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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/

(2)

Modelos Estocásticos para el Precio Spot y del Futuro de Commodities con Alta Volatilidad y Reversión a la Media

! " #

$ % &'" ('')

Resumen

* ++ ,+ - + . / +

%0 * + 0 1 2 + 3* + - %

- - ++ 0" 3 " 0 4 5 - % - 0 %

3 " + 0 + " + & 3 1

Abstract

6 - 3 ++ 0 / 6 6 0 % + 7 6

+ 6 6 01 8 - 3 + - 6 - 3 6

++ 0" 3 " - - 6 - 6 ++ 0 3 "

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+ ;

(3)

&

Modelos Estocásticos para el Precio Spot y del Futuro de Commodities con Alta Volatilidad y Reversión a la Media

1. Introducción

+ + / - ++

-+ -+ - / - - % 1 # - /

-++ +- + H 1 +% " 6 + /

- ++ - * + 0 1 #

" / - - " 6 /

1 # + + 3

- % - - ++ 01 # + + - + /

- ++ 0 - * + 0 + 1

+ +H 6 , . 0

, .1 # - * /

-+ , .1 + " / - + H

% ++ 1

+ - % % &" ++ + 0 - /

+ + - + - - 1 "

-3 * 3 0 + " 0 % % / - "

-- + 6 * " / + * + 1

0 67 ,&))B." E +% " 6 " 0 + ,&))?." 0

,&))?." 6 + * + - ++ 1

$ + 0 +- ++ 1 D

0 ,('')." % = % & -

+-, . (''@ 0 (''I 3 ++ , "

+ " 0 ." = , " - * . 0 + H , % " .1 + %

/ ++ - + 0 J (''I / (''@1

+ (''@ - * &?1(K 0 3 + /

/ ++ " 0 (''I 3 :?1<K" +

-1

+ 0 % ! % ++ 6 + + 1

# G " 0 +- ,(''B. + - ,

. & 3 0 * + 0 1 #

6 + + % + "

0 / 6 - + - + " 6 - +

" 0 + / - + - - 1 #

(4)

(

Tabla 1. Volatilidades de Commodities

++ 0 (''@ (''I

::1BK ?'1?K

:&1@K B&1BK

0 ((1:K B(1IK

B?1&K B?11BK

* ()1(K ?(1<K

% :(1BK B(1&K

$ &<1<K ()1:K

# H + 1 # * (

+ 3* + - - - / ++ 0"

-+ 0 - " - - 3 " 0 - 4 5

, - * +- . - % - - ++ 0" 1 # * :

- + - 3* + " / + /

- + H + % ++ 1 2 + + /

3* + - - 0 3 = 3* +

" > - - - * = + 1 * B 0 1

2. Valuación en el Mundo Real

# - - - - + - - H % +

G " " 0 +- ,(''B.1 3 3 + /

+ - 1 # + +

+ 1 " - - - - + -

-++ 0

S

t" "

t

Y

t

F

f

e

S

=

,

.

(1)

t

S

- - ++ 0 * , F ,t.

+-+ " 0

Y

t - - H " + + / H -

-- "

t t

t t

t

Y

dt

t

dW

Y

dt

t

dW

dY

=

α

,

'

.

+

σ

,

.

=

α

+

σ

,

.

(2)

* ,(. - * + - 9 ,$ A 6 % 9."

0 * + - " 0 * α >

(5)

:

-

Y

t - " - % "

dY

t - " 0 - "

Y

t

% 6 , - .1 "

Y

t + 0 "

+% -

Y

t " - % "

dY

t "

0 - "

Y

t % ! 6 , - .1 + *

,&. /

Y

t "

S

t F ,t.1

* - $ A 6 % 9 + * *

3 6 + = - + 1 + H

-.

,

t

σ

0

dW

t + + + % 7 H 1 3 *

H

Y

t % *

S

t / % + ,

-+- . , .1&

% *

S

t" + + L"

t t t

t t t

t t t

t

dW

Y

S

t

dt

Y

S

t

Y

S

Y

t

S

dS

+

+

=

,

.

,

.

(

&

( (

(

σ

σ

α

t t t

t

S

F

t

t

dt

t

S

dW

dt

t

F

d

S

,

.

,

.

(

&

.

,

.

,

(

σ

σ

α

α

+

+

+

=

(3)

+ /

Y

t

=

S

t

F

,

t

.

" / % - !

Y

t * ,&." 0

/ "

t t F t F dt

t F d

∂ ∂

= , .

. , & .

,

+ + / , . , .

( & . , &

.

, ( t F t

dt t F d t

p +

  

+

= σ

α " >- * ,:. / "

(

t

)

t t t

t

p

t

S

S

dt

t

S

dW

dS

=

α

,

.

+

σ

,

.

(4)

M + +% %

r

t

=

S

t , * /

r

t =

H % " +- + % > ." - -

-+ L * ,B."

&

% / 3 + 3 * ,&. /

(6)

B

(

)

t

t t t t t t t t t t t t t

dW

S

S

S

t

dt

S

S

S

t

S

S

S

S

t

p

t

S

S

d

dr

+





+

+

=

=

,

.

,

.

(

&

.

,

( ( ( (

σ

σ

α

(

p

t

r

t

)

,

t

.

dt

,

t

.

dW

t

(

&

.

,

σ

(

σ

α

+





=

(5)

3 + +H - *

S

t" +

.

,

.

,

.

,

(

&

.

,

.

,

(

F

t

dt

t

F

d

t

t

p

t

α

σ

α

β

=

=

+

" % + "

[

t

t

r

t

]

dt

t

t

dW

t t

dW

t

dt

r

dt

t

dr

.

,

.

,

.

,

.

,

σ

α

β

σ

α

β

+

=

+

=

# D + >- * + + 91 + + 3 +

- $ A 6 % 9 + ;"

[

t

]

t

t

r

t

t

r

m

=

α

β

,

.

=

β

,

.

α

"

dm

t

=

α

dr

t

d

β

,

t

.

% + "

[

t

]

t t

t

dm

d

t

m

dt

t

dW

dr

,

.

,

.

(

&

β

σ

+

=

+

=

2 - !

dm

t % + "

(

)

t

r t

t

m

t

dt

t

dW

dm

=

α

+

β

,

.

+

ασ

,

.

(6)

* ,<. + * * 3 6 + =

- + 1 " - + 6 +%

% "

X

t

m

t

e

t

α

=

" 0 - + + L

X

t - H

* ,<." % "

(

)

t

t t t t t t t t t t t t

dW

m

X

t

dt

m

X

t

m

X

t

m

t

X

e

m

d

dX

+





+

+

=

=

,

.

,

.

(

&

.

,

.

,

( ( (

(

σ

ασ

α

β

α

(7)

? t t t

dW

e

t

t

d

e

α

β

,

.

+

ασ

,

.

α

=

(7)

+ * ,@."

t t h t h t s s h t t h t t t t s h t

t

d

m

e

e

d

s

s

e

dW

m

e

m

e

α α

α α

α

=

β

+

ασ

=

+

+ + + +

, .

.

,

.

,

.

,

2 - !

m

t+h * 0 /

m

t

=

α

r

t

β

,

t

.

% + "

.

,

.

,

.

,

, . . ,

h

t

r

dW

e

s

s

d

e

e

m

m

t h t h

t s s h t h t t s h t h t h

t

=

+

=

+

+

+ +

+ +

+ α

α

β

α

σ

α

α

β

2 - !

r

t+h * " % + "

+ − + − −

+ − + − − +

=

+



+



h t t h t t s h t h s h t h t h

t

r

e

s

e

dW

s

t

h

t

e

e

d

s

r

,

.

, .

,

.

&

β

,

.

β

,

.

, .

β

,

.

α

σ

α α α

α

+ − + −

− +

+

+

=

t h

t s s h t h t h t h

t

e

r

r

e

s

e

dW

r

α α

σ

,

.

α, . (8)

# "

+ − + − − + + + +

=

=

+

h t t s s h t h t t h t h t h t h

t

r

S

F

S

F

e

s

e

dW

r

,

.

α

σ

,

.

α, .

2 /

r

t

=

S

t 0 /

r

t

=

F

t" 0 - !

S

t+h / "

t e t h t h t

F

S

F

S

h α − + +

=

.

,

t h s

t dW s h t e s

e

. , . , − + −

σ+ α

(9)

>- * ,). H 1 # 6 6 / +

s

dW

% * - % % +

,

dW

s

,

'

"

ds

..

/

(8)

<

# 6 H - % % - * +H >-

S

t+h 1 #+- +

- / H % / 3 + "

+ − + −

=

t h

t s

s h t

dW

e

s

X

σ

,

.

α , .

+ + " 3 + "

h

e

t t h t s

F

S

F

t

h

t

A

α





=

+

.

+

,

" * ,). - % + " H

- * >- - - ++ 0"

x s h

t

A

t

h

t

e

S

+

=

,

+

.

(10)

$% + * ,&'. / - - ++ 0

S

t+h %

" - + H + 0

6 3 * + + % * - % %

+ " - - + % + - 0 X1

'

.

,

.

,

, .

=





=

t+h − + −

t s

s h t

dW

e

s

E

X

E

σ

α

+ − + −

=

+

=

t h

t

s h t

t

h

t

v

ds

e

s

X

Var

,

.

σ

(

,

.

(α, .

,

"

.

+ " % + /

X

,

'

"

v

,

t

+

h

"

f

..

+- + + / F (." / +- / +

- - ++ " 0 / 0 * ,&." 0 /

.

,

t

σ

" / + - H -

-++ 0" 0 / 0 * ,(." 1 + A,t N h,

(9)

@

h e

t t h t s

F

S

F

t

h

t

A

α





=

+

.

+

,

h h e

e

t

F

S

−α − −α

=

&

(11)

+ − + −

=

+

t h

t

s h t

ds

e

s

t

h

t

v

,

"

.

σ

(

,

.

(α , .

.

,

.

&

,

(

( (

X

Var

e

h

=

=

− α

α

σ

(12)

" 0 >- * ,&&. * ,&'. % + "

X e

e t X

h

t

A

t

h

t

e

S

F

e

S

+

=

,

+

"

.

=

−αh &− −αh (13)

% + / "

..

"

,

"

'

,

v

t

h

f

X

+

* ,&:. 3* + - % - - ++ 0

*

S

t+h + 1 + >- * ,&:. / - %

-- ++ 0 - +-

S

t+h " % + -

-- +-

S

t " 0 + 1 + " - +

/ % O 6 D >- *

,&(.1 $% + / v,t N h, t. !* - +- - / 6 +

+ / +

σ

,

t

.

0 - / +

+- 6 1

(10)

I

M 3 * + + % * - % %

+ " - % " % /

[ ]

. , ( & .

,X Var X E

X

e

e

E

=

+ " 0

>- ,&&. 0 ,&(." -

S

t+h H - "

[

]

e

e

v

e

t

t

h

t

h h

F

S

S

S

E

(

&

&

α

α −

+

=

(14)

h t

S

+ H - "

[

]

(

e e

)

[

v v

]

t

t h

t

S

S

F

e

e

S

Var

+

=

−αh &− −αh ( (

(15)

+ / = % H r "

-3 7 - 3 " ,

S

t , t, t Nh.1

# 3 3 7 3 "

(

)

[

]

(

)

rh rh

t rh

t

t

F

S

t

t

h

K

e

F

S

t

t

h

e

Ke

V

=

"

"

+

=

"

"

+

P - ! 3 1

+% = 3 + - " "

[

]

[

]

rh rh

t h t rh

t h

t

t

E

S

K

S

e

E

S

S

e

Ke

V

=

+

=

+

h t

S

+ - 3 - ++ 01

+% >-

V

t " % + "

(

)

[

]

rh rh

t h t rh

rh

t

t

t

h

e

Ke

E

S

S

e

Ke

S

F

"

"

+

=

+

"

(

)

, " .

( &

&

"

"

t h t v h h

e

e

e

t

t

t

t

h

S

F

S

F

+ −

=

+

α α (16)

# * ,&<. + / - 3 ++ 0

(11)

)

3 + " 3 / +- % %

, % ! . + - 0 3 ++ 1 * ,&<.

" - -

+-, - 3 . % 0 " % % 1 #

- + 6 +- 0 - - 0 3 0

+- * ,&<.1 # - % " >- * /

-3 + - - - 3 "1

! " #

+ E 9 Q 6 ,&)@:." * +

3* + - - * - +- ,4 5. % -

-++ 01

% + / - 3 * " - - + "

(

)

[

(

t h

)

t

]

rh

t

t

t

h

e

E

S

K

S

S

c

"

"

+

=

+ >

+

"

'

" P - ! 0 + " = H

% 0 + " 0

S

t+h H - * ,&:.1

(

)

[

(

)

t

]

x rh

t

t

t

h

e

E

Ae

K

S

S

c

"

"

+

=

+ >

"

'

( &

I

I

=

(17)

6 +

I

&

I

( + "

∞ −

=

A K

dx v x rh

e

v

A

K

Ae

I

(

&

(

(

(

&

π

.

,

d

(

Ke

rh

=

(18)

> −

+ − +

=

v A K u

du v

v u v

u rh

e

v

e

Ae

I

. , ( &

&

(

(

&

π

.

,

&

( &

d

e

Ae

rh v

(12)

&'

+ H " 3 + % + 3* +

-- 0 %0 - - ++ 0 +

3 * + "

(

S

"

t

"

t

h

)

I

&

I

(

c

t

+

=

.

,

.

,

& (

( &

d

Ke

d

Ae

vrh

rh

=

(20)

"

v

A

K

d

&

=

+

(21)

K

A

d

(

=

(22)

% + + / "

−∞

− −

=

y b

a x

dx

e

v

y

(

. ,

(

&

.

,

π

% * + + + a R ' 0 b Rv1

3. Valuación 3eutral al Riesgo

3.1 Proceso Propuesto para Obtener Precio Spot

+ / * ,&." -

S

t RF , t.e

Y

t " >- * / 6 +

- + - - ++ 01 * ,(." + + / 6 +

- % ! " - -

Y

t , ." 0 +H

% / - D + 3 /

+ 1 # - -

Y

t - H * +

- 9 ,$ A 6 % 9." 0 * +

-" 0 *

α

> '1 + ,

-+- ." H -

σ

,

t

.

" 0

dW

t + + + % 7

H 1 3 * H

Y

t % *

S

t / %

+ , - +- . , .1 % / , . 0 3

- - ++ 1

* + E 9 Q 6

,&)@:." / % + + % ! -

(13)

&&

M 0 P - ,&)@)." - 3 + - % !

1 " = + " % + 3 + * ,(.

" 0 - + + / -

S

t "

+ - ,% ! . 3 * - "

-= % r1

+ * ,(." H - " 0 + + / - +

/ "

t

t

Y

dt

t

dW

dY

=

α

+

σ

,

.

t

Y

% / + " - +

/ + %0

Y

t " - % " +

- * +- 4 5"c Rc,

Y

t , t." 0 + L" $% + "

t t

t t

t

dW

Y

c

t

dt

Y

c

t

Y

c

Y

t

c

dc

+





+

=

,

.

,

.

(

&

( (

(

σ

σ

α

(23)

$ + >- dc / 0 H H - "

dc

=

c

cdt

+

σ

c

cdW

t (24)

" ,(:. 0 ,(B. - + % / "





+

=

( ( (

.

,

(

&

&

t t

t c

Y

c

t

Y

c

Y

t

c

c

α

σ

(25)

0

t c

Y

c

t

c

=

&

σ

,

.

σ

(26)

+ % ! " % + 3 + - 3

- - +

dW

t" / D 3 - - 3 1

2 % + - 3 ( " H ( - +- " c& R

&,

Y

t " ." 0 ( R (,

Y

t " .1 + + " % + D+ /

(14)

&(

+ c& R c&,

Y

t " ." 0 6 + +

Θ

( D+ /

+ ( R (,S

Y

t " . - 3 1 # - 3 H

- 1 # - % "

( ( &

&

c

c

t

=

Θ

+

Θ

Π

" +% - 3 " % 3

+ " H - "

( ( &

&

dc

dc

d

Π

t

=

Θ

+

Θ

t

dW

c

t

c

t

dt

c

dt

c

.

,

,

.

,

.

.

,

Θ

& & &

+

Θ

( ( (

+

Θ

&

σ

& &

+

Θ

(

σ

( (

=

6 + 3 /

t

dW

c

t

dt

c

dc

&

=

& &

+

σ

&

,

.

&

0 /

t

dW

c

t

dt

c

dc

(

=

( (

+

σ

(

,

.

(

% + "

..

,

.

,

,

.

,

..

,

.

,

,

.

,

( &

(

( & (

( &

&

( & &

t

t

C

dt

c

t

t

t

C

dt

t

c

d

t

σ

σ

σ

σ

σ

σ

=

Π

dt

r

t t

Π

=

..

,

.

,

,

.

,

..

,

.

,

,

.

,

( &

(

( &

( &

&

( &

t

t

C

dt

c

t

r

t

t

C

dt

t

c

r

t t

σ

σ

σ

σ

σ

σ

=

# + = + " + >- * "

t

t

t

r

r

t

t

t

.

,

.

,

.

,

.

,

( & ( & (

&

σ

σ

=

σ

σ

"

% / "

λ

σ

σ

=

=

.

,

.

,

(

(

& &

t

r

t

r

t t

+ 3 + - 3 6 + " - +

λ

+ "

c t

c

r

σ

λ

=

(27)

(15)

&: ( ( ( (

.

,

&

.

,

(

&

&

t t t t t

Y

c

t

c

r

Y

c

t

Y

c

Y

t

c

c





+

=

σ

σ

α

λ

2 - ! >- * " % + * 3

-"

'

.

,

(

&

.

,

( ( (

=

+

c

r

Y

c

t

Y

c

t

Y

c

Y

t

c

t t t t

t

λσ

σ

α

= + % + "

'

.

,

(

&

.

,

( ( (

=

+

+

c

r

Y

c

t

Y

c

t

Y

t

c

t t t

t

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 t

σ

γ

α

2 / + / * ,(." + + / - + / "

t t

t

Y

dt

t

dW

dY

=

α

+

σ

,

.

3 + >- * "

dY

t

=

α

,

γ

Y

.

t

dt

+

σ

,

t

.

dW

t (30)

+ / 6 0 % / - * ,:'. / %

- * ,(."

γ

0

dW

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"

(16)

&B ( ( (

.

,

(

&

.

,

t t t t t t t t t t

dW

Y

S

t

dt

Y

S

Y

S

Y

t

S

dS

+





+

+

=

α

γ

σ

σ

(

.

,

.

,

(

&

.

,

.

,

&

t t t

t

S

F

t

t

dt

t

S

dW

dt

t

F

d

S

σ

σ

α

γ

α

α

+

+

+

+

=

(31)

+ /

Y

t Rln

S

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 Rln

S

t - - - + L

* ,:(."

[

]

( ( ( (

.

,

.

,

(

&

.

,

t t t t t t t t t t t t t t

dW

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

t

σ

t

dt

σ

t

dW

t

ρ

α

+

+

=

(33)

3 + +H - *

S

t" +

[

[

α ρ γ

]

σ

]

α αγ

β = + − − , . = , .+ , .+

( & .

, ( F t

dt t F d t r

t t 1

" % + "

.

,

.

,

t t

t

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

(17)

&?

[

,

.

]

,

.

&

t t t

t

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 - + + L

X

t - H

* ,:B." % "

[

]

( ( ( (

.

,

.

,

(

&

.

,

.

,

t t t t t t t t t t t t

dW

m

X

t

dt

m

X

t

m

X

t

m

t

X

e

m

d

dX

+





+

+

+

=

=

α

α

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 α

α α

α

β

ασ

+

+ + + +

=

+

=

, .

.

,

.

,

.

,

2 - !

m

t+h * 0 /

m

t+h

=

α

r

t

β

,

t

."

% + "

+

+ + + − + − − +

=

=

+

=

+

h t

t 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 h

t

r

e

s

e

dW

t

h

t

e

e

d

s

r

,

.

, .

&

β

,

.

β

,

.

, .

β

,

.

α

σ

α α α

α

+

+ − + − + − − + −

+

+

+

=

t h

t 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 α

α

α, .

γ

,

.

α

σ

,

.

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 h

t

r

r

r

e

e

s

e

ds

s

e

d

W

r

,

.

α

α

α, .

γ

,

.

α

σ

,

.

α, .

+ + − − + +

=

+

=

t h

t s h t h t t h t h

t

F

S

F

e

e

s

e

ds

(18)

&<

. ,

.

,

s

h t

t

s h t

W

d

e

s

+ − + −

+

σ

α

+ / D + >- * 6 + /

r

t R ln

S

t0 /

t

t

F

r

=

2 - !

S

t+h * + / "

h t

S

+ R

h 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

α

σ

α

α

α

(37)

>- * ,:@. H 1 # 6 6 /

+

dW

s % * - % % +

,

dW

s

,

'

"

ds

.."

/ H +% = % * - % % + 1

# 6 H - % % - * +H >-

S

t+h1 #+- +

- / H % / 3 + "

. ,

.

,

s

h 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

(19)

&@

+ − + −

=

+ − + −

=

+





=

t h

t

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 " 1

6 + >- * ,(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."

.

&

,

.

,

.

,

h

h t

t

s

e

ds

e

s

h

t

e

e

e

α

α

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 h

a

t

h

t

e

x

S

te

F

e

e

x

h

h α

α −

+

=

+

=

&

.

"

,

> - / >- * ,B'. 6 - = +

1

. &

, e h

e

γ

− −α

(20)

&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

α

α

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

α

α

σ

=

3.3 Precio del Futuro (o Forward) del Commodity

# " - 3 - -

-3 " % ! - " 0 + / = %

H r " - 3 7 - 3 1

" >- * ,B&." % + 3* + >-

-- 3 ++ 0 " % "

[

]

, " .

( & . & , &

.

"

"

,

S

t

t

t

h

E

S

t h

S

t

F

t

S

te

F

e

e

e

e

v t h t

F

+

=

+

=

=

−αh − −αh γ − −ah + (43)

$% + / * ,B:. = >- * ,&<. / -

-3 ++ 0 "

[

]

, " .

( & . & , &

.

"

"

,

S

t

t

t

h

E

S

t h

S

t

F

t

S

te

F

e

e

e

e

v t h t

F

+

=

+

=

=

−αh − −αh γ − −ah +

> - / >- * ,B:. 6 - = + ,& .

ah e

e

γ − − 1

(21)

&)

% + /

S

t+h % * - % % + 0 /

[

t h t

]

t

t

t

t

h

E

S

S

F

S

F

,

"

"

+

.

=

+

=

" - - 3

F

t +% =

- + 1 # - + " % + /

F

t % *

- % % + 1 3* + E 9 ,&)@<. - , +- '" /

. % 3

F

t 0 N6" - " /

+ "

[

(

)

'

]

. ' ,

"

+ >

.

"

'

,

t

e

E

F

K

O

F

c

=

r tt

[

(

& (

)

'

]

. ' ,

.

,

.

,

.

"

'

,

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

t

t

t

h

A

t

h

t

e

e t h s e ds

e

v t h t

F

h t t s + + −

+

=

+

+ α

γ

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 α

γ

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 + + − + − +

=

+ − + α

α

α

α

γ

(45)

" P - ! " = H % " F,', t N

h. H - * ,B?." , . % * - % % + H

+ "

F

' 3 + * '" 0

d

&0

d

( H - "

h

t

h

t

k

h

t

f

d

+

+

+





+

=

& ( & &

(

.

,

.

"

'

,

σ

σ

(

.

,

.

"

'

,

( & &

h

t

h

t

k

h

t

f

+

+

+





+

=

σ

(22)

('

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

&( + - 3

F

t1

3* + E 9 + /

F

t % * - % % + " 0

H -

σ

&(,t N h.1 3 " D + >- *

-

σ

&(- ,t Nh.1 + % /

( &

σ

,t Nh.

H

d

&0

d

(" / + * ,B?. - "

( &

σ

,t Nh. RV arTln ,t, t Nh. U

F

'V (48)

V arTln ,t, tNh. U

F

'V" + - + ETln

F

t U

F

'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 α

α

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

α

α

,

.

,

.

"

[

]

=

+

+

[

]

+

+

[

]

t h t s h t h t h h t t

F

X

E

ds

e

s

t

e

F

S

e

F

F

S

E

'

,

.

,

.

'

α

α

α

α

γ

References

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