Financial Computation

(1)

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

0 +& ' +12 & + 3 +' 4'& 5 #& 6

7 89:;< => ? @A ? B CCD < E FGH I J 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ _`a bcd e f ! " +& ' +1 % ! 1 g& 7 h &# 5 +& 5 6 i 2 j k & ' ! & +1! . 1* # . # & 5 +& 5 6 # & %& ' - l m + j n . ' . ' & ' 6 # & %& ' - o p j # # 6 q r ' +#1 # & %& ' - s ) ' # 6 # + ! . 1 6 t r . 1 % +1 & k # +& ' 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B u ev ew ed x ey z {" j n +" j| 7 " " } ~ } + ! k &# . - 4'& 5 #& 6 7 m# 6 6 } h h z $ * 3 +# + 5& +1 k+ 1 + ¡£¢ ¤ ¥ ¡¦§ ¨ § ¤ ¡ § ¨ § ¤ © ª z m" k &1 6 j . +' ." ' "k &1 6 j . + # 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =D « j ' m#* 6 6 # 6 $% j 1 6 +' . r # ' +' ). & ' 5 6 &. ' 3 +# # +' 6 } m#* 6 6 # r # ' 1 6 j! !6 ' 7 ¬' ). 1 6 j 1 +6 / & %6 j #® 1 +%¯ 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = A

(2)

`a a c d b be c a ) & 6" ' 3 # j 7 * g % 11 ' ! ' 1 6 j " # &61¯ 16 + 5 6 & ' j 1 k+ # * % ! " +& '

* &# . «& 1 j! 1 | & k '&

f q l q f s f q 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = e y` w ey v bae c c e z 2 # +% k+ 1 $ 1 5k+ 1 & ' &! $" % j + 6 +' . +' & . . + 6 +'& 11 + & ' 6 z ) ' # +% k+ 1 p j # 3 & 6 +' & . . + 6 ¬& 6 +' & ' 6 ( % # & +' # 6 & ' 6 ¬ # +% j 6 ¬' +1 7 & % +1 + # g&! +& ' 6 ¬ # g&! +& ' +1 -# & j ! 6 r ' +# 1 6 &! " 1 +& ' } % 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =

5

10

15

20 x

2000

4000

6000

8000

10000

12000

14000

16000

Time

x

2 x

3

2 x

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = % e y` w ey v bae c & x e z $ +% & 6" "6 +11 7 ' +' & 6"6 z ) % " 1 .k +'& 6"6 * # 1 ' -n . + .' g . n & ' % ! 6%" # & & 6 # + j n . ' . ' . # & 5 +& 5 6 z ( # & 6 j % +1% " 1 +& ' ¬ # j + 6) * h ' . 6 } 3 j # & 6 j ' "! k # * #

& . 6" ' & 1! + " &#

7 6 1 + # ¬ i n 7 +# 6 % " # & 7 j + 6 ! # j +' i q o+ i , f -. + 7 6 $ + # * # + i n 7 +# 6%" &# 7 j + 6 , o+ f /. ' . 6 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0

(3)

! " # " ! 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = $ % &' () *+ ', -) & 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ C . # ew /w c ev 0a cy w v 1 ew d $ cd d x e z f 2 m j j & 6 6 3 4 5 * ® +% j&1 # f t s f 2 l q z f 2 o 6! . # ' #* &1 j # 7 r 2 m * r ¯+# &3 z f 2 q 6 j + & +1 ¬ 6 6 m&# %& ' -r . 1 ¬ m r * 2 # 7 ' # } $ j +# } | & ' ' # f 2 f q f 2 t l } +' . r &6 6' z f 2 q 6 j 6 %& ' ! ¯+# 6 j 7 j & 6 6 * !$ + " 1 6 ' +' . / + ! + z f 2 s 6 j # 7 * & ' # & %& ' -* ® 1 +%¯ f 2 i t f 2 2 o +' . $% j 1 6 z f 2 s 6 # 6 ' ' 3& ' 6#"! ' 6 +' . # & %& ' -! j . 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@ @ . # ew /w c ev d 1 / c y e 1 0 cy a w v 1 ew d à ew y 7 8 9:; <= >? @AB C AD EGFAH @I J 8 9 ; KML 8 9NO PQ R B S TU V J 8 9 8W L 8 9XY PQ Z HU [? BS \? J 8 9 ; : L 8 9 9X P^] 7 8 9 Y ; <= B _ <> \A` a S B> @ < J^b : Q c de f [ Q @ > \] P Z HU Bg T >? AD <A IH T h < > Z > @B Z> U < T< >AB Jî @II Zj g < P^] 7 8 9X; <=R @kZ <e U <> B S ` @ U > < a Z >@ U > @ H TU > @ V B Z H> @ \ T B \ S T > < l iZ \m S < J e in P TU o @U H>@ p qr R r f d] 7 8 9; K <= e U B>@ U @ > J fr s f P Z HU ` Z DTUB Z ` @ < J e i n P^] 7 8 9N; <=B Z@I >AT UZ IH Z Z > Z g < @ J [AH H P Z HU s [ < J f a a I @ P ] 7 8 9 O ; <= e i n s [Z UH j A > S < 8 h W h : ] 7 8 9 9 ; <= t TU H Au <: ] 8 J n T \ B A <AD > P Z U H t AB I H t T H @ t @ g Jî @ B UB @ < h j @ @ P^] 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ B

(4)

" a "cy ` w y e y . `a z / & ' +' %& +1 j # & 6 & ' # & %& ' - z r + !j +& % +1k +%¯ -# " ' . 6 z &# 5 +& 5 6 % " # & & 6 z m&# %& ' -! . 1 6 z , 6 %& ' +1 -# & ! j &6 ' # & %& ' -' ' +' %& +1& ' 6#!" ' 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@D & - ) ) ('- 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @A z ¬1 -# & j ! 6 +# # %& 6 # % ." # 6 j + % +' k " # ' . & ' % ! " # # -# + ! 6 z 4' % ! " k+ 1 # k 1! 6 6 & j 6 # -#! + j + 5& '' '& 1 6 ) 6 & j 6 # -#! + k " -* # z ! " k+ 1 # k 1! 6 ) ' # +% k+ 1 # k 1 ! 6 2 # +% k+ 1 # k 1! 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@ bae c z $ +# &3 j + 6 * k &+ 6%

# +& ' 6 3& j% j &3 11k

+ 6"!6 . ¯ + ' " '& * &! z 2 j +1 ' "! k # * j 6 # +& ' 6 & 6 j +1 3 #¯ . ' k 7 +' +1 -# & ! j & 6% ! " +& ' +1 % ! 1 g& 7 z 2 j 6 +% % ! 1 g& 7 & 6 j ! + " ' * ! ! # 7 6 +% " 6 .k 7 +' +1 -# & ! j z ' % ' # + ' j k+ 6 # +% % ! 1 g& 7 * +' +1 -# & j ! & ' 6 + . * & 6 . + & 1 .&! 1 ! ' +& ' z ! 1 g& 7 & 6 + -. -"& . +' +1 -# & ! j 6 #" ' '& ' -&! 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @

(5)

.y a a c xy z ' & 6 . # j 6 +#% j +1 -# & ! j ' f t z 2 j 3 # 6 n % + 6 % ! 1 g& 7 & 6 % ! +# & 6 ' 6 z 2 j # +# # +& ' 6k & 6 . 6% ! +# & 6 ' z « % +# ' 1 7 + k " j + 6 7 ! & % -# 3 j # + ' j g +% ' "! k # * # +& ' 6 $ j % ! 1 g& 7

* ! & +' & +'& '

- j 1 & 6 6 "k "!6 .k 7 j % ! 1 g& 7 * j k . 7 * j 1 z 2 j % ! 1 g& 7 & 6 j ' % 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ % . b aw c " v w & e w x " cd d be aed f h i ' n * " ' . 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ 0 z | 6 +' *. # j 6 & * j # k 1! 0"! k # * !1 ' 6 } ' "! k # * % + 6 j 3 6 } % z

| & ' +# &! &* j % !

1 g& 7 & 6 z

" + .# +& % &! &* j % !

1 g& 7 & 6 z

"k & % &! &* j % !

1 g& 7 & 6 z , g ' ' & +1 &! &* j % ! 1 g& 7 & 6 h ! " # z $" # 1 7 ' ! & +1&* j % ! 1 g& 7 & 61 6 6 j +' g ' ' & +1 k " & j -j # j +' +' 7 1 7 ' ! & +1 z ) & 6 6 6 & k 1 * # +' g ' ' & +1 n &! +1 -# & ! j # * !# 3 11 ' 7 & % +1 & ' " 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@ $ . d cy x d x e a c d `a $ ew v w d x e z ¬ # ." % & ' * j #" ' '& ' -&! * # ! f 6 o 6 & 6' + 6 6 & -'& ' % +' + 6 j +* # ! f j o j z ® " & j &6 6 3# ' - « j +&* 7 " j + 5 f } 6 % " &# & 6 # & % « j + &* 7 " ! " 6 ! +% # + & ' . + .1 & ' 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B C

(6)

- , - & & , - ' ' - , 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =B@ " e c e cd e 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = BB z ) * & ' # 6 & 6% ! " ' . . &! 6 # +' ' "! } / m f " f z ¬ 6 +' . f & ' , f } / m " 3 j # h s f t h t z ) * j +' ' " +1& ' # 6 # + & 6 * # 7 +# 6 +' . * # j * 11 &3 ' - 7 +# 6 } j / * ! f &3 11k @ " @" B " B 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =BD # xcaed . b aw c " y v w $ # d 1 $ # z 2 j m * j % + 6 j 3 $ $ $ " + &! 6 f h & 6 $ f % $ f % '& & & $ " f % " z ! " .k 7 j +1 -# & j ! ' h o & ' &! 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = BA

(7)

. b w c a " v w b ` a cd $ # f % f f $ * 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =B " e 1 e / e "cd 1 0 w d ew y `u be z

2 & j &6 . &+ 6

" ( # ' # f s t q f t i s & ' f t f 2 z ) & 6 j! 6 6 %& ' 6 6 & k 1 & ' ! # 6 * j k+ 6 1" ' "! k # * &+# ! j & % # +& ' 6 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B e 1y z 2 j !# 7 & 1 . . ' 6 j # " # ' * & ' 5 !6 ' z 2 3 &3 . 1 7 " 6 . 7 & 1 . 6 +# j k ' . "& 5 +1 ' 7 & 1 . ® , { +' . j! # -+ - "& 5 +1 ' 7 & 1 . r , { ® , { % # # 6 ' . 6 j & ', f ' h i j + " + 6 m &3 j / 3 j ' h r , { % # # 6 ' . 6 j & ' , f ' h i j + " + 6 m &3 j / 3 j ' f h 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =B % a d ew d b au e v au e` w d u u z

) & 6 j& ' # 6 # + 3& j% j

" + 6 +'& ' 5 !6 ' 6 m 3 & j& 6 # & % } $ f % $ f % $ f %

& & &

$ " f % " z ) + 6 6 "! 6 +11 % + 6 j 3 6 +# # & ' 5 6 . + j # + + 6 j & ' # ' +1 # + * # " # ' z 2 j k+ 5* ! # " 1 &+ 6 j* " ' . +& ' " ' 3& j% j # & %& ' -! j . 1 -& 6 +# k "& 1 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B 0

(8)

` ew c x b a e " 1y v w c e b1y z $ 1 5 % * # % f } 3 j # % " $ f % & 6 j! +#¯ # & % z 2 j* " ' % & ' %

& ! 6 ' '& %& '

% &* $ , * # +11 z ( ' % +" '& " 6 1" & ' g& 6 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =B $ " e ae ! d u "y d a e " 1 z $ +# &3 j +' # 6 + # g&! +& ' + # * z 2 j ' " z « j ' % ! " & ' - 7 & 1 . 6 } " $ f " 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =D C

x

k

x

k 1

x

y

f x

( )

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =D @ & b cd & y a e y v d b cd e w ` a c d y z 2 j 0 3 ' n + j 6 '! j .% +' k g ' . . & j -j # . &! ' & 6 ' 6 z| % k j j + # g&! +& ' j 6 1 " & ' * j 3 6 &! " 1 +' " 6 " +& ' 6 } % % 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =D B

(9)

& b cd & y a e y v d b cd e w ` a c d y x d `x b 1 e 1 z 2 j f 6 + # g&! +& ' " % " 6 +& ' 6 6 j * 11 &3 ' -1 & ' +# " +& ' 6 } 3 j # " " +' . % " % " % z 2 j k+ 5 j + 6 +" '& " 6 1" & ' * # " % " 3 j ' j h+ ! h + # & g& 6 & ' 5 #& k 1 z $ " % " " % % " 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =DD e e b ` d / d 1y z " ' # + z m +# 5 +1" } + & . +! + " &# 7 z . ' 6 j +# 5 +1" +' . $ . ' 6 j % " ' z + 6 j 3 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = DA x ! " #$ % 8 '& () # % 8 *& () ! 8 + % 8 '& () & ( % 8 *& ( ) , z ' "! k # * % + 6 j 3 6 z '!" k # * + 7 ! ' 6 # 7 +# z +' ' " +1 # + % ! " ' . . &! 6 # +' '!" z $ -* 3 j ' -& 6 j +' ' " +1 % " ' # + z m&# % % +' k % ! " &. ' f &! 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =D ) & *. ( - , / ) ' - ' - 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =D

(10)

z 1 +& &1 7 ! +"6 # 6 j 3k ' . # & % 6 # 6 ' . & ' # 6 # + % j +' - 6 z ) & ¯6 7 j &# ¯ 6! +' + - ! ' * & ' # 6 n # + n

6 ' & 6& 5 6 % " &# & 6

z ¬ 6"!6 1 5 1 n % " ' k ' . 6 j# " -j " z ' ' #

& % 5 1 +& &1

7

+ 6 j 6 ' & 6& & 5

7 * j #% ' + - # & % % j +' - % j +' - 6 & '& ' # 6 # + 6 } 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =D % $w c x e # a b a c bc v / d 1y z 2 j # & % 5 1 +& &1 7 * +% " ' k ' .& 6 $ * % $ * % f % "" f % $ * % f % "" f % f % 3 j # & 6 j +# 5 +1" } +' . $& 6 j % " ' + 7 ! ' # # & . z ( ' % % ! " k+ 1 & ' % ' 6 +' &! z / # k ' . 6 &3 j " ! k . . . & ' 6 } , 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =D 0 x` b ` wa c d z 2 j r +% +" 1 + 7 ." # +& ' r & 6 + 3 & -j . + 5 # + - * j &! 6 +' + 6 6 6% + 6 j 3 6 z 2 j 3& -j 6 +# j % + 6 j 3 6 m 6& . & 5 . .k 7 j + 6 6 6 # & % } r f " $ f % z 2 j r +% +" 1 + 7 ." # +& ' } & ' # & . 6 } & 6 " +1 r f % * % h 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =D $ v / d 1y z 2 j r * +% " ' k ' .& 6 r f " $ f % f % " i z +' k &! 6 1 & ' . r - f % f % " f % % - -% f % " f % 3 j # -& 6 j # & . % " ' # + z 2 j r * + # n % "

' k ' .& &6 6 ! # ! + " &#

7 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = A C

(11)

z , " +& ' 6 h ' i 2 +' . i ' l j 1 . ' 1 7 &* j % " ' $ } j +# 5 +1 " } +' . j ! + " # & 7 +# +11 & ' . ' . ' * j 7 & 1 . % z 2 j +& 6 }

&* j % + 6 j 3& &6 ' .

' . ' * 7 & 1 . 6 z 2 6 & j 6 & ' } 6 " 6 j! +#¯ 7 & 1 . . %&1 ' 6 z 2 j r &3 11k 1 ' - j ' . z

/ # 6 % " &# & 6 3 j !6 + " &#

7 +% " +11 7 . % # + 6 6 + 6 + # "6 1 } j r ! + 7 +% " +11 7 . % # + 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =A @ 0 ! a "cd v z 2 j r j +&6 6 &#

-& ' & '! +"6&# '

- j 1 '

- j * &! +

k ' .& ' 5 !6 ' & 6 " 6 +' & . '

- z ® " 7 " " 6 & j + 3 + 7 + 7 " # # & 1 z 2 j r 6 j " 1 .k 6 '! & +' 1 7 +! 6 +"6&# ' - # & % 5 1 +& &1 7 z r +' 7 } &* ' ! 6 } ." # +& ' n # 1 + . ! # & ' 1 - 7 % +' ' k % ! # j ' . . j # &3 6 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =A B 1 c e 1 ` wa c d z

r & '. . ." # &+ ' & 6 .' ' . + 6

* % r f % l z ® 7 2 + 7 1 # g +' & 6 ' } ) + z ® j r +' . ! . &' . ." # +& ' +# + 6 7 % +1 % " 1 + 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = A D a e x c e ` wa c d z ¬

-' # +1 ' "! &# % +1* !# " 1 *+ # 5 1 +& &1

7 & 6 j % & 5 ." # +& ' } " % " % o & 6 j # & % &* j 7 & 1 &. 6 . % # + 6 .k 7 % " & 6 j # & % &* j 7

& 1 .& &6 ' % # + 6 .k

7 % & 6 j

& '& & +1

# & % } % & 6 j

& '& & +1

7 & 1 . % & !6 6 +11 z p ' % +' % ! " j % & 5 ." # +& ' * " 6 k+ " +' 7 ' ' +' %& +1& ' 6 #"! ' 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = AA

(12)

z ) & ! 6 6 " *6 " 1 3 j # 7 & 1 . % j +' - 6 +1 # j % + 6 j 3

# 6 % " &# & 6 3 j 6 % + 6 j 3& 6 6 % !

1 g j + &! 6 1 * !# " 1 + 6 +# " ' + 5& +1 k+ 1 z ¬' +1 # ' +& 5& 6 " 6 " % z 2 & j 6 & 6 ! # % ' ! & % +1 k " 1 6 6 +% % " # + 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =A d ae c z ' 5 g& 7 & 6 . ' ' . + 6 & ' # & . 6 % f q z / # +k ' . &3 j 6

& & 5 % ' 5 g&

7 }

j

#

& % &# 6 ! 6 #

* # + # + . %&1 ' j +'& * +11 *6 # + # + & ' % # + 6 *

" +1 ! + -'& " . z ( ' % } k 3 ' 3 k ' . 6 &3 j j !6 + ." # +& ' } j ' &3 j + & j -j # % ' 5 g& 7 & ! 6 # 5 +1" k+ 1 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =A z 2 j + # g&! +& ' * ) + 3 # ¯ 6 * # 6 ! +11 7 & 1 . % j +' - 6 z 2 &! # 5 " '& * # 1 +# -# 7 & 1 . % j +' - 6 } " 6 ) % f % f h % f % + % f h + + % 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = A %

0.02

0.04

0.06

0.08 Yield

50

100

150

200

250 Price

9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = A 0

(13)

( ' (+ ', + ( ) % &' ( ' ' 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =A $ ew a & w` a x` w e v ad ew a ey au ey z ( 3 . & ' # 6 # + 6% j +' - 3 & j ! + " # & 7 z 2 j 6 * 7 & 1 . 6 ! + " &# 7 * # k ' .*6 !# 6 j !# 6 #" % " # 2 j k ' . 6 ! " 6k * " +1 " +1 & 7 2 j 7 . & # 6 1 1 7

& ' j& # ! # 6 ! + " &#

7 z 2 j ! # 6#" % " # & *6 " ' !. + ' +1 j 5 +1" +& ' * ' g . n

& ' % ! 6 % " &# & 6

9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = C

0

5

10

15

20

25

30 Year

1

2

3

4

5

6

7 Yield (%)

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = @ ew a & w` a x` w e v ad ew a ey au ey x d `x b 1 e 1 z 3 * ' #* # 6 g % 1" & 6 5 1 7 j 7 & 1 . 6 * # n % " ' k ' . 6 z ¬ 1 6 7 & 1 . 6 ! + " # & 7 + -+ & ' 6 ! + " # & 7 z ¬ 7

& 1 . % " # 5& 6% ' 6 #" % *. # ! k ' . 6 # +& .'

-' +# +# 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B

(14)

& a au ey z 2 j n # & . 6 # + & 6 j 7 & 1 . ! + " &# 7 * +' n # & . # n % " ' k ' . z 2 j m * ' . 11 +# # & .*6 # ! ' 3& 6 f z 2 j ' n # & . 6 # + & 6% +11 . j 6 j # # + z ¬ 6 # + % " # 5 & 6 + 1 * 6 # + 6 + -+ & ' 6 ! + " # & 7 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = D & a u a e cy x ` d a e a " 1 b z ¬ 1 5 1 n % " ' k ' . j + 6 j # & % " $ f f " s z 2 & j 6 # & %& ' -! j &. ' % # # + &6 '* !# +& '* # ! j ! # 6#" % " # z & %6 " ' +% j % + 6 j 3 + j % # # 6 ' & . ' -6 # + 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = A x ` a a x y z ) ' -' # +1 } +' 7 # & ¯ 6 1 6 6 6 % " &# 7 j +& 5' -+% + 6 j 3 $ $ $ " 6 j " 1 . j + 5 ! + +#¯ # & % * " $ ¬k 5 } f } f h } +# % +11 . . & 6 % " ' * +% # 6 & 6 j m * ' . 11 +# # & .*6 # ! ' 3 z 2 j . & 6 % " ' * +% # 6 +# * '& ' # 1 + . * ! # +

% ' & ' " " *6 " ' % & ' % +11 . j & . %6 " ' * " ' % & '

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = a w a x cd & a au ey vw c e b1 ` w e z $ +# 3 & j j 6 j # # + f 0 j + 6 j # n # ! 2 # +"6 # & 6 +# # n % " ' k ' . 6 z ! " h * # ! j 3 n # & . % " ' k ' . # & % k 7 6 1& 5' - $ f f $ f f h 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T =

(15)

a w a x cd & a au ey v w c e b1 ` w e x ad cd` e 1 z ) ' ." % & 5 1 7 } 3 +# -& 5 ' j ! +#¯ # & % * j n # & . % " ' k ' . +' . f h f z 2 j ' % +' k % ! " *. # ! , s } # + . k 1 3 } " $ f f " z 2 j +1 #" ' '& ' -&! & 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = % a w a x cd & a au ey vw c e b1 ` w e x d `x b 1 e 1 f f * f f f * f h i $ 1 5 $ + $ f * f * # f 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0 z ' 6 & . # + k ' . 3 & j j % + 6 j 3 $ $ $ " +' . 6 11 & ' -* # z $

& ' % &# ¯ &6 ' !6 6 " 6k % !

' 6 + . } " $ f z 2 j 6 +& % 6 # + .& 6 j ! + " ' "6 % j j + " $ f z +' k % ! " .k 7 j 0 3 ' n + j 6 '! j . 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = $ + & * &' ' ' -' -, ) & , ' 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = C

(16)

2 j # +# j# ¯& ' . 6 * &1 6 1 & 6 } . + ! ' &1 6 } +' . 6 +& &6 % 6 ® ' +

! & ' & 6# +&1

f t l f t t f p' . + &j 6 + # + - . 7 } k " + ! & 1&1 ' . + j 6 +# + 6 +& 6 & % 6 * $ +1 & ' f t s 2 f 2 o i 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = @ ed a y z

2 j 5 &+# +' % * + # +' !. 5 &+# k+ 1 & 6 .' ' . + 6

+# z

2 j % 5 &+# +' % k 3 ' # +' !. 5 &+# k+ 1 6 +' . & 6

5 3 j # +' . +# j! +' 6 * +' . } # 6 % & 5 1 7 z +' !. 5 &+# k+ 1 6 +' . +# " ' % # # 1 + .&* 5 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = B e a `a z

¬ # +' !. 5 &+# k+ 1 j + 6 j ' ! # +1 & . 6&#k " & ' &3 j

! +' +' . 5 &+# +' % &* & 6 # k k+& &1 7 . ' & 6 7 * " ' % & '& 6 ! # B ! B # * h z 2 & j 6 & 6 g # 6 6 .k 7 z

2 j 6 +' . +# . ' ! # +1 & . 6&#k " & ' j + 6 # ! +'

}

" '&

5 &+# +' %

}

+' . j & . 6&# k " & '* " ' % & '

m# k f h B 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = D ed ew a c d v d c w c a e w b cy a w c ` a c d y z | k " '&* ! # 1 7 . & 6&# k " . 5 # f 6 j + m# k * # f z + .1 7 .# + 3 3 6 + ! 1 6 +' . * # ! " ' & 1 h f h f f z 2 j ' - h f +' . - h f +# & ' . ' . ' 6 +' . +# . ' ! # +1 5 &+# k+ 1 6 3 j # - h 1' * 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = A

(17)

z| +# & ' . ' . ' +' ." '&* ! # 1 7 . & 6&#k " . 5 # f z ¬ &! 6 1! j . -' # + j 6 +' . +# . ' !# +1 5 &+# k+ 1 & 6 % +1% " 1 + q z ¬1 3 + 7 6k 1! + 7 " # # +' !. '!" k # -' # + # 1 +6 & ' 6 + . } % j %¯ 7 " # # -#! + '6 # 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = ed ew a c d v / c w c a e w b cy a w c `a c d y z m& +# 6 * ' ! # +11 7 .

& 6&#k " . 5 &+# k+ 1 6 &3 j % # # 1 +& '

% +' k -' # + . z| +' . k & ' . ' . ' 6 +' . +# . ' !# +1 5 +# & + k 1 6 z 2 j ' f

+# j . & 6# . # +' !. 5 &+# k+ 1 6 &3 j

+# +# +' . 5 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = " e d w b a cy w c` a c d z

¬ # +' !. 5 &+# k+ 1 & 6 &6 + . j + 5 + 1

-' !# +1 . & 6 # & k " & ' &* 1' j + 6 + ' !# +1 . & 6 # & k " & ' z | +' . z 2 j! +' +' . 5 &+# +' % * +# " B +' . " B B f t # 6 % & 5 1 7 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = % ' -) & - , 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0

(18)

2 j 6&*j 3 +# . & ' 6 + 6 j % ' # * -# +& 5 7 * ' ' +' % r # ' ( r& 11 # f 2 h i h 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = $ bb y d 1 $ `a y z ¬ % +11

-& 5 &6 6 j 1 . # j &#

-j k " 7 + ' "! k # * j " ' . #1 7 & ' -+ 6 6 k 7 + 7 & ' -+ 6&#¯ # & % z ¬ " -& 5 6 & 6 j 1 . # j # & -j 6 11 + ' "! k # * j " ' . #1 7 & ' -+ 6 6 * # j 6 # &¯ # & % z ¬' ! k . . . & ' j + 6 k # + . . +1 ' -3 & j j " ' . #1 7 & ' -+ 6 6 z ( 3 # & % & ' 6 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = %C x e z « j ' +% +11& 6 g #%& 6 . } j j 1 . # + 7 6 j 6&#¯ # & % & ' g % j +' - * # j 6 %¯ z « j ' + " & 6 g #%& 6 . } j j 1 . # # % & 5 6 * # ! j 3&# # j 6&#¯ # & % & ' g % j +' - * # j 6 %¯ z ¬' & ' % +' k g #%& 6 . # & # j g & # +& ' . + +# 1 7 g #%& 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = %@ . ew c x d d 1 ` w e d z ¬ ! &# % +' & ' 6% +' k g #%& 6 . + +' 7 &! " j g & # +& ' . + z , " # +' & ' 6% +' ' 1 7 k g #%& 6 . + g & # +& ' z ¬' ! ¬ &# % +' & ' & 6 3 # j + 1 + 6 +! 6 " % j + 6 +'

j # &3 6& . ' & % ,+1 " #

+' & ' k % "+ 6 * j +#1 7 g #%& 6* + " # 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = %B

(19)

d ed caed d aed c d y z $ % +11 5 +1" z " 5 +1 " z 6&#¯ # & % z 6 %¯ # & % z . & 5 & . ' . 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = %D $ z 2 j + 7 * +% +11 + g

& # +& ' & 6 $

! +g z 2 j + 7 * + " + g

& # +& ' & 6

! +g z ¬ +' 7 &! k * # j g & # +& ' . + } 3 % +11 ! +g

j & ' &# ' & 6% 5 +1" * +% +11

z ¬ +' 7 &! k * # j g & # +& ' . + } 3 % +11 ! +g j & ' # & ' 6 & % 5 +1 " * + " z / & ' & . ' -+' & ' 6 5 +1" + +' 7 &! k * # g & # +& ' & 6 ! + + #& ' 11 % " +1k # ¯ + j# " -j 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = % A

20

40

60

80 Price

Long a put

10

20

30

40

50 Payoff

20

40

60

80 Price

Short a put

-50

-40

-30

-20

-10

Payoff

20

40

60

80 Price

Long a call

10

20

30

40 Payoff

20

40

60

80 Price

Short a call

-40

-30

-20

-10

Payoff

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = %

80

85

90

95

100

105

110

115 Stock price

0

5

10

15

20 Call value

80

85

90

95

100

105

110

115 Stock price

0

2

4

6

8

10

12

14 Put value

9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = %

(20)

y " c c 1 ed 1y z , g % j +' - n # + . . 6 %¯ & ' 6 +# ' % + 6 j . & & 5 . ' . n # % . 2 j & ' % ' # +% & 6 ' + . " 6 . * # % + 6 j . & 5 & . ' . 6 z 2 j 6 %¯ # & % * +11 6k 7 +' ! + " ' # " -j1 7 " +1 j +

! " ' * j % + 6 j & . & 5 . ' . +&6

- 6 g n . & & 5 . ' . 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = %% ( '- ( - & ' -) &. ( - , - & 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = %0 z 2 j ' n +#k& # + - # & ' %& 1 6 + 7 6 j # 6 j " 1 .k ' * # 1" ' % j z ) "6 1 & 6 j +# -"! ' * # & ' # & %& ' - z ¬ &# ¯ 6 1 6 6 +#k & # + - # " '& 7 & 6 ' j + } 3 & j " +' 7

& '& & +1& ' 5 !6 '

} -' # + 6' ' ' -+& 5 # " # ' 6" ' . # +11 %& #% "! 6 +' % 6 +' . 6 & & 5 # " # ' 6" ' . # !6 z ) ' +' 6 %& ' ! +#¯ } 6 " % j # " '& & 6 . ' g& 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = %$ u e b a c e _ a c d $w c x ey z 2 j 6 # 1 +& ' 6 j 1 . # -+# .1 6 6 * j

# k k+& &1 &6 %

! . 1* # 6 %¯ # & % 6 z ¬ 6"6 ! } + ! ' - j # j & ' -6 } j + j # +# ' # +' 6 +% & ' 6% 6 6 #! +# -& ' # " & # ! ' 6 } k # # 3 & ' -+' . 1 ' . & ' -+# + 5 + & 1 + k 1 + j # & 6 ¯ 1 6 6 & ' # 6 # + } & ' # 6 # + 6 +# ' ' ' -+& 5 } +' . j # +# ' +#k & # + - # " '& & 6 z 2 j " n % +11 +# & 7 $ m 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0 C

(21)

' -) &. ( - , - & ) * 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = 0@ " e a&a e cd z 2 j ' n +#k & # + - # & ' %&

1 & &6 ' "6 6 %& '

& ' . 3' j g +% & ' 5 +1" &3 j " * " # j # + 6"!6 & ' 6 ' j

# k k+& &1 &6 % k j +& 5 # * 6 %¯

# & % 6 z p ' ! + # k 6 +% 1 & 6 j + & 6 ! 6 + # & 6 ¯ n + . " 6 . & ' # 6 # + & 6 ' . . . & 6 % " ' j & ' 6 + 7 z ® # ¯ + j# " -j%! + & ' f 2 s i 3 j '® 1 +%¯ f 2 i t f 2 2 o +' . $% j 1 6 &3 j j 1 * # ! r # ' " k &1 6 j . j& # % 1 k # + . & ' # & %& ' -! .1 z ' 3 ' + 6 j ® 1 +%¯ n $% j 1 6 & ' # & %& ' -! . 1 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0B z $ % +11 5 +1" z " 5 +1" z 6 # &¯ # & % z 6 %¯ # & % z , j % ' & ' " " 16 7 % ! " ' . .&# ¯ 6 1 6 6# + # # & . z -# 6 6 # " # ' z $ +#* # ! j & . %6 # n

&! k & ' ! & +1! .1

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = 0D / cd c b _ a c d $w c xcd 1 e b / _ $ z

2 &! & 6& . %6 # +' !. +"6 # &. '

# & . 6 z ) * j % " # # ' 6 %¯ # & % & 6 } & % +' - 3 & j # k k+& &1 7 +' . &3 j # k k+& &1 7 f } 3 j # f +' . ) '* +% } ! " 6 j 1 . # " 1 " +#k & # + - z $ & g

& % 6 * & '* ! # +& ' "6 6 % . !# & ' j

& ' 5 +1" k + 6 . ' +#k& # + -% ' & 6 . # +& ' 6 } } } } } +' . j ' "! k # * # & . 6 g & # +& ' 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0 A

(22)

S

Su

q

1 q

Sd

9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = 0 bb d d c c 1 ed 1 $ cd & a x & c d be $ ew c 1 z 2 j g

& # +& ' . + & 6 ' 1

7 ' # & .* # ! ' 3 z $ & 6 j #

& % + &! ' &* j 6 %¯

# & % ! 5 6 z $ & 6 j # & % + &! ' &* j 6 %¯ # & % ! 5 6 z 1 +#1 7 } $ ! +g $ ! +g 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0

C

Cu= max( 0, Su X )

q

1 q

Cd = max( 0, Sd X )

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = 0 % bb $w c x cd cd _ d e $ ew c 1 z $ " + #* &1 * 6 j +# 6 * 6 %¯ +' . . 11 +# &6 ' # & ¯ 6 1 6 6k ' . 6 2 & j 6% 6 6 « % +11 j j . -# +& # . 1 + z 2 j 5 +1" * & j 6

#* &1 + &! ' & 6& j #

# z j 6 +' . "6 % j j + j #* &1 # 1 & % + 6 j + 7 * j % +11 } $ $ 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = 0 0

(23)

x _ e 1 x `x 1 e 1 z $ 1 5 j k+ 5 " +& ' 6 k & +' $ $ 2 $ $ f z ® 7 j ' n +#k & # + - # & ' %& 1 } j , " # +' % +11 6 j " 1 . % 6 j 6 + ! + 6 j " & 5 +1 ' #* &1 } $ z ¬ 6 $ $ } j "& 5 +1 ' #* &1 & 6 +1 5 # . 1 ' - 6

& & '& ' 6 %¯ 6

9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = 0$ . ew c x d bb $w c x cd cd _ d e $ ew c 1 z

( + 5 % ' & 6 . #&! ! & . + g #%& 6

z $ ! +g « j ' } j % +11 6 j " 1 . ' k

g #%& 6 &!. ! & . + 1

7 « j ' } j & ' 6 j " 1 .k

g #%& 6 &!. ! & . + 1

7 z / # ' ' n . & & 5 . ' . n + 7 & ' -6 %¯ 6 } +# 1 7 g #%& 6& 6' &! +1 } 6 $ 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = $C $ `a $w c xc d c d _d e $ ew c 1 z m " 6% +' k 6 &! & 1 +#1 7 # & % . z 2 j . 1 *+ # j " & 6 * } 3 j # ! +g ! +g z| ! # z 2 j, " # +' " & 6 3 # j z 2 j ¬ ! &# % +' " & 6 3 # j ! +g 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = $@ $y `e 1 $w ac b c z *¬ # 6 " k 6 & " & ' +' . # +# # +' - ! ' } $ $ f f z 3&# , f f + 6 $ f $ 3 j # z ¬ 6 f } & ! + 7 k & ' # # . + 6 + # k k+ & &1 7 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = $B

(24)

u cy ` a e w b $w ac bc z 2 j g % . # + * # " # '* # j 6 %¯ & 6 " +1 j # & ¯ 6 1 6 6# + " ' . # + 6 f z & ¯ 6 n ' " # +1& ' 5 6 # 6% +# ' 1 7 + k " g % . # " # ' 6 z 2 j g % . # + 6 * # " # ' * +11 6 % " # & & 6 ! " 6k j #

& ¯ 6 1 6 6# + 3 j '& ' 5 6 # 6 +# &# ¯ 6

n ' " # +1 z / # & j 6# + 6 ' } & 6% +11 . j &# ¯ 6 n ' " # +1 # k k+& &1 7 z 2 j 5 +1 " * +' & '& 6 j g % +& ' * & 6 . & 6 % " ' . * " " # + 7 & ' + # & 6 ¯ n ' " # +1 % ' ! 7 z $ j # + " 6 *. # & . %6 " ' & ' - j / & 6 j &# ¯ 6 1 6 6 # + & ' +&# ¯ 6 n ' " # +1 % ' ! 7 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = $D / c d c b a cy w c` a c d z

' j k& ' ! & +1 & . 6&#k " & ' &3 j

+#! + # 6 +' . k 7 f " f " + f

& & &

h+ f &3 j j % ' 5 ' & ' f z $" 6 7 " 6 6 +% & ' &! 6 3 & j k & ' - j # k k+ & &1 7 * - & ' -j + . 6 z 2 j ' & 6 j # k k+& &1 7 * - & ' - j + . 6 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = $ A _ a c d d d c c 1 ed 1 $ cd & a x `a b c $ ew c 1 z ' & 6 . # +% +11 &3 j 3 #

& . 6#! & +'& '

-k * # g & # +& ' z 4 ' . # j k & ' ! & +1! . 1 } j 6 %¯ % +' + ¯ ' j# 6 6 & k 1 # & % 6 + &! 3 } } +' . 0 j + j # % ! k& ' 6 z ¬ +' 7 ' . } j ' g 3 6 %¯ # & % 6 ' 1 7 . ' . ' j % " # # ' # & % } ' j # & % 6 * +#&1 # &! 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = $

S

Su

Sd

Suu

Sud

Sdd

9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = $

(25)

_ a c d d d c c 1 ed 1 $ cd & a x ` b a c $ ew c 1 x a d cd` e 1 z| $ k j % +11 6 5 +1" + &! 3 &* j 6 %¯ # & % & 6 z 2 j" 6 } $ ! +g z $ +' . $ % +' k % +1% " 1 + . +' +1 -" 61 7 } $ ! +g $ ! +g 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ = $%

C

Cu

Cd

Cuu= max( 0, Suu X )

Cud = max( 0, Sud X )

Cdd = max( 0, Sdd X )

9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = $0 _ a c d d d c c 1 ed 1 $ cd & a x `a b c $ ew c 1 x ad cd` e 1 z 2 j % +11 5 +1 " 6 + &! ' % +' k k + & ' .k 7 + 1 7 & ' - j 6 + ! 1 -& % $ $ f $ f h $ $ f $ z 1 + 6% +' k . &# 5 .* # ! , 2 ' t 2 z / # g! + 1 } j . 1 + + $ & 6 $ $ * 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ = $ $ _ a c d d d c c 1 ed 1 $ cd & a x ` b a c $ ew c 1 x d `x b 1 e 1 z « ' 3 # +% j j % " # # ' # & . z ¬' " & 5 +1 ' #* &1 * 6 j +# 6 * 6 %¯ +' . ! # & 6 ¯ 1 6 6k ' . 6% +' k 6 " * # j % +11 j + % 6 6 $ $ } # 6 &* j 6 %¯ # & % - 6 } # 6 z 2 j 5 +1" 6 * +' . % +' k . &# 5 .* # ! , 6 2 f ' t 2 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ C C

(26)

x e

z

$

& ' % j % +11 &3 11 ' k g #%& 6 . + &! ' 5 '&* &

& 6! ¬ &# % +' } $ +' . $ z 2 j #* # } ! J 8 + P J 8 + P + ! + + , z $ j % +11 + -+

& ' &3 11 ' k g #%& 6 . +

# 6 ' } +' . $ $ f $ 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ C@ / x ! w 1 d `1a x c d ew e b z 2 j k+ 5 g # 6 6 & ' % +1 % " 1 + 6 $ * # ! j 3 6 " % % 6 6 # ' . 6 $ +' . $ +' . ' ' k 7 ' . z 2 j !6 + % ! " +& ' j + ' 6 + $ +' . $ } } + 6 . ! ' 6# + &. ' , f h ' 2 2 z 2 & j 6 # % " # 6 & 5 # % ." # & 6% +11 .k +%¯ 3 +# .& ' ." % & ' z 0 3 } $ " +1 6 W J 8 + P J 8 + P J 8 P ! ` Z k % ; + ) W J 8 + P ` Z k J ; + P J 8 + P ` Z k % ; + ) J 8 P , 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ CB / x ! w 1 d `1a x c d x ad cd` e 1 z ) ' j n # & . % + 6 } $ " " f " + ! +g " " 2 j 5 +1" * +% +11 ' + ' ' n . & & 5 . ' . n + 7 & ' -6 %¯ & 6 j g % . & . %6 " ' . + 7 + g

& # +& ' & ' +

# & 6 ¯ n ' " # +1 % ' ! 7 z 2 j 5 +1" * , + " # +' " & 6 " " f " + ! +g " " 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@ CD u cy e `a w b $w c xc d a e " 1 b z , 5 # 7 . &# 5 +& 5 % +' k # & % . +&6* j % ' ! 7 3 # # & ¯ 6 n ' " # +1 z / # + , " # +' n 6 7 1 . # & 5 +& 5 3 & j j # ! & ' +1 + 7 * " ' % & ' } & 6 5 +1 " & 6 " ! +' 6 j g % +& '& 6 ¯ + ' " ' . # j # & ¯ 6 n ' " # +1 # k k+& &1 7 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ C A

(27)

" e / c d c b _ a c d $w c x cd $ w ` b z| k j! & '&! !" '!" k # * " 3 +# . # & % ! 5 6 * # j % +11 ' '& 6 j& ' j! ' 7 z $ & 6 j ! 6 +11 6 ' ' ' -+& 5& ' -# "6 % j j + " } # 1' * " 1' * z ( ' % } $ " " f " " " 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ C ` ew c x b bey z ¬ ' ' n . & 5 & . ' . n + 7 & ' -6 %¯ & 6 6 11 & ' -* # ! f q z f o +' . o z f t h i h # # & . z ' 6 & . # + , " # +' % +11 ' j & 6 6 %¯ 3 & j f o +' . i z 2 j % +11 5 +1" & 6 ! t o q 2k 7 k +%¯ 3 +# &. ' ." % & ' z ¬1 6 j m * j g % . + 7 + g & # +& ' } i 2 + i l i i + l l f f h t o q 2 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ C

160

540 (0.343)

180 (0.441)

60 (0.189)

20 (0.027)

Binomial process for the stock price

(probabilities in parentheses)

360 (0.49)

120 (0.42)

40 (0.09)

240 (0.7)

80 (0.3)

85.069 (0.82031)

390

30

0

0 Binomial process for the call price

(hedge ratios in parentheses)

235 (1.0)

17.5 (0.25)

0 (0.0)

141.458 (0.90625)

10.208 (0.21875)

9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@ C % ` ew c x b b ey x d a cd ` e 1 z r& 6 # & %& ' -1 + . 6 +#k & # + - # ' 6 z $" 6 j

& ' & 6 6 1&1 '

-* # ! 2& ' 6 + . z $ 11 j % +11* # ! 2 +' .& ' 5 6 ! t o q 2 & ' j # 1 & % +& ' - #* &1 3 & j t h i f 6 j +# 6 * 6 %¯ # " & # .k 7 . 1 + z ® # # 3 t h i f+ f q t o q 2 l q f t q . 11 +# 6 z 2 j * " ' . j + # ! + & ' 6 } 2 t o q 2 l 2 i f . 11 +# 6 } & 6 j +#k& # + - # ' + 6 3 3 & 11 6 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ C 0

(28)

` ew c x b bey x ad cd` e 1 2 &! f z $" 6 j 6 %¯ # & % ! 5 6 ! h l z 2 j ' 3 . 1 &+ 6 2 q h o z ® " 7 2 q h o t h i f t o 2 l ! # 6 j +# 6 + j % 6 * t o 2 l+ h l h q h o q . 11 +# '6 ' +' % .k 7 k # # &3 ' - z k ' 3 +1 6 h q h o q l q f t q+ f h s q l h i h . 11 +# 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ C$ ` ew c x b b ey x d a cd ` e 1 2 &! h z $" 6 j 6 %¯ # & % 1" ' - 6 ! f h z 2 j ' 3 . 1 &+ 6 h o z $ 11 2 q h o h o q o q h o 6 j +# *6 # +'& ' % ! * q o q h o+ f h s t s o . 11 +# 6 z 4 6 j & 6 & ' % ! # ." % j .k s q l h i h+ f h s t s o f h o . 11 +# 6 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ @ C 2 &! i j % + 6 * &# 6 & ' - # & % z 2 j 6 %¯ # & % ! 5 6 ! f t z 2 j % +11 3 3# ' '& 6 j &6 ' j! ' 7 z / # + 1 6 6 * f t f o i . 11 +# 6 } % 1 6 " j 6 & & ' k 7 & j # k " 7 & ' -k +%¯ j % +11 # k " 7 & ' -+ 6 j +# * 6 %¯ * # .&1 5 # 7 z / & ' +' %& '

- & j 61 6 6 &3 jk # # &3 '

-k &# ' -6 j +1 .k f h o+ f h i l o . 11 +# 6 z ) & 6 # + & .k 7 6 11 & ' - j h o 6 j +# 6 * 6 %¯ * # h o+ f t l o . 11 +# 6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@@ @ ` ew c x b b ey x d x b ` 1 e 1 2 &! i j % + 6 * . %&1 '& ' - # & % z 2 j 6 %¯ # & % ! 5 6 ! q z 2 j % +11 3 3 # & 6 3 # j1 6 6 z $ 11 j h o 6 j +# 6 * 6 %¯ * # + +1 * h o+ q f o . 11 +# 6 z 4 6 & # + 7 j .k * f h o+ f h f o . 11 +# 6 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @@ B

(29)

z

2 j® p m r&!

1

& 6 j k& ' ! & +1 # +1

-# & ! j j + + 1

& 6k +%¯ 3 +# .& ' ." % & '

z 2 j +1 # " ' '& ' -&! & 6 z 2 j! ! # 7 # "& #! ' & 6 z 2 # & % , " # +' " 6 } 6 &! 1 7 # 1 +% j + 7 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ @D

C[2][0]

C[2][1]

C[2][2]

C[1][0]

C[1][1]

C[0][0]

p

max

0 ,

2 Sud

X

c

h

max

0 ,

2 Su d

X

c

h

max

0 ,

3 Su

X

c

h

max

0 ,

3 Sd

X

c

h

1 p

9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ @A ! w a 1 " e / b x & x " bey $ w ` b z ¬ 6 j ' " ! k # * # & . 6 } } & ' % # + 6 6 } j 6 %¯ # & % # +' - 6 5 # 5 # 1 +# -# ' "! k # 6 * 6 6 & k 1 5 +1" 6 } +' . # + . & ' - + ¯ 6 1 +% ' +# 1 7 % ' & ' " " 61 7 z ¬ # # % &+1k # +& ' * j! . 1 +#! + #! 6 ¯ + 6 j ® p m r % ' 5 # - j % ' & ' " " 6 n &! ! . 1 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@@ ! w 1 a " e / b x & x " b ey $ w ` b x d a cd ` e 1 z | . ' j &! g & # +& ' * j & ' ! +"6 # &. ' 7 +# 6 z | k j % ' & ' " " 61 7 % ! " ' . . +' ' " +1 # + z &m %¯ " " f h f h z p j # % j& % 6 +# 6 6 & k 1 6 g z 2 j &# ¯ 6 n ' " # +1 # k k+& &1 7 ! + 7 1 & " & 6 . f z 2 j # k 1! 6& . 6 + +# 3 j ' , * 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @@

(30)

! w a 1 " e / b x & x " bey $ w ` b x a d cd` e 1 z ¬ 6 1' +

# +% j 6 j ' !# +1 & . 6&#k " & ' &3 !j +'

1' +' . 5 &+# +' % z j + 6 + 1 -' ! # +1 . & 6 # & k " & '& ' j 1 &! & 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ @ % ! w a 1 " e / b x & x " bey $ w ` b x d `x b 1 e 1 $ 1' * * h 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ @ 0 / _ $ d 1 / b x & x " b ey 1 e b z 2 j® 1 +%¯ n $% j 1 *6 ! # " 1 + ' . '6 5 +#! + # 6 } } } } +' . z ® & ' ! & +1 # +1 -# & j ! 6 + ¯ 6 & g& ' " 6 } } } } } +' . z 2 j % ' ' % & ' 6 +# " " * z 2 j k & ' ! & +1 # +1 -# & ! j 6% ' 5 # -# + 6 ' k+ 1 7 * +6 z p 6%& 11 +& ' 6% +' k 1 &! & ' + .k 7 j " . & %& " 6% j& % 6 * +' . 6 g 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@@ $

5

10

15

20

25

30

35 n

11.5

12

12.5

13 Call value

0

10

20

30

40

50

60 n

15.1

15.2

15.3

15.4

15.5 Call value

9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @B C

(31)

z 1 +& &1 7 & 6 j 6 1 +# + ! # ' . & # % 1 7 k 6 # 5k+ 1 z 2 j® 1 +%¯ n $% j 1 *6 ! # " 1 +% +' k " 6 . % ! " j ! ¯+# 6

& '& ' * j 5 1 +& &1

7 $ 1 5* # -& 5 ' j & ' # & % } } } } +' . 3 & j ' "! &# % +1 ! j . 6 ( 3 k+ " ¬ ! &# % +' & ' 6 z

2 & j 6 5 1 +& &1

7

& 6% +11 . j &!

1

& . 5 1 +& &1

7 z )! 1

& . 5 1 +& &1

7 & 6 * ' #* # # . & j 6 &# % +1 5 1 +& &1 7 & ' # +% & % 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ B@ / cd c b w e e . b aw c " y v w . ew c x d `a$ y z , +# 1 7 g #%& 6 j + 6 k % ' 6 & . # . z 2 j k& ' ! & +1 # +1 -#

& ! j 6 +# 6 &3 j j !# & ' +1

+ 7 !6 +g " +' . + 1 & 6k +%¯ 3 +# . & ' ." % & ' z ¬ +% j& ' # ! . & + ' . } & % j %¯ 6 * # +#1 7 g #%& 6 k 7 % ! +# & ' - j + 7 &* g #%& 6 . 3 & j % ' & ' " +& ' 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ BB _ a c d y d a & x a" $ y c c 1 ed 1y z , +#1 7 g #%& 6 ! " 6k % ' 6 & . # . z m#

#& ' +1 & . & 5 . ' .

+ 7 " ! . 1& 6 # +% k+ 1 6 g

2 j & . & 5 . ' .! + " ' & 6 +% ' 6 +'

# #& ' * j # 5& +&1 ' -6 %¯ # & % z ) ' -' # +1 } j % # # + & . & 5 . ' . 1 & % 7 & 6 +% ! 1 g & 6"6 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@BD ad cd` ` y c c 1 ed 1 c e b1y z & 5 & . ' . 6 +# m + & . % ' & ' " " 61 7 ¬ # g&! + 6 +k # + . n k + 6 . 6 %¯ ! ¯+# #* &1 z 2 j + 7 ! ' * +% ' & ' " " 6 . & 5 & . ' . 7 & 1 . + # + # ." % 6 j -# 3 j # + * j 6 %¯ # & % k 7 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @BA

(32)

z 2 j ® 1 +%¯ n $% j 1 6 * ! # " 1 + 6 j 1 . 3 & j # 1 +% .k 7 $ f i f i 3 j # 1' * * h 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ B a d cd` ` y c c 1 ed 1 c e b1y x d `x b 1 e 1 z 2 #" ' k & ' ! & +1 # +1 -# & ! j 6 } & %¯ j &# ¯ 6 n ' " # +1 # k k+& &1 7 + 6 ! # f l 3 j # * z 2 j +' . #! & +' " ' % j +' - . z p j # j +' j % j +' - & ' , f l } k & ' ! & +1 # +1 -# & j ! 6 6 + 7 j 6 + ! 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ B ' & )- & ) ' -) & ) ( 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@B % / w w c ew _ a c d y z 2 j& # + 7 . ' . 6 ' 3 j j # j " ' . #1 7 & ' -+ 6 6 6 # & % # +% j 6 +% # & +' # & % 1 5 1 z ¬¯ ' %¯ n "

& ' & 6 +' # & . ' +#

7 , " # +' & ' 3 j & % j % + 6 6 g& 6 &* j k +#&# # & 6 # +% j .k 7 j # & % * & 6" ' . #1 7 & ' -+ 6 6 z ¬ % +11¯ ' %¯ n "

& ' & 6 !6 &! 6% +11 . +

. 3' n +' . n " & ' &* z ¬ " ¯ ' %¯ n "

& ' & 6 !6 &! 6% +11 . +'

" n +' . n " & ' 3 j ' , 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @B 0

(33)

/ w w c ew _ a c d y x d `x b 1 e 1 z ¬¯ ' %¯ n & '

& ' % ! &6 ' g& 6 ' % &* +% # & +'

k +#&# #& 6 # +% j . z ¬ . 3' n +' . n & ' & ' & 6 +% +11¯ ' %¯ n & ' & ' j + % ! &6 ' g& 6 ' % ' 1 7 3 j ' j k +#&# #& 6# +% j . +' . z ¬' " n +' . n & ' & 6 + " ¯ ' %¯ n & ' & ' j + % ! 6 & ' g& 6 ' % ' 1 7 3 j ' j k +#&# #& 6# +% j . +' . , z

/! # " 1 + 6 g& 6* # +11¯ & ' . 6 * k +#&# #

& ' 6 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ B $ / cd c b w e e . b w c a " y z ® +#&# # & ' 6% +' k # & % .k 7 k & ' ! & +1 # +1 -# & ! j 6 z ® 1 3 & 6 * # j . 3 ' n +' . n " & '

0 H

9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ D C

H

8

16

4

32

8

2

64

16

4

1

4.992

12.48

1.6

27.2

4.0

0

58

10

0

0 X

0.0

t } q } l } f h o } h } +' . o ® +%¯ 3 +# .& ' ." % & ' $ o+ $ o+ $ * f h o 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@D @ / c d c b w e e . b aw c " y x d `x b 1 e 1 z ® " % ' 5 #

-' % & 6 # # +& % k % +" 6 & 6' + +

# & % 1 5 1 ' j # 6 1 ' ' g + - 2 7 & % +11 7 } j k +#&# # j + 6 k + . " 6 . k + + # & % 1 51 z ( ' % j +1 -# & j ! 6 +# " 6 1 6 6 & ' # +% & % z $ 1" & ' 6 &3 11k # 6 ' . 1 + # 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ D B

(34)

100

150

200

250

300

350

400 #Periods

3

3.5

4

4.5

5

5.5 Down-and-in call value

9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ DD $ a " e ed 1aed ew ca c ey z| " . ' j # & % 6 * j " ' . #1 7 & ' -+ 6 6 5 # j &1* * j & ' z & 6 j ¯ ' 3' # & % + &! # z " & 6 j # & % + g & # +& ' z 2 j 6 +' . +# . , " # +' % +11 j + 6 + # ! & ' +1 5 +1 " . ' . & ' -' 1 7 ' j 1 +6 # & % } ! +g " z ) 6 5 +1" j" 6 . ' . 6 ' 1 7 ' j " ' . #1 7 & ' -+ 6 6 6 ! # & ' +1 # & % # -+# .1 6 6 * j 3& - 6 j # 9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ DA $ a " e ed 1aed ew ca c ey x ad cd` e 1 z ) ' % ' # +6 } 6 ! . &# 5 +& 5 6 +# + j n . ' . ' & ' j + j & # # ! & ' +1 + 7 6 . ' . % # & & % +11 7 ' j + j 6 z 2 j +# & ! j & % + 5 # + - n # + % +11 j + 6 + ! # & ' +1 5 +1" -& 5 ' k 7 ! +g f f " z 2 j + 5 # + - n # + " 6 ! # & ' +1 5 +1" & 6 -& 5 ' k 7 ! +g f f " 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@D $ a " e ed 1 ed a ew c a c ey x d x b ` 1 e 1 z ¬ 5 # + - n # + & ' 6 +# +1 6 % +11 . ¬& 6 +' & ' 6 z 2 j 7 +# " 6 * " 1 j . -& ' - 1 6 * #' !# 6 j + 3 & 11! + ¯ + 6 # + ! * " # % j + 6 6 5 # + &! # & . 2 j % 6 6 +# &1¯ 1 7 k &1 '¯ . j + 5 # + - # & % 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ D

(35)

. ew e u a e _ a c d y z ¬ 5 # + - n # + & ' 6 +# ' # & " 6 1 7 j +# . # & % z 2 j k& ' ! & +1 # * # j + 5 # + - 6 . 6' % ! k& ' 6 f i t z ¬ 6#& + -j* # 3 +# . +1 -# & ! j & 6 ' "! # + j h " # & % + j 6 * # +' n # & .k & ' ! & +1 # +' . j ' + 5 # + - j + 7 6 z ® " j g ' ' & +1 % ! 1 g& 7 ! ¯ + &6 &! # +% & % +1 z ¬ 6 + # "6 1 } j r ' +#1 ! j . +' . + # g&! +& ' +1 -# & ! j 6 +# !6 * j +1 # ' +& 5 61* 9 K B CC DL ;M > NO PQR S TPQ F E P P U V :W;XT H TY W XT Z XT W[ \W=>: ] E L T ^ =@ D %

S

Su

Sd

Suu

Sud

Sdu

Sdd

p

1 −

p













+

₋

=













+

₋

=













+

₋

=













+

₋

=

(

)

−

+

=

(

)

−

+

=

(

)

−

+

=

p

1 −

p

1 −

p

1 −

p

1 −

p

1 −

p

9 B C C DL ;M > NO PQR S T PQ F P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ D 0 z ® + 6 . ' j® p m r z ' & 6 . # + ' . + &! 3 & j j " ' . # 1 7 & ' -+ 6 6 # & % " +1 z 0 + ! "6 % j + ' . z 2 j #" ' '& ' -6 "! + & j 6' . j + 6 + ! +g&! "! 5 +1" * f

& & &

'& & & f " f f f 9 K B CC DL ;M > NO PQR S T PQF E P P U V :W;XT H TY W XT Z TX W[ \W=>:] E L T ^ =@D $ . w c a c d . b aw c " v w .y c d _ a c d y x ad cd` e 1 z

& & 5 . & j 6 5 +1" k

7 f +' . % +11& z $ &! & 1 +#1 7 } j #" ' '& ' -6 "! j + 6! + & '&! "! 5 +1" * f '& & & '& & & f " f f f z

& & 5 . & j 6 5 +1" k

7 f +' . % +11& z +' . +# # " ' '& ' -+ 5 # + - 6 9 B C C DL ;M > NO PQR S T PQF P P V T :W;X H TY T W Z T X [ X W \ => ] W : L T = @ A C