Proceedigs of the 2008 Witer Simulatio Coferece S. J. Maso, R. R. Hill, L. Möch, O. Rose, T. Jefferso, J. W. Fowler eds. REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS Guagwu Liu L. Jeff Hog Departmet of Idustrial Egieerig ad Logistics Maagemet The Hog Kog Uiversity of Sciece ad Techology Clear Water Bay, Hog Kog, Chia ABSTRACT We revisit the stochastic mesh method for pricig America optios, from a coditioig viewpoit, rather tha the importace samplig viewpoit of Broadie ad Glasserma 997). Startig from this ew viewpoit, we derive the weights proposed by Broadie ad Glasserma 997) ad show that their weights at each exercise date use oly the iformatio of the ext exercise date therefore, we call them forward-lookig weights). We also derive ew weights that exploit ot oly the iformatio of the ext exercise date but also the iformatio of the last exercise date therefore, we call them biocular weights). We show how to apply the biocular weights to the Black-Scholes model, more geeral diffusio models, ad the variace-gamma model. We demostrate the performace of the biocular weights ad compare to the performace of the forward-lookig weights through umerical experimets. INTRODUCTION The pricig of America optios is oe of the challegig problems i fiacial egieerig. By the term America optios, we refer to derivative securities which ca be earlyexercised at a fiite umber of dates prior to the maturity. They are sometimes called Bermuda optios. To price a America optio usig Mote Carlo simulatio, oe may formulate it as a dyamic programmig problem, ad the approximate the value of the America optio backwards recursively. To approximate the value of the optio at each exercise date, Tsitsiklis ad Va Roy 999) ad Logstaff ad Schwartz 200) use a regressio approach by employig a sequece of basis fuctios, ad Broadie ad Glasserma 997) desig a stochastic mesh method. I this paper, we focus o the stochastic mesh method. Basically the stochastic mesh method approximates the optio value by usig weight fuctios which explore the iformatio cotaied i the simulatio, e.g., the desity iformatio. Alog this lie of research, Avramidis ad Hyde 999) cosider the the efficiecy improvemet of the method, ad Avramidis ad Matziger 2004) show the covergece of the stochastic mesh estimators. Other subsequet work icludes Broadie, Glasserma, ad Ha 2000) ad Broadie, Glasserma, ad Jai 997). A key feature of the stochastic mesh method is how to derive the weight fuctios. Broadie ad Glasserma 997) take a importace samplig viewpoit ad derive weights of each exercise date based o the iformatio of the ext exercise date. Therefore, we call them forwardlookig weights. I this paper we revisit this problem, ad cosider it from a coditioig viewpoit. From this viewpoit, we ca derive the same weights of Broadie ad Glasserma 997). Furthermore, we ca also derive ew weights that use ot oly the iformatio of the ext exercise date but also the iformatio of the last exercise date. Therefore, we call them biocular weights. To illustrate how to apply the biocular weights, we study how to apply them to the Black-Scholes model ad more geeral diffusio models. We compare these two weights for the Black-Scholes model through some simple ad prelimiary umerical experimets. The umerical results show that the forward-lookig weights have smaller variaces, but the biocular weights have smaller biases. We also demostrate how to apply the biocular weights to the variace-gamma model. Note that the forward-lookig weights are difficult to implemet for this model sice they require a large amout of computatioal effort. A simple umerical study shows that the biocular weights work well for the variace-gamma model. The rest of the paper is orgaized as follows. I Sectio 2 we review some prelimiary kowledge o pricig America optios ad the stochastic mesh method. The i Sectio 3 we aalyze the problem from a coditioig viewpoit, ad derive the forward-lookig ad biocular weights. I Sectio 4 we cosider several examples to illustrate how to apply the forward-lookig ad biocular weights, followed 978--4244-2708-6/08/$25.00 2008 IEEE 594
by umerical study i Sectio 5. We coclude the paper i Sectio 6. 2 PRELIMINARIES Let S t deote the price at time t of the uderlyigasset whose price dyamics follows a Markov process o R d. Suppose that 0 = t 0 < t <...<t m = T are exercise opportuities also called exercise dates), i.e., the America optio ca be exercised at t i for ay i {0,,...,m}. Without loss of geerality, we assume thatt i+ t i = τ for all i = 0,,...,m. We write S i for S ti for simplicity of otatio. Moreover, suppose that idepedet sample paths of {S 0,S,...,S m } are geerated, deoted by {S 0,S j,...,s m} j the j-th sample path. Let L i x) deote the payoff fuctio of the America optio from exercise at date t i whe S i = x, ad V i x) deote the value of the optio at date t i whe S i = x. The a backwards recursio algorithm for pricig the America optio ca be expressed as V m x) = L m x) V i x) = max L i x),e rτ EV i+ S i+ ) S i = x] ), i = 0,,...,m, where the expectatio is take uder the risk-eutral measure, r is the risk-free iterest rate. For simplicity we oly cosider the iterest rate as a costat, while it ca be exteded to more complicated models of iterest rates. The the price of the America optio at time 0 is V 0 S 0 ). For i = 0,,...,m, let H i x) be the holdig value of the optio at date t i whe S i = x, i.e., H i x)=e rτ EV i+ S i+ ) S i = x]. The the major difficulty of pricig the America optio reduces to how to estimate the holdig value H i x) for ay state x. Broadie ad Glasserma 997) propose a stochastic mesh method to price the America optio. The key feature of the method is that for ay x, it evaluates H i x) by exploitig all the odes at time t i+, i.e., S i+,...,s i+. Essetially they choose a appropriate weight fuctio wi,x,s i,s i+ ) such that H i x) ca be estimated by H i x)=e rτ V i+ S j i+ ) wi,x,sj i,sj i+ ), j= where V i+ x) =max{l i+ x), H i+ x)}. The key issue of the stochastic mesh method is how to choose a appropriate weight fuctio wi, x,s i,s i+ ). Broadie ad Glasserma 997) aalyze this problem from a importace samplig viewpoit. Oe of the weight fuctios they suggest is f i x,s i+ ) w 2 i,x,s i+ )= j= f is j i,s i+), where f i x,y) is the trasitio desity from S i = x to S i+ = y. 3 ESTIMATING THE HOLDING VALUE H i x) Note that H i x) = e rτ EV i+ S i+ ) S i = x] E e rτ V i+ S i+ ) ] ε 0 + E ] ) ] E e rτ V i+ S i+ ) E ], whe ε is small. Based o this expressio, a estimator of H i x) ca be where H ε i x) = e rτ j= ) V i+ ε S j i+ w ε x,s j i ), w ε x,s j i )= {x ε S i x+ε } k=, {x ε S k x+ε } or more geerally, w ε x,s j i )= K S j i x ε )/ k= S k ) ] K i x by the kerel method Bosq 998) where K is a kerel desity fuctio, e.g., the stadard ormal desity fuctio. To esure the covergece of the kerel estimators, by Bosq 998), we eed to select ε such that ε 0 ad ε as. I the above kerel estimator, V ε i+ x) = max L i+ x), H ε i+ x)) ad V ε mx) =L m x). A advatage of this estimator is that it does ot require ay desity iformatio, but oly the sample paths S j i, i =,2,...,m ad j =,2,...,. However, its performace is typically poor sice it essetially exploits oly the iformatio i the S j i s which are close to x. Geerally, kerel estimators have a rate of covergece of ε ) /2 ε which is slower tha the typical /2. Basically, the weight fuctio wx,s j i ) is crucial to the performace of the estimator. Ituitively, with further iformatio, e.g., the desities, oe may obtai better weights, ad hece better estimators for H i x), which have faster rate 595
of covergece. For istace, the rate of covergece of the estimators of Broadie ad Glasserma 997) is /2. I the followig two subsectios, we apply coditioig approach to Equatio ) to icorporate more iformatio i the weight fuctios, ad derive estimators that have better rate of covergece. 3. Forward-Lookig Weights Note that by coditioig o S i+,wehave E H i x) ε 0 E ε 0 E ε 0 e rτ V i+ S i+ ) ] E ] E e rτ V i+ S i+ ) E ] e rτ V i+ S i+ )E Si+ ]) ]) Si+ E ]. With some regularity coditios, we ca take the limit ito the expectatios. The we have where H i x)=e e rτ V i+ S i+ )wi,x,s i+ ) ], E wi,x,s i+ )=lim ε 0 {x ε Si x+ε} ] S i+ E ]. Suppose that the trasitio desity of S i+ give S i = x ad the margial desity of S i are available, deoted by f i x, ), ad f i, ) respectively. Whe they are smooth, we have wi,x,s i+ ) ε 0 2ε E ] S i+ lim ε 0 2ε E ] ε 0 x+ε 2ε x ε f i,u) f iu,s i+ )du f i +,S i+ ) x+ε lim ε 0 2ε = f i,x) f ix,s i+ ) f i +,S i+ ) x ε f i,u)du f i,x) = f ix,s i+ ) f i +,S i+ ), where the third equality follows from the mea value theorem. The above weight wi,x,s i+ ) ivolves two desity fuctios, f i x, ) ad f i+, ). I practice f i x, ) is usually kow or ca be calculated, sice it is actually the trasitio desity which is used to geerate the sample pathes of the uderlyig asset price. However, the explicit expressio of f i +, ) is ofte ukow or ca ot be easily calculated, except for some simple models, e.g., S t followig geometric Browia motio. Whe the explicit expressio of f i+, ) is ukow, we may estimate it by usig the trasitio desities. Sice f i +,v)=e f i S i,v)], the f i +,S i+ ) ca be approximated by a sample mea, i.e., f i +,S i+ )= k= f i S k i,s i+ ). Therefore, we obtai two weights, deoted by w ad w 2 respectively, w i,x,s i+ ) = w 2 i,x,s i+ ) = f i x,s i+ ) f i +,S i+ ), 2) f i x,s i+ ) j= f is j i,s i+). 3) We refer to these two weights as forward-lookig weights, sice they are obtaied by coditioig o S i+, the sample paths i the ext exercise date. I fact, the forward-lookig weights derived here are special cases of the weights i Broadie ad Glasserma 997). As show i Broadie ad Glasserma 997), the weight w i,x,s i+ ) may lead to estimator whose variace grows expoetially with the umber of exercise opportuities, while w 2 i,x,s i+ ) ca avoid this problem. Geerally speakig, the weights i Broadie ad Glasserma 997) exploit the iformatio of the ext exercise date, ad they are obtaied from a importace samplig viewpoit, rather tha the coditioig viewpoit i our aalysis. I their work the weights ca be geerally expressed as wi,x,s i+ )= f i x,s i+ )/g i+ S i+ ), where g i+ ) is the desity of S i+ from which the mesh poits S j i+ s are actually geerated. Emphasis should be give to that f i x,s i+ ) is the trasitio desity uder risk-eutral measure while the margial desity g i+ ) may ot be uder risk-eutral measure. Sice the choice of g i+ ) is crucial to the performaces of the estimators, Broadie ad Glasserma 997) suggest a good choice of g i+ ), g i+ u)= k= f is k i,u), which is called average desity fuctio. The the weight fuctio becomes w 2 i,x,s i+ ). Ituitively, the average desity fuctio is equivalet to geeratig idepedet paths of S t ad the forgettig the path to which each S j i belogs see, e.g., Broadie ad Glasserma 997 or Avramidis ad Hyde 999). For more details of the weights i Broadie ad Glasserma 997), oe is referred to Glasserma 2004) for a comprehesive overview. 596
3.2 Biocular Weights Notice that the forward-lookig weights are obtaied by coditioig o the iformatio of the ext exercise date. Now we take oe step further: what if we coditios o ot oly the iformatio of the ext exercise date, but also the iformatio of the last exercise date? Motivated by the usage of Browia bridge samplig i Mote Carlo methods, hopefully we may obtai ew weights. We describes how we ca do so. Sice these ew weights use the iformatio o both sides of the curret exercise date, we refer to them as biocular weights. By coditioig o S i ad S i+, we have, H i x) E e rτ E ]) V i+ S i+ ) Si,S i+ ε 0 E ] E e rτ V i+ S i+ )E ]) Si,S i+ ε 0 E ]. With some regularity coditios we ca take the limit iside the expectatio. The H i x) = E e rτ V i+ S i+ ) wi,x,s i,s i+ ) ], where wi,x,s i,s i+ ) ε 0 2ε E ] {x ε Si x+ε} Si,S i+ lim ε 0 2ε E ]. Let f i i,i+,v,v 2 ) deote the coditioal desity of S i give S i = v ad S i+ = v 2, ad we assume that it is a smooth fuctio. The by the mea value theorem, wi,x,s i,s i+ )= f i i,i+x,s i,s i+ ). f i,x) For may models, the expressio of f i i,i+,v,v 2 ) is kow or ca be approximated based o the bridge samplig techiques. For istace, if S t follows a geometric Browia motio, the f i i,i+,v,v 2 ) ca be obtaied usig the result for Browia bridge. It ca also be calculated or approximated i other models, e.g., the variace-gamma model. Sice the margial desity f i, x) is typically ukow except for some very simple models of S t, we may use f i i,i+,v,v 2 ) to estimate it. Note that f i, x)=e f i i,i+,s i,s i+ ) ], the f i,x) ca be ubiasedly estimated by ˆf i,x)= k= The we obtai two ew weights: f i i,i+ x,s k i,sk i+ ). w i,x,s i,s i+ ) = f i i,i+x,s i,s i+ ), 4) f i,x) f i i,i+ x,s i,s i+ ) w 2 i,x,s i,s i+ ) = k= f i i,i+x,s k 5) i,sk i+ ). Note the deomiator of w i,x,s i,s i+ ) is exactly the margial desity f i,x), while i w 2 i,x,s i,s i+ ) it is replaced by a average. As we have discussed for the forward-lookig weights w i,x,s i+ ) ad w 2 i,x,s i+ ), usig the margial desity may lead to estimator whose variace grows expoetially with the umber of exercise opportuities, while the use of a average ca avoid this problem. We cojecture that the biocular weights have the similar properties, ad we ideed observe this pheomeo i umerical experimets. Therefore, we recommed to use w 2 i,x,s i,s i+ ) whe both ca be implemeted. 4 EXAMPLES I this sectio we use several examples to illustrate how the forward-lookig ad biocular weights ca be applied. We first cosider the Black-Schole model where the uderlyig asset follows a geometric Browia motio, ad the geeral diffusio models. At the ed we cosider the variacegamma model, which is a Lévy process. For the Black-Scholes model, both forward-lookig ad biocular weights ca be derived, while for geeral diffusio models, forward-lookig weights ca be derived but biocular weights eed to be approximated. Forward-lookig weights ca also be derived for the variace-gamma model, but it is ot practical to implemet them sice they are expressed i terms of expectatios ad hece require to be evaluated usig extra simulatios which may be computatioally itesive. However, biocular weights with explicit forms ca be derived for the variace-gamma model, which ca be implemeted practically. 4. Black-Scholes Model Suppose that the price of the uderlyig asset follows a Geometric Browia motio uder the risk eutral measure, i.e., ds t =r δ )dt + σdb t, S t 597
where r is the risk-free iterest rate, δ is the divided rate, σ is the volatility, ad B t is a stadard Browia motio. The we have S i = S 0 e r δ 2 σ 2 )t i +σb ti, ad by elemetary calculatio, f i u,v) = f i,x) = v ) vσ τ φ σ log µ σ /2)τ] ) 2, τ u ) x xσ φ t i σ log µ σ 2 /2 ) ]) t i, t i S 0 where φ ) deotes the stadard ormal desity. Plug the trasitio desity f i u,v) ad the margial desity f i,x) i Equatios 2) ad 3), the the forwardlookig weights ca be obtaied. To cosider the biocular weights, we eed to first obtai the coditioal desity of S i give S i ad S i+. To do so, we use the result of a Browia bridge. Coditioigo B i ad B i+, B i is a Browia bridge where B i represets B ti for simplicity of otatio. Particularly, B i /2 B i + B i+ ]+ τ/2 Z see, e.g., Avramids ad L Ecuyer 2006), where Z follows a stadard ormal distributio, ad the operator stads for equivalece i distributio. The coditioig o S i ad S i+, we ca easily obtai that S i S i S i+ e σ τ/2 Z. The by some simple algebra, we have f i i,i+ x,v,v 2 )= xσ τ/2 φ σ τ/2 log x ). v v 2 Therefore, the biocular weights of Equatios 4) ad 5) ca also be applied. 4.2 Geeral Diffusio Models Suppose that the price of the uderlyig asset follows the diffusio process: ds t = µt,s t )dt + σt,s t )db t. We use Euler scheme to discretize S t Glasserma 2004). Uder the scheme, S i+ = S i +µt i,s i )τ +σt i,s i ) τ Z i+, i = 0,,,m, where {Z,Z 2,,Z m } are idepedet stadard ormal radom variables. To simplify the otatio, we let µ i S i ) ad σ i S i ) deote µt i,s i ) ad σt i,s i ) respectively. For geeral diffusio models where the drift µ ad volatility σ deped o t ad S t, it is easy to derive the forward-lookig weights sice the trasitio desity f i x, ) ca be calculated based o the discretizatio scheme, while it is ot easy to derive the biocular weights sice the coditioal desity f i i,i+2,v,v 2 ) is ot easy to obtai. However, based o the discretizatio scheme, we are able to derive explicit expressios for approximatios of the biocular weights. We describe how this ca be doe. Recall that coditioigo B i ad B i+, B i /2 B i + B i+ )+ τ/2 Z, where Z is a stadard ormal radom variable. The B i B i 2 B i+ B i )+ τ/2 Z. 6) By Euler scheme, whe the step size τ is small, approximately we have S i = S i + µ i S i )τ + σ i S i )B i B i ], 7) S i+ S i + µ i S i )2τ + σ i S i )B i+ B i ]. 8) The combiig Equatios 6), 7) ad 8) together we have, coditioig o S i ad S i+, S i 2 S i + S i+ ]+σ i S i ) τ/2 Z. The by some simple algebra, we have f i i,i+ x,v,v 2 )= x σ i v ) τ/2 φ 2 v ) + v 2 ) σ i v ). τ/2 Moreover, the trasitio desity f i x, ) ca be easily obtaied: ) u x f i x,u) = σt i,x) τ φ µti,x)τ σt i,x). τ The pluggig f i x,u) ad f i i,i+ x,v,v 2 ) i Equatios 3) ad 5) respectively, we obtai the forward-lookig weight ad the biocular weight. 4.3 Variace-Gamma Model The forward-lookig ad biocular weights work ot oly for the diffusio processes, but also for some other models. I this example, we cosider the variace gamma model. Followig the otatio i Avramidis ad L Ecuyer 2006). Let Bt; θ, σ) be a Browia motio with drift parameter θ ad variace parameter σ. Let Gt; µ,ν) be a gamma process idepedet of Bt; θ,σ), with drift µ > 0 ad volatility ν > 0. The G0; µ, ν) =0, the process G has idepedet icremets, ad the icremets follow a Gamma distributio, i.e., Gt + δ ; µ,ν) Gt; µ,ν) Γδµ 2 /ν,ν/µ) for t 0 ad δ > 0. 598
A variace gamma process with parameters θ,σ,ν) is defied by X = {Xt)=Xt; θ,σ,ν)=bgt;,ν),θ,σ),t 0}, which is obtaied by subjectig the Browia motio to a radom time chage followig a gamma process with parameter µ =. The the risk eutralized asset price process S t is S t = S 0 exp{ω + r δ )t + Xt)}, where r is the risk-free iterest rate, δ is the divided rate, ad the costat ω = log θν σ 2 ν/2)/ν is chose so that the discouted value of a portfolio ivested i the asset is a martigale. I particular, ES t )=S 0 expr δ )t]. The we require θ + σ 2 /2)ν <, which esures that ES t ) < for all t > 0. We assume that this requiremet is satisfied i this example. To aalyze this model, we first review two schemes of simulatig the variace gamma process. The first oe is simulatig it as Gamma time-chaged Browia motio, while the secod oe simulatig it via a Browia bridge. For details of these schemes, oe is referred to Fu 2007) ad Avramidis ad L Ecuyer 2006). With the first scheme, we will derive the trasitio desity f i x, ), while with the secod scheme we obtai the coditioal desity of S i give S i, S i+. The the weights obtaied by pluggig these desities i Equatios 3) ad 5) ca be applied. We first look at the scheme of simulatig variace gamma process as Gamma time-chaged Browia motio. For simplicity of otatio, we let X i ad G i deote Xt i ) ad Gt i ) respectively from ow o. We idepedetly geerate G i := G i+ G i accordig to a Gamma distributio ad Z i from a stadard ormal distributio, which are idepedet of the past r.v.s. Particularly, let Γa, b) deote the Gamma distributio with shape parameter a ad scale parameter b, ad a, b) the ormal distributio with mea a ad variace b, the G i Γτ/ν,ν), ad Z i 0,). The we have X i+ = X i + θ G i + σ G i Z i. By simple algebra we obtai the trasitio desity of S i+ give S i = x: f i x,u) = 0 uσ y φ = E W uσ W φ logu/x) ω + r δ )τ θ y σ y logu/x) ω + r δ )τ θw σ W ) γy)dy )], where γ ) is the desity of the radom variable which follows a Γτ/ν, ν) distributio, ad the expectatio is take over W. The we ca use f i x,u) to obtai forward-lookig weights. The variace-gamma process ca also be simulated via Browia bridge. I particular, give X i, X i+, G i ad G i+, X i ca be simulated by a two-step algorithm. Let β a,b) deote the beta distributio with parameters a ad b, the i the first step, we geerate Y β τ/ν,τ/ν), ad let G i = G i +G i+ G i )Y. 9) The i the secod step, we geerate Z 0,Gi+ G i )σ 2 Y ), ad let The we have X i = YX i+ + Y)X i + Z. 0) X i = YX i+ + Y)X i + σ G i+ G i )Y Z = YX i+ + Y)X i +Y Z, where Z is a stadard ormal radom variable idepedet of Y, ad Y = σ Y Y)G i+ G i ). With the above bridge samplig scheme, we derive the coditioal desity of S i give S i, S i+, G i ad G i+ by simple algebra. Specifically, where f i i,i+ x,s i,s i+,g i,g i+ ) = 0 xσ φy)gy)dy y y)g i+ G i ] ] = E Y xσ φy ), Y Y)G i+ G i ] φy)=φ log ) +2y )w + r δ )τ σ y y)g i+ G i ], x S y i+ S y i gy) is the desity of the radom variable Y which follows a β τ/ν, τ/ν) distributio, ad the expectatio is take for Y. The the correspodig biocular weight ca be obtaied. So far we have derived the forward-lookig ad biocular weights followig exactly the aalysis i the previous sectios. For these weights, though the trasitio desity f i x,s i+ ) ad the coditioal desity f i i,i+ x,s i,s i+,gt i ),Gt i+ )) ca be estimated by ruig extra Mote Carlo simulatios, it may ot be easy to implemet i practice because of the huge computatioal 599
effort required. Fortuately, we may obtai other weights which are much easier to implemet, i the light of the coditioig viewpoit of the weights. This viewpoit provides us some flexibility i choosig the coditioig quatities. By coditioig o some appropriate quatities we may obtai weights that are practically applicable. We illustrate how we ca do so. Rather tha coditio oly o S i,s i+,g i,g i+ ), we additioallycoditioo G i. The similar to the previous aalysis, a biocular weight ca be expressed as: w i,x,s i,s i+,g i,g i+,g i ) ε 0 E ] S i,s i+,g i,g i+,g i lim ε 0 E ] f c i,x,s i,s i+,g i,g i+,g i ) = E f c i,x,s i,s i+,g i,g i+,g i )], ) where f c i,x,s i,s i+,g i,g i+,g i ) is the coditioal desity of S i give S i,s i+,g i,g i+,g i ). Usig Equatios 9) ad 0), by elemetary algebra we ca obtai f c i,x,s i,s i+,g i,g i+,g i )= xσ φ U i ), l i where G i = G i+ G i, ad U i = p i = G i G i + G i, l i = G i G i G i + G i, )] log x/ S p i i+ S p i i +ω + r δ )2p i )τ σ. l i 5 NUMERICAL STUDY I the previous sectio we have show how to applied the forward-lookigad biocular weights for several examples. To illustrate the performaces of these weights, we coduct umerical experimets for the Black-Scholes model ad the variace-gamma model. We cosider a America call optio uderlyig a asset followig the Black-Scholes model, i.e., the uderlyig asset price follows a geometric Browia motio. The optio expires i three years ad ca be exercised at ay of 0 equally spaced exercise opportuities. The payoff upo exercise at t i is S i K) +, with K = 00 ad S 0 = 00, volatility σ = 0.2, iterest rate r = 5%, ad divided yield δ = 0%. We have kow that the price of this America call optio is 7.98, obtaied from a biomial lattice see age 469 of Glasserma 2004). We use it as a bechmark value to test the performaces of differet weights. We coduct Table : Results of forward-lookig ad biocular weights for the Black-Scholes model 500 000 500 2000 forward mea 8.28 8.3 8.075 8.048 Var 0.86 0.090 0.05 0.037 bias 0.30 0.5 0.095 0.048 biocular mea 8.28 8.05 8.026 8.007 Var 0.276 0.28 0.085 0.063 bias 0.48 0.07 0.046 0.027 000 replicatios to estimate the error of the estimators. We observed that the estimators usig weights w ad w which ivolve margial desities, have large errors the stadard deviatios ca be i a order of 0 4 while the true value is 7.98). These large errors are due to some extreme large observatios occasioally. This pheomeo coicides with the proof i Broadie ad Glasserma 997) that use of margial desities i weights may lead to estimators whose variaces grow expoetially with the umber of exercise opportuities. The we maily compare the estimators usig the weights w 2 ad w 2. The compariso results are preseted i Table, where we show the mea, variace ad bias of the estimators correspod to forward-lookig weight ad biocular weight respectively. From the table we ca see that biocular weight has smaller bias while the the forward-lookig weight has smaller variace. We also cosider a America put optio uder the variace-gamma model, to illustrate the performace of the stochastic mesh method usig the weight w i,x,s i,s i+,g i,g i+,g i ). I the experimets, we let T = 0.566, r = 5.4%, δ =.2%, σ = 20.72%, ν = 0.5022, θ = 0.2290, S 0 = 369.4 ad K = 200. These settigs are cited from Hirsa ad Mada 2003). We let m = 0 ad the value of the America optio at curret time is approximately 35.56. By usig the biocular weights as i Equatio ), the umerical results of the stochastic mesh estimator are summarized i Table 2. From the table we ca see that ad stadard deviatio stdev) of the estimator decreases as the sample size icreases. 6 CONCLUSIONS I this paper we revisit the stochastic mesh method from a coditioig viewpoit. Based o this ew viewpoit, biocular weights of the stochastic mesh method are derived, which exploit the iformatio o both sides of the curret exercise date. Though biocular weights may ot be superior to the existig forward-lookig weights, they ca be applied 600
Table 2: Results of the biocular weights for the variacegamma model 500 000 500 2000 mea 40.40 39.53 39.28 38.97 stdev 4.43 3.28 2.63 2.2 to some models, e.g., the variace-gamma model, where the forward-lookig weights may ot be applied efficietly. For future research, it would be iterestig to compare the forward-lookig weights ad the biocular weights for high dimesioal problems, to examie whether the extra iformatio has beefits i the estimatio. ACKNOWLEDGMENTS This research was partially supported by Hog Kog Research Grats Coucil grat CERG 63907. REFERENCES Avramidis, A. N., ad P. Hyde. 999. Efficiecy improvemets for pricig America optios with a stochastic mesh. Proceedigs of the 999 Witer Simulatio Coferece, 344-350. Avramidis, A. N., ad P. L Ecuyer. 2006. Efficiet Mote Carlo ad quasi-mote Carlo optio pricig uder the variace Gamma model. Maagemet Sciece, 52:930-944. Avramidis, A. N., ad H. Matziger. 2004. Covergece of the stochastic mesh estimator for pricig America optios. Joural of Computatioal Fiace, 74). Bosq, D. 998. Noparametric Statistics for Stochastic Processes. Secod Editio. Spriger, New York. Broadie, M., ad P. Glasserma. 997. A stochastic mesh method for pricig high-dimesioal America optios. PaieWebber Workig Papers i Moey, Ecoomics ad Fiace #PW9804, Columbia Busiess School, New York. Broadie, M., P. Glasserma, ad Z. Ha. 2000. Pricig America optios by simulatio usig a stochastic mesh with optimized weights. I Probabilistic Costraied Optimizatio: Methodology ad Applicatios, ed S. Uryasev. Kluwer Academic Publishers, Norwell, Mass. Broadie, M., P. Glasserma, ad G. Jai. 997. Ehaced Mote Carlo estimators for America optio prices. Joural of Derivatives, 5):25-44. Fu, M. 2007. Variace-Gamma ad Mote Carlo. I Advaces i Mathematical Fiace, ed Fu et al. Birkhauser, Bosto. Glasserma, P. 2004. Mote Carlo Methods i Fiacial Egieerig. Spriger, New York. Hirsa, A., ad D. B. Mada. 2003. Pricig America optios uder variace gamma. Joural of Computatioal Fiace, 72):63-80. Logstaff, F. A., ad E. S. Schwartz. 200. Valuig America optios by simulatio: a simple least-square approach. Review of Fiacial Studies, 4:3-47. Tsitsiklis, J., ad B. Va Roy. 999. Optimal stoppig of Markov processes: Hilbert space theory, approximatio algorithms, ad a applicatio to pricig highdimesioal fiacial derivatives. IEEE Trasactios o Automatic Cotrol, 44:840-85. AUTHOR BIOGRAPHIES GUANGWU LIU is a Ph.D. studet i the Departmet of Idustrial Egieerig ad Logistics Maagemet at The Hog Kog Uiversity of Sciece ad Techology. His research iterests iclude simulatio methodology ad fiacial egieerig. His e-mail address is <liugw@ust.hk>. L. JEFF HONG is a assistat professor i idustrial egieerig ad logistics maagemet at The Hog Kog Uiversity of Sciece ad Techology. His research iterests iclude Mote-Carlo method, sesitivity aalysis, fiacial egieerig ad simulatio optimizatio. He is curretly a associate editor of Operatios Research, Naval Research Logistics ad ACM Trasactios o Modelig ad Computer Simulatio. His e-mail address is <hogl@ust.hk>. 60