T - Prism - Université Paris 1 Panthéon-Sorbonne

Cahiers de Recherche PRISM-Sorbonne
Pôle de Recherche Interdisciplinaire en Sciences du Management
The Traditional Hedging Model Revisited
with a Non Observable Convenience Yield
Constantin MELLIOS
Professeur, Université Paris 1 Panthéon-Sorbonne PRISM
Pierre SIX
Rouen Business School
CR-10-34
PRISM-Sorbonne
Pôle de Recherche Interdisciplinaire en Sciences du Management
UFR de Gestion et Economie d’Entreprise – Université Paris 1 Panthéon-Sorbonne
17, rue de la Sorbonne - 75231 Paris Cedex 05 http://prism.univ-paris1.fr/

Cahiers de Recherche PRISM-Sorbonne 10-34
The Traditional Hedging Model Revisited with a Non
Observable Convenience Yield
Constantin Mellios a , Pierre Six b, ∗
___________________________________________________________________
Abstract :This article addresses the issue of hedging a constrained position in the spot
(storable) commodity with futures contracts when, in particular, the convenience yield is not
observable and is estimated by using the continuous-time Kalman-Bucy method. We extend
the relevant literature when the investors operate under incomplete information and study its
impact on optimal demands. The latter depend crucially on the investor’s initial beliefs. The
speculative and mean-variance positions in the futures contract are the unique positions
capturing the effect of the incomplete information and are strongly affected by the initial
value of the estimation error. We achieve a decomposition allowing investors to asses the
impact of both the state variables and the initial estimation error on optimal demands.
Finally, a higher initial value of the estimation error exacerbates the Samuelson effect.
JEL Classification: G11; G12; G13
Keywords: Incomplete Information; Optimal Dynamic demand; Convenience Yield; Commodity
Futures Prices; Market Prices Of Risk; Interest Rates.
___________________________________________________________________
__________________________
a
University of Paris 1 Panthéon-Sorbonne, PRISM, 17, place de la Sorbonne, 75231 Paris Cedex 05, France.
Tel: + 33(0)140462807; fax: + 33(0)40463366. E-mail address: constantin.mellios@univ-paris1.fr
b
Rouen Busines School, Economics and Finance department, 1, rue du Maréchal Juin, 76825 Mont Saint Aignan Cedex.
Tel.: +33(0)232821718. E-mail: pierre.six@rouenbs.fr.
This paper received the Outstanding Derivatives Paper Award, Eastern Finance Association, Miami, 2010.
0

1. Introduction
In recent years, storable commodity prices’ sharp rises have been followed by abrupt decreases.
Consequently, investors seek ways to hedge the risk associated with the fluctuations of these prices.
Commodity futures contracts are well-adapted instruments to hedge this risk the more so as there exist
numerous liquid contracts written on a wide range of commodities. This topic is therefore of great
interest for academics as well as for practitioners. This paper presents a continuous-time model to
determine optimal demands for risky assets by a constrained investor 1 with a fixed position in the spot
commodity, who uses futures contracts as hedging instruments. It allows one to revisit the Traditional
Hedging Model (THM hereafter) (Adler and Detemple, 1988b) by studying the specific case of
commodities characterized by an unobservable convenience yield.
Storable commodities (agricultural, metals, energy etc.) differ from other conventional assets
as they are produced and consumed. In particular, holding commodity inventories provide some
services by avoiding interruptions (in) and guarantying the continuity of the production process. The
net flow of these services, when expressed as a rate, is called the convenience yield defined by
Brennan (1991) as “the flow of services accruing to the owner of the physical inventory, but not to the
owner of a contract of a future delivery”. A parallel can be drawn between the convenience yield and
the dividend on a stock. However, the convenience yield is an abstract concept, non-observed in the
market. For pricing and hedging purposes, therefore, the non-observability of the convenience yield,
as is widely recognized in the literature (see, for instance, Schwartz, 1997), adds an extra difficulty.
One way to circumvent this difficulty is the cost-of-carry model. It is at the heart of the theory of
storage (Kaldor, 1939; Working, 1948; Brennan, 1958) and established a no-arbitrage relation between
futures prices, spot commodity prices, the interest rate and the net convenience yield. Gibson and
Schwartz (1990) suggested to compute the implied convenience yield by using the cost-of-carry
formula. Unfortunately, Carmona and Ludkovski (2004 a) showed that not only the implied
convenience yield is not consistent with the forward curve, but also very different values of the
1 A constrained investor cannot freely trade on the underlying spot asset.
1

convenience yield are obtained for different futures contracts maturities. To remedy this flaw, our
objective in this paper is to take directly into account the unobservable character of the convenience
yield.
Many models examined the dynamic asset allocation with futures contracts (see, among
others, Ho, 1984; Stulz, 1984; Adler and Detemple, 1988a, b; Duffie and Jackson, 1990; Briys et al.,
1990; Duffie and Richardson, 1991; Duffie and Stanton, 1992; Lioui and Poncet, 2001). The investor’s
optimal futures demand consists of three terms (see Merton, 1973; Breeden, 1979): a mean-variance
speculative term, a pure hedge minimum-variance element related to the non-traded position and à la
Merton-Breeden hedging components reflecting how to hedge against a stochastic opportunity set. The
number of the Merton-Breeden hedging terms is equal to the number of the unspecified state variables
included in the investment opportunity set.
Although these models have rigorously tackled optimal demands with futures contracts, they
suffer, with regard to commodities, from two drawbacks. First, the random behavior of the
convenience yield, resulting in a stochastic investment set, is not modelled 2 . Yet for non-myopic
investors a Merton-Breeden term is neglected. Moreover, empirical studies (see, for instance, Fama
and French, 1987, 1988; Besembinder and Chan, 1992; Khan et al., 2007; Hong and Yogo, 2009) have
shown that the convenience yield is a crucial variable in predicting commodities returns. Second, the
models above assume that the variables describing the investment opportunity set are observable.
However, the convenience yield is not observable and investors operate under incomplete information.
Dothan and Feldman (1986), Detemple (1986), Gennotte (1986), Feldman (1989), Xia (2001) and
Lundofte (2006) among others, investigated, in a dynamic framework, the optimal asset allocation in a
partially observable economy. To the best of our knowledge, the case of hedging with futures
contracts with an unobservable convenience yield has not been examined in the relevant literature yet 3 .
2 A few exceptions explicitly modeling the random behavior of the convenience yield, but in a fully observable economy and
in different contexts, are Hong (2001) and Bertus et al. (2009).
3 Carmona and Ludkovski (2004 b) address the pricing of commodity derivatives with a partially observable convenience
yield.
2

Following Schwartz (1997), model 3, and Casassus and Collin-Dufresne (2005), three
imperfectly correlated state variables - the spot commodity price, the instantaneous riskless rate and
the convenience yield - are supposed to explain the dynamics of the futures price. In addition,
according to the empirical evidence on predictability, risk-premiums in commodities are time-varying
(see, for example, Liu, 2007). Market prices of risk depend on the state variables and are hence
stochastic. As the convenience yield is not observable, investors estimate it by observing the spot price
and the short-rate and by using the continuous-time Kalman-Bucy method. The three state variables
follow a (conditionally) Gaussian distribution and the investor’s initial beliefs about the true value of
the convenience yield are normally distributed. The estimation procedure has two main consequences 4 .
First, the (conditionally) Gaussian filter results in a fully observable Markovian market, where the
convenience yield is replaced by its estimate. It follows that futures prices and optimal demands are
functions of the estimate and of the estimation error or filtering error 5 . Especially, optimal demands
depend on the investor’s prior beliefs. Second, since the investor cannot trade on the spot commodity,
the market is dynamically incomplete. However, in the informationally equivalent economy, as the
space of the sources of uncertainty is reduced to two, the investor faces a dynamically complete
market.
To derive optimal demands for risky assets, an investor maximizes the expected constant
relative risk aversion (CRRA) utility function of his (her) final wealth by following the no-arbitrage
martingale approach (Karatzas, Lehoczky and Shreve, 1987; Cox and Huang, 1989). Inspired, for
instance, by Rodriguez (2002), Stoikov and Zariphopoulou (2005), Björk et al. (2008) and Detemple
and Rindisbacher (2009), we suggest an appropriate change of a martingale measure specific to the
CRRA utility function - the CRRA-forward probability measure 6 . Under this measure the Merton-
4 Interest readers should refer to Feldman (2007) who provides an in-depth discussion on incomplete information and on
these two points especially.
5 In the (conditionally) model framework, the conditional distribution of the unobserved variable given the information
provided by the observations is determined by a sufficient statistic: the conditional mean. The estimation error is
deterministic (Liptser and Shiryaev, 2001a,b; chapters 11,12).
6 In contrast to these papers, we operate a change of a martingale measure under which asset prices are martingales and not a
change of an equivalent measure.
3

Breeden terms reduce to two terms. The first, as in Lioui and Poncet (2001), Munk and
Sorensen (2004) and Detemple and Rindisbacher (2009), is associated with the risk of the short-rate
and hedges the changes in a discount bond with a maturity that matches the investor’s horizon. The
second addend, results from the stochastic prices of risk, and is couched in terms of an investor
specific discount bond as it depends on the investor’s risk aversion, horizon and prior beliefs 7 . This
decomposition allows us to give an economic interpretation to our results and to suggest new useful
expressions of optimal demands.
By isolating the orthogonal risk of the spot commodity, i.e. the part of its risk that is not
correlated with the risk of the short-rate, we are able both to derive the speculative and minimumvariance
proportions for each asset and to study the impact of the incomplete information. Our results
show that these components related to the futures contract capture the effect of the estimation error.
Those of the discount bond are affected only through the components of the futures contract. Contrary
to the traditional THM, the pure hedge term is investor dependent as it involves the investor’s initial
beliefs. The orthogonal risk presents another advantage. It leads to a decomposition of the investor
specific bond for each and every state variable. An investor may then accurately measure the impact of
the state variables on optimal demands. This impact is assessed through the sensitivity of the investor
specific bond on the state variables, which depends on the initial estimation error. As a consequence,
all these hedging elements are influenced by this error.
An illustration applied to the copper market shows that the initial estimation error has a much
stronger effect on the mean-variance and the minimum-variance demands than on the investor specific
bond demand. Moreover, the futures price volatility exhibits the Samuelson effect (this volatility
increases as the futures contract approaches its maturity date), which is amplified for a higher initial
value of the estimation error.
The remainder of the paper is organized as follows. Section 2 describes the economic
framework. Section 3 analyzes optimal demands. Section 4 provides an illustration of the results of
7 In the Detemple and Rindisbacher (2009) paper, this component is expressed in a less intuitive way as a function of the
density of the forward probability measure.
4

section 3 in the case of the copper market. Section 5 concludes and offers possible extensions. All
proofs are available from the authors upon request.
2. The economic framework
Consider a continuous-time frictionless economy. Uncertainty in the economy is described by a
complete filtered probability space ( , F , Θ,P)
filtration Θ ≡ { F t
: t ∈[ 0,
T ]}
, where [ ,T ]
Ω endowed with a (right) continuous non-decreasing
0 is a fixed time interval, T is the finite horizon of the
economy, and
F T
≡ F . P is the historical probability measure. All (continuous-time) processes
described below are assumed to be adapted. ( ) ′
* * *
correlated Brownian motions defined on ( , F , Θ,P)
* *
ρ ≡ dz dz dt , i ≠ j
ij
it
motions.
jt
with , { S,r,δ}
zSt
, zrt
, zδ t
, stands for a 3-dimensional vector of
Ω where ' denotes the transpose and
i ∈ is the constant correlation coefficient of the Brownian
In our model, the state variables that describe the economy are in the spirit of those of
Schwartz (1997) and Casassus and Collin-Dufresne (2005). There are three imperfectly correlated
variables: the spot commodity price,
yield, δ
t
. They are governed by the following stochastic processes:
S
t
, the instantaneous riskless rate, r
t
, and the (net) convenience
dSt
S
dr
t
t
*
[ r + σ λ ] dt + σ dz X ≡ ln( S ),
S ≡ > 0
+ δ dt =
S
(1a)
t
t
S
[ ( t)
− αr
] dt + σ dz
*
, r ≡ r
=
r
t
r rt 0
St
S
St, t
t 0
θ (1b)
[ θ −δ
] dt + σ dz
*
, δ ≡ δ
d δ
t
= κ
δ t
δ δt
0
(1c)
σ i
and λ i
(.) represent the constant strictly positive volatility of the state variables and the market price
of risk associated with the state variables, respectively.
λ =
0
+ + , λ
rt
= λr
0
+ λrrrt
and
St
λS
λSX
X
t
λS
δδt
λδ t
= λδ
0
+ λδδδ
( t)
. λ
S 0, λSX
, λSδ
, λr
0,
λrr
, λδ
0
and λδδ
are constants. The convenience yield and the spot
price are related through inventory decisions (Routledge, Seppi, and Spatt 2000). To take into account
this relation, (see, for instance, Brennan, 1958; Dincerler et al. 2005), the price of risk associated with
the (log) of the spot price process is an affine function of the level of both the (log) of the spot price
5

and the convenience yield. The prices of risk associated with the short-rate and the convenience yield
are affine functions of their own level.
The short rate follows a process such that the model incorporates the initial term structure of
interest rates (see Hull and White, 1990; Heath et al., 1992). In particular, its drift depends on a
2
deterministic function θ t)
{ αf
( t)
+ ∂ f ( t)
+ σ D
α
( t + σ λ }
describes the initial forward yield curve,
with respect to x and ( )
r
(
0 t 0 r 2
)
r r 0
2
y
= , where α is a constant, f ( )
∂
x
stands for the partial derivative (column gradient vector)
D u
denotes the deterministic function D u
( y)
( 1−
exp( 2uy)
)
0 t
1
= for two
2u
2
−
scalars u and y. Empirical studies (see Fama and French, 1988 ; Brennan, 1991) reported that the
convenience yield should be specified by a mean-reverting process. It follows then a mean-reverting
process with a constant long term mean, θ
δ
and a constant speed of mean reversion k.
The investor does not observe the convenience yield but draws inferences about it from his (her)
observations of the spot price of the commodity and the interest rate. It is assumed that (s)he views the
prior distribution of δ as Gaussian with given mean
m( 0) ≡ m and variance e( 0) ≡ e . The agent
estimates the unobserved stochastic process of the convenience yield {δ(t): t ∈ [0,T]} given noisy
observations of the joint processes {S(t), r(t): t ∈ [0,T]}. The agent utilizes the information available,
F
S , r
t
( S , r , u ≤ t)
= σ 0 ≤
u
u
, to derive the optimal estimate of the state variable, δ(t). The conditional
, r
mean is defined as follows: m(
t)
E[ δ ( t)
F ( t)
] S
≡ E [ δ ( t)
]
error) is given by ( e,
t)
= E ( δ ( t)
− m(
t)
)
= , and the estimation error (the mean square
2 , r
[ F ( t)
]
S
t
ε . m
t
and ε ( e,
t)
obey the following system of
differential equations (Liptser and Shiryayev, 2001 a, b, chapters 11-12) – detailed computation is
available from the authors upon request:
dm
t
⎡ κ ⎤ ⎡
⎤
Sδ
κ
Sδ
= κ[ θ − mt
] dt + ⎢σ
ρS
0
+ ρS
ε ( e,
t)
⎥dzSt
+ ⎢σ
ρr
0
+ ρr
ε ( e,
t)
⎥dz
(2b)
δ δ
ε
δ
ε
rt
⎣ σ
S ⎦ ⎣ σ
S ⎦
ε
∞
− ε
2
ε ( e,
t)
= ε
∞
−
(2c)
e −ε
2
1−
exp( ∆t)
e −ε
∞
6

where κ ≡ σ λ −1
∆ ≡ b 2 − 4ac ,
ρ − ρ ρ ρrδ
− ρSδ
ρSr
1
ρSr
ρ ≡ , ρr
0
≡ , ρ
2
S ε
≡ , ρ
2 r ε
≡ − and
2
− ρ
1− ρ
− ρ
Sδ
rδ
Sr
S δ S Sδ
,
S 0
2
1−
ρSr
− b + ∆
ε
2
≡ ,
2a
2 2
[ − ρ − ρ − 2ρ
ρ ρ ]
2
σ δ
1
S 0 r 0 Sr r 0 S 0
− b − ∆
ε
∞
≡
with
2a
1
Sr
c ≡ . The innovation processes, dz
St
and
2
Sr
1
Sr
⎡κ ⎤
S
a ≡ − δ
⎡ σ ⎤
⎢ ⎥ ,
δ
b ≡ −2 ⎢κ
+ κ
Sδ ρS
0 ⎥ and
⎣ σ
S ⎦ ⎣ σ
S ⎦
dz
rt
, are given by
*
dz
rt
= dz rt
and
dz
St
1
=
σ
S
⎡dSt
⎢
⎢⎣
St
⎡dSt
− E⎢
⎢⎣
St
F
S , r
t
⎤⎤
⎥⎥
⎥⎦
⎥⎦
and are observable Brownian motions (see Liptser and Shiryaev; 2001
a,b ). ε ≡ ε ( ∞,
e)
denotes the steady state value of the estimation error which does not depend on e.
∞
The system of equations (1a) to (1c) can be written under the equivalent fully observable
economy as follows:
dS t
+ mtdt
=
2
(3a)
S
dr
t
t
dm
[ rt
+ σ
S
( λS
0
+ λSX
X
t
+ λSδ mt
)] dt + σ
S
[ ρSrdzrt
+ 1−
ρSr
dzwt
]
[ θr
t − rt
] dt σ
rdzrt
= β ( ) +
(3b)
t
⎡
⎤
2 κ
Sδ
= κ[ θ − mt
] dt + σ ρr
dzrt
+ ⎢σ
1−
ρSr
ρS
+
( e,
t)
⎥
0
ε dz
(3c)
δ δ δ
δ
2
wt
⎢
⎣
σ
⎥
S
1−
ρSr
⎦
Liptser and Shiryaev (2001 a, b) shown that the information generated by the state variable
process in the original economy, {S(t), r(t), δ(t): t ∈ [0,T]}, is equivalent to the information generated
by the observable process {S(t), r(t), m(t): t ∈ [0,T]}. Thus, the investor makes his (her) financial
decisions as if (s)he faces the fully observable Markovian market given by (3a-c) (see Feldman, 2007).
This has an impact on the covariances between the state variables. Although the conditional
covariance between the short rate and the conditional mean is not modified, that between the (log) spot
price and the conditional mean is a function of the estimation error:
Cov
( dX ; dm ) σ σ ρ dt κ ε ( e,
t dt
= . This covariance has two addends. The first is equal to the
t t t S δ Sδ
+
Sδ
)
covariance in the original economy. To this addend, as a consequence of the estimation procedure, a
second component is added involving the estimation error. The impact of the latter on the covariance
depends on the sign of κ
S δ
≡ σ
SλSδ
−1. When κ S δ
> 0 , the filtering error has a positive effect and vice
versa.
7

It is worth pointing out that in the original economy three Brownian motions govern the
dynamics of the state variables, while in the informationally equivalent economy two sources of
uncertainty, the innovation processes, z
St
and z
rt
, determine the evolution of the three state variables.
For reasons that will become clear later, since the innovation processes are correlated,
dz
St
can be
written in the following manner:
dz
St
= ρ dz + 1−
ρ dz , where z
wt
is a Brownian motion
Sr
rt
2
Sr
wt
uncorrelated with z
rt
. It follows that, as can be seen in equation (3c), the effect of the estimation error
on the conditional mean is isolated and appears through z
wt
.
In order to obtain compact formulae, the equations above can be expressed in a matrix form.
Let Y [ X r m ] ′ , λ [ λ λ ] ′ and [ dz dz ] ′
t
=
t t t
t
=
rt wt
dz be the vectors of the state variables, the
t
=
rt wt
prices of risk and the Wiener processes respectively. Then:
dY
t
′
= µ , , (4a)
[ ( t)
− µ
YYt
] dt + σ
Y
( t e) dzt
λ = λ 0
+ λ Y , (4b)
t
Y
t
where
′
σ =
S
⎡ 1 2⎤
⎢σ
S
λs
0
− σ ⎥
µ = ⎢ βθ ( 2 S
( t)
) ⎥
r
t
⎢
⎥
⎢ κθ δ
⎥
⎢⎣
⎥⎦
2
[ σ
S
ρSr
σ
S
1−
ρSr
]
′
,
⎡−
σ λ −1 − κ
⎡ ′
⎤
σ ⎤
S SX
Sδ
S
⎢ ⎥
µ =
⎢
⎥ ′
Y
⎢
0 β 0
⎥
, ( ) ⎢
′
σ ⎥
Y
t,
e = σ
r
with
⎢ ⎥
⎢⎣
0 0 κ ⎥
′
⎦
⎢σ
m(
t,
e)
⎥
⎣ ⎦
, σ = [ 0]
, σ ( e,
t)
′
= [ σ ρ σ ( t,
e)
]
r
σ r
m
δ rδ
mw
⎡λr
0 ⎤
. λ
0
= ⎢ ⎥ and
⎣λw0
⎦
⎡ 0
λY
= ⎢
⎣λ
wX
λ
λ
rr
wr
0 ⎤
λ
⎥
wm ⎦
are such that
λ
wX
λ
ρSr
≡ , λwr
≡ − λ
2
rr
,
1− ρ
SX
2
1−
ρSr
Sr
λ
wm
λ
≡ and
Sδ
2
1−
ρSr
λ
w0
≡
λ
S 0
1−
ρ
2
Sr
−
ρ
Sr
1−
ρ
2
Sr
λ . We denote by
r 0
⎡
⎤
2 κ
Sδ
σ
mw(
t,
e)
≡ ⎢σ
δ
1−
ρSr
ρS
0
+ ε ( t,
e)
⎥ the
2
⎢⎣
σ
S
1−
ρSr
⎥⎦
sensitivity of the estimate of the convenience yield to
dz
wt
.
There are in the economy a locally riskless asset, the savings account, such that:
⎧ ⎫
= ⎨∫ t β ( t)
exp r(
s)
ds⎬
, with initial condition β ( 0) = 1, and three risky traded assets. The spot
⎩ 0 ⎭
commodity whose dynamics are given in equation (3a), a discount bond with maturity
T
B
, whose
8

price, at time t,
0 ≤ t ≤ TB
, is Bt
( r,
TB
) ≡ Bt
( TB
) , and a futures contract written on a commodity with
maturity
T
H
, whose price, at date t,
0 ≤ t ≤ T H
≤ T , is denoted H ( Y , e,
T ) ≡ H ( e,
T ). The price
B
t
t
H
t
H
dynamics of the last two securities can be written as follows:
dBt
( TB
)
= [ rt
−σ
B(
t,
TB
) λrt
] dt −σ
B(
t,
TB
) dzrt
, (5a)
B ( T )
t
dH
H
B
t
( e,
TH
)
( e,
T )
t
H
[ σ
Hw( t,
e,
TH
) λwt
+ σ
Hr
( t,
e,
TH
) λrt
] dt + σ
Hr
( t,
e,
TH
) dzrt
+ σ
Hw( t,
e,
TH
) dzwt
= , (5b)
where ( t,
T ) ≡ σ D ( T − t)
Hr
2
σ σ ( t , e,
T ) σ h ( t,
e,
T ) 1−
ρ + σ ( t,
e) h ( t,
e,
T )
B
B
r
α
B
( t e,
T ) σ h ( t,
e,
T ) ρ + σ h ( t,
e,
T ) σ δ
ρ h ( t,
e,
T )
σ ≡ +
are such that:
,
H S X H Sr r r H
rδ
m
Hw
H
≡ and
S
X
H
H
Sr
mw
m
H
⎡∂thX
⎢
⎢
∂thr
⎢⎣
∂thm
( t,
e,
TH
)
( t,
e,
TH
)
( t,
e,
T )
H
⎤ ⎡ 0
⎥ ⎢
⎥
=
⎢
−1
⎥⎦
⎢⎣
1
0
α
0
( t,
e,
TH
)
( t,
e,
TH
)
( t,
e,
T )
( TH
, e,
TH
) ⎤ ⎡1⎤
( T )
⎥
=
⎢
H
, e,
TH
⎥ ⎢
0
( ) ⎥ ⎢ ⎥ ⎥⎥ TH
, e,
TH
⎦ ⎣0⎦
σ
mw(
t,
e)
λwX
⎤⎡hX
⎤ ⎡hX
σ ( ) +
⎥⎢
⎥ ⎢
mw
t,
e λwr
σ ρr
λrr
⎥⎢
hr
⎥
,
⎢
h
. (5c)
δ δ
r
κ + σ ( , ) ⎥⎦
⎢⎣
⎥⎦
⎢
mw
t e λwm
hm
H ⎣hm
Moreover, we denote by σ ( t e,
T ) σ ( t,
e,
T ) 2 + σ ( t,
e,
T ) 2
contract.
H
,
H
Hw H Hr H
≡ the volatility of the futures
We consider the case of a constrained investor who is endowed with a fixed position θ
S
in the
spot commodity, which means that the investor cannot freely trade the spot commodity. In contrast,
(s)he can freely invest in the discount bond and the futures contract, and in the risk-free asset. We
denote by θ
βt
, θ
Bt
and θ
Ht
the units invested in the risk-free asset, the zero-coupon bond and the
futures contract, respectively. Investor’s time t wealth is then given by:
where
t
t
t
S
t
Bt
t
( TB
) M
t
W = θ β
β + θ S + θ B + . (6)
M
t
is the margin account, since the investor trades futures contracts. The expression of
given by (Duffie and Stanton, 1992):
M
t
=
t
∫
0
M
t
is
t
⎧ ⎫
exp⎨∫
rv
dv⎬θ HudHu
( TH
)
(7)
⎩ u ⎭
The investor, endowed with an initial wealth W(0), has an investment horizon
T I
< T
, a
constant relative risk aversion γ > 0 and seeks to maximize his (her) CRRA utility function of
terminal wealth. Under the complete information economy, there are two sources of uncertainty and,
9

as a consequence of the investor’s fixed position in the spot commodity, two risky traded securities.
Therefore, the constrained investor faces a dynamically complete financial market. It is worth pointing
out that in the original economy, there are three sources of uncertainty and two traded assets. As a
result, the market is not complete for the constrained investor. The market completeness is directly
related to incomplete information. The martingale approach for complete markets, developed by
Karatzas et al. (1987) and Cox and Huang (1989), can then be applied to determine the constrained
investor’s optimal asset allocation by solving the following static program:
1−γ
⎡W
⎤
TI
max Εt
⎢ ⎥
WTI
⎢⎣
1−
γ ⎥⎦
(8a)
s.t.
Wt
G
t
⎡W
= Et
⎢
⎢⎣
G
T
T
I
I
⎤
⎥
⎥⎦
(8b)
where [ ⋅ F ] ≡ [] .
Ε denotes the expectation, under P, conditional on the information, F t , available at
t
E t
time t and
t
t
= ⎧
= ⎨∫
+ ∫ + ∫
t
β ( t)
1 2 '
G( t)
exp ru
ds λu
λu
dz
ξ ( t)
⎩ 2
0 0
0
u
⎫
⎬, with G(0) = 1, represents the numéraire or
⎭
optimal growth portfolio such that the value of any admissible portfolio relative to this numéraire is a
martingale under P (see Long, 1990; Merton, 1990; Bajeux-Besnainou and Portait, 1997).
stands
for the norm in R 2 and ξ (t)
is the Radon-Nikodym derivative of the so-called, unique, risk-neutral
probability measure Q equivalent to the historical probability P , such that the relative price (with
respect to the savings account chosen as numéraire), of any risky security is a Q-martingale (see
Harrison and Pliska, 1981).
3 Optimal demands under incomplete information
Within the financial market described above, the solution to the static problem (8), which is a standard
Lagrangian optimization problem, allows us to derive the optimal wealth at time t:
where
W
t
* −1 γ
t t t
t
gγ
( T ) B ( T )
= k G B
, (9)
I
γt
I
gγ
[{ { G
} ]
k is a Lagrange multiplier, ( T ) E G B ( T )
B = is and ≡1− γ
γt
I
t
t
t
I
TI
g . ( )
γ
1
B γ
can be
t
T I
written as follows:
10

B
γt
gγ
[{ { G
} ]
( T ) = E G B ( T )
I
t
t
t
2
⎧ gγ
exp⎨−
⎩ 2
I
TI
where [ ] ′
B σ
t
∫
0
= B(0,
T )
λ −σ
( u,
T )
σ t,
T ) ≡ ( t,
T ) 0 .
(
I B I
u
B
gγ
I
I
2
2
⎡ ⎧ gγ
Et
⎢exp⎨−
⎢⎣
⎩ 2
du − g
γ
t
t
∫
λ −σ
( u,
T )
'
∫[ λu
−σ
B
( u,
TI
)]
0
0
u
B
I
2
⎫⎤
dz(
u)
⎬⎥
⎭⎥⎦
⎫
du⎬
⎭
The computation of the optimal wealth reduces to that of B γ t
( T I
). Its computation may be
simplified by making an appropriate change of numéraire. Jamishidian (1987, 1989) was the first to
use a discount bond as a numéraire and introduced the forward martingale measure equivalent to P.
This measure was applied to asset allocation by Lioui and Poncet (2001), Munk and Sorensen (2004)
and Detemple and Rindisbacher (2009). In the spirit of Rodriguez (2002), Stoikov and Zariphopoulou
(2005) and Björk et al. (2008), we operate a change of probability measure specific to CRRA utility
functions. The Radom-Nikodym derivative defines the probability measure
Geman et al., 1995):
( g γ )
P equivalent to P (see
( gγ
)
dP
dP
S , t
t
F
[ λu
−σ
B(
u,
TI
)]
2 t
t
⎪⎧ g
2
γ
′
≡ exp⎨−
∫ λu
−σ
B(
u,
TI
) du − gγ
∫
dz
⎪⎩
2
0
0
u
⎪⎫
⎬
⎪⎭
By using Baye’s rule for conditional expectations:
B
γt
( T )
I
T
( ) ⎡ ⎪⎧
I
⎪⎫
⎤
g g
2
γ γ
= Et
⎢exp⎨−
∫ λu
−σ
B ( u,
TI
) du⎬⎥
, (10)
⎢⎣
⎪⎩ 2γ
t
⎪⎭ ⎥⎦
where
( g
E ) γ
[] . stand for the expectation under
t
( g γ )
P conditional on
S r
F , t
.
B γ t
( T I
) is a function of the investor’s risk aversion coefficient, horizon, and initial beliefs. As
it has the functional form of a discount bond, it will be called the investor specific discount bond. Its
dependence on the investor’s initial beliefs distinguishes the expression of B γ t
( T I
) from that in a fully
observable economy. It is stochastic because of the stochastic character of the market prices of risk. It
is worth pointing out that B γ t
( T I
) is not a traded financial asset but it can be duplicated, in our
complete market, by existing traded securities. As the expectation in equation (10) involves a
quadratic function, following the literature of the term structure of interest rates (see Dai and
11

Singleton, 2000, 2002; Ahn et al., 2002), B γ t
( T I
) can be expressed as an exponential quadratic function
of the state variables:
⎧
' 1 ' ⎫
B
t
( TI
) = exp⎨b
( t,
e,
TI
) + b ( t,
e,
TI
) Yt
+ Yt
b ( t,
e,
TI
) Yt
⎬ , (11a)
γ 0γ
1γ
2γ
⎩
2
⎭
where b γ
( t,
e,
T ) is a deterministic function unnecessary to the hedging analysis and b γ
( t,
e,
T ) ,
0 I
b γ
( t,
e,
T ) are functions solving the following system of ordinary differential equations:
2 I
∂ b
2γ
( t,
e,
T ) − b ( t,
e,
T ) µ ( t,
e) − ( t,
e) b ( t,
e,
T ) + b ( t,
e,
T ) Σ ( t,
e) b ( t,
e,
T )
t 2γ I 2γ
I Yγ
µ
Yγ
− a
∂ b
− a
1γ
= 0
2x2
′
2γ
'
( t,
e,
TI
) − µ
Yγ
( t,
e) b1
γ
( t,
e,
TI
) + b2γ
( t,
e,
TI
)
γ
( t,
e,
TI
) + b2γ
( t,
e,
TI
) ΣY
( t,
e) b1
γ
( t,
e,
TI
)
( t,
TI
) = 02
t 1γ µ
with the terminal conditions b
2γ ( T e,
) = 0 2 2
and
1
( T , e,
) = 02
I
, T I x
I
2γ
I
T I
I
Y
2γ
I
1 I
, (11b)
, (11c)
b γ
. 02
x 2
and 02
are a 2-dimensional
matrix and a 2-dimensional vector of zeroes respectively. ( t, e) ≡ σ ( t,
e) σ ( t e)
Y Y
Y
,
′
Σ is the variancecovariance
matrix of
Y , ( t e,
T ) µ ( t)
− g σ ( t,
e)
′[ λ − σ ( T )]
t
'
µ
γ
,
I
≡
γ Y
0 Bt I
and µ
Yγ ( t,
e)
≡ µ
Y
+ gγσ
Y
( e,
t)
λY
are the two terms of the expected changes in
Y
t
under
( g γ )
gγ
′
P . a t,
T ) ≡ λ [ λ −σ
( t,
T )]
and
1γ
(
I
Y 0 B I
γ
a
g
γ
2 γ
≡ λY
λY
. Applying Itô’s lemma to expression (6) and by the self-financing property, we obtain
γ
′
the dynamics of the investor’s wealth 8 :
dW t
rt
t
(
St S t
( e TH
TB
)
t
) ⎤dt
+ (
St S
+
t
( e TH
TB
)
t
) ′ dzt
W ⎢⎣
⎡ ′
= + λ π σ + σ , , π π σ σ , , π , (12)
⎥⎦
t
⎡σ
B( t,
TB
) σ
Hr
( t,
e,
TH
)
with the initial condition W(0), ( )
( ) ⎥ ⎤
σ
t
e,
TH
, TB
= ⎢
and π
t
≡ [ π
Bt
π
Ht
]
⎣ 0 σ
t
te,
TH
⎦
' .
S
W
π ≡ B T W and
π
St
≡ θS
t t
is the fixed proportion invested in the spot commodity,
Bt
θBt
t
(
B
)
t
Ht
Ht
t
( e TH
) Wt
π ≡ θ H , represent the proportions invested in the two traded risky securities respectively.
In what follows, we shall distinguish the speculative or mean-variance proportion,
MV
π
t
, the minimum-
8 Investor’s utility function satisfies the Inada conditions. Consequently, optimal wealth is positive and (12) is a well defined
equation.
12

variance proportion,
that,
MPR
t
mV
π
t
, from the hedging proportion,
B γ
.
π , related to the investor specific bond, ( )
t
T I
π , related to the discount bond ( )
IR
t
B and
The dynamics of optimal wealth are obtained by applying Itô’s lemma to equation (9). It is
well-known that the diffusion vector of optimal wealth, σ
W*
, is sufficient to compute the optimal asset
t
allocation:
1
σW* = λt
+ g σ
B(
t,
TI
) σ
B
( e,
TI
)
t
γ
+
(13)
γt
γ
Optimal proportions invested in risky assets are derived by equating the diffusion part of the selffinanced
portfolio (12) to that of optimal wealth (13) (Karatzas et al., 1987). By rearranging terms, we
obtain the following decomposition of the traded risky asset proportions (a detailed calculation is
available from the authors upon request):
t
T I
π = π + π + π + π , (14a)
t
MV
t
mV
t
IR
t
MPR
t
( TB
) − rt
( ) ⎥ ⎤
e,
TH
⎦
MV
⎡ ⎤
MV
π
⎡
≡ ⎢ ⎥ =
1 −
µ
Bt
Bt
π
t
Σ( t,
e,
TH
, TB
)
1
MV
⎢ , (14b)
⎣π
Ht ⎦ γ
⎣ µ
Ht
π
mV
t
⎡π
≡ ⎢
⎣π
mV
Bt
mV
Ht
⎤
⎥ = −Σ
⎦
−1
( t,
e,
TH
, TB
) σ ( t,
e,
TH
, TB
) σ Sπ
St
′
, (14c)
π
π
IR
t
MPR
t
IR
⎡π
⎤
Bt
≡ ⎢ = g
IR ⎥ γ
Σ
⎣π
Ht ⎦
⎡π
≡ ⎢
⎣π
MPR
Bt
MPR
Ht
⎤
⎥ = Σ
⎦
where µ
Bt( T B
), Ht( e, T H
)
−1
( t,
e,
T , T ) σ ( t,
e,
T , T ) σ B(
t,
T )
H
B
−1
( t,
e,
T , T ) σ ( t,
e,
T , T ) σ ( e,
T )
H
B
H
H
B
B
′
′
Bγt
I
I
, (14d)
, (14e)
µ are the instantaneous expected returns of the two traded risky assets and
′
Σ denotes the variance-covariance matrix of the traded assets.
( t , e,
T , T ) ≡ σ ( t,
e,
T , T ) σ ( t,
e,
T , T )
H
B
H
B
H
B
σ e,
T ) is the diffusion vector of e,
T ) and is obtained by applying Itô’s lemma to
Bγt
(
I
(11a): σ ( e,
T ) σ ( t,
e)
Bγt
I
B γ t
(
I
∂Y
Bγ
t
( e,
TI
)
=
Y
, with ∂
Y
B
γ t
( e,
TI
) Bγ
t
( e,
TI
) = b1 γ
( t,
e,
TI
) + b2
γ
( t,
e,
TI
) Yt
.
B ( e,
T )
γt
I
Optimal demand (equation 14a) admits the traditional decomposition. The first component is
the speculative fund evolving stochastically over time because of the random character of the market
13

prices of risk. As expected, it is a decreasing function of the risk aversion coefficient. The other three
terms are hedging terms. The first is usually qualified as the preference-free minimum-variance term,
since it does not depend on the investor’s risk aversion. It serves to hedge the risk associated with the
fixed position in the spot commodity. The other two hedging addends involve the investor’s attitude
IR
vis-à-vis the risk. The role of π
t
is to hedge against random fluctuations in the instantaneous return
of the discount bond B
t
( T I
), which is a function of the instantaneously riskless rate. The second
MPR
component, π
t
, offsets the risk generated by the investor specific bond. It arises because the prices
of risk are stochastic.
Two main differences distinguish our model from other existing models (see, among others,
Ho, 1984; Stulz, 1984; Adler and Detemple, 1988a, b; Duffie and Jackson, 1990; Briys et al., 1990;
Duffie and Richardson, 1991; Lioui et al., 1996). First, concerning the last two hedging components,
in these models there are as many hedging terms, called Merton-Breeden terms, as state variables. In
our model, only two hedging ingredients emerge. This result is in the spirit of Lioui and Poncet (2001)
and Detemple and Rindisbacher (2009). However, compared to these papers,
MPR
π
t
, in particular, is
expressed in a more intuitive and convenient way. Contrary to Lioui and Poncet paper, we provide a
solution to B γ
e,
T ) and we specify its volatility σ e,
T ) . With regard to Detemple and
t
(
I
Bγt
(
I
MPR
Rindisbacher model, we express π
t
in terms of the investor specific bond and not in terms of the
more abstract density of the forward measure. Second, in contrast to the above (complete information)
models, the intertemporal demand for risky assets depends on the investor’s prior beliefs. This means
that the minimum-variance ingredient especially cannot be implemented by regression analysis.
Indeed, under complete information, this element is common to all investors who agree on the second
moments of the asset price distribution (see, for instance, Anderson and Danthine 1981; Adler and
Detemple 1988b).
The decomposition given by equations (14) is in line with standard results. However, it does
not allow one to determine the demand for each risky asset, to isolate the effect of the estimation error
on these demands and to assess the impact of the state variables as well. Let us recall that it is possible
to disentangle the effect of the estimation error on the conditional mean (see equation 3c) by
14

decomposing the Brownian motion z
St
in two uncorrelated Wiener processes ( z rt
and z
wt
). It follows
that the futures price dynamics contain a random part involving z
wt
and a diffusion term as a function
of the estimation error. Moreover, optimal demands (14a-e) depend on the variance-covariance matrix
of the traded assets Σ ( t , e,
T , T ) ≡ σ ( t,
e,
T , T ) σ ( t,
e,
T , T )
H
B
H
B
′
H
B
, which can be partitioned in a adequate
way. Indeed, Anderson and Danthine (1981) and Adler and Detemple (1988b) showed that the
variance-covariance matrix can be partitioned so that the mean-variance demand of some assets can be
expressed as a function of the mean-variance demand of other assets. Liu (2007) generalized this
separation result to the total demand in traded assets. We suggest a partition of the diffusion matrix
that is suited to our framework. To this effect, we introduce into the financial market a synthetic asset 9
perfectly correlated with the orthogonal source of risk z
wt
:
dH
H
wt
wt
( e,
TH
)
= µ
Hwt
( e,
TH
) dt + σ
Hw( t,
e,
TH
) dzwt
, (15)
( e,
T )
H
where, by absence of arbitrage opportunities µ Hwt
( e, T H
) r t
+ σ Hw
( t,
e,
T H
) λ wt
= , and
λ
wt
2
[ λ − ρ λ ] − ρ
= is the price of risk related to z
wt
.
St
Sr
rt
1
Sr
In Propositions 1 and 2 below, we take advantage of the synthetic asset to provide more
insightful expressions of the minimum-variance, the speculative and the hedging components.
Proposition 1.
a) The mean-variance component can be written as follows:
π
MV
Ht
( e,
TH
) −
( t,
e,
T ) 2
1 µ
Hwt
rt
= , (16a)
γ σ
Hw
H
( t,
e,
TH
, TB
)
( t,
T )
1 µ ( T ) − r Σ
= π . (16b)
MV Bt B t HB
π
Bt
−
2
2
γ σ
B( t,
TB
) σ
B B
b) The minimum-variance component can be written as follows:
( t,
e,
TH
)
( t,
e,
T )
mV HwS
π
Ht
2
σ
Hw H
St
MV
Ht
Σ
= −
π , (17a)
9 This asset can be duplicated, in our complete market, by a portfolio of the risk-free asset and the two traded risky assets.
15

( t,
e,
TH
, TB
)
( t,
T )
Σ ( t,
T ) Σ
= −
−
π . (17b)
mV BS B
HB
π
Bt
π
2 St
2
σ
B
( t,
TB
) σ
B B
{ S;
B;
H }
Σ , k , j ∈ ; stands for the covariance between assets k, j k ≠ j .
kj
H w
mV
Ht
Σ
σ
HwS
Hw
2
( t,
e,
TH
) 1−
ρSrσ
S
=
2
( e,
t;
T ) σ ( e,
t;
T )
H
Hw
H
,
Σ
HB
σ
( t,
e,
TH
, TB
)
2
( t,
T )
B
B
σ
Hr
= −
σ
( t,
e,
TH
)
( t,
T )
B
B
ΣBS
( t,
TB
)
and =
σ
B
2
( t,
T ) σ ( t,
T )
B
ρ σ
B
Sr
S
B
.
Proof. Available from the authors upon request.
Despite their difference in interpretation and use, the mean-variance and the minimumvariance
demands have two common features. First, they are expressed in a simple recursive manner
and consist of a fund specific to the futures contract and a fund related to the discount bond.
MV
π
Ht
and
mV
π
Ht
depend on the first two moments of the synthetic asset and are thus directly related to
zwt
underscoring the importance of the decomposition of z
St
. In other terms, these elements are
MV
directly related to the orthogonal source of risk of the estimate of the convenience yield. π
Ht
reflects
the investor’s expectations about the orthogonal source of risk, while
mV
π
Ht
hedges against the
orthogonal risk affecting the fixed position in the spot commodity. The speculative and minimumvariance
proportions of the discount bond are associated with the risk of the short rate. However, since
MV
the futures contract is correlated with the short rate, these proportions are adjusted by π
Ht
and
respectively, weighted by the usual variance/covariance ratios. Second, Proposition 1 underlines the
relation between the investor’s prior beliefs and the demand in the futures contract. Indeed, as
mV
π
Ht
explained above, the estimation error influences the estimate of the convenience yield through
z
wt
.
MV
π
Ht
and
mV
π
Ht
are devoted to this source of uncertainty. It follows that the mean-variance and
minimum-variance demands of the futures contract are functions of the estimation error and thus of the
MV
mV
investor’s initial prior beliefs. π
Bt
and π
Bt
are impacted by the estimation procedure only through
MV
mV
π
Ht
and π
Ht
.
The mean-variance and minimum variance proportions are functions of volatilities and of
covariances. Moreover, the investor’s position, short or long, depend on the sign of the fixed position
MV
and of that of the covariances. Proposition 1 shows that π
Ht
and
π are functions of σ ( t , e,
)
mV
Ht
Hw
T H
16

and are likely to exhibit the same pattern as that of this volatility, which dominates the futures price
volatility. Especially, as will be seen later, futures price volatility evolves according to the Samuelson
effect, which suggests that the volatility of futures prices for contracts close to maturity exceeds that of
MV
mV
more distant contracts. This effect implies that π
Ht
and π
Ht
are increasing functions, in absolute
values, of the futures contract maturity. The mean-variance and minimum variance demands in the
MV
mV
bond are modified by π
Ht
and π
Ht
weighted by
Σ
−
HB
σ
( t,
e,
TH
, TB
)
( t,
T ) 2
have a positive impact on the futures contract, the covariance ( t , e,
T , T )
B
B
. As interest rates are expected to
Σ should be negative
MV
mV
implying that this ratio should be positive. Therefore, the sign of π
Ht
and π
Ht
will determine their
MV
mV
impact (negative or positive) on π
Bt
and π
Bt
.
We turn now to the study of the last two hedging elements in equation (14). A direct
calculation of (14d) gives:
π = 0 , (18a)
π
IR
Ht
σ
( t,
TI
)
( t,
T )
IR
B
Bt
= gγ
, (18b)
σ
B B
The hedging demand stemming from the stochastic behavior of the return of the discount bond with
HB
H
B
maturity
T
I
reduces to the proportion,
IR
π
Bt
, invested in the bond that provides insurance against
interest rate risk. It is independent of the estimation error. Its expression is given in (18b) confirming a
well-known result, that is it is proportional to the ratio of the volatilities of two bonds with maturity T
I
and T
B
respectively (see Lioui and Poncet, 2001; Munk and Sorensen, 2004). Given that investors are,
in general, more risk averse than the Bernoulli investor and that
T < T , this proportion is positive and
I
B
less than one.
MPR
The component π
t
hedges against the risk generated by the investor specific discount bond.
It arises from the need to hedge against the stochastic prices of risk, which they are functions of the
state variables. Equation (14e) involves the diffusion term of ( )
t
T I
B γ
. By definition, σ e,
T )
depends on the volatilities of the state variables. It can then be written in the following manner:
Bγt
(
I
17

σ
( e,
T ) + σ rΨ
( e,
T ) + σ m(
t,
e)
Ψ ( e T )
Bγt
( , TI
) σ X Ψγ
Xt I
γrt
I
γmt
e = , , (19)
where Ψ ( e, T ) = ∂ B ( e,
T ) B ( e,
T ), i ∈{ X , r m}
B γ
e,
T ) .
t
(
I
it I i γt
I γt
I
,
γ
are the components of the gradient vector of
MPR
π
t
can be decomposed into three terms for each and every state variables:
I
π = π + π + π , (20)
MPR
t
MPRX
t
MPRr
t
MPRm
t
The following proposition is devoted to these terms.
Proposition 2.
The optimal hedging proportions generated by the investor specific bond may be decomposed for each
and every state variable.
a) The risk associated with the (log) spot commodity price is hedged by the futures contract and the
discount bond.
π
MPRX
Ht
Σ
( t,
e,
T )
( e,
T )
HwS H
= Ψ
2 γXt
I
, (21a)
σ
Hw( t,
e,
TH
)
( e,
T )
( t,
e,
TH
, TB
)
( t,
T )
Σ ( t,
T ) Σ
= Ψ −
π . (21b)
π
MPRX BS B
HB
Bt
2 Xt I
σ
B( t,
TB
)
γ σ
2
B B
MPRX
Ht
MPRr
b) The risk associated with the interest rate is hedged by the discount bond ( π = 0 ).
π
Σ
( t,
T )
( e,
T )
MPRr Br B
Bt
= Ψ . (22)
2 γrt
I
σ
B( t;
TB
)
c) The risk associated with the estimate of the convenience yield is hedged by the futures contract and
the discount bond.
π
Σ
( t,
e,
T )
( e,
T )
MPRm Hwm H
Ht
= Ψ
2 γmt
I
, (23a)
σ
Hw( t,
e,
TH
)
( e,
T )
( t,
e,
TH
, TB
)
( t,
T )
Σ ( t,
T ) Σ
= Ψ −
π . (23b)
π
MPRm Bm B
HB
Bt
2 mt I
σ
B( t;
TB
)
γ σ
2
B B
MPRm
Ht
Σ , i ∈{ X ; r;
m} , j { B;
H } stands for the covariance between the assets { }
i , j
∈
w
Ht
B; and the state
H w
Σ
; .
σ
variables { X r;
m}
Br
B
( t,
TB
)
= −
2
( t;
T ) D ( t T )
B
α
1
,
B
,
Σ
σ
Hwm
Hw
( t,
e,
TH
)
=
2
( t,
e,
T ) σ ( t,
e,
T )
H
σ ( t,
e)
Hw
mw
H
,
ΣBm(
t,
TB
)
= −
σ
B
σ δ
ρr
δ
2
( t;
T ) D ( t,
T )
B
α
B
.
Proof. Available from the authors upon request.
18

Proposition 2 seems to suggest a decomposition à la Merton-Breeden as the optimal hedging
proportions are derived for each and every state variable. At a first sight, this result may be in
contradiction with the decomposition given in equations (14) in terms of two bonds. However, a
preliminary remark is in order. Thanks to the change of probability measure mentioned above, the
Merton-Breeden hedging terms reduce to two addends involving two bonds whatever the number of
the state variables. In order to study how the state variables and incomplete information affect optimal
demands, the latter are couched in terms of each variable. An important difference with the familiar
Merton-Breeden terms is that the decomposition in Proposition 2 preserves the advantages of
equations (14) since it depends on the characteristics of the two bonds.
In the same manner as for the minimum-variance and mean-variance elements, these hedging
terms are expressed in a convenient recursive way. The futures contract serves to hedge the orthogonal
risk of the estimate of the convenience yield and directly captures the effect of the estimation error.
The latter is also captured through Ψ ( e, ), which, given that ( )
γit T I
B γ
has the functional form of a
discount bond, admit a simple economic interpretation. It represents the sensitivity of the investor
discount bond on the three state variables (see also Wachter, 2002). In particular, ( e, )
t
T I
Ψ measures
γmt T I
the sensitivity of B γ t
( T I
) to changes in the estimate of the convenience yield and thus the impact of the
incomplete information.
Proposition 2 has another two advantages. First, it allows an investor to assess the impact of
each state variable on optimal hedging demands and to decide which of the state variables are worth to
be comprised in the investment opportunity set. This assessment is based on ( e, )
Ψ weighted by the
γit T I
covariance/variance ratios. Second, Proposition 2 shows which risky assets should be used and in what
proportions in order to hedge the risk of the investor specific bond related to the state variables. To
gain intuition on how state variables affect Ψ ( e, )
γit T I
, let us rewrite expression (10) in a convenient
way separating the price of risk related to the orthogonal risk from that associated with the short rate
risk:
B
γt
( ) ⎡ ⎪⎧
g gγ
2
2
( TI
) Et
⎢exp⎨−
[ λwu
+ λru
−σ
B
( u,
TI
) ]
T I
⎪⎫
⎤
γ
= ∫
du⎬⎥
(10b)
⎢⎣
⎪⎩ 2γ
t
⎪⎭ ⎥⎦
19

The sign of Ψ ( e, ) and ( e, )
γXt T I
Ψ can be examined through the effect of X(t) and m(t) on λ
wt
,
γmt T I
while the analysis of the sign of ( e, )
Ψ is more complicated since r(t) impacts both λ
rt
and
γrt T I
Note that for more risk-averse investors than the logarithmic investor, γ ≥1
and g ≥ 0
γ
λ
wt
.
. Therefore, as
the exponential involves a quadratic function, when
λ is positive (negative), the sign of Ψ ( e, )
wt
γXt T I
should be the opposite (same) to that of X(t ) and m(t) on
λ
wt
. The same reasoning applies to
Ψ ( e, ) . Unfortunately, it is not possible to directly deduce the sign of Ψ ( e, )
γmt T I
γrt T I
. However, as the
correlation of the short rate and the risky assets, the spot commodity in particular, is negative, we
guess that its sign is negative.
4. Illustration
This section provides an illustration of our model in the case of the copper market. We simulate how
speculative and hedging demands react to the investor’s horizon, to the state variables changes and to
the investor’s initial beliefs. The parameters values are based on those estimated by Schwartz (1997)
and Casassus and Collin-Dufresne (2005). To focus on the influence of the convenience yield and the
spot price on optimal demands, we run our simulations for a flat (forward) initial term structure of the
risk-free rate. The values of α and the initial forward yield curve, f ( ) , are computed so that θ (∞ )
and β are respectively equal to the values of the long term mean (3%) and speed of mean-reversion
(0.2) of the risk-free rate provided in Casassus and Collin-Dufresne (2005). Table 1a gathers the
values of the parameters used in our simulations.
[INSERT TABLE 1a ABOUT HERE]
The parameters values are set so that we can reproduce some stylized facts characterizing
commodities. Spot commodity prices and the convenience yield (see Bessembinder et al., 1995) as
well as the short rate exhibit mean reversion, so that α > 0 and k > 0. As Casassus and Collin-Dufresne
(2005) pointed out, mean-reversion in the state variables is reinforced by that of prices of risk. Thus,
0 t
r
λ
SX
<
rr
0 and λ < 0 . To assess the impact of the state variables, we consider three scenarios allowing
us to take into account backwardation (more frequently observed in commodity markets) and contango
20

situations. For each scenario, the short-term rate is equal to its long-term mean in order to focus our
analysis on the specific impact of the spot price and the convenience yield. The first scenario describes
a situation where the (log) spot price and the estimate of the convenience yield are equal to their longterm
mean. For the second (third) scenario, called backwardation (contango), the spot price is 10%
above (below) its long term mean and the estimate of the convenience yield is equal to 9.5% (3.5%).
The three scenarios are shown inTable 1b.
[INSERT TABLE 1b ABOUT HERE]
The filtering error steady state value can be computed using the values in Table 1a: ε ∞ =0.59%.
We then study the effect of the initial value of the estimation error for high values above (below) the
steady state, e = 1% > ε ∞ ( e = 0.1% < ε ∞ ). When examining optimal demands as a function of the
futures contract maturity, we let this maturity vary between 0 and 18 months, since copper contracts
are usually available for maturities up to 24 months (less liquid contracts). Investor’s horizon also
varies in the same interval. Throughout the analysis, we set the bond maturity equal to seven years,
T B =7, and, for simplicity, the constrained position equal to 1, π = 1. Finally, we consider the case of
a more-risk averse investor than the log-utility (myopic) investor, and we retain a value of γ = 3 . For
compactness of notation, we drop out the parentheses from the quantities we study.
[INSERT FIGURES 1a AND 1b ABOUT HERE]
Figures 1a,b display σ H and σ Hw as a function of T H , respectively. Figure 1a shows the
decreasing pattern of the futures contract volatility as a function of its maturity, i.e. the Samuelson
effect. This effect is more pronounced for a high initial estimation error. Moreover, the higher the
initial error, the lower the volatility of the futures price. This can be explained as follows. For the
values used in this illustration, κ S
> 0 , which implies that e has a positive impact on σ
mw
and thus on
δ
σ H . However, mean reversion in the market prices of risk outweighs this positive effect resulting in a
decreasing σ H as a function of e. As the maturity of the futures contract increases, this phenomenon is
amplified. Figure 1b reveals that the behavior of σ H is essentially dictated by that of σ Hw . Indeed,
additional results, not reproduced here, show that, as expected, σ Hr is positive, but its value is low and
decreases from around 3.2% to around 2.4%.
St
21

[INSERT TABLE 2 ABOUT HERE]
Table 2 displays the covariance/variance ratios in propositions 1 and 2 as a function of T H . As
these ratios determine the investor’s position (short or long) as well as its magnitude, it is interesting to
examine their evolution over time. As expected, Σ HwS /σ Hw 2
is positive and since it is inversely
proportional to σ Hw , from the analysis above, it follows that it is an increasing function of both the
futures contract maturity and the initial value of the estimation error. Their influence is substantial.
Indeed, when e = 0.1%, this ratio increases by 78.33% between T H = 3 months and T H = 18 months.
When e = 1%, the rise in this ratio, for the same period, is 91.4%. Moreover, the longer the maturity
the higher the impact of e on this ratio. In contrast, Σ HB /σ 2 B is negative and, as discussed above, much
less sensitive than Σ HwS /σ 2 Hw to T H and e.
[INSERT TABLE 3 ABOUT HERE]
Let us now analyze how futures contracts maturity affects the minimum-variance and meanvariance
proportions (see tables 3 and 4 respectively). For future reference, we denote by
Σ
≡ π , l ∈ { mV , MV , MPR,
MPRX , MPRm}
the part of the bond demand related to the
l
HB
π
B,
H
−
2
σ
B
l
H
futures contract demand. Since π = 1, the minimum-variance futures proportion is equal to -Σ HwS
St
/σ 2 Hw and mimics its behavior. The minimum-variance futures hedge demand is an opposite position to
the spot commitment. The pure hedge of the discount bond is a little more complicated. The spot
ΣBS
commodity is negatively correlated with the interest rate so that
2
σ
B
is negative and
Σ
−
BS
π
S
σ 2
B
is
positive. However, as seen above,
mV
π
B
exacerbates the effect of the first term.
mV
is adjusted by second term π
B,H
, which is also negative and
[INSERT TABLE 4 ABOUT HERE]
Table 4 displays the mean-variance proportions for the three scenarios considered in Table 1b.
The impact of T H and the initial estimation error on
MV
π
H
is similar to that on
mV
π
H
. The term
1 µ
Bt(
TB
) − rt
γ σ
B
( t,
T ) 2
B
is independent on T H and e. It is the same across all scenarios and is equal to 128.3%.
Since Σ HB /σ 2 MV
B is negative, π
B,H
adds to the above term resulting in a substantial speculative demand,
22

MV
π
B
, for the discount bond. When the spot price is above its long-term mean, speculative demands are
higher than those when the spot price is below its long-term mean. This result, may, at a first sight, be
seen as counter-intuitive. However, it can be explained by mean-reversion in the market prices of risk.
The higher the spot price, the lower λ wt , and the higher the estimate of the convenience yield, the
higher λ wt . However, the spot price effect dominates that of the estimate.
[INSERT TABLE 5 ABOUT HERE]
MV
Table 5 performs a similar analysis as for π
H
and
the investor’s horizon.
π of , i ∈{ X , r m}
mV
H
Ψ γ it
,
as a function of
Ψ
γit
are, in absolute value, increasing functions of T I and the results confirm the
intuition about their sign explained below Proposition 2. In addition, a similar explanation to the case
of
MV
π
H
and
mV
π
H
can be put forward to understand the behavior of the sensitivities in the three
MV
scenarios. Finally, in contrast to π
H
and
a negligible effect on
value of e.
mV
π
H
, a change in the value of the initial estimation error has
Ψ
γit
. The sensitivities depend more on the level of the state variables than on the
[INSERT TABLE 6 ABOUT HERE]
Table 6 exhibits the hedging components related to the investor discount bond and generated
by the random behavior of the prices of risk. These demands depend on both T H and T I . However, to
simplify their understanding, without loss of generality, we set T H =T I . As is demonstrated in
Proposition 2, π is a weighted average of the sensitivities Ψ
Xt
and Ψ
mt
. The weighs are equal to
MPR
Ht
the variance covariance ratios Σ HwS /σ Hw
2
and Σ Hwm /σ Hw 2 , which, in our case, are positive. In contrast,
γ
γ
Ψ
γXt
and Ψγ
mt
MPR
have an opposite sign and therefore the two terms in π
Ht
partially offset each other.
MPR
This explains the low values of π
Ht
for short maturities especially and the change of the position
(long or short) for e =0 .01%. π is a function of Ψ
Xt
, Ψ
mt
and Ψ
rt
. To study this term turns out to
be more complex than
π
MPR
Bt
B
( t;
T )
MPR
π
Ht
Σ
BX
( t,
T )
=
σ
MPR
Bt
γ
MPR
. However, write π
Bt
as follows facilitates its interpretation:
( e,
T )
Σ
( t,
T )
( e,
T )
B
Br B
Bm B
Ψ
2 γXt
I
+ Ψ
2 γrt
I
+ Ψ
2 γmt
,
B
σ
B
( t;
TB
)
σ
B
( t;
TB
)
Σ
γ
( t,
T )
γ
MPR
( e T ) −π
I
B,
Ht
23

As Σ BX /σ B
2
=-55.8%, Σ Br /σ B
2
=-17.5% and Σ Bm /σ B
2
=-56.0%, the first term, in the equation above, is
negative, while the second and third components are positive. Moreover, the first term partially
MPR
MPR
compensates the third one and given that π
B,Ht
takes low values, π
Bt
is, for a large part, driven by
the second element. With regard to the impact of e, an inspection of Table 5 reveals, however, that
Ψγrt
is insensible to e. Indeed, a quick calculation shows that the difference
MPR
MPR
MPR
MPR
π ( e = %) − π ( e 0.1% ) is similar to the difference ( e = 1 %) − π ( e = 0.1% )
B, Ht
1
B,
Ht
=
π .
Bt
Bt
5. Concluding remarks
In this article we have studied the Traditional Hedging Model (Adler and Detemple, 1988 a,b), i.e. the
issue of hedging a fixed cash asset position with correlated futures contracts, for commodity markets
when the convenience yield is not observable and is estimated given the information conveyed by the
spot commodity and the short-rate by using the continuous-time Kalman filter. The estimate of the
convenience yield and the estimation error directly affect the mean-variance and minimum-variance
demands in futures contracts. Moreover, the initial value of the estimation error has a heavy impact on
these demands. The positions in the discount bond are impacted by the estimation procedure only
through the demand in futures contracts. We achieve a decomposition of the hedging component
related to the stochastic prices of risk by distinguishing the effect of each and every state variable.
Contrary to the other two demands, these hedging elements are not very sensitive to the initial
estimation error.
Our framework can be extended in order to take into account richer dynamics of the state
variables including, for example, jumps in the state variables (see Jeanblanc et al. 2009) and/or other
utility functions (recursive utility, loss aversion, for instance). In the paper, we assume that investors
use the information coming from interest rates and spot commodity prices to infer the convenience
yield. However, investors can have access to private information regarding the value of the
convenience yield through their business activity. This information would result in an additional
source of uncertainty. It would then be optimal for the investor to use two futures contracts as hedging
instruments.
24

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29

Figures
0.24
0.22
ε=0.01
ε=0.001
0.24
0.22
ε=0.01
ε=0.001
0.2
0.2
0.18
0.18
σ H
0.16
σ Hw
0.16
0.14
0.14
0.12
0.12
0.1
0.1
0 0.5 1 1.5
Futures contract time to maturity (T H
in years)
Fig. 1a. Futures price volatility as a function of the
futures contract time to maturity. This figure plots σ H as a
function of T H for maturities varying from 0 to 1.5 years,
and for e=0.1% (dash line) and e=1% (solid line).
Parameters values are from Table 1a.
0.08
0 0.5 1 1.5
Futures contract time to maturity (T H
in years)
Fig. 1b. Futures contract sensitivity to the spot price
orthogonal risk as a function of the futures contract time to
maturity. This figure plots σ Hw as a function of T H for
maturities varying from 0 to 1.5 years for e=0.1% (dash
line) and e=1% (solid line). Parameters values are from
Table 1a.
30

Tables
Table 1a: State variables dynamics parameters for the copper market
σ S λ S0 λ SX λ Sδ σ r α f 0 (t) λ r0 λ rr θ δ Κ σ δ ρ rδ ρ Sδ ρ Sr
2 2 . 8 % 2 2 . 3 4 - 5 . 0 3 8 . 4 2 1 % 6 % 5 . 2 8 % 0 . 2 - 1 4 6 . 4 6 % 1 . 1 2 0 % 0 . 1 6 0 . 7 0 . 1 4
Table(1a) displays the parameters values for the dynamics (1 a-c) in the case of the copper market.
Table 1b: State variables values for the three market scenarios
Long Term Mean Scenario Backwardation Scenario Contango Scenario
X 4.498 4.59 4.39
r 3% 3% 3%
m 6.46% 9.5% 3.5%
Table (1b) displays the state variables values for the long term mean,
backwardation and contango scenarios
Table 2. Covariance/variance ratios as a function of the futures contract maturity
e
Σ HwS /σ 2 Hw (%) Σ HB /σ 2 B (%) Σ Hwm /σ 2 Hw (%)
T H (years) T H (years) T H (years)
0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5
0.1 % 115.8 132.0 166.8 206.5 -48.5 -44.5 -41.4 -41.0 72.3 82.4 104.1 128.9
1 % 119.9 140.3 182.8 229.5 -48.6 -44.8 -41.8 -41.5 94.3 110.4 143.8 180.6
Table 2 displays the covariance/variance ratios, Σ HwS /σ Hw 2 , Σ HB /σ B 2 and Σ Hwm /σ Hw 2 , as functions of the
futures contract time to maturity varying from 0.25 to 1.5 years, and for e=0.1 % and e=1%. Parameters values
are given in table 1a.
Table 3. Minimum-variance proportions as a function of the futures contract maturity
e
mV
mV
π (%)
π (%)
π (%)
mV
Ht
T H (years) T H (years) T H (years)
0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5
0.1% -115.8 -132.0 -166.8 -206.5 -74.3 -77.7 -91.4 -112.1 -0.4 -3.8 -17.5 -38.2
1 % -119.9 -140.3 -182.8 -229.5 -77.2 -83.1 -101.2 -125.9 -3.3 -9.2 -27.3 -52.0
B,
Ht
Bt
Table 3 displays the minimum variance proportions π , π
mV
H
mV
B,H
andπ as functions of the futures contract time to maturity
varying from 0.25 to 1.5 years, and for e=0.1% and e=1%. Parameters values are given in table 1a.
mV
B
31

Table 4. Mean-variance proportions as a function of the futures contract maturity
e
e
e
MV
MV
π (%)
π (%)
π (%)
MV
Ht
T H (years) T H (years) T H (years)
0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5
Long Term Mean Market
0.1 % 49.5 56.4 71.0 87.2 24.0 25.1 29.4 35.8 152.3 153.4 157.7 164.1
1 % 51.3 59.9 77.5 96.4 24.9 26.8 32.4 40.0 153.2 155.1 160.7 168.3
Backwardation Market
0.1 % 11.3 12.9 16.3 20.0 5.5 5.7 6.7 8.2 133.8 134.0 135.0 136.5
1 % 11.7 13.7 17.8 22.1 5.7 6.1 7.4 9.2 134.0 134.4 135.7 137.4
Contango Market
0. 1 % 97.5 111.1 139.8 171.7 47.3 49.4 57.9 70.5 175.6 177.7 186.2 198.8
1 % 100.9 117.9 152.7 189.9 49.1 52.8 63.9 78.7 177.4 181.0 192.2 207.0
B,
Ht
Bt
Table 4 displays the mean-variance proportions π , π and
Table 5.Investor specific bond sensitivities to state variables as a function of
the investor’s horizon
MV
H
MV
B
MV
π
B,H
as a function of the futures contract
time to maturity varying from 0.25 to 1.5 years for the three scenarios identified in table 1b. For each
scenario e=0.1% and e=1%. Parameters values are given in table 1a and γ=3.
e
e
e
(%)
γXt
Ψ Ψ (%)
Ψ (%)
T I (years) T I (years) T I (years)
0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5
Long Term Mean Market
0.1 % 7.0 12.1 19.0 23.4 -19.2 -36.6 -67.1 -93.2 -11.1 -18.0 -25.4 -28.7
1 % 6.8 11.6 18.0 22.3 -19.2 -36.4 -66.9 -93.0 -10.7 -17.2 -24.0 -27.1
Backwardation Market
0.1 % 1.9 3.7 7.0 9.7 -17.5 -34.2 -65.1 -92.5 -3.0 -5.4 -8.9 -10.9
1 % 1.8 3.5 6.6 9.2 -17.5 -34.2 -65.0 -92.4 -2.9 -5.2 -8.4 -10.3
Contango Market
0. 1 % 13.4 22.5 33.6 40.0 -21.4 -39.5 -69.6 -94.3 -21.1 -33.5 -45.5 -50.3
1 % 12.9 21.4 31.8 38.0 -21.2 -39.2 -69.2 -93.9 -20.4 -31.9 -43.0 -47.5
γrt
γmt
Table 5 displays the sensitivities Ψ , Ψ and
X
r
Ψ
m
as a function of the futures contract time to maturity
varying from 0.25 to 1.5 years for the three scenarios identified in table 1b. For each scenario e=0.1% and
e=1%. Parameters values are given in table 1a and γ=3
Table 6. The investor specific discount bond proportions as a function of investment
horizon and futures contract maturity
MPR
MPR
π (%)
π (%)
π (%)
e
e
e
MPR
Ht
B,
Ht
T H =T I (years) T H =T I (years) T H =T I (years)
0.25 0.5 1 1.5 0.25 0.5 1 1.5 0.25 0.5 1 1.5
Long Term Mean Market
0.001 0.1 1.2 5.2 11.3 0.1 0.5 2.2 4.6 5.7 10.2 17.5 24.0
0.01 -1.9 -2.7 -1.6 2.1 -0.9 -1.2 -0.7 0.9 4.6 8.3 14.4 19.9
Backwardation Market
0.001 0.0 0.4 2.3 5.9 0.0 0.2 1.0 2.4 3.7 7.1 13.5 19.3
0.01 -0.5 -0.8 0.0 2.6 -0.2 -0.3 0.0 1.1 3.4 6.6 12.4 17.8
Contango Market
0.001 0.3 2.1 8.6 17.7 0.1 0.9 3.6 7.3 8.2 14.0 22.5 29.6
0.01 -3.7 -5.1 -3.6 1.4 -1.8 -2.3 -1.5 0.6 6.1 10.4 16.9 22.4
Bt
Table 6 displays the market price of risk proportions π , π and
MV
H
MV
B
MV
π
B,H
as a function of the
futures contract time to maturity and investment horizon varying from 0.25 to 1.5 years for the
three scenarios identified in table 1b.T H =T I . For each scenario e=0.1% and e=1%. Parameters
values are given in table 1a and γ=3.
32

Liste des cahiers de recherche du PRISM – ANNEE 2010
N° Titre Auteurs
CR-10-01 Vers une approche constructionniste de la valorisation de
l’entreprise
J.-J. Pluchart,
L. Barbara
CR-10-02 « Tu pousses le bouchon un peu trop loin, Maurice ! » Vers un
repérage des leviers publicitaires influençant les enfants.
Application au domaine alimentaire
P. Ezan, M. Gollety,
N. Guichard,
V. Nicolas-Hémar
CR-10-03 De la nécessité de prendre en considération simultanément les
différents contextes sociaux des enfants pour comprendre leur
comportement alimentaire
P. Ezan, M. Gollety,
N. Guichard,
V. Nicolas-Hémar
CR-10-04 La valeur économique du brevet « bloquant » A. Bami, G. A. Shiri
CR-10-05 Etude exploratoire sur l’effet des dimensions de la couleur des
sites de marque sur les préférences des enfants
H. Ben Miled-Chérif,
N. Bezaz-Zeghache
CR-10-06 Gestion tribale de la marque et distribution spécialisée : le cas
Abercrombie & Fitch
J.-F. Lemoine,
O. Badot
CR-10-07 Le knowledge management et la communication financière de
l’entreprise : principes, leviers et mise en œuvre
J.-J. Pluchart,
S. Ayoub
CR-10-08 The impact of social and environmental variables on S.-E Gadioux
performance: a panel data application to international banks
CR-10-09 Les relations entre la performance sociétale et la performance S.-E Gadioux
financière des organisations : une étude empirique comparée
des banques européennes et des banques non européennes
CR-10-10 Le changement organisationnel des Entreprises Socialement J.-J. Pluchart,
CR-10-11
CR-10-12
CR-10-13
Responsables (ESR)
L’influence de la familiarité et de l’intérêt pour la catégorie de
produit sur l’innovativité et le comportement innovateur du
consommateur : une approche modérée par le bouche à oreille
L'impact conjoint des variables situationnelles et individuelles
sur les réactions du consommateur face à un nouveau produit:
d'une étude exploratoire qualitative à un essai de modélisation
Coordination, engagement et RSE au cœur de la quête
managériale du changement perpétuel
D. Gnanzou
A. Sellami
A. Sellami
O. Uzan,
B. Condomines
CR-10-14 De l’homo métis à l’homo academicus J.-J. Pluchart
CR-10-15
L’opérationnalisation du succès de carrière : intérêts et limites
E. Hennequin
des méthodologies actuelle
CR-10-16 Quelle réussite professionnelle pour les enseignantschercheurs
E. Hennequin
?
CR-10-17 La confiance institutionnelle en question J.-J. Pluchart
CR-10-18 L’impact des mécanismes internes de gouvernement de H. Ben Ayedl’entreprise
sur la qualité de l’information comptable
Koubaa
CR-10-19 La gouvernance des entreprises socialement responsables J.-J. Pluchart
CR-10-20 Les mécanismes de régulation du marché du coaching : les H. Cloet
conventions
CR-10-21 How R&D Competition affects Investment Choices T. Lafay, C. Maximin
CR-10-22 Les stratégies d’innovation dans le commerce indépendant de J.-F. Lemoine
proximité
CR-10-23 Optimal dynamic demands in commodity futures markets with a
stochastic convenience yield
C.Mellios
P. Six
CR-10-24 Comment le site internet d’une enseigne modifie le R. Vanheems
comportement de ses clients en magasin
CR-10-25 Les violences familiales : une problématique organisationnelle ? E. Hennequin
N. Wielhorski
CR-10-26 Etat de l’art sur l’harmonisation vie privée – vie professionnelle
des salariés
S . Kilic

CR-10-27
CR-10-28
Les mécanismes de régularisation du marché du coaching : les
conventions
La diffusion de la fraude en entreprise, le cas de la collusion
tacite
H. Cloet
P. Jacquinot
A. Pellissier-Tanon
S. Strtak
JJ. Lemoine
CR-10-29 Agent virtuel et confiance des internautes vis-à-vis d’un site
Web
CR-10-30 Peut-on encore lancer un programme de recherche en
management stratégique ?
CR-10-31 Equivalent Risky Allocation S. Plunus
R. Gillet
G. Hubner
CR-10-32
CR-10-33
CR-10-34
Voluntary financial disclosure, introduction of IFRS and the
setting of a communication policy: An empirical test on SBF
French firms using a publication score
RSE et développement des populations pauvres du sud : Une
analyse relative aux OMD comme cadre d’analyse
macroéconomique du développement.
The Traditional Hedging Model Revisited with a Non Observable
Convenience Yied
JJ. Notebaert
S. Edouard
A. Gratacap
H. De La Bruslerie
H. Gabteni
D. Gnanzou
C.Mellios
P. Six