Munich Personal RePEc Archive
Stochastic models for the spot and future
prices of commodities with high volatility
and mean reversion
García de la Vega, Victor Manuel and Ruiz-Porras, Antonio
Universidad de Guadalajara, CUCEA
10 October 2009
Online at
https://mpra.ub.uni-muenchen.de/23177/
Modelos Estocásticos para el Precio Spot y del Futuro de Commodities con Alta Volatilidad y Reversión a la Media
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$ % &'" ('')
Resumen
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3 " + 0 + " + & 3 1
Abstract
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+ 6 6 01 8 - 3 + - 6 - 3 6
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6 7 9 0" %0 & 3 + 1
Clasificación JEL: &(" &:
Palabras Clave: " " *
Keywords: 7 " 9 7 " +
2 * ; 1 9 <&" (" 1 6 - - " => " 21 1
&&?@'" => 1 ; ,?(A??. ::''A?B<'" #A+ ; 1 C3 1 +
2 * ; 3= @))" * " & " D E "
1 1 B?&''" F - - " G " => 1 ; ,?(A::. :@@'A::''" #> 1 ?()&" #A+ ;
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Modelos Estocásticos para el Precio Spot y del Futuro de Commodities con Alta Volatilidad y Reversión a la Media
1. Introducción
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Tabla 1. Volatilidades de Commodities
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t , t, t Nh.1# 3 3 7 3 "
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d
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3. Valuación 3eutral al Riesgo
3.1 Proceso Propuesto para Obtener Precio Spot
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t RF , t.eY
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" 0 - + + / -
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t "+ - ,% ! . 3 * - "
-= % r1
+ * ,(." H - " 0 + + / - +
/ "
t
t
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dt
t
dW
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=
−
α
+
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,
.
t
Y
% / + " - +/ + %0
Y
t " - % " +- * +- 4 5"c Rc,
Y
t , t." 0 + L" $% + "t t
t t
t
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c
t
dt
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c
t
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c
Y
t
c
dc
∂
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+
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∂
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∂
−
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∂
=
,
.
,
.
(
&
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(
σ
σ
α
(23)$ + >- dc / 0 H H - "
dc
=
ccdt
+
σ
ccdW
t (24)" ,(:. 0 ,(B. - + % / "
∂
∂
+
∂
∂
−
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∂
=
( ( (.
,
(
&
&
t t
t c
Y
c
t
Y
c
Y
t
c
c
α
σ
(25)0
t c
Y
c
t
c
∂
∂
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σ
,
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σ
(26)+ % ! " % + 3 + - 3
- - +
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t" / D 3 - - 3 12 % + - 3 ( " H ( - +- " c& R
&,
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t " ." 0 ( R (,Y
t " .1 + + " % + D+ /&(
+ c& R c&,
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t " ." 0 6 + +Θ
( D+ /+ ( R (,S
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t " . - 3 1 # - 3 H- 1 # - % "
( ( &
&
c
c
t
=
Θ
+
Θ
Π
" +% - 3 " % 3
+ " H - "
( ( &
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dc
dc
d
Π
t=
Θ
+
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t
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t
c
t
dt
c
dt
c
.
,
,
.
,
.
.
,
Θ
& & &+
Θ
( ( (+
Θ
&σ
& &+
Θ
(σ
( (=
6 + 3 /
t
dW
c
t
dt
c
dc
&=
& &+
σ
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.
&0 /
t
dW
c
t
dt
c
dc
(=
( (+
σ
(,
.
(% + "
..
,
.
,
,
.
,
..
,
.
,
,
.
,
( &
(
( & (
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&
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t
t
C
dt
c
t
t
t
C
dt
t
c
d
tσ
σ
σ
σ
σ
σ
−
−
−
=
Π
dt
r
t tΠ
=
..
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.
,
,
.
,
..
,
.
,
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,
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(
( &
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t
t
C
dt
c
t
r
t
t
C
dt
t
c
r
t tσ
σ
σ
σ
σ
σ
−
−
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=
# + = + " + >- * "
t
t
t
r
r
t
t
t
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,
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( & ( & (&
σ
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λ
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,
((
& &
t
r
t
r
t t+ 3 + - 3 6 + " - +
λ
+ "
c t
c
r
σ
λ
=
−
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t t t t tY
c
t
c
r
Y
c
t
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c
Y
t
c
c
∂
∂
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∂
+
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∂
−
∂
∂
=
σ
σ
α
λ
2 - ! >- * " % + * 3
-"
'
.
,
(
&
.
,
( ( (−
=
∂
∂
+
∂
∂
−
∂
∂
−
∂
∂
c
r
Y
c
t
Y
c
t
Y
c
Y
t
c
t t t tt
λσ
σ
α
= + % + "
'
.
,
(
&
.
,
( ( (−
=
∂
∂
+
∂
∂
−
−
+
∂
∂
c
r
Y
c
t
Y
c
t
Y
t
c
t t tt
α
σ
λσ
α
6 3 + "
α λσ
γ =− ,t. (28)
" 0 >- * ,(I. * 3 - - "
% + * 3 - "
(
)
,
.
'
(
&
( ( (−
=
∂
∂
+
∂
∂
−
+
∂
∂
c
r
Y
c
t
Y
c
Y
t
c
t t t tσ
γ
α
2 / + / * ,(." + + / - + / "
t t
t
Y
dt
t
dW
dY
=
−
α
+
σ
,
.
3 + >- * "
dY
t=
α
,
γ
−
Y
.
tdt
+
σ
,
t
.
dW
t (30)+ / 6 0 % / - * ,:'. / %
- * ,(."
γ
0dW
t 1 >- *γ
H 3 *,(I. 0 - + % / >- * 0
λ
" + + / 6 + 3* ,(@.1 2 % + /
λ
3 + - +-%
Y
t1 +λ
- 3 *Y
t 0 t"- - 3 H - " + + /
λ
1 %t
dW
+ + + % 7 H " *-6 + " - %
dW
t / - 1- % 3* + >- - - - ++ 0
S
t"&B ( ( (
.
,
(
&
.
,
t t t t t t t t t tdW
Y
S
t
dt
Y
S
Y
S
Y
t
S
dS
∂
∂
+
∂
∂
+
∂
∂
−
+
∂
∂
=
α
γ
σ
σ
(
.
,
.
,
(
&
.
,
.
,
&
t t tt
S
F
t
t
dt
t
S
dW
dt
t
F
d
S
σ
σ
α
γ
α
α
+
+
−
+
+
=
(31)+ /
Y
t RlnS
t Aln F,t." / % - !Y
t *t
t
F
t
F
dt
t
F
d
∂
∂
=
,
.
.
,
&
.
,
+ + /,
.
,
.
(
&
.
,
&
(t
F
t
dt
t
F
d
t
+
+
=
σ
α
ρ
>- * ,:&. /"
dS
t=
α
S
t,
ρ
,
t
.
+
γ
−
S
t.
dt
+
σ
,
t
.
dW
t (32)M + +% %
r
t RlnS
t - - - + L* ,:(."
[
]
( ( ( (.
,
.
,
(
&
.
,
t t t t t t t t t t t t t tdW
S
S
S
t
dt
S
S
S
t
S
S
S
t
S
t
S
S
d
dr
∂
∂
+
∂
∂
+
∂
∂
−
+
+
∂
∂
=
=
α
ρ
γ
σ
σ
[
[
]
(,
.
]
,
.
(
&
.
,
t
γ
r
tσ
t
dt
σ
t
dW
tρ
α
+
−
−
+
=
(33)
3 + +H - *
S
t" +[
[
α ρ γ]
σ]
α αγβ = + − − , . = , .+ , .+
( & .
, ( F t
dt t F d t r
t t 1
" % + "
.
,
.
,
t tt
t
dt
r
d
t
dW
dr
=
β
−
α
+
σ
[
β
,
t
.
dt
−
α
r
t]
dt
+
σ
,
t
.
dW
t=
# +- / D + >- * + 9" 0 -
-+ +H + 3 + >- * - $ A
6 % 9" - " 3 +
m
t=
α
r
t−
β
t=
−
[
β
,
t
.
−
α
r
t]
1
&?
[
,
.
]
,
.
&
t t tt
dm
d
t
m
dt
t
dW
dr
β
σ
α
+
=
−
+
=
2 - !
dm
t % + "
dm
t=
−
[
α
m
t+
β
′
,
t
.
]
=
dt
+
ασ
,
t
.
dW
t (34)* ,:B. + * * 3 6 + =
- + 1 " - + 6 +% % "
t t
t
m
e
X
=
α 0 - + + LX
t - H* ,:B." % "
[
]
( ( ( (.
,
.
,
(
&
.
,
.
,
t t t t t t t t t t t tdW
m
X
t
dt
m
X
t
m
X
t
m
t
X
e
m
d
dX
∂
∂
+
∂
∂
+
∂
∂
′
+
+
∂
∂
=
=
αα
β
α
σ
ασ
(35)
+ * ,:?."
t t h t t h t h t s s h t lt h t t s s
s
e
e
d
s
s
e
dW
m
e
m
e
m
d
α αα α
α
β
ασ
∫
+∫
∫
+ + + +−
=
+
−
=
, ..
,
.
,
.
,
2 - !
m
t+h * 0 /m
t+h=
α
r
t−
β
,
t
."
% + "
∫
+∫
+ + − + − − + − − +=
=
−
+
=
−
+
h tt s t h
h t t s h t s h t h t h
t
m
e
e
d
s
e
s
dW
r
t
h
m
α α, .β
,
.
α
α, .σ
,
.
α
β
,
.
2 - !
r
t+h * " % + "[
]
∫
+ − + − −∫
+ − + − − +=
+
+
+
−
−
h t t h t t s h t h S s h t h t ht
r
e
s
e
dW
t
h
t
e
e
d
s
r
,
.
, .&
β
,
.
β
,
.
, .β
,
.
α
σ
α α αα
∫
+∫
+ − + − + − − + −+
−
+
+
=
t ht s h t t s h t s h t h t h t h
t
e
r
r
e
e
s
e
ds
s
e
d
W
r
α αα
α, .γ
,
.
ασ
,
.
α, . (36)" . , . , α λσ
γ s =− s
# "
∫
+∫
+ − + − + − − + +−
=
−
+
+
h t t s h t t s h t s h t h t t h t ht
r
r
r
e
e
s
e
ds
s
e
d
W
r
,
.
αα
α, .γ
,
.
ασ
,
.
α, .∫
+ + − − + +−
=
−
+
=
t ht s h t h t t h t h
t
F
S
F
e
e
s
e
ds
&<
. ,
.
,
sh t
t
s h t
W
d
e
s
∫
+ − + −+
σ
α+ / D + >- * 6 + /
r
t R lnS
t0 /t
t
F
r
=
2 - !
S
t+h * + / "h t
S
+ Rh e
t t h t
F
S
F
α −
+
. ,
.
,
.
,
.
,
s h
t t
s h t h
t
t
s
W
d
e
s
ds
e
s
h
t
e
e
e
−
+
∫
+ − + −+
∫
γ
ασ
αα
α
(37)
>- * ,:@. H 1 # 6 6 /
+
dW
s % * - % % +,
dW
s≈
,
'
"
ds
.."
/ H +% = % * - % % + 1
# 6 H - % % - * +H >-
S
t+h1 #+- +- / H % / 3 + "
. ,
.
,
sh t
t
s h t
W
d
e
s
X
=
∫
+σ
−α + −+ + " 3 + "
h e
t t h t
F
S
F
t
h
t
A
α −
=
+
"
.
+,
" * ,:@. - - + "
X
ds
e
s
h
t
e
h
t
A
t
h
t
e
e
S
h t
t
s
∫
+
=
+
+
−
+
α
γ
α
α
,
.
,
.
.
"
,
(38)$% + * ,:I. / - - ++ 0
S
t+h %" - + H + 0
6 3 * + + % * - % %
+ " - - + % + - 0 X1
'
.
,
.
,
, .=
=
∫
t+h − + −t s
s h t
dW
e
s
E
X
&@
∫
∫
+ − + −=
+ − + −=
+
=
t ht
s h t h
t
t s
s h t
t
h
t
v
ds
e
s
dW
e
s
Var
X
Var
,
.
σ
,
.
α, .σ
(,
.
(α, .,
"
.
+ " % + /
X
≈
,
'
"
v
,
t
+
h
"
t
..1
3 H " + + / F , ." / +- / +
- - ++ 0 * " 0 / 0
* ,&." t
Y
t
F
t
e
S
=
,
.
" 0 /σ
,
t
.
" / +- H - - ++ 0" 0 / 0 *
,:'."
dY
t=
α
,
γ
−
Y
t.
dt
+
σ
,
t
.
dW
t " 16 + >- * ,(I." % "
α
λσ
γ
=
−
,
t
.
α
λσ
−
=
(39)+ A,t N h, t. 0 v,t N h, t. / - + % >- ,&&. 0 ,&(."
- + 1
6 - + - +- 3 * ,:I.1 * / 6 % + /
.
,
t
σ
" D >- * ,:)."α λσ
γ = − 1 # - + 6
+- 3 * - + >- * ,:I."
.
&
,
.
,
.
,
hh t
t
s
e
ds
e
s
h
t
e
e
e
αα
γ
γ
α
α
−+
−
+
−
=
∫
" * ,:I. - +- 3 + "
X ds e s h
t e h
t
A
t
h
t
e
e
S
h t t
s
∫
+
=
+ + − +
α
γ
α
α
, . , ..
"
,
X e
e
e
t
h
t
A
,
+
"
.
,&− −ah .=
γ
X e
e e
t
F
e
e
S
− h &− − h ,&− −ah .=
α α γ (40)* ,B'. 3* + >- - - ++ 0 1
+ + " % + / * ,B'. = >- * ,&:. /
-- - ++ 0 "
S
t ha
t
h
t
e
xS
teF
ee
xh
h α
α −
− −
+
=
+
=
&
.
"
,
> - / >- * ,B'. 6 - = +
1
. &
, e h
e
γ
− −α&I
3* + >- -
S
t H - % 3* +- 3 ++ 0 * :1:1 M 3 *
+ + % * - % % + 0 6
>- ,:@. 0 ,&&." + *
. &
, .
, .
, t h h
t
s
e ds
e s h
t e
e
e
αα
γ γ
α
α − + + − −
=
∫
"
-h t
S
+ H - "
[
]
. " , ( & . & ,
& e e v t h t
e t t
h
t
S
S
F
e
e
S
E
+=
−αh − −αh γ − −ah + (41)"
S
t+h H - "[
]
e e e[
v v]
t t
h
t
S
S
F
e
e
e
S
Var
+=
,
−αh &− −αh.
( (γ ,&− −ah. (−
(42)+ * ,:). /
α λσ
γ = − 0 >- * ,&(. / v,t N
h, t. R
,
&
.
(
( (
h
e
v
αα
σ
−
−=
3.3 Precio del Futuro (o Forward) del Commodity
# " - 3 - -
-3 " % ! - " 0 + / = %
H r " - 3 7 - 3 1
" >- * ,B&." % + 3* + >-
-- 3 ++ 0 " % "
[
]
, " .( & . & , &
.
"
"
,
S
tt
t
h
E
S
t hS
tF
tS
teF
ee
ee
v t h tF
+
=
+=
=
−αh − −αh γ − −ah + (43)$% + / * ,B:. = >- * ,&<. / -
-3 ++ 0 "
[
]
, " .( & . & , &
.
"
"
,
S
tt
t
h
E
S
t hS
tF
tS
teF
ee
ee
v t h tF
+
=
+=
=
−αh − −αh γ − −ah +> - / >- * ,B:. 6 - = + ,& .
ah e
e
γ − − 1&)
% + /
S
t+h % * - % % + 0 /[
t h t]
tt
t
t
h
E
S
S
F
S
F
,
"
"
+
.
=
+=
" - - 3F
t +% =- + 1 # - + " % + /
F
t % *- % % + 1 3* + E 9 ,&)@<. - , +- '" /
. % 3
F
t 0 N6" - " /+ "
[
(
)
']
. ' ,"
+ >
.
"
'
,
t
e
E
F
K
O
F
c
=
−r t− t−
[
(
& ()
']
. ' ,
.
,
.
,
.
"
'
,
t
h
d
K
d
F
F
e
r t+
−
=
− −(44)
- H 6 + >- *
* ,B:. -
F
,
S
t"
t
"
t
+
h
.
=
F
t1 >- * 3 H+ " % "
. " , ( & . , . ,
.
"
,
.
"
"
,
S
tt
t
h
A
t
h
t
e
e t h s e dse
v t h tF
h t t s + + −∫
+
=
+
+ αγ
α
α
. " , ( & . , .,t h s e ds v t h t e
e
t t h
t
e
e
F
S
F
h t t s h + + − +
∫
=
+ − α αγ
α
α
. " , ( & . , . , ., S F e t h s e ds v t h t e F
e
e
h t t s t t h h t + + − + − +∫
=
+ − + αα
α
α
γ
(45)
" P - ! " = H % " F,', t N
h. H - * ,B?." , . % * - % % + H
+ "
F
' 3 + * '" 0d
&0d
( H - "
h
t
h
t
k
h
t
f
d
+
+
+
+
=
& ( & &(
.
,
.
"
'
,
σ
σ
(
.
,
.
"
'
,
( & &h
t
h
t
k
h
t
f
+
+
+
+
=
σ
('
h
t
h
t
k
h
t
f
d
+
+
−
+
=
& ( & ((
.
,
.
"
'
,
σ
σ
(
.
,
.
"
'
,
( & &h
t
h
t
k
h
t
f
+
−
+
+
=
σ
σ
=
d
&−
σ
&t
+
h
(47)+ /
σ
&( + - 3F
t13* + E 9 + /
F
t % * - % % + " 0H -
σ
&(,t N h.1 3 " D + >- *-
σ
&(- ,t Nh.1 + % /( &
σ
,t Nh.H
d
&0d
(" / + * ,B?. - "( &
σ
,t Nh. RV arTln ,t, t Nh. UF
'V (48)V arTln ,t, tNh. U
F
'V" + - + ETlnF
t UF
'V1 "[
]
[
]
,
"
.
(
&
.
,
.
,
''
v
t
h
t
ds
e
s
h
t
e
F
e
F
S
E
e
F
F
F
E
e
h t t s t h t h h t t+
+
−
+
−
+
=
∫
+ − − + α αα
α
α
γ
2 >- * ,:I. 0 *
h e t t h t
F
S
F
t
h
t
A
α −
=
+
"
.
+,
" % + / "
h e h t h t t t
F
S
F
S
α −
=
−−
e
t
s
e
ds
x
e
e
h t t s∫
+−
α
γ
αα
,
.
,
.
"
[
]
=
++
−[
−−
−]
+
−
∫
−+
[
]
t h t s h t h t h h t tF
X
E
ds
e
s
t
e
F
S
e
F
F
S
E
',
.
,
.
'α
α