Optimal Payoffs under State-dependent Constraints

Abstract

Most decision theories including expected utility theory, rank dependent
utility theory and the cumulative prospect theory assume that investors are
only interested in the distribution of returns and not about 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 is often at odds with the needs of
many investors. We introduce a framework for portfolio selection that permits
to deal with state-dependent preferences. We are able to characterize optimal
payoffs in explicit form. Some applications in security design are discussed in
detail. We extend the classical expected utility optimization problem of Merton
to the state-dependent situation and also give some stochastic extensions of
the target probability optimization problem.

Full text

Optimal Payoffs under State-dependent Constraints
C. Bernard∗
, F. Moraux†
,L.R¨uschendorf‡and S. Vanduffel§
January 20, 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 is often at odds with the needs of many
investors. We introduce a framework for portfolio selection that permits to deal
with state-dependent preferences. We are able to characterize optimal payoffs
in explicit form. Some applications in security design are discussed in detail.
We extend the classical expected utility optimization problem of Merton to the
state-dependent situation and also give 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.
†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
(the CNRS research center) and IAE de 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
The study of optimal investment strategies is usually based on the optimization of an
expected utility, a target probability or some other (increasing) law-invariant mea-
sure. Assuming that investors have law-invariant preferences is equivalent to suppos-
ing 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 the case for example with
expected utility theory, Yaari’s dual theory, rank-dependent utility theory, mean-
variance optimization and cumulative prospect theory. Clearly, an optimal strategy
has some distribution of terminal wealth and it must be the cheapest possible strategy
which attains this distribution. Otherwise it is possible to strictly improve the objec-
tive and to contradict its optimality. Dybvig (1988) was first to study strategies that
reach a given return distribution at lowest possible cost. Bernard, Boyle and Van-
duffel (2014) call these strategies cost-efficient and provide sufficient conditions for
cost-efficiency. In a fairly general market setting they show that the cheapest way to
generate a given distribution is obtained by a contract whose payoff has perfect nega-
tive correlation with the pricing kernel (see also Carlier and Dana (2011)). The basic
intuition is that investors consume less in states of an economic recession because
these are more expensive to insure. This feature is also explicit in a Black-Scholes
framework where 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 the optimization criteria and the resulting payoffs un-
der 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, who may consider sources of background risk when making investment de-
cisions. In other words, two payoffs with the same distribution do not necessarily
present the same “value” for an investor. Bernard and Vanduffel (2012) show that
insurance contracts can usually be substituted by financial contracts that have the
same payoff distribution but are cheaper. The existence of insurance contracts that
provide protection against specific events show that they must present more value for
an investor than financial payoffs which do not have 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 in “normal” conditions.
This paper makes several further theoretical contributions and highlights valu-
able applications in portfolio management and security design. First, we clarify
the setting under which optimal investment strategies must necessarily exhibit path-
independence. These findings complement Cox and Leland (1982), (2000) and Dyb-
vig’s (1988) seminal results and further enforces the important role of path-independence
in traditional optimal portfolio selection.
As main contribution, we introduce a framework for portfolio selection that per-
mits to also consider the states in which income is received. More precisely, it is
2

assumed that investors target some distribution for their terminal wealth and addi-
tionally aim at maintaining a certain (desired) interaction with a random benchmark.
For example, the investor may want his strategy to be unrelated to the benchmark
when it decreases while following it under normal conditions. We are able to charac-
terize optimal payoffs explicitly (Theorems 3.2 and 3.4) and show that they become
conditionally increasing functions of the terminal value of the underlying risky asset.
This setting is also well-suited to solving several investment problems of interest.
A main contribution in this part of the paper is the extension of the classical result of
portfolio optimization under expected utility (Cox and Huang (1989)). More precisely,
we determine the optimal payoff for an expected utility maximizer under a dependence
constraint, reflecting a desired interaction with the benchmark (or background risk)
(Theorem 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 (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 used in the financial market. We
show how to substitute highly path-dependent products by weakly path-dependent
and less involved contracts that we call ”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. “Twins” are defined as
payoffs that depend on two asset values only. We show that they are optimal for state-
dependent preferences. In Section 4, we discuss applications to improving security
designs. In particular, we propose several improvements in the design of geometric
Asian options. In Section 5, we give an extension of the fundamental result on the
optimal design of portfolios under the expected utility paradigm where we specify the
interaction with the benchmark. In this context, we also generalize Browne (1999)
and Cvitani`c and Spivak (1999) results on target probability maximization. Final
remarks are presented in Section 6. Most proofs are given in the Appendix.
1 Framework and notations
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).
It consists of a bank account Bpaying a constant risk-free rate r>0, so that B0
invested 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
3

S=(St)0tT.We assume that St(0 <t<T) has a continuous distribution FSt.
The no-arbitrage price1at time 0 of a payoff XTpaid at time T>0isgivenby
c0(XT)=E[ξTXT],(1)
where (ξt)tis the state-price process ensuring that (ξtSt)tis a martingale. Based
on standard economic theory, we assume throughout this paper that state prices are
decreasing with asset prices2, i.e.
ξt=gt(St),t0,(2)
where gtis decreasing. The functional form (2) for (ξt)tallows presenting our results
on optimal portfolios using (St)tas a reference which is practical3. Assumption (2)
is satisfied by many popular pricing models, including the CAPM, the consumption-
based models and by exponential L´evy markets where the market participants use
Esscher pricing (Vanduffel et al. (2008)).
The Black–Scholes model can be seen as a special case of the latter setting. As
we use it to illustrate our theoretical results we recall its main properties. In the
Black–Scholes market, the price process (St)tsatisfies
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(STx)=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 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. Extending the current setting
to a multidimensional market is straightforward and is discussed in the Final Remarks
(Section 6).
1The 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.
2See for instance Cox, Ingersoll and Ross (1985) as well as 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 Ta positive
expected payoff unconditionally, and non-negative expected payoffs conditionally on ξT.
3However, a deeper feature of the results and characterizations we present is that optimality of a
payoff XTis tied to its (conditional) anti-monotonicity with ξT.See also the final remarks in Section
6.
4

2 Law-invariant preferences and optimality of path-
independent payoffs
In this section, investors have law-invariant (state-independent) preferences. This
means that they are indifferent between two payoffs having the same payoff distri-
bution (under P). In this case, any random payoff XT(that possibly 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 these investors. Recall that a payoff is
path-independent ifthereexistssomefunctionfsuch 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 (usual)
assumption that preferences are increasing, any path-dependent payoff can be strictly
dominated by a path-independent one which is increasing in the risky asset4.
Note that results in this section are closely related to original work of Cox and
Leland (1982), Dybvig (1988), Bernard, Boyle and Vanduffel (2014) 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 is a path-independent alterna-
tive with the same price, which is at least as good for investors with law-invariant
preferences. Thus, such an investor needs to consider path-independent payoffs only.
All other payoffs are indeed redundant in the sense that they are not needed to op-
timize 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 given 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 next section shows that
the dominance becomes strict as soon as preferences are increasing.
2.2 Optimality of path-independent payoffs
The following basic result originally due to Dybvig (1988) and presented more gen-
erally in Bernard, Boyle and Vanduffel (2014) shows that any path-dependent payoff
4This dominance can easily be implemented in practice as all path-independent payoffs can be
statistically replicated with European call and put options as shown for example by Carr and Chou
(1997) and Breeden and Litzenberger (1978).
5

admits a path-independent alternative which is strictly cheaper5This result implies
that investors with increasing law-invariant preferences may restrict their optimiza-
tion strictly to the set of path-independent payoffs, when making investment decisions.
Theorem 2.2 (Cost optimality of path-independent payoffs).Let Fbe a cdf. The
following optimization problem
min
XT∼Fc0(XT)(5)
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)),(6)
where F−1is the (left-continuous) inverse of the cdf Fdefined as
F−1(p) = inf {x|F(x)p}.(7)
This theorem can be seen as an application of the Hoeffding–Fr´echet bounds recalled
in Lemma A.1 presented in the appendix. A given payoff XTcan be optimized by
redistributing wealth levels XT(ω) across the different states ωof the economy such
that they become ordered with the stock price ST(ω). The payoff (6) generating a
given terminal distribution Fat minimal price is called cost-efficient by Bernard,
Boyle and Vanduffel (2014). It is obviously increasing in ST. Theorem 2.2 implies
that both properties (cost-efficiency or increasingness in ST) are actually equivalent.
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 where optimal payoffs, derived
using various techniques, turn out to always exhibit this property.
Corollary 2.4 (Optimal payoffs for increasing law-invariant preferences).For any
payoff YTat price cwhich is not almost surely increasing in STthere exists a path-
independent payoff Y∗
Tat price cwhich is a strict improvement for any investor
with increasing and law-invariant preferences. A possible choice is given by Y∗
T:=
F−1(FST(ST)) + (c−c∗
0)erT where c∗
0denotes the price of (6).Y∗
Thas price cand is
almost surely increasing in ST.
Indeed assume that the investor considers some payoff YT(with price cand cdf F)
which is not almost surely increasing in ST. Then the payoff Y∗
T=F−1(FST(ST)) +
(c−c∗
0)erT is strictly better than YT. This is because it consists in investing an amount
c∗
0<cin the cost-efficient payoff (distributed also with F) and placing the remaining
funds c−c∗
0>0 in the bank account which is a strict improvement of the payoff YT.
5Similar optimality results as in Theorem 2.2 have been given in the class of admissible claims
XTwhich are smaller than Fin convex order in Dana and Jeanblanc (2005) and Burgert and
R¨uschendorf (2006).
6

3 Optimal payoffs under state-dependent prefer-
ences.
Optimal contracts chosen by law-invariant investors do not offer protection in times
of economic scarcity. In fact, due to the observed monotonicity property with ST
the lowest outcomes for an optimal (thus cost-efficient) payoff exactly occur when
the stock price STreaches its lowest levels. More precisely, denote by f(ST)acost-
efficient payoff (with a increasing function f)andbyXTanother payoff such that
both are distributed with Fat maturity. Then, f(ST) delivers low outcomes when
STis low and it holds6for all a0that
E[f(ST)|ST<a]E[XT|ST<a].(8)
Let Fbe the distribution of a put option with payoff XT:= (K−ST)+.Bernard,Boyle
and Vanduffel (2014) show that the payoff of the cheapest strategy with cdf Fcan be
computed as in (6). It is given by X∗
T=(K−aS−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 Fwhereas the first “ordinary” put strategy (with payoff XT) is actually
the most expensive way to do so. These payoffs interact with STin a fundamentally
different way, as one payoff is increasing while the other is decreasing in ST. A put
option protects the investor against a declining market where consumption is more
expensive, whereas the cost-efficient counterpart X∗
Tprovides no protection at all but
rather emphasizes the effect of a market deterioration on the wealth received.
The use of put options is a signal that many investors do care about states of the
economy in which income of investment strategies is received. In particular, they may
seek strategies that resist against declining markets or, more generally, that exhibit
a desired dependence with some source of background risk. More evidence of state-
dependent preferences can be found in the mere existence of insurance contracts.
As explained in the introduction, for each insurance contract there exists a financial
contract that has the same payoff distribution but is cheaper to buy (Bernard and
Vanduffel (2012)). Yet people buy insurance indicating that insurance payments,
which provide protection conditionally upon the occurrence of adverse events, present
more value for an investor than financial payoffs which do not have this feature.
Hence, in the rest of the paper, we consider investors who exhibit state-dependent
preferences in the sense that they seek a payoff a payoff XTwith a desired distri-
butionand additionally a desired dependence with a benchmark asset AT.Inother
words, they fix the joint distribution Gof the random couple (XT,A
T).The optimal
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
6We give here a short proof of (8). It is clear that the couple (f(ST),1ST<a) has the same
marginals as (XT,1ST<a ) but E[f(ST)1ST<a]E[XT1ST<a ] because f(ST)and1ST<a are anti-
monotonic (from Lemma A.1).
7

of state-independent preferences that we discussed in the previous section. In what
follows we consider as benchmark (or background risk) the underlying risky asset
or any other asset in the market, considered at final or intermediate time(s). We
voluntarily organize the rest of this section similarly as Section 2 so that the impact
of state-dependent preferences on the structure of optimal payoffs is clear.
Remark 3.1.One can use a copula as a device to model the interaction between payoffs
and benchmarks. The joint distribution Gofthecouple(XT,A
T) can be written using
a copula C. From Sklar’s theorem, G(x, a)=C(FXT(x),F
AT(a)), where Cis a copula
(this representation is unique for continuously distributed random variables). It is
then clear that the optimization problem (9) can also be formulated as
min
XT∼F,
C(XT,AT)=C
c0(XT)
where “C(XT,AT)=C” means that the copula between the payoff XTand the bench-
mark ATis C.
3.1 Sufficiency of twins
In this paper any payoff that writes as f(ST,A
T)orf(ST,S
t) is called a twin. We
first show that in our state-dependent setting for any payoff there exists a twin,
which is at least as good. When also assuming that preferences are increasing we find
that optimal payoffs writes 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.
The next 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,A
T)is assumed to have a joint density with respect
to the Lebesgue measure. Then, there exists at least one twin f(ST,A
T)with price
c:= c0(f(ST,A
T)) having the same joint distribution Gwith AT.
This theorem does not cover the important case where STis playing the role of
the benchmark (because (ST,S
T) has no density). This interesting case is considered
in the next theorem.
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,
where (ST,S
t)with 0<t<T is assumed to have a joint density with respect to the
Lebesgue measure. Then, for any 0<t<T there exists at least one twin f(St,S
T)
with price c=c0(f(St,S
T)) having a joint distribution Gwith ST. For example, for
some t∈(0,T),
f(St,S
T):=F−1
XT|ST(FSt|ST(St)).(10)
8

The proofs for Theorems 3.2 and 3.3 are in Appendix A.3 and A.4. In particular an
explicit construction of f(ST,A
T) can be found in the proof of Theorem 3.2. 
Theorems 3.2 and 3.3 essentially state that all investors who care about the joint
distribution of terminal wealth with some benchmark7only need to consider twins.
This result is a direct extension to Proposition 2.1. All other payoffs are useless in
the sense that they are not needed for these investors per se.
Note that in Theorem 3.3, tcan be chosen freely between 0 and T(strictly) and
hence there is an infinite number of twins f(St,S
T) all having the joint distribution G
with STand the same price8. The question then raises how to select the optimal one
among them. A natural possibility is to determine the optimal twin XT=f(St,S
T)
by imposing some additional criterion. For example, one could then define the best
twin XTas the one that minimizes
E(XT−AT)2.(11)
Because all marginal distributions are fixed, the criterion (11) is equivalent to max-
imizing the correlation between XTand AT. We use this criterion in one of the
applications.
3.2 Optimality of twins
Next we investigate the optimality of twins. The following main result of Section 3
extends Theorem 2.2 to the state-dependent case.
Theorem 3.4 (Cost optimality of twins).Assume that (ST,A
T)has joint density
with respect to the Lebesgue measure. Let Gbe a bivariate cumulative distribution
function. The following optimization problem
min
(XT,AT)∼Gc0(XT) (12)
has an almost surely unique solution X∗
Twhich is a twin of the form f(ST,A
T),almost
surely increasing in STconditionally on AT,and given by
X∗
T:= F−1
XT|AT(FST|AT(ST)).(13)
The proof of Theorem 3.4 is given 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 (13) that optimal payoffs become path-dependent and
conditionally (on AT) increasing in STWe 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)
7The benchmark can be ATwhere (AT,S
T) is continuously distributed as in Theorem 3.2 or it
can be STas in Theorem 3.3.
8To see this, recall that the joint distribution between the twin f(St,S
T)andSTis 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,S
T) with such a property have the same price E[ξTf(St,S
T)].
9

Corollary 3.5 (Cheapest twin).Assume that (ST,A
T)has joint density with respect
to the Lebesgue measure. Let Gbe a bivariate cumulative distribution function. Let
XTbe a payoff such that (XT,A
T)∼G.ThenXTis a cheapest payoff if and only if,
conditionally on AT,XTis (almost surely) increasing in ST.
The proof of Corollary 3.5 is given 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 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 ATand
ST, then we use Theorem 3.2 to construct a simpler contract which is “at least
as good”and 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 with
various examples. We start with the Asian option with fixed strike followed by the
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)+.(14)
Here Kdenotes the fixed strike and GTis the geometric average of stock prices from
0toTdefined as,
ln(GT):= 1
TT
0
ln (Ss)ds. (15)
We can now apply the material exposed above to design products that improve upon
YT.
10

Use of cost-efficiency payoff for investors with increasing law-invariant
preferences. By applying Theorem 2.2 to the payoff YT(14), 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+
(16)
where d=S1−1
√3
0e1
2−√1
3μ−σ2
2T. This is also the payoff of a power call option,
whose pricing is well-known. One obtains:
c0(Y∗
T)=S0e(1
√3−1)rT+( 1
2−1
√3)μT −σ2T
12 Φ(h1)−Ke−rT Φ(h2) (17)
where
h1=ln S0
K+(1
2−1
√3)μT +r
√3T+1
12 σ2T
σT
3
,h
2=h1−σT
3.
While the above results can also be found in Bernard et al. (2014) they are worth
considering here for the purpose of comparison with what follows. Note that letting
Kto zero provides the cost-efficient payoff equivalent to the geometric average GT.
A twin, 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,S
T)
such that
(ST,R
T(t)) ∼(ST,G
T).(18)
By definition, this twin preserves existing dependence between GTand ST.Compared
to the original contract, however, it is weakly path-dependent. Interestingly, both the
call option written on RT(t) and the call option written on GThave thus the same
joint distribution with ST. More formally, one has
ST,(RT(t)−K)+∼ST,(GT−K)+.(19)
(RT(t)−K)+is therefore a twin equivalent to the fixed strike (continuously moni-
tored) geometric Asian call (as of 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(20)
where tis freely chosen in (0,T). Details on how (10) becomes (20) are given in
Appendix B.1 9. Equality of joint distributions exposed in (19) implies that the call
option written on RT(t) has the same price as the original fixed strike (continuously
monitored) geometric Asian call (14). The time−0 price of both contracts is therefore
c0((RT(t)−K)+)=S0e−rT
2−σ2T
12 Φ( ˜
d1)−Ke−rT Φ( ˜
d2) (21)
where ˜
d1=ln(S0/K)+rT /2+σ2T/12
σ√T/3and ˜
d2=˜
d1−σT/3 (see Kemna and Vorst (1990)).
9The formula (20) is based on the expression (10) for a twin depending on Stand ST.Note
that there is no uniqueness. For example, 1 −FSt|ST(St) is also independent of STwe can thus
also consider HT(t):=F−1
XT|ST(1 −FSt|ST(St)) as suitable twin (0 <t<T) satisfying the joint
distribution as in (18). 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.
11

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)isgivenby
cov (ln (RT(t)) ,ln (GT)) = σ2
2T
2+√t√T−t
2√3
and, by construction of RT(t), standard deviations of ln (RT(t)) and ln (GT) are both
equal to σT
3. So maximizing the correlation coefficient is equivalent to maximizing
the covariance and thus of f(t)=(T−t)t. This is obtained for t∗=T
2and the
maximal correlation ρmax between ln(RT(t)) and ln(GT)is
ρmax =3
4+√3(T−t∗)t∗
4T=3
4+√3
8≈0.9665,
reflecting 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)+.(22)
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 approximated
numerically because the distribution of the difference between two lognormal distri-
butions is unknown.
If investors care about the dependence with ST, by applying Theorem 3.3, one
finds twins F−1
YT|ST(FSt|ST(St)) as functions of Stand ST.Similarly as in the previous
subsection, it is equal to
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+
.(23)
Details can be found in Appendix B.2.
Finally, if investors care about the dependence with GT, then it is possible to
construct a cheaper twin because the payoff (22) is not conditionally increasing in
STand therefore can strictly be improved using Theorem 3.4 and computing the
expression (13). The reason is that we can improve the payoff (22) by making it
cheaper while keeping dependence with GT.Hence, we invoke Theorem 3.4 to exhibit
another payoff XT=F−1
YT|GTFST|GT(ST)such that
(YT,G
T)∼(XT,G
T)
12

but so that XTis strictly cheaper. After some calculations, we find that XTwrites
as
XT=GT−aG3
T
ST+
(24)
where a=eμ−σ2
2T
2
S0.Details can be found in Appendix B.3.
Finally, one can easily assess to which extent the twin (24) 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−σT
3 (25)
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−σT
3
(26)
where d=σ2T
12 −μT
2
σ√T
3
which we need to compare numerically to (25). For example, when
μ=0.06,r=0.02,σ=0.3andT= 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 the formulas (25) and (26). The proofs for these are given in Appendix B.4.
5 Portfolio management
This section provides several contributions to portfolio management. We first derive
the optimal investment for an expected utility maximizer who has a constraint on the
dependence with a given benchmark. Next, we revisit optimal strategies for target
probability maximizers (see Browne (1999) and Cvitani`c and Spivak (1999)), and we
extend this problem in two different directions by adding dependence constraints and
by considering a random target. In both cases, we derive analytical solutions that
are given by twins.
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 a increasing utility function u(·). Assuming u(·)is
concave is equivalent to assuming investors are risk averse in the sense that for a
given budget they prefer a sure income above a random one with the same mean. In
13

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, u(·)is
strictly decreasing, limxau(x)=+∞and limxbu(x)=0.Consider the following
portfolio optimization problem
max
E[ξTXT]=W0
E[u(XT)]
The optimal solution to this problem is given by
X∗
T=[u]−1(λξT) (27)
where λis such that EξT[u]−1λX∗
T=W0.
Note that the optimal EUT payoff X∗
Tis decreasing in ξTand thus increasing in
ST(illustration of the results in Section 2), also pointing out the lack of protection of
optimal portfolios when markets go down. To account for this, we give the investor
the opportunity to maintain a desired dependence with a benchmark portfolio (for
example representing the financial market). This extends earlier results on expected
utility maximization with constraints, for example by Brennan and Solanki (1981),
Brennan and Schwartz (1989), He and Pearson (1991a),(1991b), Basak (1995), Gross-
man and Zhou (1996), Sorensen (1999) and Jensen and Sorensen (2001). They mostly
study 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 different 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)] (28)
where ATis a benchmark (which could be for instance the portfolio of another man-
ager in the market). In Theorem 2.1 (page 327) of Boyle and Tian (2007), the optimal
contract X∗
Tis derived explicitly (under some regularity conditions ensuring feasibility
of the stated problem) and it is an optimal twin.
This also follows from our results. Assume that the solution to (28) exists, and
denote it by X∗
T.ThenletGbe the bivariate cdf of (X∗
T,A
T). The cheapest way to
preserve this joint bivariate cdf is obtained by a twin f(AT,S
T) 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.
With a similar reasoning, this result also holds when there are several probability
constraints involving the joint distribution of terminal portfolio value XTand the
14

benchmark AT. The next theorem extends Theorem 5.1 and the above literature by
considering an expected utility maximization problem where the investor is fixing the
dependence with a benchmark, entirely. This amounts to specifying upfront the joint
copula of (XT,A
T). Assume that the copula between XTand ATis specified to be
C, i.e. C(XT,AT)=C. Let us denote by C1|ATthe conditional distribution of the first
component given AT(or equivalently given FAT(AT)).
Theorem 5.2 makes use of the projection on the convex cone
M↓:= {f∈L2[0,1]; fdecreasing},(29)
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 ϕ=πM↓(ϕ) the projection
of ϕon M↓.ϕcan be interpreted as the best approximation of ϕby a decreasing
function for the || · ||2norm.
Theorem 5.2 (Optimal payoff in EUT with dependence constraint).Given a utility
function u(·)as in Theorem 5.1, consider the following portfolio optimization problem
max
c0(XT)=W0
C(XT,AT)=C
E(u(XT)) .(30)
Assume that (AT,S
T)has a joint density. With UT=FST|AT(ST)and ZT=C−1
1|AT(UT),
let HT=E(ξT|ZT)=ϕ(ZT), and define 
HT=ϕ(ZT),whereϕbe the projection of ϕ
on M↓. Then the solution to the optimization problem (30) is given by

XT=[u]−1λ
HT(31)
where λis such that EξT[u]−1λ
HT=W0.
The proof of Theorem 5.2 is given in Appendix C.1. 
Remark 5.3.InthecasethatHT=E(ξT|ZT) is decreasing in ZT,weobtainas
solution to (30)

XT=[u]−1(λHT).(32)
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 finding the optimal element in this class.
Remark 5.4.The determination of the isotonic approximation ϕof ϕis a well-studied
problem (see Theorem 1.1 in Barlow et al. (1972)). ϕis the slope of the smallest
concave majorant SCM(ϕ)ofϕ, i.e. ϕ=(SCM(ϕ)). In Barlow et al. (1972) the
projection on M↑is given as slope of the greatest convex minorant GCM (ϕ)ofϕ.
Fast algorithms are known to determine ϕ.
15

Example
We illustrate Theorem 5.2 by an example. Let W0denote the initial wealth and let
AT=Stfor some (t<T). Consider for Cthe Gaussian copula with correlation
coefficient ρ∈−1−t
T,1.Then we find that

XT=[u]−1λe−θρ√t+√(1−ρ2)(T−t)WT(33)
where λis such that EξT
XT=W0and WTis a function of STand Stgiven by
WT=1−ρ2ln(ST
St )−(μ−σ2
2)(T−t)
σ√T−t+ρln( St
S0)−(μ−σ2
2)t
σ√t.
The proof of (33) can be found in Appendix C.2.
5.2 Target probability maximization
Target probability maximizers are investors who, for a given budget (initial wealth)
and 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 shows that it is optimal to buy a digital option
written on the risky asset. We show that their results follow from Theorem 2.2 in a
more straightforward way.
Proposition 5.5 (Browne’s original problem).Let W0be the initial wealth and let
b>W
0erT be the desired target10. The solution to the following target probability
maximization problem
max
XT0,c0(XT)=W0
P[XTb]
is given by the payoff
X∗
T=b1{ST>λ}
where λis given by E(ξTX∗
T)=W0.
The proof of this proposition is given in Appendix C.3. In a Black–Scholes market
one easily verifies that λ=exp(r−σ2
2)T−σ√TΦ−1W0erT
b.
A target probability maximizing strategy is essentially an all-or-nothing strategy.
Intuitively, investors would perhaps not be so attracted by the design of the optimal
payoff maximizing the probability to beat a fixed target. The obtained wealth solely
depends on the ultimate value of the underlying risky asset making it highly depen-
dent on final market behavior, and thus prone to unexpected and brutal changes
in the value of the underlying. A first extension concerns the case of a stochastic
benchmark, so that preferences become state-dependent.
10If bW0erT then the problem is not interesting since an investment in the risk-free asset allows
reaching a 100% chance to beat the target b.
16

Theorem 5.6 (Target probability maximization with a random target).Let W0be
the initial wealth. Consider the following portfolio optimization problem,
max
XT0,c0(XT)=W0
P[XTAT]
Let ATbe a random target such that (AT,S
T)has a density. The solution to the
problem is given by
X∗
T=AT1{ATξT<λ}
where λis implicitly given by EξTAT1{ATξT<λ}=W0.
The proof of this proposition is given in Appendix C.4. 
A second extension assumes a fixed dependence with a benchmark in the financial
market. We now consider the problem of an investor who for a given budget aims
at maximizing the probability that the final wealth achieves some fixed target while
preserving a certain dependence with a benchmark.
Theorem 5.7 (Target probability maximization with a random benchmark).Let W0
be the initial wealth and b>W
0erT the desired target for final wealth. Consider the
problem,
max
XT0,c0(XT)=W0,
C(XT,AT)=C
P[XTb]
Assume the pair (AT,S
T)has a density and define as in Theorem 5.2
UT=FST|AT(ST)and ZT=C−1
1|AT(UT).
Then the solution to the optimization problem is given by
X∗
T=b1{ZT>λ}
where λis determined by bP(ZT>λ)=W0.
The proof of this result is given in Appendix C.5. 
Example The result in Theorem 5.7 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},(34)
with α=T−t
t(1−ρ2)ρ−1 and ln(λ)=r−σ2
2(αt +T)−σ√kΦ−1W0erT
bwith
k=(α+1)
2t+(T−t). The proof for this example is given in Appendix C.6.
17

6 Final remarks
In this paper we characterize optimal strategies for investors who target a known
wealth distribution at maturity (as in the traditional setting) and additionally aim at
maintaining a desired interaction with a random benchmark. We show that optimal
contracts are weakly path-dependent (depending at most on two underlying assets)
and we are able to characterize them explicitly. Throughout the paper we have as-
sumed that the state-price 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
to still provide explicit representations of optimal payoffs. However, the optimality is
then no longer related to path-independence properties.
In a Black–Scholes market the state-price ξtis the inverse of the value of a unit
investment in a constant-mix strategy at time t, where a fraction θ
σis invested in the
risky asset and the remaining fraction 1 −θ
σin the bank account. Indeed the value
process (S∗
t)tof such strategy evolves according to
dS∗
t
S∗
t
=θ
σμ+1−θ
σrdt +θdZt.
Hence, for S∗
0= 1 one finds that S∗
t=eθ
σμt+(1−θ
σ)rt−θ2
2t+θZtand ξt=1/S*
t.Itiseasy
to prove that this strategy is optimal for an expected log-utility maximizer and in
the literature it is also referred to as the growth-optimal portfolio11 (GOP). 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 growth-optimal portfolio. Hence our results
are also valid in their setting where now the growth optimal portfolio is taken as the
reference. Path-independence properties will then be expressed in terms of the GOP.
The case of multidimensional markets described by a price process (S(1)
t,...,S(d)
t)t
is essentially included in the results given in this paper assuming that the state-
price 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. 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). Dealing with this case in more concrete form and establishing the
results in this paper under weaker regularity assumptions are left for future research.
11Its origins can be traced back to Kelly (1956).
18

AProofs
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)E[XY ]EF−1
X(U)F−1
Y(U).(35)
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.DenoteV=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,
c
−∞
ea+byfY(y)dy =ea+bE(Y)+ b2
2var (Y)1
2πvar(Y)c
−∞
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. 
19

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)andξTare 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),c
0(Z∗
T)].
If c=c0(X∗
T)thenX∗
Tis a solution. Similarly, if c=c0(Z∗
T)thenZ∗
Tis a solution.
Next, let c∈(c0(X∗
T),c
0(Z∗
T)) and define the payoff fa(ST) with a∈R,
fa(ST)=F−1[(1 −FST(ST))1STa+(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 (5) 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.Considernext
apayoffXTand note that F−1
XT|AT(U)∼XT.We find that
c0(XT)=E[XTξT]=E[E[XTξT|AT]]
EEF−1
XT|AT(U)ξTAT=EF−1
XT|AT(U)ξT,(36)
where the inequality follows from the fact that F−1
XT|AT(U)andξTare conditionally
(on AT) anti-monotonic and using (35) in Lemma A.1 for the conditional expectation
(conditionally on AT). Similarly, one finds that
c0(XT)EF−1
XT|AT(1 −U)ξT.
20

Next we define the uniform (0,1) distributed variable,
ga(ST)=(1−FST(ST))1STa+(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,A
T)givenas
fa(ST,A
T)=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,A
T)andZ∗
T=f1(ST,A
T) almost
surely. The same discussion as in the proof of Proposition 2.1 applies here. When
c=c0(X∗
T)thenX∗
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),c
0(Z∗
T))
then the continuity of c0(fa(ST,A
T)) with respect to aensures that there exists a*
such that c:= c0(fa*(ST,A
T)). Thus, fa*(ST,A
T) is a twin with the desired joint
distribution Gwith ATandwithcostc. 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,S
T)begivenas
f(St,S
T):=F−1
XT|ST(FSt|ST(St)).
Using Lemma A.2 again, one finds that (f(St,S
T),S
T)∼(XT,S
T)∼G. This also
implies,
c0(f(St,S
T)) = E[f(St,S
T)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.Letthe
twin X∗
Tbe given as
X∗
T=F−1
XT|AT(U).
Invoking Lemma A.2 again, (X∗
T,A
T)∼(XT,A
T)∼G. Moreover,
c0(XT)=E[XTξT]=E[E[XTξT|AT]]
EEF−1
XT|AT(U)ξTAT
=EF−1
XT|AT(U)ξT=c0(X∗
T)
where the inequality follows from the fact that F−1
XT|AT(U)andSTare conditionally
(on AT) comonotonic and using (35) in Lemma A.1 for the conditional expectation
(conditionally on AT). 
21

A.6 Proof of Corollary 3.5
LetusfirstassumethatXTis a cheapest twin. By Theorem 3.4, XTis (almost
surely) equal to X∗
Tas defined by (13) which is, conditionally on AT, increasing
in ST. Reciprocally, we now assume that XT=f(ST,A
T) 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 (12) and thus a cheapest twin. 
B Security design
B.1 Twin of the fixed strike (continuously monitored) geo-
metric Asian call option
Expression (10) allows us to find twins satisfying the constraint (18) 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,σ
2t1−t
T,
and thus
FSt|ST(St)=Φ⎛
⎜
⎜
⎜
⎝
ln StS
t
T−1
0
S
t
T
T
σtT −t2
T
⎞
⎟
⎟
⎟
⎠.
Furthermore, the couple (ln (GT),ln (ST)) is bivariate normally distributed with mean
and variance for the marginals 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.Forthe
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 ,(37)
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 (20) is then straightforward to derive.
22

For choosing a specific twin among others, we suggest to maximize ρ(ln RT(t),ln GT).
First, we calculate,
cov ln ST,1
TT
0
ln (Ss)ds=1
TT
0
cov (ln ST,ln (Ss)) ds
=σ2
TT
0
(s∧T)ds =σ2T
2.
Furthermore, by denoting a=1
2−1
2√3T−t
t,b=T
t
1
2√3t
T−tand c=1
2−1
2√3t
T−t,
equation (20) 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
TT
0
ln (Ss)ds+ccov ln ST,1
TT
0
ln (Ss)ds
=b1
TT
0
cov (ln St,ln Ss)ds +cσ2T
2
=bσ2
TT
0
(s∧t)ds +cσ2T
2=bσ2
Tt
0
sds +T
t
tds+cσ2T
2
=bσ2
Tt2
2+t(T−t)+cσ2T
2=bσ2t
TT−t
2+cσ2T
2
=T
t
1
2√3t
T−t
σ2t
TT−t
2+1
2−1
2√3t
T−tσ2T
2
=σ2
2√3t
T−tT−t
2+1
2−1
2√3t
T−tσ2T
2
=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
=
σ2
2T
2+√t√T−t
2√3
σ2T
3
=3
4+√3(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 (37) that,
ln(GT)|ln(ST)∼Nln S1
2
0S1
2
T,σ2T
12 .
23

Therefore YT=(GT−ST)+has the following conditional cdf
P(YTy|ST=s)=Φ⎛
⎝ln(s+y)−ln S1/2
0s1/2
σ√T
2√3⎞
⎠1y0
Then
F−1
YT|ST(z)=S1
2
0S1
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 (23). 
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
S1
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
⎞
⎟
⎟
⎟
⎠.(38)
Furthermore, YT=(GT−ST)+has the following conditional cdf,
P(YTy|GT=g)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1ifyg,
Φ⎛
⎜
⎜
⎝
ln⎛
⎝
g3/2
S
1
2
0
⎞
⎠+1
4μ−σ2
2T−ln(g−y)
σ√T
2
⎞
⎟
⎟
⎠if 0 yg,
0ify<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 (38) for FST|GT(ST)) derived above, then gives rise to
expression (24). 
B.4 Derivation of the prices (25) and (26)
Price (25)
Let us observe that,
(GT−ST)+=GT1−ST
GT+=S0eY1−eZ+,
24

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
2var Q(Y|Z)−eEQ(Y|Z)+ 1
2var Q(Y|Z)+Z+.
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)var
Q(Y)=3
4
σ2T
3=σ2T
4.
Hence,
EQ(GT−ST)+=S0EQerT
4+1
2Z−erT
4+3
2Z+
=S00
−∞
erT
4+1
2ZfZ(z)dz −S00
−∞
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 QNr−σ2
2T
2+σ2T
6,σ2T
30
−S0erT QNr−σ2
2T
2+σ2T
2,σ2T
30
=S0erT
2−σ2T
12 Φ⎛
⎝−r−σ2
2T
2−σ2T
6
σ2T
3
⎞
⎠−S0erT Φ⎛
⎝−r−σ2
2T
2−σ2T
2
σ2T
3
⎞
⎠
Choose f=−rT
2+σ2T
12
σ√T
3
to obtain (25).
Price (26)
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
2var Q(Y|Z)−ceEQ(Y|Z)+ 1
2var Q(Y|Z)+Z+.
25

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+
=S0ln(c)
−∞
erT
2−σ2T
8+1
2ZfZ(z)dz −S0cln(c)
−∞
erT
2−σ2T
8+3
2ZfZ(z)dz,
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, we find,
EQGT−aG3
T
ST+
=S0QNσ2T
6,σ2T
3−ln(c)exp rT
2−σ2T
12 
−c.S0QNσ2T
2,σ2T
3−ln(c)exp rT
2+σ2T
4
=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 ϕdenote the projection of ϕon the cone M↓
defined as in (29) with respect to L2(λ[0,1]). Then we define 
XTand k(·)by
u(
XT):=λϕ(ZT),
i.e. 
XT=[u]−1(λϕ(ZT)) =: k(ZT) with λsuch that EξT
XT=E[ϕ(ZT)k(ZT)] =
1
0ϕ(t)k(t)dt =ϕ·k=W0.
By definition, 
XTis increasing in ZTsince [u]−1is decreasing and ϕis decreasing
(it belongs to M↓). As a consequence 
XTis increasing in ST, conditionally on AT.
For any YT=h(ZT) with a increasing function h,wehavebyconcavityofu
u(YT)−u(
Xt)u(
XT)(YT−
XT)=λϕ(ZT)(h(ZT)−k(ZT)).
Thus, we obtain
E[u(YT)] −E[u(
XT)] λ1
0ϕ(t)(h(t)−k(t))dt =λϕ·(h−Ψ( ϕ)),(39)
26

where Ψ( ϕ)=[u]−1(λϕ)=kis increasing and Ψ(t)=[u]−1(λt) is decreasing.
Now we use some properties of isotonic approximations (see Barlow et al. (1972))
and obtain
ϕ·(h−Ψ( ϕ)) = ϕ·((−Ψ)( ϕ)−(−h))
=ϕ·(−Ψ)(ϕ)−ϕ·(−h)(cf. Theorem 1.7 in Barlow et al. (1972))
=ϕ·(−h)−ϕ·(−h)both claims have price W0
=(ϕ−ϕ)·(−h)0
by the projection equation (cf. Theorem 7.8 in Barlow et al. (1972)) using that
−h∈M↓. As a result we obtain from (39) that
E[u(YT)] E[u(
XT)],
i.e. 
XTis an optimal claim. 
C.2 Proof of formula (33)
We apply Theorem 5.2. Let W0denote the initial wealth and let AT=Stfor some
(t<T). From Lemma A.3 we find,
ln(ST)|ln(St)∼Nln ST
St−μ−σ2
2(T−t),σ
2(T−t).
Hence,
FST|St(ST)=Φ⎛
⎝ln ST
St−μ−σ2
2(T−t)
σ√T−t⎞
⎠.(40)
Since Cis a Gaussian copula,
C1|St(x)=⎡
⎢
⎢
⎢
⎣
Φ−1[x]−ρlnSt
S0−μ−σ2
2t
σ√t
1−ρ2⎤
⎥
⎥
⎥
⎦.
This implies,
ZT=C−1
1|St(FST|St(ST)) = Φ [WT],
where WTis a function of STand Stgiven by
WT=1−ρ2⎛
⎝ln ST
St −μ−σ2
2(T−t)
σ√T−t⎞
⎠+ρ⎛
⎝ln St
S0−μ−σ2
2t
σ√t⎞
⎠.
27

Recall that from (4), ξT=αTST
S0−βwhere αT=expθ
σμ−σ2
2T−r+θ2
2T,
β=θ
σ,θ=μ−r
σ. It follows that
HT=E(ξT|ZT)=E(ξT|WT)=δe−βcov(ln(ST),WT),
for some δ>0.Hence we find that
HT=δe−θρ√t+√(1−ρ2)(T−t)WT.
Note that the conditions on the correlation coefficient imply that HTis decreasing in
WTand thus HTis decreasing in ZT.The optimal contract thus writes as
X∗
T:= [u]−1λe−θρ√t+√(1−ρ2)(T−t)WT(41)
where λis such that E[ξTX∗
T]=W0.
C.3 Proof of Proposition 5.5
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}.Weshow
that P(ST∈A0∪A1) = 1 must hold. Assume P(ST∈A0∪A1)<1sothat
P(ST∈A2∪A3)>0.Define
Y=⎧
⎨
⎩
f∗(ST)forST∈A0∪A1,
0forST∈A2,
bfor ST∈A3.
Then we observe that Y=f∗(ST)onA0∪A1and Y<f
∗(ST)onA2∪A3.Since
P(ST∈A2∪A3)>0 also Q(ST∈A2∪A3)>0 because Pand the risk neutral
probability Qare equivalent. Hence c0(Y)<W
0.Next we define Z=b1ST∈C+Y
wherewehavechosenC⊆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 5.6
The target probability maximization problem is given by
max
XT0,c0(XT)=W0
P[XTAT].
Assume that there exists an optimal solution X∗
Tto this optimization problem. There
are three steps in the proof.
28

1. The optimal payoff is of the form f(ST,A
T).
2. The optimal payoff is of the form AT1h(ST,AT)∈A.
3. The optimal payoff is of the form AT1ATξT<λ∗for λ∗>0.
Step 1: We o bserve t hat X∗
Thas some joint distribution Gwith AT.Theorem 3.2
implies there exists a twin f(AT,S
T) such that (f(AT,S
T),A
T)∼(X∗
T,A
T)∼Gand
c0(f(AT,S
T)) = c0(X∗
T)=W0. Therefore P(f(AT,S
T)AT)=P(X∗
TAT)and
P(f(AT,S
T)0) = P(X∗
T0) = 1. Thus f(AT,S
T) is also an optimal solution.
Step 2: This is similar to the proof of Proposition 5.5, applied conditionally on
AT.Define the sets A0={s, f (AT,s)=0},A1={s, f (AT,s)=AT},then P(ST∈
A0∪A1|AT) = 1 and therefore P(ST∈A0∪A1) = 1. Thus there exists a set Aand
a function hsuch that
f(AT,S
T)=AT1h(ST,AT)∈A.
Step 3: Define λ>0 such that
P(h(ST,A
T)∈A)=P(ATξT<λ).
Observe that 1h(ST,AT)∈Aand 1ATξT<λ have the same distribution and that in addition,
ATξTis anti-monotonic with 1ATξT<λ. Therefore by applying Lemma A.1 one has that
c0(AT1ATξT<λ)=E[ATξT1ATξT<λ]E[ATξT1h(ST,AT)∈A]
and therefore the optimum must be of the form AT1ATξT<λ∗where λ∗>λis deter-
mined such that c0(AT1ATξT<λ∗)=W0.
C.5 Proof of Theorem 5.7
The target probability maximization problem is given by
max
XT0,c
0(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,A
T).
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.
29

Step 1: We o bserve t hat X∗
Thas some joint distribution Gwith AT.Theorem 3.2
implies there exists a twin f(ST,A
T) such that (f(ST,A
T),A
T)∼(X∗
T,A
T)∼G
and c0(f(ST,A
T)) = c0(X∗
T)=W0. Therefore P(f(ST,A
T)b)=P(X∗
Tb)and
P(f(ST,A
T)0) = P(X∗
T0) = 1. Thus f(ST,A
T) is also an optimal solution.
Step 2: This is similar to the proof of Proposition 5.5. 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,A
T)=b1h(ST,AT)∈A.
Step 3: Define λ>0 such that
P(h(ST,A
T)∈A)=P(ZT>λ).
Observe that b1h(ST,AT)∈Aand b1ZT>λ have the same joint distribution G with distri-
bution AT. Therefore, Theorem 3.4 shows that,
c0(b1ZT>λ)c0(b1h(ST,AT)∈A).
Hence, b1ZT>λ∗where λ∗such that c0(b1ZT>λ∗)=W0is the optimum. 
C.6 Proof of formula (34)
We know th a t b1ZT>λ∗where λ∗is such that c0(b1ZT>λ∗)=W0is the optimal solution.
We find that (see also the proof for (33)),
ZT=C−1
1|St(FST|St(ST))
=Φ
⎡
⎣1−ρ2⎛
⎝ln ST
St −μ−σ2
2(T−t)
σ√T−t⎞
⎠+ρ⎛
⎝ln St
S0−μ−σ2
2t
σ√t⎞
⎠⎤
⎦.
It is then straightforward that X∗
T=b1{Sα
tST>λ∗}is the optimal solution, with αand
λgiven by
α='T−t
t(1 −ρ2)ρ−1
λ=exp
r−σ2
2(αt +T)−σ(α+1)
2t+(T−t)Φ−1W0erT
b.

30

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