

u0(b

XT) := λbϕ(ZT),

i.e. b

XT= (u0)−1(λbϕ(ZT)) =: k(ZT) with λsuch that E[ξTb

XT] = E[ϕ(ZT)k(ZT)] =

R1

0ϕ(t)k(t)dt =ϕ·k=W0. By deﬁnition, b

XTis increasing in ZTsince (u0)−1is

decreasing and bϕis decreasing (it belongs to M↓). As a consequence b

XTis increasing

in ST, conditionally on AT. For any YT=h(ZT) with a increasing function h, we

have by concavity of u

u(YT)−u(b

Xt)6u0(b

XT)(YT−b

XT) = λbϕ(ZT)(h(ZT)−k(ZT)).

Thus, we obtain

E[u(YT)] −E[u(b

XT)] 6λZ1

0bϕ(t)(h(t)−k(t))dt =λbϕ·(h−Ψ(bϕ)),(50)

where Ψ(bϕ) = (u0)−1(λbϕ) = kis increasing and Ψ(t) = (u0)−1(λt) is decreasing.

Now we use some properties of isotonic approximations (see Barlow et al. (1972))

and obtain

bϕ·(h−Ψ(bϕ)) = bϕ·((−Ψ)( bϕ)−(−h))

=ϕ·(−Ψ)(bϕ)−bϕ·(−h)(see Theorem 1.7 in Barlow et al. (1972))

=ϕ·(−h)−bϕ·(−h)both claims have price W0

= (ϕ−bϕ)·(−h)60

by the projection equation (see Theorem 7.8 in Barlow et al. (1972)) using that

−h∈M↓. As a result we obtain from (50) that

E[u(YT)] 6E[u(b

XT)],

i.e. b

XTis an optimal claim.

30

C.2 Proofs of equations (36) and (38) in the example of sub-

section 5.1

We apply Theorem 5.1 to an investor with a power-utility. Then,

X?

T(η)=(u0)−1(λξT) = (λξT)−1

η(51)

where λis chosen to meet the budget constraint, i.e.

E[ξT(u0)−1(λξT)] = EhξT(λξT)−1

ηi=λ−1

ηEξ1−1

η

T=W0(52)

Since ξT= exp −rT −1

2θ2T−θZT, we ﬁnd that λ−1

η=W0exp n−r1−η

ηT−1

2θ2T1−η

η1

ηo

and

X?

T(η) = (λξT)−1

η=W0e−r(1−η

η)T−1

2θ2T(1−η

η)1

η−1

ηhθ

σµ−σ2

2T−r+θ2

2TiST

S0θ

ση

,

which can be simpliﬁed to ﬁnd (36).

Next, we apply Theorem 5.2 with AT=St, for some tsuch that t < T . From

Lemma A.3 we know

ln(ST)|ln(St)∼ N ln (St) + µ−σ2

2(T−t), σ2(T−t)

so that

FST|St(ST) = Φ

ln ST

St−µ−σ2

2(T−t)

σ√T−t

.

Because Cis a Gaussian copula, one has

C1|St(x) = Φ

Φ−1[x]−ρlnSt

S0−µ−σ2

2t

σ√t

p1−ρ2

and

C−1

1|St(y) = Φ

p1−ρ2Φ−1[y] + ρ

ln St

S0−µ−σ2

2t

σ√t

.

This implies

ζT=C−1

1|St(FST|St(ST)) = Φ [$T],

where $Tis a function of STand Stgiven by

$T=p1−ρ2

ln ST

St−µ−σ2

2(T−t)

σ√T−t

+ρ

ln St

S0−µ−σ2

2t

σ√t

.(53)

31

Since ξT=αTST

S0−βwhere αT= exp θ

σµ−σ2

2T−r+θ2

2T,β=θ

σand

θ=µ−r

σ(from (4)), one has

HT=E(ξT|ζT) = E(ξT|$T) = δe−βcov(ln(ST),$T)$T,

for some δ > 0 and we ﬁnd

HT=δe−θρ√t+√(1−ρ2)(T−t)$T.

Note that conditions on the correlation coeﬃcient imply that HTis decreasing in $T

and thus HTis decreasing in ZT.The optimal contract thus writes as

b

XT:= (u0)−1λe−θρ√t+√(1−ρ2)(T−t)$T,(54)

where λis chosen to meet the budget constraint.

When the investor has a power-utility, i.e., u(x) = x1−η

1−ηso that (u0)−1(x) = x−1

η

we ﬁnd that equation (54) reads as

b

XT(η) := λ−1

ηe1

ηθρ√t+√(1−ρ2)(T−t)$T(55)

and the budget constraint (i.e., EhξTb

XT(η)i=W0) requires that

Ee−rT e−θ2

2T−θZTλ−1

ηexp 1

ηθρ√t+p(1 −ρ2)(T−t)$T=W0,

where we have used the expressions for ξTand b

XT(η). We ﬁnd that

λ−1

η=W0erT eθ2ρ√t+√(1−ρ2)(T−t)21

η−1

2η2.

The optimal solution is then derived by using this expression into (55).

C.3 Proof of Proposition 5.6

Assume that there exists an optimal solution to the target probability maximization

problem. It is a maximization of a law-invariant objective and therefore it is path-

independent. Denote it by X∗

T:= f∗(ST).Deﬁne A0={x|f∗(x) = 0},A1=

{x|f∗(x) = b},A2={x|f∗(x)∈]0, b[}and A3={x|f∗(x)> b}. We show

that P(ST∈A0∪A1) = 1 must hold. Assume P(ST∈A0∪A1)<1 so that

P(ST∈A2∪A3)>0.Deﬁne

Y=

f∗(ST) for ST∈A0∪A1,

0 for ST∈A2,

bfor ST∈A3.

Then we observe that Y=f∗(ST) on A0∪A1and Y < f ∗(ST) on A2∪A3.Since

P(ST∈A2∪A3)>0 also Q(ST∈A2∪A3)>0 because Pand the risk neutral

32

probability Qare equivalent. Hence c0(Y)< W0.Next we deﬁne Z=b1ST∈C+Y

where we have chosen C⊆A2∪A0such that c0(b1ST∈C) = W0−c0(Y).Since

P(ST∈C)>0 one has that P(Z>b)>P(Y>b) = P(f∗(ST)>b).Hence Z

contradicts the optimality of f∗(ST). Therefore P(ST∈A0∪A1)=1.Hence f∗(ST)

can take only the values 0 or b. Since it is increasing in STalmost surely (by cost-

eﬃciency) it must write as

f∗(ST) = b1ST>a,

where ais chosen such that the budget constraint is satisﬁed.

C.4 Proof of Theorem 5.7

The (random) target probability maximization problem is given as

max

XT>0,c0(XT)=W0

P[XT>B].

Assume that there exists an optimal solution X∗

Tto this optimization problem. There

are three steps in the proof.

1. The optimal payoﬀ is of the form f(ST, B).

2. The optimal payoﬀ is of the form B1h(ST,B)∈A.

3. The optimal payoﬀ is of the form B1BξT<λ∗for λ∗>0.

Step 1: We observe that X∗

Thas some joint distribution Gwith B. Theorem 3.2

implies there exists a twin f(B, ST) such that (f(B, ST), BT)∼(X∗

T, B)∼Gand

c0(f(B, ST)) = c0(X∗

T) = W0. Therefore P(f(B, ST)>B) = P(X∗

T>B) and

P(f(B, ST)>0) = P(X∗

T>0) = 1. Thus f(B, ST) is also an optimal solution.

Step 2: This is similar to the proof of Proposition 5.6, applied conditionally on

B. Deﬁne the sets A0={s, f (B, s) = 0},A1={s, f (B, s) = B},then P(ST∈

A0∪A1|B) = 1 and therefore P(ST∈A0∪A1) = 1. Thus there exists a set Aand a

function hsuch that

f(B, ST) = B1h(ST,B)∈A.

Step 3: Deﬁne λ > 0 such that

P(h(ST, B)∈A) = P(BξT< λ).

Observe that 1h(ST,B)∈Aand 1BξT<λ have the same distribution and that in addition,

BξTis anti-monotonic with 1BξT<λ . Therefore by applying Lemma A.1 one has that

c0(B1BξT<λ) = E[BξT1BξT<λ]6E[BξT1h(ST,B)∈A]

and therefore the optimum must be of the form B1BξT<λ∗where λ∗> λ is determined

such that c0(B1BξT<λ∗) = W0.

33

C.5 Proof of Theorem 5.8

The target probability maximization problem is given by

max

XT>0, c0(XT)=W0,

C(XT,AT)=C

P[XT>b]

Assume that there exists an optimal solution X∗

Tto this optimization problem. There

are three steps in the proof.

1. The optimal payoﬀ is of the form f(ST, AT).

2. The optimal payoﬀ is of the form b1h(ST,AT)∈A.

3. The optimal payoﬀ is of the form AT1ZT>λ∗for λ∗>0.

Step 1: We observe that X∗

Thas some joint distribution Gwith AT.Theorem 3.2

implies there exists a twin f(ST, AT) such that (f(ST, AT), AT)∼(X∗

T, AT)∼G

and c0(f(ST, AT)) = c0(X∗

T) = W0. Therefore P(f(ST, AT)>b) = P(X∗

T>b) and

P(f(ST, AT)>0) = P(X∗

T>0) = 1. Thus f(ST, AT) is also an optimal solution.

Step 2: This is similar to the proof of Proposition 5.6. Deﬁne the sets A0=

{(s, t), f (s, t) = 0},A1={(s, t), f(s, t) = b},then P(ST∈A0∪A1) = 1.Thus

there exists a set Aand a function hsuch that

f(ST, AT) = b1h(ST,AT)∈A.

Step 3: Deﬁne λ > 0 such that

P(h(ST, AT)∈A) = P(ZT> λ).

Observe that b1h(ST,AT)∈Aand b1ZT>λ have the same joint distribution Gwith distri-

bution AT. Therefore, Theorem 3.4 shows that,

c0(b1ZT>λ)6c0(b1h(ST,AT)∈A).

Hence, b1ZT>λ∗where λ∗such that c0(b1ZT>λ∗) = W0is the optimum.

C.6 Proof of formula (45)

We know that b1ZT>λ∗where λ∗is such that c0(b1ZT>λ∗) = W0is the optimal solution.

We ﬁnd that

ZT=C−1

1|St(FST|St(ST))

= Φ

p1−ρ2

ln ST

St −µ−σ2

2(T−t)

σ√T−t

+ρ

ln St

S0−µ−σ2

2t

σ√t

.

34

It is then straightforward that X∗

T=b1{Sα

tST>λ∗}is the optimal solution, with αand

λgiven by

α=sT−t

t(1 −ρ2)ρ−1

λ=S1+α

0exp r−σ2

2(αt +T)−σp(α+ 1)2t+ (T−t)Φ−1W0erT

b.

35

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