Ultimate Implementation and Analysis Ultimate Implementation and - PowerPoint PPT Presentation

Ultimate Implementation and Analysis Ultimate Implementation and Analysis of the AMO Algorithm of the AMO Algorithm for Approximate Pricing for Approximate Pricing of European- -Asian Options Asian Options of European Akiyoshi Shioura Akiyoshi Shioura (Tohoku University) (Tohoku University) joint work with T. Tokuyama joint work with T. Tokuyama

Summary of This Talk of This Talk Summary � o p t i o 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 o p e a n - A s i a n o p t i o n o n b i n o m i a l m o d e l - - - d i f f i c u l t t o c o m p u t e a c c u r a t e l y ⇒ a p p r o x i m a t i o n � A i n g w o r t h , M o t w a n i & O l d h a m ( S O D A 0 0 ) t i m e : O ( k n ) , a b s o l u t e e r r o r O ( n X / k ) 2 � O u r A l g o r i t h m : t i m e : O ( k n ) , a b s o l u t e e r r o r O ( X / k ) 2 n , X : p r o b l e m p a r a m e t e r s , k : t i m e - e r r o r t r a d e o f f p a r a m .

Option Option � o p t i o n : r i g h t t o s e l l ( o r b u y ) s o m e f i n a n c i a l a s s e t ( e . g . , s t o c k ) a t s o m e p o i n t i n t h e f u t u r e ( e x p i r a t i o n d a t e ) f o 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 o 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 o m t h e f l u c t u a t i o n o f s t o c k p r i c e

Payoff of Option Payoff of Option E x a m p l e : o p t i o n t o b u y a s t o c k o f G o o g l e I n c . a t t h e y e a r - e n d a t $ 2 0 0 � s t o c k p r i c e g o e s u p t o $ 2 2 0 a t t h e y e a r - e n d ⇒e x e r c i s e o p t i o n t o b u y t h e s t o c k a t $ 2 0 0 ⇒s e l l i t f o r $ 2 2 0 ⇒ g a i n $ 2 0 （ p a y o f f ) � s t o c k p r i c e g o e s d o w n t o $ 1 7 0 ⇒d o n o t e x e r c i s e o p t i o n ⇒ p a y o f f = $ 0 P a y o f f o f E u r o p e a n O p t i o n : ( S ‒ X ) ＝m a x { S ‒ X , 0 } + （ Ｓ ： s t o c k p r i c e a t e x p i r a t i o n d a t e , Ｘ ： s t r i k e p r i c e ）

European- -Asian Option Asian Option European � p a y o f f o f E u r o p e a n - A s i a n o p t i o n d e p e n d s o n a v e r a g e o f s t o c k p r i c e A d u r i n g w h o l e p e r i o d p a y o f f :( A ‒ X ) ＝m a x { A ‒ X , 0 } + S : s t o c k p r i c e ( S - X ) = 0 + strike price Ｘ A : a v e r a g e o f ( A - X ) > 0 + s t o c k p r i c e t i m e s a f e a g a i n s t f l u c t u a t i o n o f s t o c k p r i c e

Computation of Option Price Computation of Option Price � p r i c e o f o p t i o n = d i s c o u n t e d e x p e c t e d v a l u e o f p a y o f f - - - n e e d t o m o d e l t h e m o v e m e n t o f s t o c k p r i c e � O u r m o d e l : b i n o m i a l m o d e l ( d i s c r e t e m o d e l ) � p r o p o s e d b y C o x , R o 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 o c k p r i c e m o v e m e n t b y a b i n o m i a l t r e e � c a n c o m p u t e e x a c t o p t i o n p r i c e b y Ｄ Ｐ

n - t h p e r i o d Binomial Model Binomial Model （ e x p i r a t i o n d a t e ） S n 2 n d p e r i o d 0 t h p e r i o d 1 s t p e r i o d S S S 2 0 1 u Ｓ 2 p r i c e g o e s u p t o u S u Ｓ w i t h p r o b . p i n i t . s t o c k u d = 1 p r i c e Ｓ u d Ｓ p r i c e g o e s d o w n d Ｓ t o d S w i t h p r o b . 1 - p d Ｓ 2 � a p a t h P = ( S , S , S , . . . , S ) f r o m t h e r o o t t o 0 1 2 n a l e a f r e p r e s e n t s t h e m o v e m e n t o f s t o c k p r i c e + ∑ ⎛ ⎞ n � p a y o f f o f E u r o p e a n - A s i a n o p t i o n = S ⎜ ⎟ − = i i 0 X ⎜ ⎟ + n 1 ⎝ ⎠

Our Problem Our Problem c o m p u t e t h e e x p e c t e d p a y o f f ⎛ ⎞ + ⎛ ∑ ⎞ n ⎜ ⎟ S ⎜ ⎟ o f E u r o p e a n - A s i a n o p t i o n − = i i 0 E ⎜ X ⎟ ⎜ ⎟ + ⎜ n ⎟ 1 o n t h e b i n o m i a l m o d e l ⎝ ⎠ ⎝ ⎠ � p a y o f f i s d e p e n d e n t o n t h e p a t h P = ( S , S , S , . . . , S ) 0 1 2 n （ p a t h - d e p e n d e n t o p t i o n ） � p a y o f f i s n o n l i n e a r w . r . t . t h e r u n n i n g t o t a l ∑ i S i ⇒ n e e d e n u m e r a t i o n o f a l l t h e p a t h s ⇒ e x p o n e n t i a l t i m e � c o m p u t a t i o n o f t h e p r i c e o f p a t h - d e p e n d e n t o p t i o n i s # P - h a r d

Approximation Algorithms Approximation Algorithms for Pricing European- -Asian Option Asian Option for Pricing European � M o n t e C a r l o M e t h o d � b a s e d o n p a t h s a m p l i n g � e r r o r b o u n d d e p e n d s o n t h e v o l a t i l i t y o f s t o c k p r i c e � O t h e r m e t h o d s � b a s e d o n h e u r i s t i c s � n o t h e o r e t i c a l e r r o r b o u n d

AMO Algorithm and its Variants AMO Algorithm and its Variants D a i , H u a n g & L y u u ( 2 0 0 2 ) ⎛ ⎞ n X a b s . e r r . : ⎜ ⎟ O O u r R e s u l t ⎜ ⎟ k ⎝ ⎠ a d j u s t # o f b u c k e t s S h i o u r a & T o k u y a m a A i n g w o r t h , M o t w a n i ( 2 0 0 4 ) & O l d h a m ( 2 0 0 0 ) ⎛ ⎞ X a b s . e r r . : t i m e : O ( k n ) 2 ⎜ ⎟ O O h t a , S a d a k a n e , ⎝ ⎠ k a b s . e r r . : O ( n X / k ) S h i o u r a & T o k u y a m a u s e b o t h i d e a s D P + b u c k e t i n g ( 2 0 0 2 ) ⎛ ⎞ 1 ⎜ ⎟ n 4 X a b s . e r r . : n d i s a p p e a r s ! ⎜ ⎟ O i n d e p e n d e n t k ⎜ ⎟ ⎝ ⎠ o f v o l a t i l i t y r a n d o m i z a t i o n （ n ： d e p t h o f b i n o m i a l t r e e , X : s t r i k e p r i c e , k : p o s i t i v e i n t e g e r ）

Exact Algorithm by DP Exact Algorithm by DP a t e a c h n o d e o f b i n o m i a l t r e e , c o m p u t e s ∑ = a l l p o s s i b l e r u n n i n g s u b t o t a l s t S i i 0 3 3 8 & t h e i r p r o b a b i l i t i e ( 8 1 3 , 1 / 8 ) 2 2 5 ( 4 7 5 , 1 / 4 ) ( 6 2 5 , 1 / 8 ) 1 5 0 1 5 0 ( 5 0 0 , 1 / 8 ) ( 1 0 0 , 1 ) ( 2 5 0 , 1 / 2 ) ( 4 1 7 , 1 / 8 ) 1 0 0 1 0 0 ( 3 5 0 , 1 / 4 ) ( 4 1 7 , 1 / 8 ) 6 7 6 7 p r o b . 0 . 5 ( 2 6 7 , 1 / 4 ) ( 3 3 4 , 1 / 8 ) ( 1 6 7 , 1 / 2 ) u = 1 . 5 ( 2 7 8 , 1 / 8 ) 4 4 p r o b . 0 . 5 ( 2 1 1 , 1 / 4 ) d = 0 . 6 7 ( 2 4 1 , 1 / 8 ) 3 0

AMO Algorithm (1) AMO Algorithm (1) � # of running subtotals can be exponential ⇒ approximate running subtotals by bucketing r o u n d u p running subtotal interval & probability 400 r u n n i n g s u b t o t a l s (400, 0.05) 400 300 (310, 0.05) & 300 300 s u m u p 300 (205, 0.15) (300, 0.47) 200 p r o b a b i l i t i e s (240, 0.12) 200 200 (285, 0.20) i n e a c h b u c k e t (200, 0.30) 100 200 (170, 0.10) (150, 0.10) 100 (100, 0.06) 100 (110, 0.10) 0 100 (80, 0.05) 0 (30, 0.01)