Generalized Control Variate Methods for Pricing Asian Options

Transcription

1 eneralized Conrol Variae Mehods for Pricing Asian Opions Chuan-Hsiang Han Yongzeng Lai March 25, 29 Absrac he convenional conrol variae mehod proposed by Kemna and Vors (199 o evaluae Asian opions under he Black-Scholes model can be inerpreed as a paricular selecion of linear maringale conrols. We generalize he consan conrol parameer ino a conrol process o gain more reducion on variance. By means of an opion price approximaion, we consruc a maringale conrol variae mehod, which ouperforms he convenional conrol variae mehod. I is sraighforward o exend such linear conrol o a nonlinear siuaion such as he American Asian opion problem. From he variance analysis of maringales, he performance of conrol variae mehods depends on he disance beween he approximae maringale and he opimal maringale. his measure becomes helpful for he design of conrol variae mehods for complex problems such as Asian opion under sochasic volailiy models. We demonsrae muliple choices of conrols and es hem under MC/QMC (Mone Carlo/ Quasi Mone Carlo- simulaions. QMC mehods work significanly well afer adding a conrol, he variance reducion raios increase o 26 imes for randomized QMC compared wih 6 imes for MC simulaions wih a conrol. 1 Inroducion Conrol variae mehods have been widely used for compuaional finance as a mean of variance reducion. Perhaps he mos successful applicaion is o evaluae coninuous-ime arihmeic-average Asian opions. Kemna and Vors [14] employed a conrol, which is a discouned counerpar geomericaverage Asian opion payoff less han is price. his mehod is efficien because he correlaion beween he arihmeic-average and he geomeric-average is high, and he counerpar geomeric-average Asian opion price has a closed-form soluion. herefore, crieria for consrucing conrol variaes are: (i highly correlaed payoff random variable wih (ii closed-form expecaion. here are sophisicaed echniques such as sraified sampling or Brownian bridge, which are developed o combine wih he conrol variae mehod in order o enhance variance reducion. See [7] and references herein for more discussions. However, for complex problems such as American syle or he case when volailiy is random, he crieria for consrucing conrol variaes may no be easy o saisfy. In his paper we invesigae he convenional conrol variae mehod proposed by Kemna and Vors Deparmen of Quaniaive Finance, Naional sing-hua Universiy, Hsinchu,aiwan 313, ROC. chhan@mx.nhu.edu.w Deparmen of Mahemaics, Wilfrid Laurier Universiy, Waerloo, Onario, N2L 3C5, Canada. ylai@wlu.ca 1

2 as a linear conrol wih a consan conrol parameer. Combining he maringale represenaion of he conrol wih anoher conrol process, raher han a parameer, we generalize he convenional conrol variae so ha he conrolled variance is reduced o zero in he opimal case. I urns ou ha he variance induced from he convenional conrol variae mehod is a consan projecion from a sochasic conrol parameer space. Using some opion price approximaion, we reduce he generalized conrol variae o a maringale conrol variae. herefore, herefore, convenional conrol variae (or maringale conrol variae can be viewed as a special case for generalized conrol variae by using a consan conrol parameer (or an approximae conrol process, respecively. he maringale conrol consruced from a more accurae price approximaion is more correlaed o he original discouned payoff, and is mean is zero. his idea links sudies of conrol variae mehods o opion price approximaions. here are numerous sudies on arihmeic-average Asian opion prices even under he Black-Scholes model. See [18, 27, 3] for example. Using a recen approximaion by Zhang [3], we can consruc a new maringale conrol for he use of variance reducion, which ouperforms he convenional conrol variae in erms of generaing smaller sandard errors. In paricular, hese maringale conrols are applicable o nonlinear problems such as American Asian opions. here is only one uncerainy induced by Brownian moion under he Black-Scholes model. Hence, maringale conrols defined on he same uncerainy are expeced o reduce he variance. We confirm his wih many numerical resuls in his paper. In he conex of sochasic volailiy models, consrucion of conrol variae is less known and sudied. Clewlow and Carverhill [1] applied a financial inuiion of dela hedging o build porfolio possibly wih oher reeks hedge as conrols in order o evaluae opion prices. hey found ha he variance obained by his conrol variae under an one-facor sochasic volailiy model [9] can be significanly reduced. By means of he leas-squares mehod [16], Poers, Bouchaud, and Sesovic [21] proposed an opimal hedged Mone Carlo mehod o price opions under he hisorical probabiliy measure. Laer, Pochar and Bouchaud [22] exended he same idea o problems ha consider shorfall risk and ransacion cos consrains. Heah and Plaen [8] and Fouque and Han [5] used differen approximae European opion prices o approximae dela porfolio under sochasic volailiy models and showed significan variance reducion. Moreover, in [3], Fouque and Han obained an asympoic resul o characerize he variance of a maringale conrol variae under sochasic volailiy models when he ime scales of driving volailiy processes are well-separaed. Based on he consrucion of hedging maringales under sochasic volailiy models, we furher invesigae he Asian opion pricing problem. We find ha using he hedging maringale as a one-sep conrol can only parially reduce uncerainies associaed wih driving volailiy processes. In conras, he convenional conrol by he geomeric-average counerpar can be shown o reduce he variance of each source of randomness, hough he counerpar opion price has no closed-form soluion. herefore, we propose a wo-sep algorihm o overcome his difficuly. All Mone Carlo mehods menioned so far are fundamenally relaed o pseudo random sequences. Inegraion mehods using he alernaive quasi-random sequences (or called low-discrepancy sequences have drawn much aenion in recen years because is heoreical rae of convergence is O(1/n 1 ε for all ε > subjeced o he dimensionaliy and he regulariy of he inegrand. hese are he so-called quasi-mone Carlo (QMC mehods. We refer o [7], [11] and [13] for furher discussions on MC/QMC mehods in applicaions of compuaional finance. We use randomized QMC o deal wih a non-smooh call payoff and a high dimension of 384 in hese experimens under sochasic volailiy models. Afer adding conrols o our esimaors, QMC mehods are able o reduce variance significanly in all cases. Even in a high dimensional regime, he conrol sars o play he role of a smooher. I can be seen from variance analysis in Secion 4.3 ha on average he flucuaion of he conrol variae is coninuous and of small order. In [5], he auhors sudy he smoohing effec by maringale conrols. An example of esimaing low-biased soluions of an American opion by leas squares mehod [16] 2

3 shows ha using QMC, including Niederreier sequence and Sobol sequence, acually gives esimaes greaer han he rue opion price. However, adding a maringale conrol for variance reducion can cause all problemaic esimaes o become low-bias and very close o rue values. he organizaion of his paper is as follows. In Secion 2, we review he convenional conrol variae mehod and generalize i in a sochasic framework. By opion price approximaions, we consruc new maringale conrols and apply hem o American Asian opion problems. In Secion 3, we inroduce muli-facor sochasic volailiy models. In Secion 4 o evaluae geomeric-average Asian opions, we apply a singular and regular perurbaion mehod o consruc a maringale conrol variae mehod, which ouperforms an imporance sampling sudied in [5]. We presen a variance analysis for a simplified perurbed volailiy model. By a combinaion of maringale conrol variae mehod for he geomeric-average Asian opion and he convenional conrol variae mehod for he arihmeicaverage Asian opion, we propose a wo-sep mehod in Secion 5. We presen numerical experimens implemened by MC/QMC mehods. We es several combinaions of conrol variae mehods wih randomized QMC mehods, including he Sobol sequence and L Ecuyer ype good laice poins ogeher wih he Brownian bridge sampling echnique. 2 Conrol Variae Mehod: Revisi and a eneralizaion We consider he esimaion of a mahemaical expecaion E{X} by Mone Carlo simulaions, where X is a one-diemnsional square-inegrable random variable defined on a probabiliy space. A conrol variae mehod is a variance reducion echnique which aims o improve he precision of he esimae obained from plain Mone Carlo simulaions [7, 17]. he basic idea of conrol variae is o choose an appropriae counerpar square-inegrable random vecor, say Y R d being cenered a zero, and muliple conrol parameers λ R d so ha he conrol variae X + λ Y is unbiased, E{X} = E{X + λ Y }, and he variance can be reduced, V ar{x} V ar{x + λ Y }. his can be done by choosing (i any mean-zero random vecor Y correlaed wih X and (ii he opimal conrol parameers, deduced from minimizing he variance of he conrol variae, λ = Σ 1 Y Y Σ XY, (1 where Σ XY denoes he covariance of X and Y. From a sraighforward calculaion, one can see ha he new variance can no be greaer han he original variance: V ar{x + λ Y } = ( 1 R 2 V ar{x} V ar{x}, (2 where R 2 = Σ XY Σ 1 Y Y Σ XY Σ XX is a generalizaion of he squared correlaion coefficien or known as he coefficien of deerminaion in regression analysis. In he single conrol parameer case, i.e. Y R, we have R = ρ, he correlaion coefficien beween random variables X and Y. When random variables X and Y are highly correlaed, eiher posiive or negaive, R 2 is close o one and a considerable reducion of variance is expeced. Because calculaing he opimal conrol parameer λ requires he exac value of E{X}, in pracice one can only use a subopimal conrol parameer λ by approximaing he righ hand side of (1 empirically hrough plain Mone Carlo simulaions. A sensiiviy analysis considering he error of variances obained from he opimal conrol variae parameer λ and is perurbaion λ ε is shown below. Lemma 1 iven any consan ε and an ideniy vecor I = (1,, 1 R d 1 such ha he subopimal conrol parameer λ ε = λ + ε I being a perurbed parameer, he gain of variance from he 3

4 subopimal conrol variae is of order ε 2 : { } V ar X + λ ε Y ( 1 R 2 V ar{x} = ε 2 V ar{i Y }. Proof: I is easy o calculae he variance of he perurbed conrol variae { } { } V ar X + (λ + ε I Y = V ar X + λ Y + ε 2 V ar{i Y } = ( 1 R 2 V ar{x} + ε 2 V ar{i Y }, where we have used he definiion in (1, Cov(X + λ Y, I Y =, and a resul in (2. Noe ha he simple variance analysis shown in (1 and (2 does no depend on he variance of muliple conrols. hey become significan when we sudy sensiiviy analysis over muliple conrol parameers as shown in Lemma 1. he variance of he sum of muliple conrols is no ofen emphasized in use of conrol variae mehods, bu we will see in Secion 5 ha i becomes imporan for a furher reducion of variance when we compare wo differen ses of conrols (one-sep versus wo-sep conrol variaes. In compuaional finance a well-known example of using he conrol variae mehod is he evaluaion of a coninuous-ime and arihmeic-average Asian opions under he Black-Scholes model. Based on he risk-neural pricing heory [24], he fair price of he Asian opion, denoed by P A, is equal o he condiional expecaion under he risk-neural probabiliy space ( Ω, F, (F, IP P A (, S, A = IE { e r( H(A F } where he underlying risky asse S is governed by a geomeric Brownian moion defined as ds = rs d + σs dw, (4 he arihmeic-average price process A is defined by A = 1 S s ds. Oher noaions are defined as follows: is he curren ime, < + is he mauriy, r is he risk-free ineres rae, σ is volailiy, W is he sandard Brownian moion, H(x is he payoff funcion saisfying he usual inegrabiliy condiion. For example if H(x = max {x K, } (x K + for he srike price K >, i is a call payoff; if H(x = max {K x, } (K x +, i is a pu payoff. he conrol variae mehod proposed by Kemna and Vors [14] o evaluae P A defined in (3 inroduces a counerpar geomeric-average price process = exp( 1 ln S s ds and a geomeric-average Asian opion price P defined by P (, S, = IE { } e r( H( F (5 such ha a conrol variae for esimaing P A is e r( H(A + λ ( e r( H( P (, S,. (6 he success of his conrol variae is aribued o a leas wo facs: he conrol e r( H( P (, S, (7 is highly correlaed o he original discouned payoff variable e r( H(A. his is confirmed by empirical ess. See for insance [7, 14] ha correlaions beween he conrol variae (7 and e r( H(A 4 (3

5 are close o 1 in many examples. From Lemma 1 a small error in he conrol parameer does no affec he variance reducion much. herefore, i is also pracical o simply use a consan conrol parameer. he counerpar geomeric-average Asian opion price P admis a Black-Scholes ype closed-form soluion and he random variable is defined on he same probabiliy space as A so ha Mone Carlo simulaions for he conrol variae become easy o implemen. 2.1 eneralized Conrol Variae By an applicaion of Io s lemma, he conrol in (7 has he following maringale represenaion ( e r P H( P (, S, = M ; where he process M ( ; is a zero-cenered maringale wih (8 M( ; = e rs (s, S s = x, s σs s dw s. (9 he convenional conrol variae mehod uilizes a conrol parameer λ o obain he minimized variance which is no zero. We now inroduce a generalized conrol variae by assuming ha he conrol parameer λ is a F - adaped process, denoed by λ, so ha a new conrol variae becomes ( e r H(A + M λ P ; Is variance condiional a ime zero is equal o { ( } V ar e r P H(A P A (, S, A + M λ ; { ( ( } PA = V ar M ; P + M λ ; { ( } PA = V ar M + P λ ; = e 2rs σ 2 IE { ( PA (s, S s, A s + λ s P (s, S s, s. (1 2 S 2 In his calculaion, we have used he lineariy of sochasic inegrals, Io s isomery, and Fubini s heorem. o eliminae he variance, he opimal conrol process, raher han a parameer, is chosen as s } ds. (11 λ = P A (, S, A / P (, S,, a.s. (12 he opimal conrol process requires he exac value of P A or is dela, i.e., P A in order o eliminae variance. hus he generalized conrol variae can be viewed as in a dynamic seing, compared wih he convenional conrol one as in a saic seing. Noe ha he convenional conrol variae mehod can only reduce he variance proporionally as shown in (1 and (2, even if he opimal λ is used. We show nex ha he opimal conrol parameer λ is simply a projecion of he opimal conrol process over he real line R, i.e, he convenional conrol variae mehod is a special case of he generalized one. 5

6 Lemma 2 For pricing Asian opions by convenional conrol variae mehod, he opimally reduced variance in (2 can be obained by solving he following minimizaion problem ( 1 ρ 2 V ar { e r H(A } { } = min V ar e r P H(A + M(λ λ R ;. Proof: From (11 he consan minimizer solving he variance above is λ = Cov ( M( P A ;, M( P ; V ar ( M( P ; such ha he opimally reduced variance is { ( IE e 2rs PA (s, S s, A s + λ P } 2 (s, S s, s σ 2 Ss 2 ds = ( 1 ρ 2 V ar { e r H(A }, where ρ denoes he correlaion of e r H(A and he conrol M( P ;. Because he geomeric-average conrol M( P ; is no perfecly correlaed wih he counerpar arihmeic-average payoff e r H(A, he variance of he conrol variae (6 can only be opimally reduced by he facor 1 ρ 2 wih he choice of a consan conrol parameer λ. 2.2 Consrucion of More Linear Conrols A pracical way of implemening he generalized conrol variae mehod for variance reducion is o approximae he arihmeic-average Asian opion price funcion P A in he opimal conrol process (12. Le P be a price approximaion o P A. A subopimal conrol is deduced: λ = P (, S, A / P (, S,. (13 Subsiuing λ ino ( (1, one can readily observe ha his generalized conrol variae is reduced o e r H(A M, P ; known as he maringale conrol variae. Hence he maringale conrol ( M P ; can be hough of as a generalizaion of he convenional conrol variae by employing an approximae conrol process (13. One can of course incorporae anoher conrol parameer in fron of he maringale conrol o gain addiional variance reducion. By his consrucion, we are able o enlarge he class of conrol variae by maringale conrols, in which approximae opion price or is dela mus be used. here has been many sudies on approximaing arihmeic-average Asian opion prices in recen decades. Our goal is no comparing variance induced from price approximaions in hese sudies. Insead, we only demonsrae he consrucion of a generalized conrol variae as follows. We employ a price approximaion wih a closed form proposed by Zhang [3] based on a perurbaion expansion on he singulariy a he diffusion coefficien. For an Asian call opion wih he srike price K, we consider only he leading order expansion in [3] as P Z (, S, A = S f(ξ,, (14 6

7 able 1: Comparisons of hree conrol variaes o esimae he arihmeic-average Asian call opion prices wih a fixed srike price while consan volailiy varies from 1% o 7 %. Oher model parameers are chosen by S = 65, K = 55, r =.6, = 1 year. In column 2, CV denoes he convenional conrol as in (7; in column 3, M( P ; denoes he equivalen maringale represenaion as in (8; in column 4, M( P Z ; denoes he new conrol as in (16. Sample means and sandard errors in parenhesis are shown in pair. Mone Carlo simulaions are implemened under he sample size N = 1 and he discreized ime sep 2. σ CV M( P ; M( P Z ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.14 where he variable ξ S e r( 1 e r( and he funcion f(ξ, τ = ξn ( ξ/ 2τ + r τ /(4τ π e ξ2. he sandard normal cumulaive disribuion funcion is denoed by N (. hus, we = (K A consruc a sochasic conrol process hen form a maringale conrol ( PZ M ; = λ = P Z (, S, A / P (, S,, (15 e rs P Z (s, S s = x, A s σs s dw s. (16 In able 1 we illusrae effecs of hree conrols, including he convenional conrol of he geomericaverage Asian opion shown in (7, is equivalen maringale conrol M( P ; as in (8, and anoher maringale conrol by Zhang s approximaion M( P Z ; as in (16. Noe ha he las maringale conrol should be undersood as a generalized conrol variae (1 using a sochasic conrol process (15 raher han a consan conrol parameer. Because he hree corresponding esimaors are unbiased, we shall focus on sandard errors. Since he firs wo conrols are equivalen, i is observed ha sandard errors are roughly of he same order. he las conrol by M( P Z ; produces he smalles sandard errors as expeced. heir empirical conrol parameers (λ are calculaed as 1.9, 1.11, and.9929 respecively. he hree compuing imes are.23, 2.36, and 1.25 seconds respecively execued under a PC Penium Hz CPU wih Malab. Due o he calculaion of sochasic inegrals, maringale conrol variae require more compuing ime because he conrol is addiive. Bu we will see in nex secion ha when a noninear siuaion appears such as in he American Asian opion pricing problem, he compuaional coss become roughly he same. We have proposed a way o consruc maringale conrol M( P ; where P is any price approximaion o P A. For insance if P = P, we recover he convenional conrol variae from (8 because he discouned process e r P (, S, is a maringale. When we choose he approximaion P = P Z, he represenaion of he equivalen process-ype conrol is no longer suiable for he use as a convenional conrol. his is because e r P Z (, S, A is no a maringale which can be seen from a probabiliy 7

8 represenaion of he ransformed funcion f(ξ, τ = E {max ( ϕ, ϕ }, (17 where he process ϕ is governed by he sochasic differenial equaion dϕ = σ r (1 e r( dw, ϕ = K e r 1 e r, S r and W is a Brownian moion. Noe ha his W can be defined on a differen probabiliy space han W in (4. his fac implies ha maringale conrols are more general han convenional conrols where concree correlaed processes mus be posulaed. So far we have discussed only he European Asian opions. We have reviewed and expanded he convenional conrol variae mehod [14] o maringale conrol variaes hrough an approximaion o he opimal conrol processs. One can readily observe ha hese conrols are in a linear form. Nex, we apply hese maringale conrols in a nonlinear form o he siuaion where early exercise is permied. 2.3 Pricing American Asian Opions: Nonlinear Conrol In his secion, we consider he American Asian opion pricing problem under he Black-Scholes model. We sudy he price of an American Asian call opion wih a fixed srike price K as an example bu oher payoffs can be reaed similarly. he American Asian call opion price is a soluion of an opimal sopping ime problem P AA (, S, A = ess sup IE { } e r (τ (A τ K + F, (18 τ wih τ being a bounded sopping ime beween he curren ime and he mauriy. o solve an opimal sopping problem by simulaion, known as a primal approach, is challenging even hough here exis mehods such as leas squares mehod[16], ec. Because he primal approach is less relaed o conrol variae, we will focus direcly on he dual approach o solve he problem (18. Using he dual formulaion [4, 23], we have an equivalen expression a ime zero: { } ( P AA (, S, A = IE sup (A K + M F, (19 inf M H 1 where maringales in he space H 1 are uniformly inegrable and cenered a zero a ime zero. Hence, given a suiable maringale, a high-biased esimae of he American Asian price P AA can be obained. I urns ou ha he opimal maringale M = M( P AA ; gives he pahwise equaliy [4] ( P AA (, S, A = sup (A K + M. he price and variance error analysis are also given in [4]. Heurisically, we like o propose some subopimal maringales using P AA price approximaion such as is European counerpar approximaion P or P Z. In able 2 we illusrae he effec of wo maringale conrols, including he European geomeric-average Asian opion counerpar M( P ; and Zhang s approximaion M( P Z ;, wih various srike prices. Noice ha he sandard errors, shown in parenhesis and MAD (he mean absolue deviaion from 8

9 able 2: Comparisons of wo maringale conrol variaes in esimaing American arihmeic-average Asian call opion price when he fixed srike price K akes value of 45, 5 and 55. Oher model parameers are S = 55, r =.1, σ = 4%, = 1 year. In column 2, M( P ; denoes he he maringale conrol as in (8; and MAD in column 3 is compued based on resuls in column 2. In column 4, M( P Z ; denoes he maringale conrol as in (16 and column 5 shows he associaed MAD. Sample means and sandard errors in parenhesis are shown in columns 2 and 4. Mone Carlo simulaions are implemened under he sample size N = 5 and he discreized ime sep 5. K M( P Z ( ( ( ( ( ( he mean are imporan facors o indicae how much he high-biased esimae deviaed from he rue price. See [4] for he relaionship beween price bound and variance error, and [23] he definiion of MAD. We observe ha all sandard errors and MADs from Zhang s approximaion are smaller han he ypical approximaion by he geomeric-average Asian opion. hese numerical resuls also suppor ha M( P Z ; is a beer conrol han M( P ;. Noe ha M( P ; = e r H( P (, S, ;, where P (, S, ; denoes he price of a geomeric-average Asian opion price wih he mauriy raher han. One can in principle replace his maringale conrol by is process represenaion, bu his does no reduce any compuaional burden. In pricing European Asian opions, he maringale a mauriy, M( P ;, is only needed for he linear conrol. he compuaional complexiy for e r H( P (, S, ; is much less han M( P ;, where he dela mus be compued along each simulaed pah. In pricing American Asian opions, he maringale conrol process { M( P ; } is required o esimae he high-biased so- luion. herefore, he compuaional complexiy for M( P ; and e r H( P (, S, ; are roughly he same. Currenly only one random source is encounered, namely he Brownian moion W. eneral maringale conrols are in he form of sochasic inegrals wih respec o ha Brownian moion. Nex we inroduce muli-facor sochasic volailiy models, where muliple Brownian moions are involved. I hen becomes hard o choose from many conrols. We shall inroduce muli-facor sochasic volailiy model and he perurbaion echnique in Secion 3. In Secion 4, he singular and regular perurbaion echnique is applied o simplify he choice of maringales for he geomeric-average Asian opions. In Secion 5, we consider using generalized conrol variaes in esimaing arihmeic-average Asian opions. A combinaion of he convenional conrol variae mehod and he maringale conrol urns ou o be a much beer choice han applying maringale conrols alone. 9

10 3 Mone Carlo Pricing under Muli-facor Sochasic Volailiy Models Under a risk-neural probabiliy measure IP paramerized by he combined marke price volailiy premium (Λ 1, Λ 2, we consider he following muli-facor sochasic volailiy models defined by ds = rs d + σ S dw (, (2 σ = f(y, Z, [ 1 dy = ε c 1(Y + g ] 1(Y Λ 1 (Y, Z d + g ( 1(Y ρ 1 dw ( ε ε [ dz = δc 2 (Z + ] δg 2 (Z Λ 2 (Y, Z d + ( δg 2 (Z ρ 2 dw ( + ρ 12 dw ( ρ 2 2 ρ 2 12dW (2, 1 ρ 2 1dW (1, where S is he underlying asse price process wih a consan risk-free ineres rae r. Is random volailiy σ is driven by( wo sochasic processes Y and Z varying on he ime scales ε and 1/δ, respecively. he vecor W (, W (1, W (2 consiss of hree independen sandard Brownian moions. Insan correlaion coefficiens ρ 1, ρ 2, and ρ 12 saisfy ρ 1 < 1 and ρ ρ 2 12 < 1. he volailiy funcion f is assumed o be smooh and bounded. Coefficien funcions of processes Y and Z, namely (c 1, g 1, Λ 1 and (c 2, g 2, Λ 2 are assumed o be smooh such ha hey saisfy he exisence and uniqueness condiions for he srong soluions of sochasic differenial equaions. Mean-revering processes such as Ornsein-Uhlenbeck (OU processes or square-roo processes are ypical examples of modeling driving volailiy processes [6, 9]. Under his seup, he join process (S, Y, Z is Markovian. iven he muli-facor sochasic volailiy model (2 under IP, he price of a plain European opion wih he inegrable payoff funcion H and expiry is defined by P ε,δ (, x, y, z = IE,x,y,z { e r( H(S }, where IE,x,y,z is a shor noaion for he expecaion wih respec o IP condiioning on he curren saes S = x, Y = y, Z = z. A basic Mone Carlo simulaion approximaes he opion price P ε,δ (, S, Y, Z a ime by he sample mean 1 N N i=1 e r H(S (i (21 where N is he oal number of sample pahs and S (i denoes he i-h simulaed sock price a ime. Variance reducion echniques are paricularly imporan o accelerae he compuing efficiency of he basic Mone Carlo pricing esimaor (21. Nex, we briefly review he consrucion of a generic algorihm, i.e. maringale conrol variae mehod proposed and analyzed by Fouque and Han [3, 5]. 3.1 Consrucion of Maringale Conrol Variaes Assuming ha he European opion price P ε,δ (, S, Y, Z is wice differeniable in sae space and once differeniable in ime, we apply Io s lemma o is discouned price e r P ε,δ, hen inegrae from 1

11 ime o he mauriy. One can use he pricing parial differenial equaion o cancel ou nonmaringale erms, and use he fac ha P ε,δ (, S, Y, Z = H(S o obain he following maringale represenaion P ε,δ (, S, Y, Z = e r H(S M (P ε,δ x 1 M 1 (P ε,δ ε y δm 2 (Pz ε,δ, (22 where subscrips denoe parial derivaives and cenered maringales are given by M (Px ε,δ = M 1 (P ε,δ y = M 2 (P ε,δ z = where he Brownian moions are W (1 s W (2 s = ρ 1 W ( s + = ρ 2 W ( s ε,δ rs P e (s, S s, Y s, Z s f(y s, Z s S s dw s (, (23 ε,δ rs P e y (s, S (1 s, Y s, Z s g 1 (Y s d W s, (24 ε,δ rs P e z (s, S (2 s, Y s, Z s g 2 (Z s d W s, (25 1 ρ 2 1W (1 + ρ 12 W (1 s + s, 1 ρ 2 1 ρ 2 12W (2 s. hese maringales M, M 1, and M 3 play he role of perfec conrols of Mone Carlo simulaions. Namely, if he maringales (23, (24, and (25 can be exacly compued, hen one can generae one sample pah o evaluae he opion price hrough (22. Unforunaely he gradien ( Px ε,δ, Py ε,δ, Pz ε,δ of he opion price appearing in he maringales is no known because he opion price P ε,δ iself is exacly wha we wan o esimae. However, one can choose an approximae opion price o subsiue P ε,δ used in he maringales (23, (24 and (25, and sill reain heir maringale properies. When ime scales ε and 1/δ are well separaed; namely, < ε, δ 1, he zeroh order approximaion of he Black-Scholes ype can be found in [6] P ε,δ (, x, y, z P BS (, x; σ(z (26 ( ε, wih is poin-wise accuracy of order O δ for coninuous piecewise payoffs. he homogenized price P BS (, x; σ(z solves he Black-Scholes parial differenial equaion wih he consan volailiy σ(z and he erminal condiion P BS (, x = H(x. Noe ha his approximaion P BS (, x; σ(z is y-variable independen, where y is he iniial value of he fas varying process Y. he z-dependen effecive volailiy σ(z is defined as he square roo of an average of he variance funcion f 2 wih respec o a limiing disribuion of Y : σ 2 (z = f 2 (y, zdφ(y = f 2 (y, z. (27 Here Φ(y denoes he invarian disribuion of he fas varying process Y where volailiy premium Λ 1 is zero, because in he drif erm of he dy equaion in (2, 1 ε erm dominaes 1 ε erm when ε 1. We use he bracke < > o represen such average. In he OU case, we le c 1 (y = m 1 y and g 1 (y = ν 1 2 wih Λ1 = such ha 1/ε is he rae of mean reversion, m 1 is he long-run mean, and ν 1 is he long-run sandard deviaion. Is invarian disribuion Φ is normal wih mean m 1 and variance ν 2 1. Please refer o [6] for deailed discussions. Because he approximae opion price P BS (, x; σ(z is independen of y, he erm M 1 (P BSy becomes 11

12 zero. In (22, he las erm associaed wih M 2 (P BSz is small of order δ. herefore, we can neglec his erm as well. We hen selec he sochasic inegral M (P BSx as he major conrol for he esimaor (21 and formulae he following maringale conrol variae esimaor: 1 N [ ] e r H(S (i N M(i (P BSx. (28 i=1 his is he approach aken in [5] where he proposed maringale conrol variae mehod is numerically superior o an imporance sampling mehod in [2] for pricing European opions. An asympoic analysis of he maringale conrol variae, shown in heorem 1 [3], guaranees ha: under OU-ype processes for modeling (Y, Z in (2 wih < ε, δ 1, he variance of he maringale conrol variae for European opions is small of order ε and δ; namely Var ( e r H(S M (P BSx = O(max{ε, δ}. (29 Moreover he financial inerpreaion of he maringale conrol erm M (P BSx = e rs P BS (s, S s; σ(z s f(y s, Z s S s dw s ( corresponds o he cumulaive cos of a dela hedging sraegy. his maringale conrol variae mehod can be easily exended o hiing ime problems such as barrier opions and opimal sopping ime problems such as American opions. Numerical resuls and variance analysis are discussed in [3]. 4 Variance Reducion for Asian Opions: eomeric-average Case A general form of payoffs for geomeric-average Asian opions (AO consiss of a fixed srike K 2, a floaing srike S, a coefficien K 1, and a geomeric-average of sock prices. For example, he price a ime of a AO call opion is defined by IE { e r( ( K 1 S K 2 + F }, where F denoes he filraion generaed by he process (S s, Y s, Z s s. he random variable denoes he geomeric average of sock prices up o ime ( 1 = exp ln S d. We inroduce he running sum process L = ln S udu whose differenial form is dl = ln S d, (3 such ha he join dynamics (S, Y, Z, L is Markovian. Hence we denoe he call price of AO by ( ( } + P ε,δ (, x, y, z, L = IE,x,y,z,L {e r( L exp K 1 S K 2. (31 We assume (S = x, Y = y, Z = z, L = L. A basic Mone Carlo simulaion consiss in generaing N independen rajecories governed by equaions (2 and (3, and averaging he discouned sample payoffs in order o obain an unbiased AO price esimaor: P ε,δ P MC = e r( N ( ( N exp k=1 12 L (k K 1 S (k K 2 +. (32

13 4.1 Maringale Conrol Variae for eomeric-average Asian Opions he consrucion of maringale conrol variaes for AO price is similar o he procedure presened in Secion 3. We firs apply Io s lemma o he discouned AO price and hen inegrae i wih respec o he ime variable. herefore, he following maringale represenaion is obained ( + P ε,δ (, S, Y, Z, L = e (exp r L K 1 S K 2 M (P ε,δ x 1 M 1 (P ε,δ ε y (33 δm 2 (P ε,δ z, where sochasic inegrals M, M 1 and M 2 are defined similarly as in (23 - (25 bu wih P ε,δ in hese maringales insead. he maringale conrol variae esimaor for he AO price is hen formulaed by P ε,δ (, S, Y, Z, L 1 N ( ( N [e r L (k + ] exp K 1 S (k K 2 M (k (PBS x, (34 k=1 provided ha he homogenized AO price PBS, as an approximaion o he rue AO price, can be easily calculaed. For he case of fixed-srike AOs. i.e. K 1 =, i is shown in [2] ha PBS (, x, L; σ(z has a closed-form soluion PBS(, x, L; σ(z (35 ( L ln x = exp + ln x + R(,, z N (d 1 (x, z, L Ke r( N (d 2 (x, z, L, where R(,, z = ( r σ2 (z ( 2 + σ 2 ( 3 (z r(, d 1 (x, z, L = ln(x/k + L ln x + (r σ2 (z/2( 2 /2 + σ 2 (z ( σ(z 3 3 ( 3 d 2 (x, z, L = d 1 (x, z, L σ(z. 3 2 ( 3 3 (36 he probabilisic represenaion of he homogenized AO price PBS (, x, L; σ(z is PBS(, S = x, L = L; σ(z = IE {e ( } + r( e L / K S = x, L = L, (37 where S and L are governed by d S = r S d + σ(z S d W, (38 d L = ln S d, respecively and Z = z. Le W denoe a Brownian moion under a probabiliy measure P, under which he condiional expecaion is defined. hese derivaion can also be found in [2]. For oher AO payoffs such as call or pu of he floaing srike, i.e. K 2 =, one can derive similar resuls. We omi hese cases o limi he lengh of his paper. 13

14 able 3: Parameers used in he wo-facor sochasic volailiy model (2. r m f m s ν f ν s ρ 1 ρ 2 ρ 12 Λ f Λ s f(y, z 1% exp(y + z able 4: Iniial condiions and Asian call opion parameers. $S Y Z L $K 2 = K years Numerical Resuls for Pricing AO We presen numerical resuls from Mone Carlo simulaions o evaluae fixed-srike AO prices in his secion. Parameers in our model are shown in able 3. Oher values (iniials condiions and opion parameers are given in able 4. Parameers chosen in hese ables are exacly he same as ones used in [2] for he purpose of comparing efficiency. Sample pahs in (34 are simulaed based on he Euler scheme o discreize equaions (2 and (3 wih ime sep =.5. he sochasic inegral M is approximaed by a Riemann sum, and he number of oal pahs are 5. As demonsraed in [2], resuls of variance reducion raios obained from an imporance sampling echnique versus he basic Mone Carlo mehod are now lised in he hird column of able 5. he variance reducion raios obained from he maringale conrol variae mehod are lised in he las column of able 5. We find ha he maringale conrol variae mehod ouperforms he imporance sampling mehod in all cases. 4.3 Variance Analysis of Perurbed Volailiy Based on he fac ha random volailiy is flucuaing around is long-run mean, we analyze he variance of a simplified model which is helpful o explain he effec of maringale conrol. Le s assume ha under he risk-neural probabiliy measure, S ε,δ is a risky asse defined by ds ε,δ = rs ε,δ d + σ ε,δ S ε,δ dw, (39 where he perurbed volailiy is σ ε,δ = σ + εg + δh, σ > denoes he effecive volailiy, ε and δ are small parameers, and perurbed funcions {g, h } are assumed o be deerminisic and able 5: Comparison of variance reducion raios for various ime scales ε and δ. V MC denoes he sample variance obained from he basic Mone Carlo mehod. V IS ( P denoes he sample variance compued by an imporance sampling wih he firs-order price approximaion P. his echnique and is several numerical variance reducion raios V MC /V IS ( P can be found in [2]. V MC+CV (PBS denoes he sample variance compued by he maringale conrol variae mehod wih he zeroh-order AO price approximaion PBS in (35. ε δ V MC /V IS ( P V MC /V MC+CV (PBS 1/ / / /

15 bounded such ha σ ε,δ >,. A geomeric-average Asian opion is defined by ( P ε,δ, S ε,δ = IE,S ε,δ { e r( H(L ε,δ }, (4 where we denoe he running sum process L ε,δ smooh and bounded. = 1 ln Sε,δ d and assume ha he funcion H is heorem 3 (Variance Analysis iven condiions described above, for any fixed iniial sae (, S ε,δ, here exis ε > and δ > small enough and a posiive consan C such ha ( V ar ( e r H L ε,δ M (P BS x C max{ε, δ}, where PBS is defined as in (37 and (38 excep ha he homogenized volailiy σ is chosen as consan. Proof: (For simpliciy, we remove subscrips under he expecaion and use IE hereafer in his heorem. aking he pahwise derivaive [7] for opion price P ε,δ wih respec o he sock price, we can apply he chain rule o obain P ε,δ S ε,δ { ( (, S ε,δ, L ε,δ = IE e r( H L ε,δ = ( {e IE r( H S ε,δ L ε,δ ln S ε,δ s S ε,δ ds F } F }. (41 Similarly, given he sae variable S ε,δ and L ε,δ, he dela of he geomeric-average Asian opion wih he consan volailiy σ has he following decomposiion P BS S (, S = S ε,δ where he consan-volailiy sock price S saisfies P ε,δ S ε,δ, L = L ε,δ = { S IE e r( H ( L } F, (42 d S = r S d + σ S dw, and we denoe running sum as L = ln S s ds. Condiional on he driving volailiy processes, he absolue difference beween and P BS S is equal o P ε,δ S ε,δ = S ε,δ (, S ε,δ IE, L ε,δ P BS S { e r( H (, S = S ε,δ, L = L ε,δ (43 (ˆLε,δ ( L ε,δ L F }, which is obained by applying he Mean-Value heorem. he inner difference condiional on F is L ε,δ ˆL = 1 = 1 ( ln S ε,δ [ u ln S d (44 σ 2 σs ε,δ2 ] u ds du ε g s + δ h s dws du. 2 15

16 he variance of he conrol variae is ( ( V ar e r H L ε,δ M (PBS x ( ( = V ar e rs P ε,δ P BS = IE C ( e 2rs P ε,δ P BS IE { ( IE { e r( H (ˆLε,δ (s, S ε,δ s 2 (s, S ε,δ s, L ε,δ s, L ε,δ s σ ε,δ Ss ε,δ dws σ ε,δ2 S ε,δ2 s ds (45 ( L ε,δ L F s } 2 F } ds. (46 Here we subsiue (43 and (44 ino (45 and C is some consan because σ ε,δ is assumed o be bounded. he nesed expecaion defined in (46 can be bounded above by { ( CIE L ε,δ L } 2 F C ( IE C max{ε, ε δ, δ}. 2 σ 2 σ s ε,δ2 F ds d + 2 { ( 2 } IE ε g s + δ h s dws F d We use he inegrabiliy of g and h and Io isomery propery o obain he esimae. he noaion C denoes some consan independen of parameers ε and δ. We herefore conclude ha he variance of maringale conrol variae is of O(ε, δ. An argumen o rea he general muli-facor model (2 will be sudied in a separae paper. 5 Variance Reducion for Asian Opions: Arihmeic-Average Case Similarly o he AO case, i is convenien o inroduce a running sum process I = S udu, or is differenial form, di = S d, such ha he join dynamics (S, Y, Z, I defined in (2 is Markovian. Under he risk-neural probabiliy measure IP he price of an arihmeic-average Asian call opion is given by { } + P ε,δ A (, x, y, z, I = IE,x,y,z,I e r( ( I K 1 S K 2 condiioning on S = x, Y = y, Z = z, I = I. We use his ype of opions as ypical examples when we discuss variance reducion of Mone Carlo simulaions. A basic Mone Carlo simulaion for pricing arihmeic-average Asian opions (AAO in shor wih N replicaions is given by P ε,δ A P MC A = e r( N N k=1 ( I (k + K 1 S (k K 2. (47, 16

17 5.1 One-Sep versus wo-sep Conrol Variae Mehods I becomes sraighforward o derive he maringale conrol variae mehod by direcly applying he maringale represenaion heorem o AAOs such ha ( + P ε,δ A (, S, Y, Z, I = e (exp r I K 1 S K 2 M (P ε,δ A x ; 1 M 1 (P ε,δ ε A y ; (48 δm 2 (P ε,δ A z ;, where sochasic inegrals M, M 1 and M 2 are defined he same as before. Similar o AO cases, one can formulae an one-sep maringale conrol variae esimaor P ε,δ A (, S, Y, Z, I 1 N N k=1 ( [e r I (k + ] K 1 S (k K 2 M (k (PBS A x, (49 wih PBS A as he homogenized AAO price under some consan volailiy, which does no have a closedform soluion. Inuiively, one may choose a price approximaion such as P in (5 or P Z in (14 o subsiue for he homogenized AAO price PBS A. Noice ha hese homogenized price approximaions are independen of he y-variable. We assume here is no correlaion beween hese Brownian moions, namely all ρ s in (2 are zero. Le P be a well-chosen homogenized price approximaion such as P or P Z. hen we can use he maringale conrol M ( P x ; insead. he variance of such one-sep conrol variae is ( ( + V ar e r I K 1 S K 2 M ( P x ; (5 = M (P ε,δ A x P x ; Q + 1 ε M 1 (P ε,δ A y ; + δ M 2 (P ε,δ A z ;, Q Q where Q denoes he expecaion of a quadraic variaion. Noice ha he second erm above has a large coefficien 1 because ε is small. his erm is compleely no affeced by he maringale conrol. ε he hird erm is negligible because δ is small. o furher reduce he one-sep conrolled variance (5, we would like o find a x, y, z-dependen price approximaion so ha each quadraic variaion can be reduced. he convenional conrol ( + e r( L K 1 S K 2 P ε,δ (, S, Y, Z (51 has such propery. his can be seen from he variance of he conrol variae ( ( ( + ( + V ar e r I K 1 S K 2 λ e r L K 1 S K 2 P ε,δ (, S, Y, Z = M (P ε,δ A ε,δ λp ; Q + 1 ε M 1 (P ε,δ ε,δ A λp ; Q + δ M 2 (P ε,δ ε,δ A λp ; Q. (52 Comparing wih he previous variance (5 reduced by one maringale conrol, he convenional conrol variae apparenly has he poenial o reduce he expecaion of each quadraic variaion, in paricular he large order erm 1 M ε 1 (P ε,δ A ;. he drawback of using he convenional conrol (51 is ha Q he AO price does no have a closed-form soluion. herefore, i is reasonable o use a wo-sep 17

18 algorihm for variance reducion: Sep 1: Esimae P ε,δ (, S, Y, Z by a maringale conrol M (PBS ; such ha P ε,δ (, S, Y, Z = IE { e r ( L (k K 1 S (k K 2 + M (P BS; Sep 2: Esimae P ε,δ (, S, Y, Z by he convenional conrol (51 such ha { ( + P ε,δ A (, S, Y, Z =IE e r I K 1 S K 2 ( ( } + λ e r L K 1 S K 2 P ε,δ (, S, Y, Z. he variance analysis for his wo-sep conrol variae mehod is he following. In Sep 1, i is implied from heorem 3 ha he variance is small of O(ε, δ. In Sep 2, i is expeced from (5 and (52 ha he variance of convenional conrol variae is smaller han a maringale conrol variae. Using resuls in able 6-13 from he nex secion, we confirm ha he variance induced by he wo-sep mehod is much smaller han he variance induced by he one-sep mehod, and he wo-sep mehod only increases a small amoun of compuing ime han he one-sep mehod. We should compare wih he oher wo-sep conrol variae mehod proposed in [2]. hese wo conrol variaes only differ in mehods o esimae he price P ε,δ, namely Sep 1. In [2], an imporance sampling is developed while in his paper, a maringale conrol variae is proposed and analyzed in Secion 4. In addiion o he accuracy gain by he maringale conrol variae as shown in able 5, we would like o noe ha he maringale conrol variae mehod is implemened under he same probabiliy measure as he esimae implemened in Sep 2. However, he imporance sampling in Sep 1 mus be implemened under some equivalen probabiliy measure so ha he pricing model are differen from he original one as in (2. herefore, here is an addiional programing advanage of our newly proposed wo-sep mehod. }. 5.2 Numerical Resuls for Pricing AAO In his secion, we will compare efficiencies of pricing Asian call opion using wo differen conrol variaes developed above (namely, he one-sep and wo-sep conrol variae mehod, combined wih Mone Carlo and quasi-mone Carlo (QMC mehods. he C++ on Unix is our programming language in he following examples. he pseudo-random number generaor used is ran2( in [28]. he approximaion mehod given in [29] is used o generae he sandard normal random variae. his mehod achieves 16 digis accuracy as claimed. In he QMC mehod, a quasi-random sequence or low discrepancy sequence (LDS is used, insead of a pseudo-random sequence. An LDS is usually more uniformly disribued over [, 1] s han a pseudo-random sequence does. here are wo classes of low-discrepancy sequences. One is called he digial ne sequences, such as Halon s sequence, Sobol s sequence, Faure s sequence, and Niederreier s (, s sequence, ec. A QMC mehod using his kind of LDS has convergence rae O( (log Ns N when esimaing he following inegral µ = f(xdx, (53 C s where C s = [, 1] s is he s-dimensional uni hypercube, and f(x is a leas inegrable. he direcion numbers from [12] are used so ha our implemenaion of Sobol sequence can generae poins of 18

19 dimension as high as he oher class is he inegraion laice rule poins. his ype of LDS is especially efficien for esimaing mulivariae inegrals wih periodic and smooh inegrands, and i has convergence rae (log Nαs O(, where α > 1 is a parameer relaed o he smoohness of he inegrand. he monograph N α by Niederreier [19] gives deailed informaion on digial ne sequences and laice rule poins, while he monographs by Hua and Wang [1], and Sloan and Joe [25] describe laice rules. L Ecuyer also made conribuions o laice rules based on linear congruenial generaor. One of he feaures of his ype of laice rule poins (referred o L Ecuyer s ype laice rule poins, LLRP, hereafer is ha i is easy o generae high dimensional LLRP poin ses wih convergence rae comparable o digial ne sequences. Deails of LLRP can be found in [15] and references herein. We will apply he LLRP as well since our problem is high dimensional. Besides he above LDS, we also apply he Brownian bridge (BB sampling echnique o our es problems. Deailed informaion abou Brownian bridge sampling can be found in [7]. o compare he efficiencies of differen mehods, we need a benchmark for fair comparison. If he exac value of he quaniy o be esimaed can be found, hen we use he absolue error or relaive error for comparison. Oherwise, we use he unbiased sample variance σ n 2 for comparison. For LDS sequences, we define he unbiased sample variance σ n 2 by inroducing random shif as follows. Assume ha we esimae µ = E[f(X], where X is an s dimensional random vecor. Le {x i } m i=1 I s = [, 1] s be a finie LDS sequence and {r j } n j=1 I s be a finie sequence of random vecors. For each fixed j, we have a sequence {y (j i } m i=1, where y (j i = x i + r j. Such a sequence sill has he same convergence rae as {x i } m i=1, which is proved in [26]. Denoe and µ j = 1 m µ n = 1 n he unbiased sample variance is (assuming n > 1 m i=1 f(y (j i, n µ j. j=1 σ 2 n = n j=1 (µ j µ n 2 n 1 = n ( n n 2 j=1 µ2 j j=1 µ j. (n 1n And he variance of he MC mehod is calculaed as usual based on he pseudo- random poin se {x i + r j } wih mn poins. hus, we can easily consruc he confidence inerval for a given confidence level. he variance reducion raio of a QMC mehod over he MC mehod is defined by variance reducion raio = he sample variance of he MC mehod he sample variance of he randomized QMC mehod. (54 In our comparisons, he sample sizes for MC mehod are 124, 248, 496, 8192, 16384, and 32768, respecively; and hose for Sobol sequence relaed mehods are 124, 248, 496, 8192, 16384, and 32768, respecively, each wih 1 random shifs; and he sample sizes for LLRP relaed mehods are 121, 239, 493, 8191, 16381, and 32749, respecively, and again, each wih 1 random shifs. We divide he ime inerval [, ] ino m = 128 subinervals. In he following ables, column 1 conains he numbers of poins. Numbers wihou parenheses in he MC column are opion values, and numbers wihin parenheses in he same column are he corresponding sandard errors. Numbers in he QMC 19

20 able 6: Comparison of simulaed Asian opion values and variance reducion raios by one-sep mehod for 1/ε = 75., δ =.1. N MC MC+CV Sobol Sobol+CV Sobol+BB Sobol+CV+BB ( ( ( ( ( ( columns are variance reducion raios. An arihmeic-average Asian call opion wih a fixed srike is considered. ( 1 he payoff variable is +. S d K We ake inpu parameers defined in able 3 and 4 as max( 1 S d K, = follows: S = $1, K = $11, r =.1 = 1%, = 1 year, m 1 =.8, m 2 =.6, ν 1 =.7, ν 2 = 1., ρ 1 = ρ 2 =.2, ρ 12 =., y = 1., z =.5, 1/ε = 75., δ =.1. In he following ables we demonsrae numerics of various variance reducions based on wo differen conrol variae mehods. In ables 6 and 7, he one-sep maringale conrol M (P ; is used o consruc he conrol variae esimaor (49, where we use P raher han PBS A. We observe ha he variance reducion raio is abou 23 for pseudo-random sequences by using his conrol variae echnique. Wihou a conrol, he variance reducion raios for Sobol sequence vary from 1.3 o 5.8 and hose for LLRP are from 1.2 o 5.5; he Brownian bridge (BB in shor sampling does no help much o improve he efficiency for he Sobol sequence, and i is a lile bi beer for he LLRP case. In BB sampling, we ry wo differen assigmens of he coordinaes of quasi-random poins o he hree sae variables: in he firs way we assign he firs 128 coordinaes o he firs variable, he second 128 coordinaes o he second variable and he las 128 coordinaes o he las variable. In he second way, we assign he coordinaes alernaively o he hree variables: coordinaes in (3i + 1h, (3i + 2h and (3i + 3h posiions are assigned o he firs, he second and he hird variable, respecively. he resuls by hese wo differen assigmens show very lile difference in magniude for his SV model. We believe ha is because he BB sampling mehod iself does no have much advanage for his complicaed SV model. See [2] for similar observaions. When combined wih conrol variae, he variance reducion raios for boh QMC sequences are increased: for he Sobol sequence hey vary from 28.2 o 76.5, and hose for L Ecuyer ype laice rule poins are from 55.6 o he CPU ime used in he simulaions are lised in ables 8 and 9, from which we observe ha simulaion imes used in he same group of mehods (wih or wihou conrol variaes are essenially he same magniude, and a mehod wih conrol variaes normally akes wice amoun of ime of he same mehod wihou conrol variaes. I should be poined ou ha he CPU ime is an approximae measure since here were also many oher jobs running when our programs were running on he main frame machine. he siuaions are very similar for oher cases. For example, ables 12 and 13 lis simulaion imes for wo-sep conrol variae mehods. We find ha he difference in compuing imes beween one-sep and wo-sep conrol variae mehods is small in general. In ables 1 and 11, he wo-sep mehod is used o consruc he conrol variae esimaor. We observe ha he variance reducion raio is abou 6 for pseudo-random sequences by using he conrol variae echnique. Wihou a conrol, variance reducion raios for Sobol sequence vary from 1.3 o 5.8 and hose for LLRP are from 1.2 o 5.5; he BB sampling acually worsen he efficiency 2

21 able 7: Comparison of simulaed Asian opion values and variance reducion raios by one-sep mehod for 1/ε = 75., δ =.1 (coninued. N LLRP LLRP+CV LLRP+BB LLRP+CV+BB able 8: Comparison of ime (in seconds used in simulaing Asian opion values by one-sep mehod for 1/ε = 75., δ =.1. N MC MC+CV Sobol Sobol+CV Sobol+BB Sobol+CV+BB able 9: Comparison of ime (in seconds used in simulaing Asian opion values by one-sep mehod for 1/ε = 75., δ =.1 (coninued. N LLRP LLRP+CV LLRP+BB LLRP+CV+BB