[PPT] - Ultimate Implementation and Analysis Ultimate Implementation and PowerPoint Presentation

SLIDE 1

Ultimate Implementation and Analysis Ultimate Implementation and Analysis

f the AMO Algorithm
f the AMO Algorithm

for Approximate Pricing for Approximate Pricing

f European
f European-
Asian Options

Asian Options Akiyoshi Shioura Akiyoshi Shioura (Tohoku University) (Tohoku University)

joint work with T. Tokuyama joint work with T. Tokuyama

SLIDE 2

Summary Summary of This Talk

f This Talk

p t i

n

： t y p i c a l f i n a n c i a l d e r i v a t i v e

p

r i c i n g E u r

p

e a n

A

s i a n

p

t i

n
n

b i n

m

i a l m

d

e l

d

i f f i c u l t t

c
m

p u t e a c c u r a t e l y ⇒ a p p r

x

i m a t i

n

A

i n g w

r

t h , M

t

w a n i & O l d h a m ( S O D A ) t i m e : O ( k n

2

) , a b s

l

u t e e r r

r

O ( n X / k )

O

u r A l g

r

i t h m : t i m e : O ( k n

2

) , a b s

l

u t e e r r

r

O ( X / k ) n , X : p r

b

l e m p a r a m e t e r s , k : t i m e

e

r r

r

t r a d e

f

f p a r a m .

SLIDE 3

Option Option

p t i

n

: r i g h t t

s

e l l (

r

b u y ) s

m

e f i n a n c i a l a s s e t ( e . g . , s t

c

k ) a t s

m

e p

i

n t i n t h e f u t u r e ( e x p i r a t i

n

d a t e ) f

r

a s p e c i f i e d p r i c e ( s t r i k e p r i c e )

g

a i n m

r

e b e n e f i t b y i n v e s t m e n t

h

e d g e r i s k f r

m

t h e f l u c t u a t i

n
f

s t

c

k p r i c e

SLIDE 4

Payoff of Option Payoff of Option

s t

c

k p r i c e g

e

s u p t

$

2 2 a t t h e y e a r

e

n d ⇒e x e r c i s e

p

t i

n

t

b

u y t h e s t

c

k a t $ 2 ⇒s e l l i t f

r

$ 2 2 ⇒ g a i n $ 2 （ p a y

f

f ) s t

c

k p r i c e g

e

s d

w

n t

$

1 7 ⇒d

n
t

e x e r c i s e

p

t i

n

⇒ p a y

f

f = $ E x a m p l e :

p

t i

n

t

b

u y a s t

c

k

f

G

g

l e I n c . a t t h e y e a r

e

n d a t $ 2 P a y

f

f

f

E u r

p

e a n O p t i

n

: ( S ‒ X )

+ ＝m

a x { S ‒ X , } （Ｓ： s t

c

k p r i c e a t e x p i r a t i

n

d a t e , Ｘ： s t r i k e p r i c e ）

SLIDE 5

European European-

Asian Option

Asian Option

p

a y

f

f

f

E u r

p

e a n

A

s i a n

p

t i

n

d e p e n d s

n

a v e r a g e

f

s t

c

k p r i c e A d u r i n g w h

l

e p e r i

d

p a y

f

f :( A ‒ X )

+ ＝m

a x { A ‒ X , } strike price Ｘ t i m e S : s t

c

k p r i c e A : a v e r a g e

f

s t

c

k p r i c e ( S

X

)

+ =

( A

X

)

+ >

s a f e a g a i n s t f l u c t u a t i

n
f

s t

c

k p r i c e

SLIDE 6

Computation of Option Price Computation of Option Price

p r i c e

f
p

t i

n

= d i s c

u

n t e d e x p e c t e d v a l u e

f

p a y

f

f

n

e e d t

m
d

e l t h e m

v

e m e n t

f

s t

c

k p r i c e O u r m

d

e l : b i n

m

i a l m

d

e l ( d i s c r e t e m

d

e l ) p r

p
s

e d b y C

x

, R

s

s & R u b i n s t e i n ( 1 9 7 9 ) r e p r e s e n t s t

c

k p r i c e m

v

e m e n t b y a b i n

m

i a l t r e e c a n c

m

p u t e e x a c t

p

t i

n

p r i c e b y ＤＰ

SLIDE 7

Binomial Model Binomial Model

a p a t h P= ( S , S

1

, S

2

, . . . , S

n

) f r

m

t h e r

t

t

a

l e a f r e p r e s e n t s t h e m

v

e m e n t

f

s t

c

k p r i c e p a y

f

f

f

E u r

p

e a n

A

s i a n

p

t i

n

=

i n i t . s t

c

k p r i c e Ｓ

u Ｓ d Ｓ d

2

Ｓ u

2

Ｓ u d Ｓ

p r i c e g

e

s u p t

u

S w i t h p r

b

. p t h p e r i

d

S 1 s t p e r i

d

S

1

n

t

h p e r i

d

（ e x p i r a t i

n

d a t e ） S

n

p r i c e g

e

s d

w

n t

d

S w i t h p r

b

. 1

p

2 n d p e r i

d

S

2

u d = 1

+ =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +

∑

X n S

n i i

1

SLIDE 8

Our Problem Our Problem

c

m

p u t e t h e e x p e c t e d p a y

f

f

f

E u r

p

e a n

A

s i a n

p

t i

n
n

t h e b i n

m

i a l m

d

e l

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +

+ =

∑

X n S E

n i i

1 p a y

f

f i s d e p e n d e n t

n

t h e p a t h P= ( S , S

1

, S

2

, . . . , S

n

) （ p a t h

d

e p e n d e n t

p

t i

n

） p a y

f

f i s n

n

l i n e a r w . r . t . t h e r u n n i n g t

t

a l ∑i S

i

⇒ n e e d e n u m e r a t i

n
f

a l l t h e p a t h s ⇒ e x p

n

e n t i a l t i m e c

m

p u t a t i

n
f

t h e p r i c e

f

p a t h

d

e p e n d e n t

p

t i

n

i s # P

h

a r d

SLIDE 9

Approximation Algorithms Approximation Algorithms for Pricing European for Pricing European-

Asian Option

Asian Option

M

n

t e C a r l

M

e t h

d

b a s e d

n

p a t h s a m p l i n g e r r

r

b

u

n d d e p e n d s

n

t h e v

l

a t i l i t y

f

s t

c

k p r i c e O t h e r m e t h

d

s b a s e d

n

h e u r i s t i c s n

t

h e

r

e t i c a l e r r

r

b

u

n d

SLIDE 10

AMO Algorithm and its Variants AMO Algorithm and its Variants

D a i , H u a n g & L y u u ( 2 2 ) a b s . e r r . : a d j u s t #

f

b u c k e t s

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ k X n O

O u r R e s u l t

S h i

u

r a & T

k

u y a m a ( 2 4 ) a b s . e r r . : u s e b

t

h i d e a s A i n g w

r

t h , M

t

w a n i & O l d h a m ( 2 ) t i m e : O ( k n

2

) a b s . e r r . : O ( n X / k ) D P + b u c k e t i n g

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ k X O

O h t a , S a d a k a n e , S h i

u

r a & T

k

u y a m a ( 2 2 ) a b s . e r r . : r a n d

m

i z a t i

n

（ n ： d e p t h

f

b i n

m

i a l t r e e , X : s t r i k e p r i c e , k : p

s

i t i v e i n t e g e r ）

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ k X n 4

1

O

i n d e p e n d e n t

f

v

l

a t i l i t y n d i s a p p e a r s !

SLIDE 11

Exact Algorithm by DP Exact Algorithm by DP

( 2 6 7 , 1 / 4 ) 1 1 5 2 2 5 6 7 4 4 1 3 3 8 1 5 6 7 3 ( 2 5 , 1 / 2 ) ( 1 6 7 , 1 / 2 ) ( 4 7 5 , 1 / 4 ) ( 3 5 , 1 / 4 ) ( 2 1 1 , 1 / 4 ) ( 1 , 1 ) p r

b

. . 5 u = 1 . 5 p r

b

. . 5 d = . 6 7 ( 8 1 3 , 1 / 8 ) ( 6 2 5 , 1 / 8 ) ( 5 , 1 / 8 ) ( 4 1 7 , 1 / 8 ) ( 3 3 4 , 1 / 8 )

a t e a c h n

d

e

f

b i n

m

i a l t r e e , c

m

p u t e a l l p

s

s i b l e r u n n i n g s u b t

t

a l s & t h e i r p r

b

a b i l i t i e s ∑ =

t i i

S

( 4 1 7 , 1 / 8 ) ( 2 7 8 , 1 / 8 ) ( 2 4 1 , 1 / 8 )

SLIDE 12

AMO Algorithm (1) AMO Algorithm (1)

# of running subtotals can be exponential

⇒ approximate running subtotals by bucketing

interval

running subtotal & probability

400 300 (310, 0.05) (205, 0.15) (240, 0.12) (285, 0.20) (170, 0.10) (150, 0.10) (110, 0.10) (80, 0.05) (30, 0.01) 300 200 200 100 100

r

u

n d u p r u n n i n g s u b t

t

a l s & s u m u p p r

b

a b i l i t i e s i n e a c h b u c k e t

400 300 (400, 0.05) 300 200 (300, 0.47) (200, 0.30) (100, 0.06) 200 100 100

SLIDE 13

AMO Algorithm (2) AMO Algorithm (2)

k : #

f

b u c k e t s a t e a c h n

d

e ⇒ e r r

r

b

u

n d ≦ m a x . v a l u e

f

r u n n i n g s u b t

t

a l / k P r

p
s

i t i

n

: r u n n i n g s u b t

t

a l i s ≧ ( n + 1 ) X a t t h e t

t

h p e r i

d

p t i

n

w i l l b e e x e r c i s e d a t t h e e x p i r a t i

n

d a t e c

n

d i t i

n

a l e x p e c t a t i

n
f

t h e p a y

f

f c a n b e c

m

p u t e d e a s i l y

∑ =

t i i

S

⇒ e r r

r

b

u

n d

f

A M O a l g

r

i t h m = ( n + 1 ) X / k

SLIDE 14

Algorithm Algorithm by by Dai et al. (2002) Dai et al. (2002)

( , ) ( 1 , ) ( 2 , ) ( 1 , 1 ) ( 2 , 2 ) ( 2 , 1 )

A M O a l g

r

i t h m ： u s e t h e s a m e n u m b e r k

f

b u c k e t s a t e a c h n

d

e

∑∑

= = n i i j ij

j i k X ) , ( ω

k k k k k k

1 1

k

1

k

2 2

k

2 1

k

2

k k e r r

r

b

u

n d s e t t h e n u m b e r

f

b u c k e t s k

i j

a t t h e n

d

e ( i , j ) f l e x i b l y

p r

b

a b i l i t y

f

r e a c h i n g n

d

e ( i , j )

e r r

r

b

u

n d

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ k X n O

a d j u s t #

f

b u c k e t s k

i j

t

m

i n i m i z e e r r

r

b

u

n d u n d e r t h e c

n

d i t i

n

∑k

i j

= k n

2

SLIDE 15

Algorithm Algorithm by by Ohta et Ohta et al. (2002)

al. (2002)

A M O a l g

r

i t h m ： a p p r

x

i m a t e r u n n i n g s u b t

t

a l s i n a b u c k e t b y r

u

n d i n g

u

p

interval

running subtotal & probability

200 100 (170, 0.30) (150, 0.10) (110, 0.20) ( 170, 0.60) ( 150, 0.60) ( 110, 0.60)

p r

b

. 1 / 3 p r

p

. 1 / 2 p r

b

. 1 / 6

c h

s

e a r u n n i n g s u b t

t

a l r a n d

m

l y a s a p p r

x

i m a t e v a l u e ( 200, 0.60)

SLIDE 16

Analysis of Ohta et al. (2002) Analysis of Ohta et al. (2002)

r e g a r d t h e b e h a v i

r
f

r a n d

m

i z e d a l g

r

i t h m a s s t

c

h a s t i c p r

c

e s s ⇒ M a r t i n g a l e e x p e c t a t i

n
f

t h e e r r

r

b y r a n d

m

c h

i

c e

f

r u n n i n g t

t

a l s a t a n

d

e = ⇒ a p p l y A z u m a ’ s i n e q u a l i t y ( 1 9 6 7 )

y) probabilit high (with O bound error ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ k X n 4

1

a n a l y s i s i s d i f f i c u l t

SLIDE 17

Our Algorithm Our Algorithm

s e t t h e n u m b e r

f

b u c k e t s k

i j

a t n

d

e ( i , j ) f l e x i b l y r a n d

m

c h

i

c e

f

r u n n i n g s u b t

t

a l

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡

∑∑

= = 2 1 n i i j ij

k j i X ) , ( O ω

e r r

r

b

u

n d a d j u s t #

f

b u c k e t s k

i j

t

m

i n i m i z e e r r

r

b

u

n d u n d e r t h e c

n

d i t i

n

∑k

i j

= k n

2

e r r

r

b

u

n d

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ k X O

a n a l y s i s i s q u i t e e a s y !

SLIDE 18