Natural volatility and option pricing

Natural volatility and option pricing

Carey, Alexander

12 January 2008

!" #$% " & " " " " ' ' ( ( & ( "" " " ) * " " ( "" & (" & +" & & & + , +

- ' ' & " ' " & , & ) .

' "" $ , $/ 0& &

' + & " " ( ( "

( " ( & " ' ) 1 "" 2 ( ) 1 & " & (

" " " - & " ( & + "" + '( " )

3 14 %5 " ( " " " " " " " " " " $& & " " ' ( + ' ( ( " + ' " " !" #$% " $ & " ( & $ & " ( & " "

" ( 5 6 7

) * 4 8 * 4

* & " $ & !" #$% " & " " " " + & " " ' & +

" " ( & " ' ) * + & " - !" # & % " / 9:70 & .

/ 9:70 + " " + + ( ( + ) * " "

" " + & " - / 99;0 + & 3 / 99;0 & - / 99;0 +

" " + ( ( + & ) !

, & + "" ' + ( + ) 4 " &

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4 - ( + "" # " &' ( + " &

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1 " ' ( ' " + & " "" " - )

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+ " & " " + & " ' "" ( " - & !" #$

% " + & " & ( + + '" &

)

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( ' " + & " ( ( , & & ' + # & )

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A "" " " " ' & ( " " (

( & )

. ' "" $ , $/ 0& &

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+ " + & " - C - & ' / 99 0 &

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" 6 + & " , & (

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+ " - ( + & ( " " + " " " " ' " " ) * + + + " + & ( + " " )

( + ( " " " " ( +

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( 0 2 " $ - ' + + ( ( & (

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) / 9970 & & + + ( &

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( ( ( - & ( + "

( "" - / 99=0 & + 3 & 3 + " / 99:0)

4 " & / & ;0 &

/ "" +- & 6 ' =0) * " & ( $ & " ( & " " - & & "

( 2 & & '" + "" + " &

'' & - " J ( ( E - & " ' )

* ' J & ( "" ) % " + + + " & ( & " ) % 7 - ( " ' & - &) % ; & &

#$ " " & ( " " " & & " + ' " " ) % @ & " ( $ & " ( & " " ) % = " & &

'' + & ( ( )

) . *C . * < . %

* + - + + " & " ) A

& ( ( 1 ) * "" # - + '

& ( & "" - ' + ++ ) 1 &

# " & % ; " " " + & (( " 2 /% 0)

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(

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* & ( ( & % / ) @ ( 0)

- + ′− & & ( & + "" & - &

+ J & ′ − ) " (

( ( " - & & & & " & +

& & + ) * + - ( "" & #

* ( " - ' $ & '+ $ "' - "" )

$⊂%∈ ( )% = )

;

- 2 ( " ) * + " ' I

" / 999 ) @ 0 ( )

1 " ' " ( " & ( & & !

+ " ( & " ( + #_′−# !

+ & & ( ∈ ) ! ' + & - &

' -" ) " ( " "" ' ' ( + / J

& 99 ) ; 0) ++ " ( & + +

( & & ! + ( " ( ' + & &

& & - " ( " )

* " ' " & ( "" ' ) 6 "

+-& ( & & + -"

α

α

< ∞ / $

"+ " 0 & & ( & & + -"

β

"

β

< ∞) " G

(

α β

)

H &

{

(

α β

)

∈

}

)

1 + " ( )

< 4<4% * 4 ) ) &

=

( )

=

( )

⊆ ∈ &

γ

∈

' =

γ

"

γ

= ∈

( ' & , ) 4- & (( " ( + ' =

γ

)

* -" (

γ

< ) "" + # -" '

& (( % % ;)

7) *C 6 A . 14 3

1 ( , - - "

(

Ω

)

& " ( +

=

[

]

#

( & &) * + # ( + & - ( "

( )

=

) * ( "" - & - ( "

-- + ) 1 + # & & + " ( ' + "

"" & + " + /C & <" # 9 0) * & +

+ J ( + - & # " &' ( (

- - " J ) 8 & + "

ξ

="

ξ

( & + -"

ξ

&

- - " + 2 " ) * & & #"

( )

1 + , 2 " + ' " + / ..0 ( + # ) *

, - - " + 2 " & & &

+ ' " " ( " /C & <" # 9 0) * - #

-"" 2 " + - '

+ # ) + ( 2 " & - C & 3 / 9:90

@

( "" ' " ) ' " .. 2 & ++ +

+ # ( .. ( ' ( , ' /A 2

< " & % ) @ 0 + " ' + '

" -" ' + & " / 99 ) 9;$ 9@0) 1 " ) & ..)

* + # + ( + $ ′ ((

ξ

' -

"

₎ − (′−)

ξ

)

( )

=

& " ( " ( ( & " * - ( + " (

() ( 1 " - - " + & ( "

.

µ

- & "

σ

∈

₎ - " ) 1 +

( ( %

/7) 0 ( =

µ

( +

σ

(

( > )

* % - & & - ' " ( "&

' ( + ' & - ( " ) "&

+ + "" & - & +

µ

&

σ

+ & "" (

µ

&

σ

) * ( & ( & "" (

, ( 2 " + ' " + ) A & ( +

" "

σ

L A 2 < " & % / )

;=$;:0)

7) ) % + & . & "

* + ( "" + & " + & ( )

! + , ) * " " ' -

σ

=

ν

( +

ν

) * # + + & " - !" # & % " / 9:70)

! & - , * " " (

(

(

)

σ

=

ν

( + (

ν

5

×

→

(

∞

)

) * (

σ

- " (

( & & & + "" ( + , ' ( "" I & - / 99;0 + & 3 / 99;0 & - / 99;0)

# . / + & ")

! ! - , * " " (

σ

=

ν

(

(

)

(

+ (

ν

5

×

*

→

(

∞

)

) 8 " # " " " " (

" " ( ( - ) * " ( + & " ( , " & - C - & ' / 99 0 & C & & / 99 0) C - & '

+ & " " " ( ( & ( ( + " " /

0) A & < / 0 & " ,

+ # & 2 + / "" ++ & & 0

- ' & ' + −∞)

. 37) ) C + #

ν

( ( + " ( (

" ( + ) * "" ( " + " )

=

( "

ν

(

0

)

& & " ( + "

0

- +

J & + )

! 1 % ( , * " " ' & - && "

( & + - & 1 ) , + " " ' &

& 1 " - - " + & ( "

& & ( ) % " " ' -

σ

= 2 2

(

(

)

(

)

2 =

κ

−2 +

υ

2

ρ

+ −

ρ

'

( -"

κ

υ

&

ρ

) * C / 9970 " " /A 2

< " & % ) ; $; 0)

. 37) ) + & " ( % - + # ( + ""

& ( & - " ( " ( 1 / 0 " & " '

+ & + ( + " ) '" $1

' + & ( " " & - " " " - ( "" / ""

σ

+ - & & ).

;) *8 . * 6 K 8 * 4

* + " " " ( ) * -E

$ , $/ 0& & & ) 8 ' 6 > +

( & & & #$ " ( ) *

& " & ' ( & ( (

" " # & + ) 1 & ( &

' ( ( " & & " " " "

" " ) * - & " ( & & & ' / 0 " & + ( " & & /-0 " "

& - ( &)

& & ( ((

ϕ τ

(

(

)

, +

τ

∈

ϕ

5 ×* → ) (( & &

+ " ( " - $

& & ) C

ϕ τ

(

(

)

+ & - + -" _τ "" )$ ' -" )

A $ , /τ 0) ! & ( (

2 " + ' " + ) + # ( + J (

/;) 0 + −τ"₎

ϕ τ

(

(

)

= )

* " ( - ( ( + " #

/ " 999 ) ==0) & + " + " ( + " &

-" ( ) * ,

-)

. 3;) ) * ( " ( + " /;) 0 " + & 3 )

* ' ( #$ " & '+ - ( & , & / 9:=0 ,

:

+ ' " + 2 & - # & " ' ( /

+ ' "" +? 0 " & & ..)

;) ) % & & . ' " K "

* & & ( & & ( "" ) %

- - " + ) 2 " 4 5 ,

λ

∈

)

λ

= +

1 " ) & ( " ) " ' % /7) 0 & '

( & & + ' " & )I + " '

(

)

λ

=

µ

−

σ

I

-/;) 0 ( = ( +

σ

( ) )

A ' & " & , ) 2 /;) 0 2 "

/;)70 (=( ,

{

(

−

σ

)

+

σ

)

}

/ " 999 ) :7$:; < ) & ) 90)

* % /;) 0 - & & - ' + ( + #

"& -E - - " + ) & ' ( + ' &

-( " ) * + "" & - & +

&

σ

+ )

* /;)70 ( + - ( & & " + & ( &

) * + J - & - ' /;)70 "

( + " /;) 0) * " ' , " & " "" -" &

+ "" ( ) 1 "" ( )

. 3;) ) ' &" ( + & ( & + & " & + ( " "

σ

& 2 " + ' " + )) ' " & (( + & " (

σ

"" & & (( " ( )

;) )

1 & (

" " " ' ) A & ( ( (

σ

'

-)

"

σ

=

σ

∈ )

* 2

σ

- & ( +

& & - - - " ( & - - " + ) &

# ( + - ' + ) "

" " & ( & & & 2 " + ' " + ))

. 3;)7) A +- ( + & " " " " & ( "" & + & -( & ( " " " 2 " " " ) *

( + & " 7) ) $7) )7 - ( " ( ( "" " " + & " 7) );)

* & ( +

σ σ

∈

₎ ) - ( &

(

)

)

"

σ σ

= ∈ & - < )

/;);0 )

σ

)

σ

=

1 " - - " + ) & ( " ) & (( "

( + &

σ

) =

σ

) ) " ' /;);0 % /;) 0 "&

/;)@0 ( = ( +

σ

( )

2 "

/;)=0 (=( ,

{

(

−

σ

)

+

σ

)

}

)

* % /;)@0 - & & - ' + ( + #

"& -E - - " + ) & ' ( + ' &

-" ( -" ) "& + ""

& - & + &

σ

+ )

* + J - + & - ' /;)=0 "

( + " /;) 0) 1 "" "" " 2 )

;)7) < " + "

* ( , & " ( + "

/;):0

(

,

{

(

)

)

}

)

)

+ −τ"

ϕ τ

(

σ

σ

= − +

/;) 0

(

,

{

(

)

)

}

)

)

+ −τ"

ϕ τ

(

σ

σ

= − + )

* ( ( + " & & & + ' " " & & & " + ' " " & - & % - ;) & ;) " ) 1 ""

- & ( + + ( + & + , (

" + + & ( < ) ) 8 ' ( + "

9

1 ! 6 7 " + ( & ' ( ( + " /;) 0

" " , ' ) & & - ( & "

" "

σ

& & " ( " ( 1 ) ) , (

. $ " + " ( , + " " - & - + " &) A " $- & " " & + / + & 0 ) *

& (( ( + + " + (

) & & " ( + " /;):0 " & + ( 5

1 ) & " "

σ

)

4( " " ( ( " " ( ""

C / 9970 + & " ( , + " ) ( " " ( % - 7) ) $

7) )7 "" & + & - " ( (( - ( )

A + - ( ( "" " " + & " 2 " " " - & + &)

1 ! & ( , 8 + + ' &

+ + " " " ( & " ' &

, ' " ( " " " &) & & ( " " ( ( % - 7) ) $7) )7 - ( & " " "

&) * " " " ( & + & & " , & " + & " &) 6 - ) & ) & & !

+ " - - " + ) & (( " "( (

' " " ( ( + " /;):0 & /;) 0 + & + "

( ) , + " ( " " " -

σ

=

ν

( "" , 9

( )

( )

+

=

( $

−

−τ

9$

(

)

₍

₎

(

)

" ( 9

ν τ

ν τ

= + +

=

−

ν τ

& $ + "

& - ( ) * " !" #$% " ( + " , " " " & - " " " )

C - + , ' ( ( " " " )

& & " # " " " " + ) * + & , & " " $" # & " ( "

"" ( " " " + "" + '( " )

@) < *C$ 4 * 4 A4 1 K4 * *

& # ( ) 1 ( & - " (

( & ( - + + & & " & & E

( () * + + + & ) 1

& ( "" "" (

" " " + "" + '( " )

& & + J - ' , '

+ ' , & $ ' & + + & " " J (

" + "

{

0

≤ ≤

}

- () * (

{

(

=

0

≤ ≤

}

, ' # " & ) 1 ""

(

,

5

×

*

→

'

(

0

)

$ & " ( & (

, ( & " + &

{

0

≤ ≤

}

) 1 ( "" ' # " )

< 4<4% * 4 @) )

,

(

)

)

, ( =" , ∈

( ' & , 7) * " 2 " " &

(

)

)

, 0 =" , ( =0 ≤ ≤ ∈ 0∈* )

. 3@) ) - - " & ( & " - & - # & A

/ 97:0 /. " 99:0) 6 " & ' + + + '

3 & - & ( + + /3 99; 0)

& , =

σ

) * , ' " +

+ ( () 1 & (

(

)

5

*

σ

×

→

∞

σ

=

,

& "" ( )

! @)

/@) 0

σ

(

(

)

="₎

σ

∈

(

0

)

") ( 0

σ

=

σ

= ≤ ≤ ∈ 0∈* )

. 3@) ) % - 7) )7 " '

,

&

σ

( + "" + & ( (

+ " ( ( " ( + 2 ,

(

0

)

&

σ

(

0

)

& & " ( + "

0

- + J & + )

. 3 @)7)

σ

& " " " " "

$ & " ( & " " ( ) * + ' "" + "" - " ( + , )

. 3 @);) * , ( (

,

&

σ

( ' < @) &

2 /@) 0 " - -" & & & " ( ( " & (

' ' - - 3 / & % ( = ) < 70)

@) " " K " "

( "" ++ & " ( + /@) 0 " - $ & " ( & " " (

σ

5

×

*

→

(

∞

)

& " " "

σ

(

(

)

* "" + ( "" ' ( ( " " " )

: + +%,) * $ & " ( & " " ( (

(

0

)

σ

=

ν

( +

ν

) * ( & + J (

σ

& "

{

0

≤ ≤

}

+ (

+ "

0

& & ) * 2 " ( ' " " "

σ

=

ν

% - 7) ) & "" " !" #$% " + & " "" & % -;)7) )

: "0 & - +%, * $ & " ( & " "

( ( +

σ

(

0

)

=

ν

(

0

)

( +

ν

5

×

→

(

∞

)

) * ( &

+ J (

σ

& "

{

0

≤ ≤

}

+ ( + " & ' 0 ) * "" " + $

& " "$& & " " ( % - 7) ) ) ' " & + " " " " + & " (

ν

- " ( ( & & &

+ "" ( + , ' ( "" )

: ! 4 +,% + ' " $ & " ( & " " I "" I ( +

σ

(

0

)

=

ν

(

0

)

( + (

ν

5

×

*

→

(

∞

)

) *

( & + ' " $& & " "

( ( % - 7) )7)

=) %8.. * % 4 %

& & ( " " " & & & ( - & 2 ) A ' " ( "" " "

+ & " " "" + + " + ( (

$ , & + ' " " " # )

A " - ( & & ' ( + " (

!" #$% " & " " " " + & " " " " " ( " " ) * & ' ' " + & " " " + ( "" ) 1

" & " & ( $ & " ( & " " -" " " & " " " & ' - " J - '

( + " ( " )

* " & "& " & + " , E +

+ $& & & & & " ' ) " "&

-' ( & !" # / 9:=0 ( + " ( ( "

' & ( & + " " + )

* ( + , && & ) 4 & (( "

+ , + & & ((

1 " &) 1 + " ( ( )

A "" + & + ( + & " & " & $ & " ( & " " + & ( ( " & + "" +

+ & + " ) + & " & ( '

+ J ) 1 , "" 2 + + "

<<

) ! ' ' + " "

γ

= (

∈ ) 1 " ( " "

1 " ( " ) * ' =

γ

1

" - - " + & ( " ( & $; ( +

( ) )

)

'

=

γ

=

/3 J & % 99 ) 79 < ) 0)

) %

γ

∈

⊂

/ " 999 ) ; 0 '

/ " 999 )77 ) 7 0)

7) %

γ

∈

' + ' " " ( " / " 999 )

; < )7:) 0 & " ( " ) A +

(

)

"

γ

="

γ

< ∞

/ " 999 ) ; $;7 < )7:) 0 2 " ( "" ( +

γ

∈

)

* ' 2 $ ' -" / " ) 0 & ( 2 $ ' -"

+ ' " " )

;) A ≤ ≤ ′

/ ) 0

"

(

'

′

−

'

)

=

"

(

′

γ

)

=

"

′

γ

=

′

"

γ

/ " 999 ) ; $;7 < )7:) 0) * ' & " ' " + " ( 5

"

γ

=" "

γ

=" =

( 2 " "& - " ( & , /! "" ' " 99@ ) ;;

* + 7;);0 & & ( "" ( + + "

γ

= ) " '

/ ) 0 "& "

(

'_′−'

)

= −′ )

* < ) "& M (> & ) A + " +

' 1 " - - " + & ( " ) !

( ) "

(

'_′−'

)

= −′ . +- ' / ) 0 "&

"

γ

′ ′

7

′

≤ < ≤ ( + "

γ

= ∈ / " 999 ) =

< )=0)

<<

& , & - ' ( + ( % - ;) ' & ") A +

+ - - " + & ) 2 " / & (

0 & $ # & + & " ) - &

, " ,

{

& − & &

}

& + " " + ' " "

& / J & 99 ) 7 ; )= & & ' ++ 0)

A + - " J ( 6 > +

/ ) 0 ) = − &

1 " - - " + ) & ( " / J &

99 ) 7 = < ) 0) ( + & ( " ,

{

& &

}

< ∞ / $ 3 0 & 2 $ ' -" + ' " " / J &

99 ) 7 $7 9 < ) @ & & ++ ) 7 90) ( "" &

&= −

λ

+7

λ

∈

& 7 2 $ ' -" + ' " "

/ ) 0 & = −

λ

/ & % ) : * + @);0) +- ' / ) 0 / ) 0 "&

& (( " ( +

/ )70 = −

λ

+ ) )

- L> " ++ / " 999 ) @; * + ) 0 & &

( = − ( (

/ );0

(

=

(

µ

−

)

(

+

σ

(

)

" ' / )70 / );0 "&

(

)

)

(

=

µ

− −

λ σ

(

+

σ

(

)

% ( + ' " " ) & I & " " " ( "

( ) I +

;

/ " 999 ) = < ):0) *

λ

=

(

µ

−

)

σ

& ' L> " ++ '

-)

( = ( +

σ

( )

<< 7

: ) A & " + " ) &

'+ $ "' - ' & - &

(

_≤

=

{

(

≤ ≤

}

)

.. 7) )% ;∈

/N0

(

( )

ω

=

(

( )

ω

′

≤ ≤

ω

∈;

ω

′∈;

) 1 ' & - & " / 999 ) 7 0) A # " (

N & ' '+ $ "' - ) * "

( "" - ( Ω N '+ $ "' - Ω) * " &

'+ $ "' - '+ $ "' - ' & - ) "" ( ""

N)

! ' ' ( + ' ' & - " (

( ( +

( )

{

}

{

5 ))) 3

}

; =

ω

∈ Ω (

ω

= ∈

Γ

≤ ≤ 3∈ &

Γ

! " ( 3/ - ' + & J 9: ) $70)

;

ω

∈ &

(

( )

ω

=

(

( )

ω

′

≤ ≤ ( +

ω

′ ∈ Ω) * " "

( )

( )

(

ω

=

(

ω

′

+ & + = ))) ₃

ω

′ ∈;) * & &

" ( N & ' '+ $ "' - )

A & + ' " & " ( "" - ( Ω

N) ; = Ω N Ω

Ω ∈ )

, " ;∈ &

ω

∈; &

(

( )

ω

=

(

( )

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