New Acceleration Schemes with the Asymptotic Expansion in Monte Carlo Simulation

Transcription

1 CIRJE-F-98 New Acceleraion Schemes wih he Asympoic Expansion in Mone Carlo Simulaion Akihiko akahashi Universiy of okyo Yoshihiko Uchida Osaka Universiy Sepember 4: Revised in June 5 CIRJE Discussion Papers can be downloaded wihou charge from: hp:// Discussion Papers are a series of manuscrips in heir draf form. hey are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 New Acceleraion Schemes wih he Asympoic Expansion in Mone Carlo Simulaion ϒ Akihiko akahashi # Yoshihiko Uchida June 5 Absrac In he presen paper, we propose a new compuaional echnique wih he Asympoic Expansion (AE approach o achieve variance reducion of he Mone-Carlo inegraion appearing especially in finance. We exend he algorihm developed by akahashi and Yoshida (3 o he second order asympoics. Moreover, we apply he AE o approximae ime dependen differenials of he arge value in Newon (994 s scheme. Our numerical examples include pricing of average and baske opions when he underlying sae variables follow Consan Elasiciy of Variance (CEV processes. ϒ he earlier version of his work eniled as Recen Developmens on Asympoic Expansion Approach in Mone Carlo Simulaion was presened a 4 Daiwa Inernaional Workshop on Financial Engineering. he auhors are graeful o Ryusuke Masuoka for compuaional assisance. # Graduae School of Economics, he Universiy of okyo. Graduae School of Economics, Osaka Universiy (uchida@econ.osaka-u.ac.jp.

3 . Inroducion In he presen paper, we propose a new compuaional echnique wih he Asympoic Expansion (AE approach o achieve variance reducion of he Mone-Carlo inegraion appearing especially in finance. In order o compue conrol variables, we uilize he analyic approximaion based on he AE in akahashi (999 and Kuniomo and akahashi (3a. We exend he algorihm developed by akahashi and Yoshida (3 o he second order asympoics. his scheme gives us more precise esimae efficienly han he second order AE when he precision of he AE is no saisfacory for pracical purpose. Moreover, we apply he AE o approximae ime dependen differenials of he arge value in Newon (994 s scheme. hrough numerical experimens, we observe remarkable acceleraion of convergence, which implies broad applicaions of our echniques. Our numerical examples include pricing of average and baske opions when he underlying sae variables follow Consan Elasiciy of Variance (CEV processes. he organizaion of his paper is as follows. In he nex secion, we review he resul of akahashi and Yoshida (5 and exend i o he higher order asympoics. Moreover, in order o demonsrae he broad usage of he AE, we propose applying he AE o Newon (994 s scheme. In secion 3, we review he ouline of he AE approach followed by deriving he pricing formulas of average and baske opions of which he underlying sae variables are described by CEV processes. In secion 4, we presen he resul of numerical experimens.. Variance reducion echnique. he Exension of akahashi and Yoshida (5 Suppose ha he d R valued processes u (, d X s y ( s u, y R follow he

4 sochasic inegral equaion : u ( ( ( u u, = s,, s,, s, ( X y y V X y ds V X y db where ( For a sochasic approximaion of < is a parameer. ( ( u, y E f X, y, an esimaor by crude Moe Carlo simulaions is expressed as N ( G( n, N = f ( X. ( N j= j Here, [ Z ] j ( j =,..., N denoe he realized value of he ih independen rial of any random variable Z, and he discreized approximaion of X based on Euler-Maruyama scheme is given by u (, (, u s, (3 ( ( ( u = η( s η s X y V X ds V X db ns where η ( s =. n From he mahemaical poin of view, we should noe ha i is no a rivial hing o jusify his ype of approximaion based on he Mone Carlo mehod, which has been ofen used in pracice. In paricular when f ( is no a smooh funcion such as cash flow funcions for opions, we need a careful discussion on is mahemaical foundaion. While akahashi and Yoshida (5 have invesigaed his problem in some deails, we shall focus on he pracical aspecs of our mehod for financial applicaions hroughou his paper. Definiion A modified new esimaor of u(, y is defined by ( ( N * ˆ ( G, n, N = Ef X (, ˆ( y f X f X. (4 N j ( ( Here, we assume ha E fˆ X, y j= is calculaed analyically. We call he While we omi he descripion of some mahemaical seing, we assume ha all he necessary mahemaical condiions o modify he formula are saisfied. [ x] indicaes he larges ineger ha is no greaer han x. 3

5 Mone Carlo simulaion which uses (4 hybrid (Mone Carlo mehod hereafer. his esimae can be explained inuiively. When he difference beween ( (, f X u y j and ( ( fˆ X Efˆ X, y j is small for each * independen copy j, hen we can expec ha he error of G (, n, N rue value u(, y can be small because errors of can be cancelled ou. hen we have ( X j minus f and ˆ ( f X j (,, (, * G n N u y j= ( ( ( { } ( { ( } N = f X, ˆ ˆ E f X f X E f X,, N j (5 and we denoe ( X as X by seing =, hen we expec ha he correlaion beween ( X ( X and X is posiively high. Hence he correlaion beween f and ˆ ( f X becomes posiively high. his ype of j j esimae in (4 could be similar o he conrol variae echnique, which has been known in he Mone Calro simulaion. he advanage of he echnique is due o he AE approach, because i is a unified mehod in a sense ha i is applicable o he broad class of processes. We noice ha i is difficul o find conrol variables whose expecaion can no be derived analyically. Furhermore, usual variance reducion echniques may use conrol variables ha could be applied o very narrow class of processes. For example, he pricing algorihm for an European average call opion which exends Example wo in akahashi and Yoshida (5 o he second order asympoics is as follows: Suppose ha he reference asse price process follows (, ( ( ( ds = rs d σ S db, S ( = S. (6 he underlying asse of an average opion a ime zero wih mauriy is 4

6 defined by ( ( A S d. (7 he price a = of an opion wih mauriy and srike price K (> is represened by ( ( r ( C = e A K. (8 A I is rewrien as ( r ( CA = e X, k, (9 where X (, S ( ( S, X (, ( ( ( A A and k A ( K. Considering he discussion of akahashi (999, we inroduce ˆ r x x C ( x = e k c f A x k <. ( Here, c, and f are obained by he AE. We shall discuss abou hese erms in he nex secion briefly. By definiion, we have ( r ( S r S = e S, A = ( e. ( r X, and X, follow ( ( ( ( dx = rx d σ X S, db, X =, (,,,, ( ( ( dx = X d, X =. (3,,, And also ( X and, ( X follow, dx = rx d σ S, db, X =, (4,,, ( ( ( dx = X d, X =. (5,,, hen, we can inroduce he modified esimaor of C A as 5

7 N ( ( ˆ ECˆ X C X C X. (6 A, A, A, N j= ˆ A ( ( X, We noice ha we can obain EC by he AE analyically. Here, X, ( and X denoe he calculaed value by he Euler-Maruyama scheme., Noice ha calculaing (4 does no require he evaluaion of j ( S, σ pah by pah while compuing ( requires. herefore, he amoun of calculaion of he presen echnique is as large as he crude Mone Carlo mehod. he algorihm shown above is based on he second order asympoic value. For oher concree applicaions using he firs order asympoic value, see akahashi and Yoshida (5.. Newon (994 s esimaor wih AE Newon (994 derives he following formula f ( X (, y = E f ( X (, y u(, X (, y σ (, X(, y db as.., (7 x where ( ( E f X, y X, y = u, X, y as... and show ha we can obain he ideal esimaor of E f ( X (, y calculaing f ( X (, y u(, X (, y σ (, X(, y db, (8 x pah by pah. Since neiher u(, X (, y nor u(, X (, y x is known, we approximae u, X, y by he AE and differeniae i wih respec o x. ha is o replace ( u, X (, y ( wih he AE formula. We call he Mone Carlo simulaion x which uses (8 Conrol Variae (Mone Carlo mehod hereafer. ( Unforunaely, since we need compue, (, u X y no only pah by pah bu also a each ime sep, his echnique consumes he larger compuaion resources. 6

8 3. Examples 3 3. Average call opion Le S be he reference asse. hen he underlying asse of an average opion a ime zero wih mauriy is defined by ( ( A S d. (9 We obain ( ( =, ( A A M d M d o where (, r ( s ( σ s s M = e S s db,, = s s s r ( s ( σ. M e S s M db he firs erm of he righ hand of ( is a deerminisic funcion. We inroduce X as ( ( ( A A X g g o(. ( g can be expressed as u ru ( s ( g = M s ds e ( Ss, s dbsdu = σ ru ( s ( = e ( S, s s s dudbs σ r ( s e ( = σ ( Ss, s dbs. r Since g is an inegral wih respec o Brownian moion whose inegrand is a deerminisic funcion, he disribuion of X approaches normal as. hen, we obain he following proposiion. ( 3 For he general discussion of he applicaion including baske and average opions, see akahashi (999. For he mahemaical validiy of AE, see Kuniomo and akahashi (3a, 7

9 Proposiion he asympoic expansion of f ( ( x follows: X, he disribuion funcion of {( cx f n[ x;, ] } X, is as Σ f ( ( x = n[ x;, Σ ] o(, (3 X x where ( r s e ( Σ = σ ( Ss, s ds r, (4 ( ( ( σ r s v rv ( s e c = e,, 3 σ Ss s Ss s Σ r (5 r ( u s r( s u e ( e σ ( S,, u u dudsdv r f = cσ. (6 See secion. in akahashi (999 for he proof. We shall consider he siuaion ha he underlying values of an average opion a ime zero depends on he reference asses price process before zero 4. We inroduce A δ, wih a posiive consan, δ, A = δ, Sudu δ. (7 δ Here, δ represens elapsed ime since conrac dae unil ime zero. And we assume ha r is consan. We can modify he payoff of he opion a ime zero wih mauriy as 4 We can use his equaion for he evaluaion of he curren value of he sanding conracs. 8

10 r = ( δ CA, δ E e A, K r = Ee Sudu K δ δ ( ( u δ δ u r = E e S du K S du δ ( ( δ u δ r = Ee Sudu K s du δ. (8 We rewrie he above formula wih κ δ, K, Assuming ha r is consan, we obain r = e E Sudu κ ( K ( δ Sudu δ =. (9 ( ( δ u δ r Ee Sudu K s du δ,, K. We only need o replace srike price wih κ δ, K, so ha we can forge he dependency of he underlying value on he reference asse before ime zero. Le C ( ( x A,, (3 δ be he price of an European average opion a ime zero wih mauriy, elapsed ime δ and price K as follows: S ( = x. ( ( e r E X k = δ C, ( ( x δ is expressed wih he srike A,, ( r A κ r δ,, K CA, δ (, x = E e ( A κ δ,, K = e E δ δ where k ( A κ δ,, K =. hen, we obain he following heorem. (3 heorem 3 When κ,, >, he asympoic expansion of he price of he average opion δ K 9

11 wih consideraion of elapsed ime, C ( ( A, δ, is as follows: r CA, δ ( = e E Sudu κδ,, K δ (3 r k e = Σ n [ k ;, Σ ] k N f k n [ k ;, Σ ] o (. δ Σ When κ,,, he price is rivial, which is as follows: δ K r CA, δ ( = e E Sudu κδ,, K δ r e = E S du δ r r e = e S κ δ r u κδ,, K (33 δ, K,. Some of he derivaion has been already discussed above. For he opion pricing formula by he AE, see secion. and 3. in akahashi ( Baske call opion he price of a European baske call opion can be derived similarly. Firs, we define a baske by n ( αisi,, ( αi i, I i= n I = S = x. (34 i= Here, S i, denoes he ih asse price a ime and αi is he amoun of he ih asse in he baske. he payoff of a baske call opion can be expressed as CB x = E e I K. (35 (, r We assume ha he price processes of risky asses can be described wih d (n d independen Brownian Moions as (, ( ( ( d ds = rs σ S db, i, i, ij i, j, j= S ( i, = x, i n αixi = x. (36 i= Since he value of he baske is he linear combinaion of risky asses, we obain he AE of he baske.

12 I S M M o i= i= i= o avoid lousy formula, we define ha 5 n n n ( ( = αi i, αi i, αi i, (. (37 ( n ( n n σ I ασ i i Si,,, ασ i i Si,,,..., ασ i id Si,, i= i= i= and We inroduce, (38 ( ( (,,, (,,,..., (,, ( (,,, σ σ S σ S σ S, (39 i i i i i id i (,, (,,..., (, σ σ S σ S σ S. (4 X i i i i i id i as follows: ( ( ( I I X = = g g o(. (4 hen, we obain he nex proposiion. Proposiion 4 he asympoic expansion of f ( ( x follows: X, he disribuion funcion of {( cx f n[ x;, ] } X, is as Σ f ( ( x = n[ x;, Σ ] o(, (4 X x where r ( s ( σ I σi Σ = e s s ds, (43 c n = α c, (44 i i, i= f f n = α f, (45 i i, i= = c Σ, (46 i, i, s 3r ru ( ( rs ( i, = σ I σ i σi σi c e e u u du e s s ds Σ. (47 5 Dash represens ranspose.

13 See secion 3. in akahashi (999 for he proof. ( ( Replacing A wih I in (5, we have ( I K r hen, calculaing E e X k heorem. his is similar o he derivaion of heorem 3. k. (48 ( ( wih proposiion 5, we obain he nex heorem 5 he asympoic expansion of he price of a European baske call opion is as follows: ( r k CB ( x, = e Σnk [ ;, Σ ] kn fknk [ ;, Σ ] o(. (49 Σ 4. Numerical Experimens We shall show he resul of numerical experimens. In his secion, we assume all he necessary coefficiens which define he underlying process. 4. European average call opion under CEV process Suppose ha S follows γ ds = rs d σ S db, S = x. (5 We shall represen his by he AE. S ( consan b as is expressed wih some posiive hen, we obain, γ ( ( ( ds = rs d b S db S = x. (5 r ( s e Σ = = rs γ b ( xe ds r r γ γ r r ( (( γ ( ( γ γ 3 b e x x e e rγ γ ( ( γ 3, (5

14 c = x r r e x x e 3 e ( r γ γ r r γ ( γ ( γ ( γ r 4γ ( x ( 7 γ x e 3 3 r 4γ r ( γ γ e x γ e γ 3 6γ 6γ r γ r ( γ ( γ γ x e x e γ r ( 4γ r r x ( γ e ( 4γ ( 3 4γ γ 3 4γ 4e γ 6γ 8γ. γ 3 6γ 6γ We se he simulaion condiions as ( (53 x=, r =.5, σ =.3S γ. For he Mone Carlo simulaion, we divide one year ino 5 ime seps. able expresses he price of he opion wih γ =.6. Column A is he rue value ha is compued by he crude Mone Carlo mehod wih 5 million rials. Column B and column C are he prices by he AE wih he firs order 6 and he second order approximaion respecively. Column D and column E are he rae of errors of he column B and C respecively, which are defined by ((he price by he AE-(rue value/(rue value. he price of he AE wih he second order approximaion is nearer o he rue value han ha wih he firs order. As for he second order approximaion, he rae of errors are sufficienly small excep 4% OM cases. In hese cases, he prices are very small and he differences beween he prices by he AE and rue values are.5599 ( year mauriy and.94 ( years mauriy, which are also very small. Overall, excep far ou of he money cases, uilizing he second order AE mehod, we can saisfy almos all he pracical requiremens of calculaion speed and accuracy. able expresses he price of he opion wih γ =.9. We can derive he similar implicaion o able. wo ables of able 3 express he performance of he simulaion algorihms wih γ =.6. For each parameer and simulaion algorihm, ( we generae 5, pahs (rials, compue he prices and ake he average, ( we repea 6 See akahashi (999 for he pricing formulae of he firs order approximaion. 3

15 aking he average imes (cases, (3 we exrac resul of he simulaion. For he upper able, each column (column A, B and C has wo sub-columns: rmse and wors sand for relaive mean square error and he bigges error among cases respecively. Column A, B and C express he resuls of he crude Mone Carlo, he hybrid Mone Carlo and he conrol variae Mone Carlo mehod respecively. he sandard variance is shown in he lower able. he rue value is he same as able. he raio of he upper able is ha beween rmses. While boh he hybrid and he conrol variae mehod Mone Carlo reduce rmse, wors and sd variance, he effec of he former is remarkabe. Using he raio of he sandard variances, we can say ha he convergence speed of he hybrid Mone Carlo mehod is 4~8 imes faser han he crude Mone Carlo mehod. We noice ha comparing column E of able wih column B of able 3 he hybrid Mone Carlo mehod gives us more precise esimae han he second order AE. able 4 expresses he performance of he simulaion algorihms wih γ =.9. We can derive he similar implicaion o able 3. We noice ha comparing column E of able wih column B of able 4 he hybrid Mone Carlo mehod improves accuracy especially in he cases of ou of he money. able 5 gives one of he mos ineresing feaures. his expresses he acual calculaion ime (seconds. We sress he imporance of he fac ha (B/(A is nearly equal o one. his implies ha he hybrid Mone Carlo mehod does require only % of exra compuaion resources. Considering he resuls of able 3 and 4, his means ha we can reduce calculaion ime dramaically uilizing he hybrid Mone Carlo mehod. Of course, even in he conrol variae Mone Carlo mehod s case, overall performance of he calculaion is improved. his shows ha he AE can be uilized broadly. 4

16 (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr %.% AM yr %.57% %OM yr %.5456% 3%OM yr %.885% 4%OM yr % -.836% %IM yrs %.9% AM yrs %.388% %OM yrs %.6486% 3%OM yrs %.83% 4%OM yrs % -.47% (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr %.64% AM yr %.3536% %OM yr %.7955% 3%OM yr %.448% 4%OM yr % -.97% %IM yrs %.396% AM yrs %.564% %OM yrs %.938% 3%OM yrs %.8% 4%OM yrs % -6.% (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able (A (B (C crude Mone Hybrid CnlVar K rmse wors rmse wors rmse wors (A/(B (A/(C %IM yr.6937% %.74%.783%.4%.35% AM yr.976% 5.59%.754%.7683%.446%.7484% %OM yr %.856%.399%.3688%.6788%.87% %OM yr %.4746%.3399%.6743%.6765%.34854% %OM yr 5.584% %.5%.556%.4543% 6.886% %IM yrs.8773% 4.933%.5%.68%.364%.379% AM yrs.6379% %.735%.97%.64%.5445% %OM yrs % %.394%.397%.3459%.7843% %OM yrs 4.97% %.666%.4548%.5975%.3455% %OM yrs 7.563% %.57%.438%.3% 3.49% , rials, cases for each parameer. (A (B (C raio be. crude Mone Hybrid CnlVar sd vars K sd variance sd variance sd variance (A/(B (A/(C %IM yr , AM yr , %OM yr , %OM yr %OM yr %IM yrs , AM yrs , %OM yrs %OM yrs %OM yrs able 3 5

17 (A (B (C crude Mone Hybrid CnlVar K rmse wors rmse wors rmse wors (A/(B (A/(C %IM yr.44463% 4.485%.57%.53%.485%.373% AM yr.34% %.778%.348%.59%.8597% %OM yr %.46%.39%.756%.89547%.549% %OM yr 5.758% %.563%.544%.385%.6893% %OM yr % % 3.63% % % % %IM yrs.8859% %.67%.989%.346%.48% AM yrs.55599% %.97%.43598%.3756%.6467% %OM yrs.969% 7.43%.443%.59779%.47983%.3366% %OM yrs %.76%.3958%.999%.7348%.748% %OM yrs 6.863% %.395%.9889%.358% 3.96% , rials, cases for each parameer. (A (B (C raio be. crude Mone Hybrid CnlVar sd vars K sd variance sd variance sd variance (A/(B (A/(C %IM yr AM yr %OM yr %OM yr %OM yr %IM yrs AM yrs %OM yrs %OM yrs %OM yrs able 4 (A (B (C # of rials Crude Mone Hybrid CrlVar (B/(A (C/(A, , , , able 5 4. European baske call opion wih n asses under CEV process 7 Le S and B be n R valued sochasic process and independen n dimensional Brownian Moion respecively. Suppose he price of he asse, follows γ i ( n ds = rs d I S σ db, S = x R,, (54 i, where r R and σ R are consan, and I( y i y O =. O y n hen, we represen S ( using some posiive consan b R n as S 7 he concree represenaions of Σ, c i, are available upon he reques. 6

18 ( ˆ λ,, γ ( ( i ( ds = rs d I b S db S = x. (55 i i Assuming ha he n n marix ˆλ has Cholesky decomposiion, le he lower riangular marix be λ, and we can modify (55 o hen, we have ( λ,, γ ( ( i ( ds = rs d I b S db S = x. (56 i i r ( s ( σ I σi Σ = e s s ds, (57 c e e u u du e s s ds. s 3r ru rs i, = σ I σi σi σi where Σ (58 σ σ n ( I ( s ασ i i i=, γ i ( λ ( i ( s ranspose of ih row of I bi( Si,, ( ( s σ ( ( s i σ i. ( Si ( ( Si = Si 4.. European baske call opion wih wo asses We suppose S follows γ i, (, i i ij i j i i σ j= ds = rs d S db S = x i =. (59 We se he simulaion condiions as ( i i ij i γ S =, r =.5, σ =.3 S, (i=,. We also suppose each asse akes 5% of he baske. And for he Mone Carlo simulaion, we divide one year ino 5 ime seps. able 6 expresses he price of he opion wih γ =.6 and ˆ λ = I d. Comparing his resul wih ha of he average opion case, he accuracy of he approximaion is slighly beer, bu we observe ha he characerisics are very similar o each oher. Overall, excep far ou of he money cases, uilizing he second order AE mehod, we can saisfy almos all he pracical requiremens of calculaion speed and accuracy. able 7 expresses he price of he opion wih γ =.9 and ˆ λ = I d. We can 7

19 derive he similar implicaion o able 6..5 able 8 expresses he price of he opion wih γ =.6 and ˆ λ =.5. We can derive he similar implicaion o able 6 and able 7. able 9,, express he performances of he simulaion algorihms wih γ =.6 and ˆ λ = I d, γ =.9 and ˆ.5 λ = I d, and γ =.6 and ˆ λ =.5 respecively. he improvemen by uilizing he hyprid Mone Carlo is remarkably. From he raio of he sandard variances, we can say ha he convergence speed of he hybrid Mone Carlo mehod is ~65 imes faser han he crude one. We also noice ha he hybrid Mone Carlo mehod improves he accuracy when he precision of he second order AE is no saisfacory. able expresses he acual calculaion ime (seconds. As in he case of he average opion, we can find he fac ha (B/(A is nearly equal o one. (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr %.54% AM yr %.749% %OM yr %.4846% 3%OM yr % -.63% 4%OM yr % % %IM yrs %.43% AM yrs %.4% %OM yrs %.53% 3%OM yrs %.668% 4%OM yrs % -.64% (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able 6 8

20 (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr % -.98% AM yr % -.566% %OM yr % -.677% 3%OM yr % % 4%OM yr % -3.% %IM yrs % -.87% AM yrs % -.86% %OM yrs % -.88% 3%OM yrs % -3.45% 4%OM yrs % -6.85% (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able 7 (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr %.94% AM yr %.377% %OM yr %.987% 3%OM yr %.568% 4%OM yr % -.63% %IM yrs %.49% AM yrs %.68% %OM yrs %.464% 3%OM yrs %.994% 4%OM yrs %.788% (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able 8 (A (B crude Mone Hybrid raio of K rmse wors rmse wors (A/(B sd vars %IM yr.5747%.43%.5469%.8375% AM yr.6549%.8957%.75%.38% %OM yr.4795% %.8494%.8538% %OM yr 3.488% %.488%.35% %OM yr % %.7739%.995% %IM yrs.863%.4745%.67%.336% AM yrs.634%.4894%.8546%.46768% %OM yrs.75587% %.7797%.663% %OM yrs.4987% %.37658%.53% %OM yrs 3.685% %.56497%.34763% , rials, cases for each parameer. able 9 9

21 (A (B crude Mone Hybrid raio of K rmse wors rmse wors (A/(B sd vars %IM yr.69977%.3948%.85%.68% AM yr.6798% %.9%.45967% %OM yr.4933% 6.488%.496%.77% %OM yr % 8.867%.7575%.337% %OM yr 4.494%.758%.4398% % %IM yrs.7374%.93463%.6848%.4436% AM yrs.9979%.6396%.7867%.7735% %OM yrs.758% %.339%.594% %OM yrs.747% 7.53%.6673%.333% %OM yrs.9657% 8.534%.955%.3947% 3.6.6, rials, cases for each parameer. able (A (B crude Mone Hybrid raio of K rmse wors rmse wors (A/(B sd vars %IM yr.6834%.478%.58%.6% AM yr.793%.9587%.843%.3467% %OM yr.9457% %.%.3463% %OM yr.4859% %.4537%.38379% %OM yr %.447%.85%.5894% %IM yrs.8736%.35756%.533%.469% AM yrs.64%.965%.776%.8374% %OM yrs.7795% %.73%.55% %OM yrs.88% %.344%.35935% %OM yrs.38% 5.88%.7853%.4656% , rials, cases for each parameer. able (A (B # of rials Crude Mone Hybrid (B/(A, , , , able 4.. European baske call opion wih five asses We suppose S follows γ i, (,,...,5 5 i i ij i j i i σ j= ds = rs d S db S = x i =. (6 We se he simulaion condiions as ( i i ij i γ S =, r =.5, σ =.3 S, (i=,,...5. We also suppose each asse akes % of he baske. And for he Mone Carlo simulaion, we divide one year ino 5 ime seps. able 3 and able 4 express he prices of he opion wih γ =.6 and ˆ λ =, and γ =.9 I d

22 and ˆ λ = I d, respecively. Comparing his resul wih ha of he wo asses case, he accuracy of he approximaion is slighly worse, bu we observe ha he characerisics are very similar o each oher. Overall, excep ou of he money cases, uilizing he second order AE mehod, we can saisfy almos all he pracical requiremens of calculaion speed and accuracy. Moreover, we noice ha i is very difficul ha calculaing he price of opions of which underlying asse has complicaed srucure. We obain i in a winkling of an eye. able 5 and 6 express he performances of he simulaion algorihms wih γ =.6 and ˆ λ =, and γ =.9 and ˆ λ = respecively. he effec of uilizing I d he hyprid Mone Carlo is very well. From he raio of he sandard variances, we can say ha he convergence speed of he hybrid Mone Carlo mehod is 4~76 imes faser han he crude Mone Carlo mehod. We also noice ha he hybrid Mone Carlo mehod improves he accuracy when he precision of he second order AE is no saisfacory able 7 expresses he acual calculaion ime (seconds. As in he case of he average opion, we can find he fac ha (B/(A is nearly equal o one. In he case of, rials, (B/(A is no near o one, because of he complexiy of he opion, we need so much pre-processing calculaion especially for he five asses case. Bu he pre-processing is done only once for one simulaion so ha he greaer number of he rials he smaller number of he (B/(A. From compuaional poin of view, his feaure is very imporan. he fac ha (B/(A is nearly equal o one for he large number rials even in he cases of very complicaed opions enables us o parallelize he compuaion wihou serious problems 8. We remain he consideraion under parallel processing for he nex research. I d 8 As in he case of boh he Inel processors and he fases super compuing processors, he rend of he improvemen of he compuaion power is due o parallerizaion. herefore, from pracical poin of view, we need o pay aenion o he scalabiliy of he parallerizaion of our algorihms.

23 (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr % -.36% AM yr % -.535% %OM yr % -.84% 3%OM yr % -6.56% 4%OM yr % -.86% %IM yrs % -.664% AM yrs % -.333% %OM yrs % % 3%OM yrs % -.356% 4%OM yrs % % (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able 3 (A (B (C (D (E K crude Mone AE s Order AE nd Order rae of diff rae of diff %IM yr % -.38% AM yr % -.56% %OM yr % % 3%OM yr % % 4%OM yr % % %IM yrs % % AM yrs % -.39% %OM yrs % % 3%OM yrs % % 4%OM yrs % % (D={(B-(A}/(A, (E={(C-(A}/(A, crude Mone : 5,, rials able 4 (A (B crude Mone Hybrid raio of K rmse wors rmse wors (A/(B sd vars %IM yr.44%.8677%.5475%.689% AM yr.3766% 3.997%.38%.47988% %OM yr 3.664% 8.543%.695%.556% %OM yr 7.796% %.839% % %OM yr % 46.53% 6.335% % %IM yrs.553%.44%.94%.695% AM yrs.686% 3.949%.595%.636% %OM yrs.5678%.353%.56%.4748% %OM yrs % 9.59%.9499%.66898% %OM yrs % %.8555% 5.43% , rials, cases for each parameer. able 5

24 (A (B crude Mone Hybrid raio of K rmse wors rmse wors (A/(B sd vars %IM yr.5538%.8845%.847%.5874% AM yr.47% %.857%.64% %OM yr % %.97%.4557% %OM yr 6.5% 7.93%.% % %OM yr % 4.339% 6.983% % %IM yrs.5988%.5555%.436%.37638% AM yrs.86% 3.346%.374%.78337% %OM yrs.898% %.784%.56676% %OM yrs %.98%.4875% 4.66% %OM yrs 5.655% %.33434% % , rials, cases for each parameer. able 6 (A (B # of rials Crude Mone Hybrid (B/(A, , , , able 7 References Kuniomo, N. and A. akahashi (3a: On Validiy of he Asympoic Expansion Approach in Coningen Claims Analysis, Annals of Applied Probabiliy, Vol.3, no.3. Kuniomo, N. and A. akahashi (3b: Applicaions of he Asympoic Expansion Approach based on Malliavin-Waanabe Calculus in Financial Problems, CIRJE Discussion Paper, CIRJE-F-45, Universiy of okyo. Newon, N. J. (994: Variance Reducion for Simulaed diffusions, SIAM Journal Applied Mahemaics, Vol.54, No.6, pp akahashi, A. (999: An Asympoic Expansion Approach o Pricing Financial Coningen Claims, Asia-Pacific Financial Markes, Vol.6, pp.5-5. akahashi, A. and N. Yoshida (5: Mone Carlo Simulaion wih Asympoic Mehod, Working Paper Series CARF-F- (CIRJE-F-335, Cener for Advanced Research in Finance, he Universiy of okyo (forhcoming in Journal of Japan Saisical Sociey. 3