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Model
ling
Ener
gy
Forw
ard
Curv
es
Sve
tlana
Boro
vkova
Free
Uni
vers
ity
ofA
mst
erda
m(V
UA
mst
erda
m)
–Typ
eset
byFoi
lTEX
–1
Ener
gy
mar
kets
•Pre
-198
0s:
regu
late
den
ergy
mar
kets
•19
80s:
der
egula
tion
ofoi
lan
dnat
ura
lga
sin
dust
ries
•19
90s:
der
egula
tion
ofel
ectr
icity
indust
ries
wor
ldw
ide
•Ener
gyis
the
wor
ld’s
larg
est
trad
edco
mm
odity
clas
s
•Cru
de
oilis
the
wor
ld’s
larg
est
com
modity
•Ener
gym
arke
tsar
eex
trem
ely
vola
tile
(annual
vola
tilit
ies:
Oil
40+
%,
NG
60+
%,
Ele
ctrici
ty10
0+%
−→co
mp.
with
15+
%fo
req
uity
indic
es)
=⇒
nee
dfo
reffi
cien
trisk
man
agem
ent
•Ener
gypr
ices
are
neg
ativ
ely
corr
elat
edto
the
stock
pric
esan
din
dic
es
=⇒
per
fect
div
ersifica
tion
tool
s
What
istr
aded
?
Phys
ical
crude
oil,
oilpr
oduct
s,N
G(s
pot
mar
kets
);fo
rwar
dco
ntr
acts
(OT
C);
Futu
res
contr
acts
onIC
E,N
YM
EX
:vo
lum
es9-
10tim
eshig
her
than
thos
ein
spot
mar
kets
!
–Typ
eset
byFoi
lTEX
–2
Futu
res
contr
act
sand
forw
ard
curv
es
•Fu
ture
s:st
andar
diz
edco
ntr
acts
for
del
iver
yof
aco
mm
odity
(e.g
.cr
ude
oil)
atdiff
eren
t
tim
epoi
nts
(exp
irie
s)in
the
futu
re.
•Price
sfo
rfu
ture
sw
ith
diff
eren
tex
pirie
s(e
.g.,
for
oil,
up
to72
mon
ths
ahea
d)
are
reco
rded
dai
ly.
•T
he
colle
ctio
nof
thes
efu
ture
spr
ices
onan
ypar
ticu
lar
day
isca
lled
the
forw
ard
curv
e.
•T
he
set{F
(t,T
),T
>t}
isth
efo
rwar
dcu
rve
prev
ailin
gat
dat
et
for
agi
ven
com
modity
ina
give
nlo
cation
;T
indic
ates
the
expiry,
orm
aturity
dat
e(m
onth
).
•T
he
forw
ard
curv
eis
the
fundam
enta
lto
olw
hen
trad
ing
com
moditie
s,as
spot
pric
es
may
be
unob
serv
able
and
option
sill
iquid
.
–Typ
eset
byFoi
lTEX
–3
Ben
efits
offo
rwar
dcu
rves
•For
war
dcu
rves
reflec
tm
arke
tfu
ndam
enta
lsan
dan
tici
pat
edpr
ice
tren
ds
•B
ench
mar
kfo
rVal
uat
ion:
Dea
lPrici
ng,
P&
L
•In
tern
alCon
sist
ency
inth
edes
kor
the
firm
with
other
der
ivat
ives
•M
ark
To
Mar
ket,
Sto
pLos
s,VaR
•T
he
forw
ard
curv
espr
ovid
eth
eca
libra
tion
ofth
em
odel
par
amet
ers
under
the
pric
ing
mea
sure
•Com
modity
por
tfol
ios
conta
infu
ture
sw
ith
diff
eren
tex
pirie
s
−→risk
expos
ure
tom
ovem
ents
ofth
een
tire
forw
ard
curv
e
•Prici
ng
ofder
ivat
ives
onfu
ture
san
dfo
rwar
ds
requires
forw
ard
curv
em
odel
s
–Typ
eset
byFoi
lTEX
–4
The
oil
price
inth
ela
sttw
odec
ades
010
0020
0030
0040
0050
0001020304050607080
$/bbl
Tra
ding
day
s si
nce
23−
06−
1988
WT
I Cru
de O
il pr
ice,
Jul
y 19
98 −
Dec
embe
r 20
06
–Typ
eset
byFoi
lTEX
–5
Oil
forw
ard
curv
es:
two
fundam
enta
lm
arke
tst
ate
sB
ack
war
dation
and
Conta
ngo
Antici
pat
edva
lue
ofth
efu
ture
spot
pric
eis
lower
(B)
orhig
her
(C)
than
the
curr
ent
one.
Influen
ced
by:
curr
ent
pric
ean
din
vento
ryle
vels,
tran
spor
tation
and
stor
age
cost
s,
supply
/dem
and,st
rate
gic
and
pol
itic
alre
ason
s,...
02
46
810
1214
1618
20051015202530
Exp
iry m
onth
s (n
umbe
red
from
now
)
$/barrel
06.0
3.20
00, b
ackw
arda
tion
mar
ket
02
46
810
1214
1618
200246810121416
Exp
iry m
onth
s (n
umbe
red
from
now
)
$/barrel
01.1
2.19
98, c
onta
ngo
mar
ket
–Typ
eset
byFoi
lTEX
–6
Changin
gfa
ceofoil
mar
ket:
arriva
lofhed
ge
funds
Fig
ure
1:W
TIO
ilfo
rwar
dcu
rve,
Mar
ch20
06
–Typ
eset
byFoi
lTEX
–7
Sep
tem
ber
2007
–Typ
eset
byFoi
lTEX
–8
Sea
sonalit
yin
com
modity
price
s
•For
oil,
seas
onal
ity
isnot
sign
ifica
nt,
since
tanke
rsar
ere
route
dto
satisf
ya
surg
eof
dem
and
ina
give
nre
gion
•Ener
gy(e
lect
rici
ty,nat
ura
lga
s,sp
ark
spre
ad)
-go
vern
edby
seas
onal
dem
and
•A
gric
ultura
lco
mm
oditie
s(w
hea
t,so
ybea
n,
soym
eal,
crush
spre
ad,
coffee
,co
coa)
-
gove
rned
byse
ason
alsu
pply
•Sea
sonal
ity
inen
ergy
orag
ricu
ltura
lsp
otpr
ices
:
wel
l-under
stood
and
easily
model
led
(e.g
.m
ean-r
ever
sion
with
seas
onal
lyva
ryin
gm
ean,
seas
onal
com
pon
ent
+au
tore
gres
sion
)
•Sea
sonal
ity
info
rwar
dcu
rves
:
much
less
studie
d;no
”exp
licit”
model
s
–Typ
eset
byFoi
lTEX
–9
Exa
mple
ofN
atu
ralG
as
forw
ard
curv
e:
010
2030
4050
6070
6.57
7.58
8.59
9.5
Mon
ths
to m
atur
ity
Sterling pence per thermN
atur
al g
as fo
rwar
d cu
rve,
Mar
ch 7
, 200
7
–Typ
eset
byFoi
lTEX
–10
Futu
res
vs.
spot
price
s:T
heo
ryofSto
rage
Cos
t-of
-car
ryre
lation
ship
(no-
arbitra
gear
gum
ents
):
F(t
,T
)=
S(t
)e[r
(t)+
w(t
)](T
−t)
(∗)
r(t
) :sp
otin
tere
stra
te,w
(t):
mar
ginal
stor
age
cost
sper
$of
spot
,per
tim
eunit.
Inpr
actice
(∗)
alm
ost
nev
erhol
ds
(e.g
.bac
kwar
dat
ion
orhum
p-s
hap
edfo
rwar
dcu
rves
):
stra
tegi
cim
por
tance
(oil)
;lim
ited
ornon
-sto
rabili
ty(a
gric
ultura
l,el
ectr
icity)
Con
veni
ence
ofhav
ing
phys
ical
com
modity
asop
pos
ite
tofu
ture
sco
ntr
act
−→co
nce
pt
ofco
nven
ienc
eyi
eld
y(t
):
F(t
,T
)=
S(t
)e[r
(t)−
y(t
)](T
−t)
-pr
emiu
m(a
spe
rcei
ved
onth
eda
yt)
earn
edby
anow
ner
ofph
ysic
alco
mm
odity
asop
posi
teto
anow
ner
ofth
efu
ture
sco
ntra
ctwith
mat
urity
T.
–Typ
eset
byFoi
lTEX
–11
Conve
nie
nce
yiel
d
•O
ften
consider
ednet
mar
ginal
stor
age
cost
s:y(t
)=
y(t
)−
w(t
).
•Con
venie
nce
yiel
dpr
opor
tion
alto
the
spot
pric
ey(S
):B
rennan
&Sch
war
tz(1
985)
.
•Sto
chas
tic
conve
nie
nce
yiel
dy(t
)=
y(t
,ω):
Gib
son
&Sch
war
tz(1
990)
,Sch
war
tz
(199
7).
•D
epen
den
ceon
t:”p
rem
ium
”to
owner
ofphys
ical
com
modity
chan
ges
with
inve
nto
ries
(sto
cks)
and
hen
ce,w
ith
agen
ts’pr
efer
ence
for
phys
ical
rath
erth
anpap
er.
•At
afixe
ddat
et,
asingl
eva
lue
ofth
epr
oce
ss(y
(t))
tfo
ral
lm
atur
itie
sis
not
com
pat
ible
with
the
hum
p-s
hap
edfo
rwar
dcu
rve
obse
rved
in20
06in
the
oilm
arke
t(a
nd
other
com
modity
mar
kets
),or
with
seas
onal
feat
ure
sof
the
forw
ard
curv
e.
–Typ
eset
byFoi
lTEX
–12
Theo
ryofSto
rage
revi
site
d
•O
ne
pos
sible
model
ling
answ
eris
toin
troduce
ate
rmst
ruct
ure
y(t
,T
)of
conve
nie
nce
yiel
ds
atdat
et,
det
erm
inistic
inth
em
aturity
argu
men
tT
and
stoch
astic
int
(Bor
ovko
va&
Gem
an,20
06,20
07)
•T
his
appr
oach
isce
rtai
nly
ben
efici
alin
the
case
ofse
ason
alco
mm
oditie
ssu
chas
nat
ura
lga
sw
her
e,as
sum
ing
today
=Ja
nuar
y20
08,y(t
,T
)sh
ould
be
diff
eren
tfo
rT
=
Sep
tem
ber
2008
orT
=D
ecem
ber
2008
.
•D
epen
den
ceof
conve
nie
nce
yiel
don
mat
urity
T(y
(t,T
)):
toem
phas
ize
seas
onal
ity
of
F(t
,T
)in
F(t
,T
)=
S(t
)e[r
(t)−
y(t
,T)]
(T−
t)
e.g.
futu
res
expirin
gat
”des
irab
le”
seas
on(e
.g.
NG
futu
res
expirin
gin
Dec
ember
)
Em
phas
izes
the
tim
e-sp
read
option
feat
ure
ofco
nve
nie
nce
.
–Typ
eset
byFoi
lTEX
–13
Forw
ard
curv
em
odel
s
One,
two
and
thre
efa
ctor
model
s:sp
otpr
ice,
conve
nie
nce
yiel
dan
din
tere
stra
te
(Bla
ck(1
976)
,G
ibso
n&
Sch
war
tz(1
990)
,Sch
war
tz(1
997)
)
Futu
res
pric
esar
eder
ived
byno-
arbitra
gear
gum
ents
:F
(t,T
)=
EQ[S
(T)|F
t].
Sea
sonal
com
moditie
s(S
oren
sen
(200
2)an
dLuci
a&
Sch
war
tz(2
002)
):
Two-
fact
orm
odel
sw
ith
seas
onal
spot
pric
ean
da
long-
term
equili
briu
mpr
ice.
Sea
sonal
ity
ente
rsth
efu
ture
spr
ice,
but
not
inan
explic
itan
dco
nsist
ent
way
.
One
step
forw
ard:
Am
in,N
g&
Pirro
ng
(199
4):
seas
onal
(but
det
erm
inistic)
conve
nie
nce
yiel
d,on
efu
ndam
enta
lfa
ctor
:sp
otpr
ice,
cost
-of-ca
rry
rela
tion
ship
.
Mai
ndra
wbac
ksof
allab
ove
model
s:
Spot
pric
eis
not
ago
od
indic
ator
ofov
eral
lst
ate
ofth
em
arke
t.
For
war
dcu
rve’
sse
ason
alfe
ature
sar
enot
take
nin
toac
count
explic
itly
=⇒
Model
sdo
not
mat
chob
serv
edfo
rwar
dcu
rves
.
–Typ
eset
byFoi
lTEX
–14
Sea
sonalco
st-o
f-ca
rry
model
:First
fundam
enta
lfa
ctor
The
aver
age
leve
lof
the
forw
ard
curv
e,or
the
aver
age
forw
ard
pric
epr
evai
ling
atdat
et:
F(t
)=
N√ √ √ √N ∏ T=
1
F(t
,T
),or
lnF
(t)
=1 N
N ∑ T=
1
lnF
(t,T
),
wher
eN
:m
axim
um
liquid
mat
urity
.
•A
ssum
e:(N
mod
12)
=0,i.e.
consider
mat
urities
up
toa
(num
ber
of)
year
(s)
−→th
atway
F(t
)is
not
seas
onal
.
•O
ther
way
sof
const
ruct
ing
anon
-sea
sonal
F(t
) ,so
the
assu
mption
can
be
rela
xed
•N
otlim
ited
tore
gula
rly
spac
edm
aturities
but
can
incl
ude
alltr
aded
liquid
mat
urities
•Can
incl
ude
all(n
otliq
uid
)m
aturities
,by
consider
ing
trad
ed-v
olum
e–wei
ghte
dav
erag
e
–Typ
eset
byFoi
lTEX
–15
Sea
sonalco
st-o
f-ca
rry
model
:Sea
sonalpre
miu
m
The
seas
onal
prem
ium
(s(M
))M
=1,.
..,1
2is
the
colle
ctio
nof
long
-ter
m–a
vera
gepr
emia
(exp
ress
edin
%)on
futu
res
expi
ring
inth
eca
lend
arm
onth
M(M
=1,..
.,12)
with
resp
ectto
the
aver
age
forw
ard
pric
eF
(t) .
•A
ssum
e(s
(1),
...,
s(1
2))
isth
edet
erm
inistic
colle
ctio
nof
12par
amet
ers;
•Req
uire
that
∑12
M=
1s(M
)=
0;
•Sea
sonal
prem
ium
isan
abso
lute
quan
tity
and
not
ara
te:
prem
ium
onfu
ture
sex
pirin
g
inJu
lyis
the
sam
ew
het
her
today
isJu
ne
orD
ecem
ber
.Pre
miu
mon
July
2008
futu
res
is
the
sam
eas
onJu
ly20
09fu
ture
s.
•Can
be
defi
ned
asa
continuou
s-tim
eper
iodic
funct
ion
(e.g
.tr
igon
omet
ric)
;how
ever
less
appr
opriat
efo
rm
onth
lyex
pirie
s.
–Typ
eset
byFoi
lTEX
–16
Sea
sonalco
st-o
f-ca
rry
model
:T
he
model
For
any
mat
urity
T,we
write
F(t
,T
)=
F(t
)e[s
(T)−
γ(t
,T)(
T−
t)] ,
(∗)
wher
eγ(t
,T
),de
fined
byth
ere
lation
ship
(∗),
isca
lled
the
stoc
hast
icco
nven
ienc
eyi
eld
netof
seas
onal
prem
ium
,fo
rm
aturity
T,as
per
ceiv
edon
the
day
t.
Sea
sonal
(mon
thly
)pr
emiu
m(o
rdisco
unt)
:in
s(T
)
Sto
chas
tic
fact
ors
influen
cing
forw
ard
pric
es:
inγ(t
,T
)
The
rela
tion
ship
(∗)
invo
lves
only
forw
ard
pric
es,hen
ceno
inte
rest
rate
s.
–Typ
eset
byFoi
lTEX
–17
Fea
ture
softh
em
odel
Rel
atio
nsh
ipto
clas
sic
conve
nie
nce
yiel
dm
odel
s:
γ(t
,T
)−
s(T
)
T−
t=
y(t
,T
)−
1 N
N ∑ K=
1
y(t
,K
)
−→γ(t
,T
)ca
nbe
inte
rpre
ted
asth
ere
lative
conv
enie
nce
yiel
dne
tof
the
(sca
led)
seas
onal
prem
ium
.
•Con
venie
nce
yiel
dγ
can
be
use
dfo
rnon
-sto
rable
com
moditie
s(e
.g.
elec
tric
ity)
,
since
spot
pric
epla
ysno
role
•If
γ(t
,T
)≡
0−→
one-
fact
orm
odel
drive
nby
F(t
)an
ddet
erm
inistic
s(T
)
•If
s(T
)=
0∀T
,th
enno
det
erm
inistic
seas
onal
ity
(e.g
.oi
l)an
dγ(t
,T
)is
the
”rel
ativ
eco
nve
nie
nce
yiel
d”−→
two-
fact
orm
odel
sim
ilar
toG
ibso
n&
Sch
war
tz(1
990)
but
with
F(t
)in
stea
dof
the
spot
pric
e.
–Typ
eset
byFoi
lTEX
–18
Dyn
am
ics
offu
ndam
enta
lfa
ctors
and
futu
res
price
s
F(t
)is
not
seas
onal
byco
nst
ruct
ion
−→ca
nbe
model
led
asa
mea
n-r
ever
sion
with
const
ant
mea
n,or
GB
M.
γ(t
,T
)is
esse
ntial
lyze
ro(o
nav
erag
e),since
allsy
stem
atic
dev
iation
sar
ein
s(T
)
−→ca
nbe
model
led
asa
mea
n-r
ever
sion
with
mea
nze
ro.
All
stoch
astic
conve
nie
nce
yiel
ds
(γT(t
))T
=1,.
..,N
are
drive
nby
the
sam
eB
row
nia
n
mot
ion,in
dep
enden
tof
the
BM
drivi
ng
the
aver
age
forw
ard
pric
e.
Sea
sonal
cost
-of-ca
rry
+dyn
amic
sof
(F(t
),γ
T(t
))=
dyn
amic
sof
(F(t
,T
))T.
Res
ultin
gfu
ture
spr
ices
F(t
,T
)ar
elo
g-nor
mal
with
inst
anta
neo
us
prop
ortion
alva
rian
ce
ξ2(t
,T
)=
σ2+
(ηT(T
−t)
)2−
2σ
ρη
T(T
−t)
–Typ
eset
byFoi
lTEX
–19
Model
estim
ation
Histo
rica
ldat
aof
dai
lyfo
rwar
dcu
rves
(F(t
,1),
...,
F(t
,12))
t=1,.
..,n
.
Est
imat
e
•th
edai
lyav
erag
efo
rwar
dpr
ice
byln
F(t
)=
1 12
∑12
M=
1ln
F(t
,M
);
•th
ese
ason
alpr
emia
(s(M
))M
,ac
cord
ing
toth
edefi
nitio
n,by
s(M
)=
1 n
n ∑ t=1
[ln
F(t
,M
)−
lnF
(t)]
,M
=1,..
.,12,
•th
est
och
astic
conve
nie
nce
yiel
dby
γ(t
,T
)=
(−ln
(F(t
,T
)/F
(t))
+s(T
))/((
T−
t)).
Mor
eth
an12
mat
urities
:ea
sily
inco
rpor
ated
,but
iffe
wer
than
12m
aturities
,th
e
unbia
sed
estim
ate
for
F(t
)is
not
avai
lable
−→a
mor
eco
mplic
ated
estim
atio
npr
oce
dure
.
–Typ
eset
byFoi
lTEX
–20
Sea
sonalpre
miu
mfo
rN
atu
ralG
as
futu
res
12
34
56
78
910
11
12
−0.2
−0.1
5
−0.1
−0.0
50
0.0
5
0.1
0.1
5
0.2
0.2
5
0.3
Cale
ndar
month
Seasonal premium in %
Seasonal pre
miu
m for
NG
futu
res
–Typ
eset
byFoi
lTEX
–21
Sea
sonalpre
miu
mfo
rel
ectr
icity
futu
res
12
34
56
78
910
11
12
−0.2
−0.1
5
−0.1
−0.0
50
0.0
5
0.1
0.1
5
Cale
ndar
month
%
Seasonal fo
rward
pre
miu
m, E
lectr
icity futu
res
–Typ
eset
byFoi
lTEX
–22
Sea
sonalpre
miu
mfo
rG
aso
ilfu
ture
s
12
34
56
78
910
11
12
−0.0
3
−0.0
2
−0.0
10
0.0
1
0.0
2
0.0
3
Cale
ndar
month
s
%
Gasoil
seasonal com
ponent
–Typ
eset
byFoi
lTEX
–23
Sea
sonalpre
miu
mfo
rsp
ark
spre
ad
12
34
56
78
910
11
12
−0.6
−0.5
−0.4
−0.3
−0.2
−0.10
0.1
0.2
0.3
0.4
Cale
ndar
month
%
Seasonal fo
rward
pre
miu
m, S
park
spre
ad
–Typ
eset
byFoi
lTEX
–24
Ter
mst
ruct
ure
ofst
och
ast
icfo
rwar
dpre
miu
mvo
latilit
ies,
Gaso
ilfu
ture
s
12
34
56
78
91
01
11
20
0.0
2
0.0
4
0.0
6
0.0
8
0.1
0.1
2
0.1
4
Ma
turity
%
Vo
latility
of
(Ga
mm
a(t
,T))
, T
=1
,...
,12
–Typ
eset
byFoi
lTEX
–25
Ter
mst
ruct
ure
ofst
och
ast
icfo
rwar
dpre
miu
mvo
latilit
ies,
Natu
ralG
as
futu
res
12
34
56
78
91
01
11
20
0.0
5
0.1
0.1
5
0.2
0.2
5
0.3
0.3
5
Ma
turity
%
Vo
latility
of
(Ga
mm
a(t
,T))
, T
=1
,...
,12
–Typ
eset
byFoi
lTEX
–26
The
seco
nd
state
variable
(sto
chast
icfo
rwar
dpre
miu
m),
for
two
month
sto
matu
rity
,G
aso
ilfu
ture
s,Jan.
2000
-D
ec.
2004
02
00
40
06
00
80
01
00
01
20
0−
0.8
−0
.6
−0
.4
−0
.20
0.2
0.4
0.6
0.8
Ga
so
il c
on
ve
nie
nce
yie
ld,
tau
=2
Tra
din
g d
ays s
ince
13
.01
.20
00
–Typ
eset
byFoi
lTEX
–27
Pro
per
ties
ofth
eco
nve
nie
nce
yiel
d
•All
obse
rved
series
(γ(t
,T))
tca
nbe
model
led
bylo
w-o
rder
auto
regr
ession
(ord
er2
-5)
−→au
tore
gres
sive
stru
cture
can
be
explo
ited
for
-fo
reca
stin
gth
est
och
astic
conve
nie
nce
yiel
d-
fore
cast
ing
mar
ket
conditio
ns
-dev
isin
gm
arke
tin
dic
ator
s-
gener
atin
gpr
ofita
ble
trad
ing
stra
tegi
es•
Con
venie
nce
yiel
dca
nbe
regr
esse
don
econ
omic
fundam
enta
lsan
dex
ogen
ous
mar
ket
variab
les,
e.g.
supply
/dem
and,vo
latilit
y,...
Mea
n-r
ever
sion
par
amet
ers
(T=
2):
elec
tric
ity
gas
gaso
iloi
la
T:
0.07
0.09
0.02
0.01
ηT:
0.16
0.10
0.05
0.04
–Typ
eset
byFoi
lTEX
–28
Rel
ationsh
ipof
γ(t
,T)
tom
arke
tin
dic
ato
rsand
econom
icfu
ndam
enta
ls
Theo
ryof
stor
age
+em
piric
alco
nsider
atio
ns
−→co
nje
cture
sab
out
the
conve
nie
nce
yiel
d:
I.
Itis
pos
itiv
ely
corr
elat
edto
the
over
all
pric
ele
vel
(giv
enby
eith
ersp
otpr
ice
orav
erag
efo
rwar
dpr
ice)
II.It
isneg
ativ
ely
corr
elat
edto
inve
nto
ries
III.It
ispos
itiv
ely
corr
elat
edto
spot
pric
e’s
vola
tilit
y
IV.
Itis
neg
ativ
ely
corr
elat
edto
the
corr
elat
ion
bet
wee
nsp
otan
dfu
ture
spr
ices
.
–Typ
eset
byFoi
lTEX
–29
Conje
cture
I:tr
ue,
espec
ially
forhig
her
matu
rities
:G
aso
il
55.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
−0.0
50
0.0
5
0.1
0.1
5
0.2
Log F
bar
Stochastic convenience yield, T=12
Sto
chastic c
onvenie
nce y
ield
, T
=12, vs a
vera
ge forw
ard
price
–Typ
eset
byFoi
lTEX
–30
Conje
cture
I:tr
ue,
espec
ially
for
hig
her
matu
rities
:N
G
51
01
52
02
53
03
5−
0.5
−0
.4
−0
.3
−0
.2
−0
.10
0.1
0.2
0.3
0.4
NG
M f
utu
res p
rice
, p
en
ce
/te
rm
Stochastic 6−month convenience yield
Sto
ch
astic c
on
v.
yie
ld v
s N
G M
price
: b
lue
: 0
1.9
7−
03
.00
, re
d:
03
.00
−0
2.0
2
–Typ
eset
byFoi
lTEX
–31
Ext
ract
ing
the
seaso
nalco
mponen
t
Sea
sonal
com
pon
ent
is”k
now
n”
mon
thly
prem
ium
−→ex
trac
tit
from
afo
rwar
dcu
rve.
Ifse
ason
ality
was
the
only
det
erm
inin
gfa
ctor
,th
enw
hat
isle
ftsh
ould
alway
sbe
flat
,but
itis
not
!=⇒
situ
atio
ns
sim
ilar
tobac
kwar
dat
ion/c
onta
ngo
arise:
12
34
56
78
9−
1.5
−1
−0.
50
0.51
Mon
ths
to e
xpiry
Sterling/MWh
De−
seas
oned
ele
ctric
ity fo
rwar
d cu
rve,
27.
06.0
1
12
34
56
78
9−
1
−0.
50
0.51
1.52
2.5
Mon
ths
to e
xpiry
Sterling/MWh
De−
seas
oned
ele
ctric
ity fo
rwar
d cu
rve,
15.
12.0
1
–Typ
eset
byFoi
lTEX
–32
Princi
palCom
ponen
tA
naly
sis
ofth
efo
rwar
dcu
rve
Case
I:in
tere
stra
tes
and
non-s
easo
nalco
mm
oditie
s(o
il)
Afo
rwar
dcu
rve
ofal
mos
tan
ysh
ape
can
be
const
ruct
edby
com
bin
ing
thre
esim
ple
shap
es:
Lev
el,Slo
pe,
Curv
ature
−→Princi
pal
Com
pon
ents
of(F
(t))
t∈N
=(F
1(t
),F
2(t
),..
.,F
N(t
))t∈
N
02
46
810
1214
1618
200
0.050.
1
0.150.
2
0.250.
3
0.35
Exp
iry0
24
68
1012
1416
1820
−0.
5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
Exp
iry0
24
68
1012
1416
1820
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
0.5
Exp
iry
First
thre
epr
inci
pal
com
pon
ents
expla
inap
prox
.99
%(!
)of
the
forw
ard
curv
e’s
variab
ility
.
(Litte
rman
&Sch
einkm
ann
’91
for
US
gove
rnm
ent
bon
ds,
Cor
taza
r&
Sch
war
tz’9
4fo
r
copper
)
–Typ
eset
byFoi
lTEX
–33
Princi
palCom
ponen
tsofdaily
retu
rns
02
46
810
1214
1618
200
0.050.
1
0.150.
2
0.250.
3
0.35
Exp
iry0
24
68
1012
1416
1820
−0.
5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
Exp
iry0
24
68
1012
1416
1820
−0.
4
−0.
20
0.2
0.4
0.6
0.81
Exp
iry
Thes
efirs
tth
ree
prin
cipal
com
pon
ents
hav
ecl
ear
econ
omic
inte
rpre
tation
,ex
pla
in95
%of
the
tota
lva
riab
ility
,ca
nbe
trea
ted
asth
em
ain
risk
fact
ors
gove
rnin
gth
efu
ture
spr
ices
’ev
olution
.
–Typ
eset
byFoi
lTEX
–34
Applic
ations
ofPCA
I.For
ecas
ting
mar
ket
tran
sition
s(b
etwee
nbac
kwar
dat
ion
and
conta
ngo
):
•T
he
seco
nd
prin
cipal
com
pon
ent
reflec
tsth
eslop
eof
the
forw
ard
curv
e
•Val
ues
clos
eto
0in
dic
ate
aflat
forw
ard
curv
e(a
nd
hen
ce,pos
sible
tran
sition
)
•D
ue
tosm
oot
htim
e-se
ries
-lik
est
ruct
ure
,it
can
be
use
dto
const
ruct
anin
dic
ator
whic
h
antici
pat
espos
sible
tran
sition
s
Bor
ovko
va,EPRM
mag
azin
e(J
une
2003
).
II.Por
tfol
iorisk
man
agem
ent
and
VaR
estim
atio
n
•First
few
prin
cipal
com
pon
ents
(ofre
turn
s)re
flec
tm
ain
risk
fact
ors
=⇒
the
num
ber
orrisk
fact
ors
isgr
eatly
reduce
d
•T
he
distr
ibution
ofpor
tfol
iore
turn
sca
nbe
appr
oxim
ated
via
the
distr
ibution
ofth
em
ain
risk
fact
ors
•In
apor
tfol
ioco
nte
xt,th
ese
risk
fact
ors
can
be
hed
ged
–Typ
eset
byFoi
lTEX
–35
Princi
palCom
ponen
tIn
dic
ato
r
”Raw
”ve
rsio
n:
proj
ection
ofth
edai
lyfo
rwar
dcu
rve
onth
ese
cond
PC
I(t
)=
N ∑ k=
1
PC
L(2
)k
Fk(t
),
wher
eP
CL
(2)
k(k
=1,..
.,N
)ar
ese
cond
prin
cipal
com
pon
ent
load
ings
ofth
efu
ture
s
pric
esse
ries
.
MA
-sm
oot
hed
vers
ion:
IM
A(t
)=
1 M
M−
1∑ i=
0
I(t
−i)
.
Choi
ceof
M:
take
afa
stan
dslow
mov
ing
aver
age.
–Typ
eset
byFoi
lTEX
–36
Applic
ation
ofth
ePC
indic
ato
rto
Bre
nt
oil
futu
res
Gen
erat
ea
”sig
nal
ofch
ange
”w
hen
the
indic
ator
ente
rsso
me
ε-n
eigh
bor
hood
ofze
ro:
010
020
030
040
050
060
070
080
090
010
00−
1
−0.
50
0.51
1.52
2.53
Day
s
$/bbl
010
020
030
040
050
060
070
080
090
010
00−
15
−10−
5051015
Day
s
$
PC
test
sta
tistic
s w
ith in
term
onth
diff
eren
ces
ε-n
eigh
bor
hood
det
erm
ined
via
the
distr
ibution
ofth
ein
dic
ator
(under
the
null-
hyp
othes
is
ofno
chan
ge),
appr
oxim
ated
byei
ther
Mon
te-C
arlo
orboot
stra
pdistr
ibution
.
–Typ
eset
byFoi
lTEX
–37
PCA
for
seaso
nalco
mm
oditie
s(e
lect
rici
ty,N
G)
Apply
PCA
todes
easo
nal
ized
forw
ard
curv
esF
(t,T
)exp
(−s(
T))
)Sec
ond
and
third
PCs
for
de-
seas
oned
FC
(firs
tPC
=le
vel)
still
reflec
tth
eslop
ean
dcu
rvat
ure
:
12
34
56
78
9−
0.4
−0.
20
0.2
0.4
0.6
0.81
Mon
ths
to e
xpiry
PC1 loadings
Firs
t prin
cipa
l com
pone
nt lo
adin
gs, d
e−ce
nter
ed, d
e−se
ason
ed e
lect
ricity
FC
12
34
56
78
9−
0.8
−0.
6
−0.
4
−0.
20
0.2
0.4
0.6
Mon
ths
to e
xpiry
PC2 loadings
Sec
ond
prin
cipa
l com
pone
nt lo
adin
gs, d
e−ce
nter
ed, d
e−se
ason
ed e
lect
ricity
FC
–Typ
eset
byFoi
lTEX
–38
Applic
ations
ofPCA
:tr
adin
g
•For
des
easo
nal
ized
forw
ard
curv
es,
situ
atio
ns
anal
ogou
sto
bac
kwar
dat
ion/c
onta
ngo
mar
kets
arise,
inte
rms
ofdev
iation
sfrom
the
”typ
ical
”se
ason
alfo
rwar
dcu
rve.
•H
igh
abso
lute
valu
esof
the
seco
nd
PC
indic
ates
whet
her
futu
res
with
shor
ter
(lon
ger)
expirie
sar
eov
erpr
iced
w.r.t.
”typ
ical
”se
ason
alpr
emiu
m
•A
gain
,use
PC
indic
ator
:th
epr
ojec
tion
ofth
edai
lydes
easo
nal
ized
forw
ard
curv
eon
the
seco
nd
PC.
–Typ
eset
byFoi
lTEX
–39
Princi
palCom
ponen
tIn
dic
ato
rfo
rel
ectr
icity
FC
Ava
lue
ofth
ein
dic
ator
far
from
zero
sign
als
sign
ifica
nt
dev
iation
from
the
”exp
ecte
d”
seas
onal
forw
ard
curv
epat
tern
:
050
100
150
200
250
−4
−3
−2
−10123
Day
s
PC1 scores
Firs
t prin
cipa
l com
pone
nt s
core
s, d
e−ce
nter
ed, d
e−se
ason
ed e
lect
ricity
FC
Can
const
ruct
profi
table
trad
ing
stra
tegi
esbas
edon
the
indic
ator
.(B
orov
kova
&G
eman
(SN
DE
2006
)).
–Typ
eset
byFoi
lTEX
–40
Concl
usions
•Ext
ract
ing
det
erm
inistic
seas
onal
ity
from
forw
ard
curv
esal
low
sto
study
feat
ure
sob
scure
dby
dom
inan
tse
ason
aleff
ects
(PCA
,co
st-o
f-ca
rry,
trad
itio
nal
term
stru
cture
model
s)
•A
vera
gefo
rwar
dpr
ice
isa
robust
iden
tifier
ofth
eov
eral
lpr
ice
leve
l,m
ore
soth
anth
esp
otpr
ice,
alth
ough
now
nee
dto
take
into
acco
unt
slop
ing
forw
ard
curv
es
•Sto
chas
tic
conve
nie
nce
yiel
dis
aquan
tity
indic
ativ
eof
mar
ket
stat
ean
dec
onom
icin
dic
ator
s;it
can
be
explo
ited
toco
nst
ruct
mar
ket
indic
ator
san
dge
ner
ate
profi
table
trad
ing
stra
tegi
es
–Typ
eset
byFoi
lTEX
–41
Per
spec
tive
rese
arch
direc
tions
•Bac
kwar
dat
ion/c
onta
ngo
-lik
epr
ofile
inse
ason
alfo
rwar
dcu
rves
•Applic
atio
ns
ofth
em
odel
toth
eder
ivat
ives
pric
ing
•Rel
atin
gth
est
och
astic
conve
nie
nce
yiel
dto
econ
omic
and
other
exog
enou
sva
riab
lessu
chas
stock
s(s
upply
),ex
trem
ewea
ther
conditio
ns(d
eman
d),
...
.
•Model
ling
the
entire
term
stru
cture
ofco
nve
nie
nce
yiel
ds,
with
anum
ber
ofso
urc
esof
unce
rtai
nty
and
vola
tilit
yfu
nct
ions
•Sea
sonal
term
stru
cture
offu
ture
spr
ices
’vo
latilit
ies
•Applic
atio
ns
ofth
em
odel
toag
ricu
ltura
lco
mm
oditie
s
–Typ
eset
byFoi
lTEX
–42
