Optimal payoffs under state-dependent preferences

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 utility 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.

Full text

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
2t+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=αtSt
S0−β
,(4)
where αt= exp θ
σµ−σ2
2t−r+θ2
2t, 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=dS1/√3
T−K
d+
,(17)
where d=S1−1
√3
0e1
2−√1
3µ−σ2
2T. 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
2T
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
YTΦlnST
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|GTFST|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=eµ−σ2
2T
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 EQGT−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 EξT(u0)−1λξT=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)−1λb
HT,(34)
where λis such that EhξT(u0)−1λb
HTi=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
σµ−σ2
2T+1
η−1
2η2θ2TST
S01
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
σµ−σ2
2ρ√t+√(1−ρ2)(T−t)2+1
η−1
2η2θ2ρ√t+√(1−ρ2)(T−t)2
×
ST
S0θ
ησ ρ√t+√(1−ρ2)(T−t)√1−ρ2
√T−tSt
S0θ
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
Ehub
XT(η)i=1
1−ηW1−η
0e(1−η)rT +1
2
1−η
ηθ2ρ√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
Ehub
XTi= ln (W0) + rT +1
2θ2ρ√t+p1−ρ2√T−t2
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 bEξT1{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−1W0erT
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 EBξT1{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 bEξT1{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−1W0erT
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
Eb1{ST>λ1}=bΦθ√T+ Φ−1W0erT
b.
By similar reasoning, we find for the expected value of the optimum of Theorem 5.8,
Eb1{Sα
tST>λ2}=bΦθαt +T
√k+ Φ−1W0erT
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=gtht(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
EF−1
X(U)F−1
Y(1 −U)6E[XY ]6EF−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
2y−(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]>EF−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)ξTATii=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)ξTATii
=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|ATFST|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, σ2t1−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
2σ2T
2, var[ln(GT)] = σ2T
3and E[ln(ST)] = ln S0+µ−1
2σ2T, 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
2T
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|STFSt|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
4µ−σ2
2T, σ2T
4!.
Hence,
FST|GT(ST)) = Φ 




ln STS
1
2
0
G
3
2
T−µ−σ2
2T
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
4µ−σ2
2T−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
4µ−σ2
2T−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)+=GT1−ST
GT+=S0eY1−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)+=S0EQEQeY|Z1−eZ+
=S0EQeEQ(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
2T
4+1
2Z
varQ(Y|Z) = (1 −ρ2) varQ(Y) = 3
4
σ2T
3=σ2T
4.
Hence,
EQ(GT−ST)+=S0EQerT
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
2T
2−σ2T
6
qσ2T
3

−S0erT Φ
−r−σ2
2T
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+
=GT1−aG2
T
ST+
=S0eY1−ceZ+
where Z= 2Y−X, Y = ln GT
S0, X = ln ST
S0, c =eµ−σ2
2T
2.Hence, with respect
to the risk neutral measure Q,
EQGT−aG3
T
ST+
=S0EQEQ(eY|Z)1−ceZ+
=S0EQeEQ(Y|Z)+ 1
2varQ(Y|Z)−ceEQ(Y|Z)+1
2varQ(Y|Z)+Z+.
We now compute,
EQ(Y|Z) = r−σ2
2T
2+1
2Zand varQ(Y|Z) = σ2T
4.
Hence,
EQGT−aG3
T
ST+
=S0EQerT
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,
EQGT−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
ηEξ1−1
η
T=W0(52)
Since ξT= exp −rT −1
2θ2T−θZT, we find that λ−1
η=W0exp n−r1−η
ηT−1
2θ2T1−η
η1
ηo
and
X?
T(η) = (λξT)−1
η=W0e−r(1−η
η)T−1
2θ2T(1−η
η)1
η−1
ηhθ
σµ−σ2
2T−r+θ2
2TiST
S0θ
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]−ρlnSt
S0−µ−σ2
2t
σ√t
p1−ρ2




and
C−1
1|St(y) = Φ 
p1−ρ2Φ−1[y] + ρ
ln St
S0−µ−σ2
2t
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
2t
σ√t
.(53)
31

Since ξT=αTST
S0−βwhere αT= exp θ
σµ−σ2
2T−r+θ2
2T,β=θ
σand
θ=µ−r
σ(from (4)), one has
HT=E(ξT|ζT) = E(ξT|$T) = δe−βcov(ln(ST),$T)$T,
for some δ > 0 and we find
HT=δe−θρ√t+√(1−ρ2)(T−t)$T.
Note that conditions on the correlation coefficient imply that HTis decreasing in $T
and thus HTis decreasing in ZT.The optimal contract thus writes as
b
XT:= (u0)−1λe−θρ√t+√(1−ρ2)(T−t)$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
ηθρ√t+√(1−ρ2)(T−t)$T(55)
and the budget constraint (i.e., EhξTb
XT(η)i=W0) requires that
Ee−rT e−θ2
2T−θZTλ−1
ηexp 1
ηθρ√t+p(1 −ρ2)(T−t)$T=W0,
where we have used the expressions for ξTand b
XT(η). We find that
λ−1
η=W0erT eθ2ρ√t+√(1−ρ2)(T−t)21
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
2t
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−1W0erT
b.

35

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