Electronic copy available at: http://ssrn.com/abstract=1930905

!"#$%

&&&''(')(*&(')+&(

,- (./+

0'1(

,'"23#!""

4#!""

Electronic copy available at: http://ssrn.com/abstract=1930905

5 !"#$6 4#!""

17((7" 8#!!9,'"!:

!

" #

$

$

%& ' ())*

" +" &

" "

+ & ,

+ - +

++

./00 0+

123/444(4*56)7()44

8+, '72"+9:";)4(<"=>:)?;*)>:((0 @

1

+

14::)1()4) A

+BC

+

" +" !

1 B4:?)C

& 7-

" #

,B4:<4C

B

C4))+

%

!

+ +

&

())<())*DE !

+ "

++BF"())*"9"

()4)C"+"&

+B"()):".&"()4)C"

B."()4)"C

""

" & B()44C

+ & ' 3 B4::)C"

,&

+

!

2

+ - +

+ + !

&"

" +

&

G""+

!+

++ !"

"++&+

!

& 2+&

+"

&"

+

+

&

!

+++

!-

());())? +

())<())* & "

+&"

! + &

&

" "

+&"%

3

7 4

&" &

+7(

B4"4C 7 ( 4 +

7( (

++

7>

!"#B());C"

+ & ,

=4 +

BC "

"++

111 )( ++ +++= t

H

t

F

ttt eDDaff B4C

tf tF 1a +

F

tD and H

tD

! +

+BC%

&

)(2 ft

F

ft

F

t rraD −= (2)

4

ttft fFr ∆=∆= log F

ftr %

! NrrN

i iftFft /)(

1

0−

= −= 2a +

++ %

+ BC

B+C%+ F

ftr

++

& +&

BC &&

BC β &"

BC&BC β

BC+

BC

Htr +

ftctHt rrr β−= B>C

tC B ttct cCr ∆=∆= log C

++

trrtrtrtrtrtr fcfcfcHρσβσσβσσ 22222 −+= B6C

2

trcσ + ctr " 2

trfσ + ftr " trr fc

ρ

ctr ftr

β +

+

5

2

tr

trrtrtr

f

fcfc

σ

ρσσβ = B;C

+

" trrtrtrtrr fcfcfcρσσσ = "+

4

!"

+

+ +

" % B6C

β "%B;C"

2

2

22)(

tr

trr

trtr

f

fc

cH σ

σσσ −= B?C

+

])(

[2

2

22

jtr

jtrr

jtr

H

jtr

HH

t

f

fc

cHbbD

−

−

−− −==σ

σσσ .,...,0 ν=j B<C

%

Hb

++

2

trHσ +

H

4# " + I F B4::4C . 7 B4::>C

6

" ( 2

trcσ

Hb

+ + BC

+++BC

+ 3 B C

"

B C+

!BC &" %

" BC BC

!

BC BC

BC

+ Hb + B+C +

!

BC " "

BC BC

&"

" BC &"

BC

#+++$

+ +$ !

&+

+ &

+ ! +

+

( 2% B?C )1( 222

tfrcrtcrtHrρσσ −=

tfrtcr

tfrcr

tfrcr σσ

σρ =

%4

7

+"

"

2

2

2

1 )]([)](1[ jtrlt

H

jtrlt

HH

t HHwSbwSbD −−−− +−= σσ B*C

vj ,...,0= mjl ,...,1+=

Hb1 Hb2

2

trHσ

+ tw +

)( 2

trct cr σ " l "

& (.)S

"+

J"

K+B4::6C"++

2+&"

& 7+

"

)( ltwS − 37 "

"

1)(]1[)( −−−

−−+= cw

ltltewS

γ B:C

c 0>γ

>

+ +

+ Hb1

>I+

tw c

γ +%B:C γ

-

8

Hb2 ! ltw − +" )( ltwS − 4

&+ Hb20 ltw − +" )( ltwS −

) &+ Hb1 !

cw lt =− " 5.0)( =−ltwS

+ Hb1 Hb2

7%B(CB*C %B4C"+

1

2

22

2

2111 )]([)](1[][ +−−−−+ ++−+−= trjtrltjtrltft

F

ftft fHHuwSwSrrr σθσθθ B4)C

211 aa=θ " Hba 1121 =θ " Hba 2122 =θ " vj ,...,0= mjl ,...,1+=

2% B4)C + +

% +

+

2

J B7" 4:?4C

+

" "

+ + + 2%

&

+ + ! %

+J

" J B

9

+C 6,&

""&%"

% B4)C + $

B4C $

+

%B4)-C"B?-CB:-C"

+"%

B44C"%+

B4"4C

7B4C

( )

2

1

2

1

22

1

2

1

2

1

;

0

0

1

1

,0

−−−−

−

++=++=

=∆

=

∆∆=

Ω

=

trftrfftrtrctrcctr

tr

tr

t

uu

uu

ttt

ttt

tr

tr

t

fffccc

f

c

crfr

frcr

f

c

huhhuh

h

hR

RH

HNu

u

uu

βαϖβαϖ

ρ

ρ

6A " ,$, B()4)C L2FJ"

)'9(]1[)(

)'6()(

/)(;,...,1;,...,0

)'10()]([

)](1[][

)11()(

1)(1

21

2122

1

1

0

21122

211211110

1

1101

1

0

1

1

−−−−−

−−

−−

−−

−

= −

−−−−

−−−−−−

=−−

=−−

−−

−−

+=

−=

=+==

++

−+−+=

+−−+++=

cwlt

jtr

jtrr

rjtr

N

i iftFft

trjtrlt

jtrltftFftft

tr

x

z

tt

y

k

kftkzctzct

lt

f

fc

jctH

fH

H

c

ewS

h

hhh

Nrrmjlvj

uhwS

hwSrrr

ucddfrrr

γ

θ

θθθ

εϕφφ

10

!

&(14::)(?1()4)

$# !B# !C79"87

I$+2!

GMF2NBGM&F2C ,

%+4 ;

"# $

7 4 9

4))

+ + & +"

I B&C

&" +

? ! &

% & 3J I 5

L"

(F3+

B4"4C B4C

+ J&

+

K+ B4::6C +

% B4)-C & B 0=j C

5,4>(;

+ ! (; "(;

11

+ + &

+

%B4)-C

3&& B4:**C

)( 1 ltwS −− % B4)-C

%

trlttrlttrlttrtrftFftft fHHHH

uwhwhwhhrrr +++++−+= −−−−−−−−−−−−3

12

152

12

1412

132

121110 ][ λλλλλλ B4(C

2111 ltrlctlt c

hrw −−−−−− = " .,...,1 mjl +=

0: 5430 === λλλH ,

+ + l < 0H J

B ]0001.0[9152.6)1033,3( =F C&B 1=l C

"27 !++,

0: 5

*

0 =λH %B4(C"+ 0: 51 ≠λH

]0000.0[8464.15)1033,1( =F J

*

"# $

+ %

++

? !B4C & 7

1

θ

! + 1

θ + "

+"""&"+ * + ++

12

tυ +

&

B 0)( =tE υ " 1)( 2 =tE υ " tυ 2

tυ

C : % B44C

4

&&

&

% B4)-C + 4) 1θ

+ +

+ F

ftr "%&

++B 8=N C

+

+ c +0

H

& + + γ

"+"

!&+

2

1−trHh +

J γ c B:-C

K+B4::6"(4>C : 1 2 G B4::>C + 4 "+"J1%I ( % & I+ # B4::(C , B 2 K+" 4::?" ?>?;C"

]1906.0[6014.1)1031,3( =F "

J

13

H

++

(

"

" 2

1−trHh

)640.0( 21 =θ " " (

"

2

1−trHh

)533.0( 22 −=θ

%%&'%%(

&

" "

!" & B()44C"

$

$

%&

())<())*

%O+B

9()4)"()44"%C

&

& + &

+

10

7 B4C )( 11 −− − ftFft rr % B4)-C

)(1101 −−

−−ttcddf !

14

"#)$

, 4 ()); ()4) !

+ # ! BC

+

+ +

)]([ˆ)](1[ˆˆ2222212 −− +−= tt wSwS θθθ B4>C

""1());'

())?" 1())<'

())*" +

+ + &

2θ

21θ "

2θ

22θ

" " "

++"

"+

!

+

2 ())?"

+ &" &

+ + "

'

+ + $

+$+

&+

+

15

+ A "

" +

""

44 ! "

())*

+ +

+ +"

+&

+ 4(

"# *$

! >

+

+

+ 1 ()); $ ' ())? " -

A+ 1 ())< $ '

())* $ " +" +

+ !"+"1

())<$1())*"+

2θ " +

&+ !"

+" % B 1 '

())*C+"

+"

44 7 & B()44" ();<();*C .& B()4)" 4?)>C 1())<F())*G&"" 4( . B4::<C + & & +" + " +" + & + & +++B++C

16

4>

!

+ 2

12ˆ

−trHhθ

+

"+

+ "

"+

*+

++ 4::)$()4) 2+&

+"&

"

+

!

+ " +

& &

&+ %

& "+"

"

+" ())<())*

+

4>!&+BC&BC β &"

P+ B 1=β C

%B>C A+1())<1())*+

17

H + &

&

&" +" % +

P+ +1'())* &

18

I " F " 4::4" I+ 2 A

,"12"?/4):4(6

I+ " # 1 F " 4::(" 5F 3&

2 ' F L +"

2+"44/46>4<(

" 9 " ()4)" A 9 ' 7

F+,"22">(/>?>><(

"9 "()44"A'+8&

9O" ! 1 , 2" 'A!/

4) 4))(HJ 66<

'31 I "7 "73 "# 1 "4::)"9+

,& !+ 7 ' 7"

1,"6;/><:>:;

2 Q " K+ " 4::?" % 7

+F"12"<6/;:<;

2 , "GL . "4::>"F !G

L"1,"6*/4<6:4<<*

,, ! " 1 " ()4)" F F 9

'+F&"12"4;*/:;4)<

1 " ()):" 8 A 9" 2 1" >)/

4<:()?

13 3 "4:?)" 7

,"+27"(</4>:4;4

. . "()4)" F&,7

9 A" 2 9"

'A!/4) 4)4?HJ ()4) ): )4*

19

.& , " ()4)" A 7 $ #- ' O"

29">*/4;:?4?)?

. 2 G " 4::<" , L 9 3" 2 9 &

F"

. . , " 7 1 " 4::>" L ' '

, ," 1 ,

5+"(*/;>;;;4

3&& " 7&&" 9 " K+ " 4:**" 3

7 + F" I&" <;/ 6:4

6::

F F 2 " ())*" I

7+"877"F()

7 1 3 " 4:?4" 7 ' 7 ,

9"2+";4/4)4(4)(;

K+ "4::6"7"2"2+7

+ F" 1 7

"*:/()*(4*

& ' " ()44" ' 7" 9+ ,&

"AI())*"29">:/();4()?4

# #" , 3 I " 4:<4" , 9 7

FF , F& 9" 1

2";>/<4<*

# , " 7 " ());" 9 '

GF&! /22+

87F&"9">6:/?64?6*

20

!

+

F ) 44(:< ) 44(:*

7 '+ ; ;6*6 ; ><4:

7& ) ><)) ) (>(>

2 . ; >?*? > 4>>?

1%I4(*4 (<;R) )))S

6>< *)>R) )))S

',>? ?:4R) )))S

>< ):;R) )))S

)1(xQ 4? *?>R) )))S

4: *>?R) )))S

)10(xQ >: ()(R) )))S

>: )>6R) )))S

)1(2

xQ << ;4:

R) )))S6) ?<4R) )))S

)10(2xQ

(*< )))R) )))S

(>? :>)R) )))S

1 ;; 4<:R) )))S

>; (??R) )))S

G/2 . 2.0','&,0 )(kQx

3JI5

&+0 )(2 kQx 3JI5 &

%+ (0 1 1# 2χ >

B2 G"4::>C +& %+ (H)4H4::)(?H)4H()4) 4)6*+ 9+%&

21

,-+++'.+/01

/1

,

2%B44C

2%B:-C

B4)-C

0φ 44 >:4B>>* *6:C

0θ ) ((:B? ;>(C

1φ ) 4))B4< 6:*C

1θ ) )>?B; ;*:C

2φ ) 4:;B(> :)>C

21θ ) ?6)B4* >*:C

3φ ) 4((B4) 6?<C

22θ ) ;>>B4; 4))C

1ϕ ) )<*B4> 6:?C

2ϕ ) 4):B4> ??<C

c ) )(?B( 64?C

3ϕ ) 4>4B4> :<*C

γ ? *4?B* >44C

ε ) **;B(*: <?>C

0d ) 4(6B>6( <>4C

1d 4 ))4B446<4 ?C

+

cϖ ) <;<B() >??C

fϖ ) :>4B(( (>:C

cα ) 4?4B4(4 (4?C

fα ) 4?4B*4 ;44C

cβ ) *>)B?:* <(<C

fβ ) *()B66: <*<C

frcruuρ ) :<>

B>>:* <C

33, 6<6* <*(

2/ tcrtcrhu 2/ tfrtfr

hu

)( tcrE ν ) )(: )( tfr

E ν ) )(:6

)( 2

tcrE ν 4 )4) )( 2

tfrE ν 4 ))::

7& ) 6(? 7& ) 66?42 . > )?* 2 . ( <;?6

)1(xQ ) <(4R) >:?S

)1(xQ ( ;**R) 4)<S

)10(xQ 4( (6>R) (?:S

)10(xQ 4> (<(R) ():S

)1(2

xQ ( <4:R) 4))S

)1(2

xQ 4 6**R) (((S

)10(2

xQ * ??)R) ;?;S

)10(2

xQ < :?<R) ?>(S

1 * ?<:

R) )>6S1

? ;(:R) )**S

1 I 6>< 4);R) )))S

1 I >?4 :**R) )))S

G/ 2/ ztztzt hu=ν " fct rrz ,= 0 %&0

I+#B4::(C

22

*+--!0

0

2%2%%&'232%%3

ctr ftr

2θ ) )*:) ) 4);*

2

12ˆ

−trHhθ ) )?(4 ) )<;>

2%2%%4'2*%2%%(

ctr ftr

2θ ) )<?? ) )<)6

212

ˆ−trH

hθ ) (>>: ) 4?;6

2%2%%453262%%(

ctr ftr

2θ ) ):;( ) )?;<

212

ˆ−trH

hθ ) 4;<4 ) 46;:

42%2%%('2*%2%%(

ctr ftr

2θ ) 4()* ) );?)

212

ˆ−trH

hθ ) (?*4 ) 46>;

23

)

-2

-1

0

1

2

3

4

2005 2006 2007 2008 2009

WTI CRUDE OIL SPOT PRICE

HEDGED POSITION VARIABILITY COEFFICIENT