Least-squares Monte Carlo for backward SDEs

Transcription

1 Least-squares Monte Carlo for backward SDEs Chrstan Bender 1 and Jessca Stener 1 December 21, 2010 Abstract In ths paper we frst gve a revew of the least-squares Monte Carlo approach for approxmatng the soluton of backward stochastc dfferental equatons (BSDEs) frst suggested by Gobet, Lemor, and Warn (Ann. Appl. Probab., 15, 2005, ). We then propose the use of bass functons, whch form a system of martngales, and explan how the least-squares Monte Carlo scheme can be smplfed by explotng the martngale property of the bass functons. We partally compare the convergence behavor of the orgnal scheme and the scheme based on martngale bass functons, and provde several numercal examples related to opton prcng problems under dfferent nterest rates for borrowng and nvestng. Keywords: Backward SDE, numercal approxmaton, Monte Carlo, opton prcng. AMS classfcaton: 65C30, 65C05, 91G20, 91G60. 1 Introducton Many prcng and optmzaton problems n fnancal mathematcs can be reformulated n terms of backward stochastc dfferental equatons (BSDEs), see e.g. the classcal survey paper by El Karou et al. (1997). These equatons are non-antcpatng termnal value problems for stochastc dfferental equatons of the form dy t = f(t, Y t, Z t )dt + Z t dw t, Y T = ξ. Here, a D-dmensonal Brownan moton W, the square-ntegrable termnal condton ξ (measurable wth respect to the fltraton generated up to tme T by the Brownan moton) and the so-called drver f are gven. The soluton tself conssts of a par of square-ntegrable adapted processes (Y, Z), such that the correspondng ntegral equaton s satsfed. 1 Saarland Unversty, Department of Mathematcs, PO Box , D Saarbrücken, Germany. bender@math.un-sb.de, stener@math.un-sb.de. 1

2 Roughly speakng, n many prcng and hedgng problems, Y t corresponds to the opton prce and Z t s related to the hedgng portfolo. In many portfolo optmzaton problems, Y t corresponds to the value process whle an optmal control can often be derved from Z t. Fnally, BSDEs can also be appled n order to obtan Feynman-Kac-type representaton formulas for nonlnear parabolc PDEs. Here Y t and Z t correspond to the soluton and the gradent of the PDE, respectvely. Wth these applcatons n mnd, the numercal approxmaton of BSDEs becomes an mportant, but challengng problem. One branch of numercal algorthms for BSDEs explots the connecton to PDEs and bascally reduces the numercal approxmaton of the BSDE to solvng the correspondng parabolc PDE numercally, see e.g. Douglas et al. (1996); Mlsten and Tretyakov (2006); Ma et al. (2009). The practcal applcablty of these algorthms may be lmted due to hgh-dmensonalty or lack of smoothness of the coeffcents. However, for low-dmensonal problem wth smooth coeffcents the PDE methods are hard to beat. Another branch of algorthms, whch s the one we dscuss n the present paper, deals drectly wth the stochastc problem. These stochastc algorthms can typcally be decomposed nto a two-step procedure. The frst step conssts of a tme dscretzaton of the BSDE. The man dffculty here s that, on the one hand, the dscretzaton qute naturally works backwards n tme, because the termnal condton s gven. On the other hand, the numercal soluton should be adapted to the fltraton (because the true soluton s so). However, the nformaton grows forwards n tme. Ths problem can be solved by projectng the soluton on the avalable nformaton n each step whle gong backwards n tme. Whle these deas can be traced back to the papers by Bally (1997) and Chevance (1997), a detaled analyss of the correspondng tme dscretzaton scheme under qute general assumptons was frst gven by Zhang (2004) and Bouchard and Touz (2004). However, projectng on the avalable nformaton means that n each tme step a condtonal expectaton must be evaluated. Gong backwards step by step one, hence, ends up wth a hgh order nestng of condtonal expectatons. As the condtonal expectaton cannot be calculated n closed form, n a second step one has to apply an approxmaton procedure for the condtonal expectatons whch can be nested wthout runnng nto explosve computatonal costs. In ths paper we wll focus on the least-squares Monte Carlo approach for estmatng the condtonal expectatons whch was made popular n fnancal mathematcs by Longstaff and Schwartz (2001) n the context of Bermudan opton prcng. It was frst appled to BSDEs and analyzed n ths settng by Gobet et al. (2005) and Lemor et al. (2006). The basc dea here s to replace the condtonal expectatons by projectons on fnte-dmensonal subspaces whch are spanned by pre-selected bass functons. The coeffcents for the projecton on the fnte-dmensonal subspaces are approxmated by the soluton of a lnear least-squares problem makng use of smulated sample paths. 2

3 After havng dscussed the tme dscretzaton step and the least-squares Monte Carlo approach, we propose the use of bass functons, whch form a system of martngales. A smlar dea can be found n Glasserman and Yu (2002) n the context of Bermudan opton prcng. For the BSDE case the use of martngale bass functons s motvated by the followng observaton: Gong backwards n tme, one actually has to evaluate three condtonal expectatons per tme step. If the approxmaton of Y at tme t +1, say, s a lnear combnaton of bass functons and these bass functons satsfy approprate condtons related to the martngale property, then two of the condtonal expectatons can be calculated n closed form. Only one condtonal expectaton whch nvolves the nonlnearty of the drver f must stll be approxmated by least-squares Monte Carlo. Based on ths observaton we suggest a smplfed verson of the least-squares Monte Carlo algorthm, when martngale bass functons are at our dsposal. An example shows how to construct such bass functons for a mult-dmensonal Black-Scholes settng, and we pont to possble extensons for more general models. We also analyze the projecton error of the new scheme based on martngale bass functons. Fnally, we present a smulaton study for the prcng problem of a call spread opton under dfferent nterest rates for borrowng and nvestng. Here we compare the orgnal least-squares Monte Carlo scheme wth the new scheme, whch explots the use of martngale bass functons. The numercal experments contan stuatons wth a small and a larger Lpschtz constant of the nonlnearty of the drver and wth optons on a sngle stock or on the maxmum of several stocks. Overall we fnd that the use of martngale bass functons mproves on the qualty of the numercal solutons n our test example and, at the same tme, sgnfcantly reduces the smulaton costs. The paper s organzed as follows: In Secton 2 we gve a revew of the least-squares Monte Carlo scheme for BSDEs. In ths secton we also refer to varous varants concernng the tme dscretzaton and the approxmaton of the condtonal expectatons whch are avalable n the lterature. Secton 3 s devoted to the new scheme based on martngale bass functons, whle the numercal experments are dscussed n Secton 4. 2 Least-squares Monte Carlo for BSDEs In ths secton we gve a revew of the least-squares Monte Carlo approach to BSDEs ntated by Gobet et al. (2005). As t s the case for most of the numercal algorthms for BSDEs, t conssts of two steps: a tme dscretzaton and a procedure for the approxmaton of (nested) condtonal expectatons. We wll dscuss both steps separately, pontng to alternatve ways for desgnng algorthms to solve BSDEs. Before we explan the tme dscretzaton step we frst ntroduce the 3

4 standng assumptons throughout the paper. The am s to approxmate a decoupled forward backward SDE of the form dx t = b(t, X t )dt + σ(t, X t )dw t, X 0 = x 0, dy t = f(t, X t, Y t, Z t )dt + Z t dw t, Y T = g(x T ). Here W t = (W 1,t,..., W D,t ), (the star denotng matrx transposton), s a D-dmensonal Brownan moton on [0, T ] and Z t = (Z 1,t,..., Z D,t ). The process X s R M -valued and the process Y s R-valued. We assume Lpschtz contnuty of the coeffcent functons n the followng sense: Assumpton 2.1. There s a constant κ such that b(t, x) b(t, x ) + σ(t, x) σ(t, x ) + f(t, x, y, z) f(t, x, y, z ) + g(x) g(x ) κ( t t + x x + y y + z z ) for all (t, x, y, z), (t, x, y, z ) [0, T ] R M R R D. Wth ths assumpton we strve for notatonal smplcty rather than for generalty. We emphasze that, for example, path dependent termnal condtons of the form Y T = Φ(X), where the functonal Φ satsfes some sutable Lpschtz condtons on the path space, can be easly ncorporated, see Zhang (2004) or Lemor et al. (2006). 2.1 Tme dscretzaton For the tme dscretzaton we consder a partton π = {t 0,..., t N } of the nterval [0, T ],.e. 0 = t 0 < t 1 < t 2 < < t N = T. We suppose that the forward SDE s already dscretzed n a sutable way by a process Xt π, t π, such that max E[ X t t π Xt π 2 ] C π (1) for a constant C 0, and (X π t, F t ) t π s Markovan. In the numercal examples n Secton 4, X s a (mult-dmensonal) geometrc Brownan moton and can, hence, be sampled perfectly on the grd π. In general stuatons, one can e.g. apply an Euler scheme on X. We now motvate a natural tme dscretzaton for the par (Y, Z), whch works backwards n tme. Denotng = t +1 t, W d, = W d,t+1 W d,t, and W = ( W 1,,..., W D, ) for t π, we wrte Y t Y t+1 f(t, X t, Y t, Z t ) Z t W. (2) 4

5 Multplyng wth a Brownan ncrement W d, for some d = 1,..., D and takng condtonal expectaton yelds, 0 = E[ W d, (Y t + f(t, X t, Y t, Z t ) ) F t ] D E[ W d, Y t+1 F t ] E[Z l,t W l,+1 W d, F t ] l=1 = E[ W d, Y t+1 F t ] Z d,t. Ths suggests that, gven Y t+1, Z t can be approxmated as Z t 1 E[ W Y t+1 F t ]. (3) In order to obtan an approxmaton of Y t, gven Y t+1, we smply take condtonal expectaton n (2) and get Y t = E[Y t F t ] E[Y t+1 f(t, X t, Y t, Z t ) F t ] E[Y t+1 f(t, X t, Y t+1, Z t ) F t ]. (4) The last approxmaton makes the approxmaton explct n tme. The heurstcs n (2) (4) lead to the tme dscretzaton (Y π, Z π ) for (Y, Z) whch was studed by Zhang (2004) and Bouchard and Touz (2004): Y π t N = g(x π t N ), Z π t N = 0, Z π t = 1 E[ W Y π t +1 F t ], = N 1,..., 0 Y π t = E[Y π t +1 f(t, X π t, Y π t +1, Z π t ) F t ], = N 1,..., 0. (5) The results n Zhang (2004) and Bouchard and Touz (2004) (see also Lemor et al., 2006) mply that, under Assumpton 2.1, the tme dscretzaton error n the L 2 -sense s of order 1/2,.e. there s a constant C (ndependent of π) such that T sup E[ Y t Yt π 2 ] + E[ Z t Zt π 2 ] C π, (6) 0 t T 0 where (Y π t, Z π t ) s the pecewse constant nterpolaton of (5). We note that Bally (1997) and Chevance (1997) were the frst to study ths type of tme dscretzaton wth a (hardly mplementable) random tme partton respectvely under strong regularty assumptons. Although the tme dscretzaton scheme n (5) s explct n tme, each tme step requres the evaluaton of condtonal expectatons, whch leads to a hgh order nestng of condtonal expectatons. The numercal approxmaton of nested condtonal expectatons s a hghly demandng problem, n partcular when the forward SDE takes values n a hgh-dmensonal state 5

6 space. We wll dscuss some aspects related to ths ssue n the next subsecton. Before dong so, we gve some remarks concernng related results on the the tme dscretzaton of BSDEs: 1. The frst lne of (4) suggests an mplct scheme for the Y -part replacng n (5) by E[Y π t +1 f(t, X π t, Y π t +1, Z π t ) F t ] E[Y π t +1 F t ] f(t, X π t, Y π t, Z π t ). Concernng the tme dscretzaton error, the convergence of ths mplct scheme s also of order 1/2, see Bouchard and Touz (2004). It requres, however, some teraton procedure to become explct n tme. The teraton can be done n each tme step (nner teraton) as n Gobet et al. (2005) or mmckng a Pcard teraton (outer teraton) as n Bender and Denk (2007) and Gobet and Labart (2010). Bender and Denk (2007) argue that the outer teraton reduces the error propagaton when the condtonal expectatons are approxmated numercally. Gobet and Labart (2010) explan how to obtan effcent control varates for the estmaton of the condtonal expectatons n a Monte Carlo settng va the outer teraton. As an alternatve method for reducng the varance, Bender and Moseler (2010) adjust the mportance samplng technque to a BSDE settng. 2. When the termnal condton g s less regular than Lpschtz contnuous, a tme dscretzaton error of order 1/2 can stll be acheved n many cases by choosng approprate, possbly non-equdstant, parttons, see Gobet and Makhlouf (2010). Under stronger smoothness condtons on the coeffcent functons b, σ, f, g the error at tme 0 Y 0 Y0 π converges to zero at a rate of 1, see Gobet and Labart (2007) who extend a related result by Chevance (1997). For a tme dscretzaton scheme of BSDEs wth jumps under Lpschtz condtons we refer to Bouchard and Ele (2008). For coupled forward backward SDEs, Bender and Zhang (2008) provde suffcent condtons to obtan a tme dscretzaton error of order 1/2 and an teratve procedure for decouplng the equaton. The case of second order BSDEs s dscussed n Bouchard et al. (2009). 3. Some frst results on the tme dscretzaton of BSDEs wth quadratc growth of the drver f n the z-varable can be found n Imkeller et al. (2010) and Rchou (2010). Imkeller et al. (2010) apply a truncaton argument and, thus, use an approxmaton va Lpschtz drvers, whle Rchou (2010) makes use of (tme-dependent) bounds on Z t. So, from a practcal pont of vew, n both cases the stuaton s, at best, 6

7 comparable wth the Lpschtz case wth a large Lpschtz constant. However, the constant C n (6) depends exponentally on the Lpschtz constant of f. So, t s no suprse that our numercal results n Secton 4 demonstrate that even n the Lpschtz case wth a large Lpschtz constant, numercal algorthms may run nto problems. 4. For reflected BSDEs a tme dscretzaton scheme related to (5) was studed by Ma and Zhang (2005) and Bouchard and Chassagneux (2008). Ther results suggests that, n general, ths scheme only converges at a rate of 1/ Approxmaton of condtonal expectatons In order to transform the tme dscretzaton scheme n (5) nto a vable numercal scheme, the condtonal expectatons must be replaced by an approxmaton procedure whch can be nested several tmes wthout runnng nto explosve costs. Dfferent technques have been suggested n the lterature ncludng: Approxmaton of the drvng Brownan moton by trees for low-dmensonal problems, see Brand et al. (2001) and Ma et al. (2002). Cubature methods, see Crsan and Manolaraks (2010), and sparse grds methods, see Gunzburger and Zhang (2010), whch rely on some smoothness assumptons. Quantzaton methods, see Bally and Pagès (2003) for reflected BSDEs and Delarue and Menozz (2006) for coupled FBSDEs. Nonparametrc kernel estmators and Mallavn Monte Carlo, as dscussed by Bouchard and Touz (2004). Least-squares Monte Carlo, whch we wll now explan n more detal. The least-squares Monte Carlo method for approxmatng condtonal expectatons was made popular n fnancal mathematcs by the Longstaff and Schwartz (2001) algorthm for the prcng of Amercan optons. More generally, t can be appled to compute condtonal expectatons of the form E[Y X] for square ntegrable random varables X and Y numercally, provded a machnery for samplng ndependent copes of the par (X, Y ) s at hand. The method bulds upon the elementary property that E[Y X] = u(x), where the functon u solves u = arg mn v E[ v(x) Y 2 ] and v runs over all measurable functons wth E[ v(x) 2 ] <. In order to smplfy ths nfnte-dmensonal mnmzaton problem, one chooses a 7

8 row vector of so-called bass functons η(x) = (η 1 (x),..., η K (x)), for some K N, and consders the K-dmensonal mnmzaton problem α (K) = arg mn α R K E[ η(x) α Y 2 ]. In a fnal step the problem can be smplfed to a lnear least-squares problem. To ths end one just replaces the expectaton by a sample mean α (K,L) 1 = arg mn α R K L L η( λ X) α λ Y 2, λ=1 where ( λ X, λ Y ), λ = 1,..., L, are ndependent copes of (X, Y ). Gven the matrx A (K,L) = 1 (η k ( λ X)) λ=1,...,l,k=1,...,k, L one has ( 1Y α (K,L) = (A (K,L) ) A (K,L)) 1 (A (K,L) ).. LY (Here, one can apply the pseudo-nverse of A (K,L), f the nverse n the prevous expresson does not exst). The least-squares Monte Carlo estmator for the condtonal expectaton u(x) := E[Y X = x] s then gven by u (K,L) (x) := η(x) α (K,L). Clearly, ths estmaton procedure has two error sources, a systematc error nduced by the choce of bass functons and a smulaton error. Gobet et al. (2005) frst suggested the use of least-squares Monte Carlo for BSDEs and analyzed the dfferent error sources. We now descrbe the algorthm proposed by Lemor et al. (2006), whch combnes the explct tme dscretzaton scheme (5) wth least-squares Monte Carlo for estmatng the condtonal expectatons. Notce frst that, due to the Markovanty of (X π t, F t ) t π, the tme dscretzaton n (5) can be rewrtten as Y π t N = g(x π t N ), Z π t N = 0, Z π t = 1 E[ W Y π t +1 X π t ], = N 1,..., 0 Y π t = E[Y π t +1 f(t, X π t, Y π t +1, Z π t ) X π t ], = N 1,..., 0. (7) Hence, there are functons y π(x) and zπ (x) such that Y π t = y π (X π t ), Z π t = z π (X π t ). These functons (y π(x), zπ (x)) are estmated recursvely by least-squares Monte Carlo. To ths end one chooses bass functons η 0 (, x) = (η 0,1 (, x),..., η 0,K (, x)) 8

9 for the estmaton of y π (x), and η d (, x) = (η d,1 (, x),..., η d,k (, x)), d = 1,..., D, for the estmaton of the dth component zd, π (x) of zπ (x). In prncple, the number of bass functons can be dfferent for each tme step and for the y- and z-part, whch we suppress for smplcty. Then, gven L ndependent copes ( λ W, λ Xt π +1 ) =0,...,N 1, λ = 1,..., L, of ( W, Xt π +1 ) =0,...,N 1, we defne ỹ π,k,l N (x) = g(x), z π,k,l N = 0, α π,k,l 1 d, = arg mn α R K L L ( λ=1 η d (, λ Xt π ) α λw d, ỹ π,k,l +1 ( λ Xt π +1 ) z π,k,l d, (x) = η d (, x) α π,k,l d,, d = 1,..., D; = N 1,..., 0, α π,k,l 1 0, = arg mn α R K L L ( η 0 (, λ Xt π ) α ỹ π,k,l +1 ( λ Xt π +1 ) λ=1 +f(t, λ X π t, ỹ π,k,l +1 ( λ X π t +1 ), z π,k,l ( λ X π t )) ) 2 ỹ π,k,l (x) = η 0 (, x) α π,k,l 0,, = N 1,..., 0. (8) Once the bass functons are chosen and the sample paths are generated, the algorthm s straghtforward to mplement, as t only requres to solve some lnear least-squares problems numercally. The L 2 -error between (ỹ π,k,l (x), z π,k,l d, (x)) and (y π(x), zπ (x)) wth respect to the law of Xt π has been analyzed by Lemor et al. (2006), Theorem 2 and Remark 1, for a sutably truncated scheme. The complete error analyss s rather techncal, partcularly because the use of the same smulated paths for estmatng all condtonal expectatons nduces a somewhat complcated dependency structure. We now roughly explan the nfluence of the dfferent error sources, but refer the nterested reader to the orgnal paper by Lemor et al. (2006) for the very detals. In order to smplfy the presentaton, we assume that the partton π of [0, T ] s equdstant wth (N + 1) tme ponts: 1. The tme dscretzaton error decreases at a rate of N 1/2, see (6). 2. The projecton error s nduced by choosng the bass functons. The squared projecton error can be bounded by a constant tmes N 1 =0 nf E[ Y π α R K t η 0 (, Xt π ) α 2 ]+ d=1 ) 2 D nf E[ Z π α R K d,t η d (, Xt π ) α 2 ]. (9) Notce, that ths expresson s the sum of the squared dstance between the tme dscretzed soluton (Y π t, Z π t ) and ts best projecton on 9

10 the bass functons. The tme dscretzed soluton and ts best projecton are both not avalable n closed form (but for trval cases). So ths error bound s stll dffcult to quantfy except for some specal classes of bass functons such as ndcator functons of hypercubes whch form a partton of the state space of X, see Gobet et al. (2005). Recall that throughout the algorthm condtonal expectatons of the form E[Yt π +1 Xt π ] are approxmated recursvely for = N 1,..., 0. The approxmaton errors n the dfferent tme steps may sum up n the worst case, whch explans the sum over tme of the projecton errors. 3. We fnally dscuss the smulaton error. The results by Lemor et al. (2006) mply that t can be bounded n terms of the number of tme ponts N (up to logarthmc factors) by N ρ/2 for ρ [0, 1], f the number of bass functons K ncreases proportonal to N δ, δ 0, and the number of smulated paths L ncreases proportonal to N 2+2δ+ρ. Here the worst contrbuton stems from estmatng the condtonal expectaton E[ W Yt π +1 Xt π ] for the Z-part, because the varance blows up when the tme partton becomes fner due to the factor W. To sum up, a fner tme partton requres a better choce of the bass functons (typcally a sgnfcant ncrease n the number of bass functons), whch n turn leads to a larger number of smulated paths. We note that the number of smulated paths must grow polynomally n the number of bass functons, whle even an exponental growth of sample paths s necessary for the Longstaff-Schwartz algorthm for prcng Amercan optons, see Glasserman and Yu (2004). Nonetheless our numercal study n Secton 4 wll exhbt some lmtatons of the algorthm, when a fne tme grd s requred. 3 Martngale bass functons In ths secton we propose the use of bass functons, whch form a system of martngales. Ths approach s n the sprt of Glasserman and Yu (2002) who appled martngale bass functons for computng dual upper bounds for Amercan optons. We frst motvate the martngale bass approach. Takng another look at the tme dscretzaton scheme (7), we notce that three condtonal expectatons must be approxmated n each tme step, 10

11 namely [ ] W E Yt π +1 Xt π, (10) E[Yt π +1 Xt π ], (11) E[f(t, Xt π, Yt π +1, Zt π ) Xt π ]. (12) We have observed n the prevous secton that estmatng the condtonal expectaton n (10), whch s related to the Z-part of the soluton, s the domnant term for choosng the number of smulated paths n order to deal wth the ncreasng varance of W. Moreover, we have seen that estmatng the condtonal expectaton n (11) leads to an unfortunate propagaton n tme of the projecton error. So, estmatng the condtonal expectaton n (12) appears to be numercally the easest of the three estmaton problems, partcularly as the multplcaton wth the tme step s expected to reduce the error. Hence, our am s to choose the bass functons n such a way that the condtonal expectatons n (10) and (11) can be computed n closed form, when Yt π +1 s replaced by a lnear combnaton of bass functons. To fx the deas, let us assume that, at tme t +1, an approxmaton ŷ π,k,l +1 (Xt π +1 ) of Yt π +1 = y+1 π (Xπ t +1 ) s already constructed and ŷ π,k,l +1 (x) s a lnear combnaton of bass functons,.e. K ŷ π,k,l +1 (x) = β k η 0,k ( + 1, x) k=1 for some β 1,..., β K R. If the bass functons form martngales n the followng sense E[η 0,k ( + 1, X π t +1 ) X π t = x] = η 0,k (, x), we can compute the condtonal expectaton of type (11) n closed form: E[ŷ π,k,l +1 (X π t +1 ) X π t ] = K β k η 0,k (, Xt π ). Smlar consderatons for the condtonal expectaton of type (10) then lead to the followng assumpton on the bass choce. Assumpton 3.1. We choose, at tme t N = T, a row vector of K bass functons η 0 (N, x) = (η 0,1 (N, x),..., η 0,K (N, x)). Then, we defne the bass functons η d (, x) = (η d,1 (, x),..., η d,k (, x)), d = 0,..., D, at the earler tme steps = 0,..., N 1 va the condtonal expectatons η 0,k (, x) = E[η 0,k (N, Xt π N ) Xt π = x] (13) [ ] Wd, η d,k (, x) = E η 0,k (N, Xt π N ) Xπ t = x, d = 1,..., D, (14) 11 k=1

12 whch we assume to be computable n closed form. The termnology martngale bass functons refers to the settng of Assumpton 3.1. Note, that by the tower property of the condtonal expectatons, we have η 0,k (, x) = E[η 0,k ( + 1, Xt π +1 ) Xt π = x], (15) [ ] Wd, η d,k (, x) = E η 0,k ( + 1, Xt π +1 ) Xπ t = x, d = 1,..., D.(16) Before we provde some examples for martngale bass functons, we frst explan how the least-squares Monte Carlo algorthm for BSDEs can be smplfed, when a set of martngale bass functons s avalable. The modfed algorthm explots propertes (15) (16). If, for the termnal condton g, the condtonal expectatons [ ] E[g(Xt π N ) Xt π Wd, = x], E g(xt π N ) Xπ t = x are avalable n closed form, one, of course, adds g to the martngale bass. Otherwse an ntalzaton step at tme t N = T s requred n order to approxmate the termnal condton g by a lnear combnaton of bass functons. Such approxmaton can e.g. be done by a least-squares Monte Carlo projecton of g on the bass: β π,k,l N 1 = arg mn β R K L L ( 2, η 0 (N, λ Xt π N ) β g( λ Xt π N )) λ=1 where here and n the followng the averagng s agan over ndependent sample copes ( λ W, λ X t+1 ) =0,...,N 1, λ = 1,..., L, of ( W, X t+1 ) =0,...,N 1. In any case, we suppose that a vector β π,k,l N R K has been chosen and η 0 (N, x) β π,k,l N s nterpreted as an approxmaton of g(x). Gven β π,k,l N the modfed algorthm computes, for = N 1,..., 0, ŷ π,k,l +1 (x) = η 0 ( + 1, x) β π,k,l +1 ẑ π,k,l d, (x) = η d (, x) β π,k,l +1, d = 1,..., D, β π,k,l 1 = arg mn β R K L L ( η 0 (, λ Xt π ) β λ=1 +f(t, λ X π t, ŷ π,k,l +1 ( λ X π t +1 ), ẑ π,k,l ( λ X π t )) ) 2 β π,k,l = β π,k,l π,k,l +1 + β. (17) The algorthm termnates at tme t = 0 wth ŷ π,k,l 0 (x) = η 0 (0, x) β π,k,l 0. 12

13 The fnal approxmaton for (Yt π, Zt π ) s gven by (ŷ π,k,l (Xt π ), ẑ π,k,l (Xt π )). We emphasze that n the modfed algorthm, by employng propertes (15) (16) of the martngale bass functons, only the condtonal expectaton of type (12) s approxmated by least-squares Monte Carlo. We now gve some examples for bass functons whch can be ncluded nto martngale bases, when the forward SDE s a (mult-dmensonal) geometrc Brownan moton. Ths stuaton corresponds to the numercal examples n Secton 4. Example 3.2. Suppose we are gven D Black-Scholes stocks, whch are for smplcty assumed to be ndependent and dentcally dstrbuted,.e. X d,t = x 0 exp{(µ σ 2 /2)t + σw d,t }, d = 1,..., D, where x 0, σ > 0 and µ R. Here, X can be sampled perfectly, and we hence wrte X nstead of X π. The martngale bass functons whch we apply for the numercal examples below are bult from ndcator functons of hypercubes, monomals, and the payoff functon of a max-call opton. For the ndcator functons of the form η a,b := 1 [a,b] = 1 [a1,b 1 ] [a D,b D ] one easly calculates, E[η a,b (X T ) X t = x] = D E[1 [ad,b d ](X d,t ) X d,t = x d ] = d=1 D N(ã d ) N( b d ), where N s the cumulatve dstrbuton functon of a standard normal and for y = a, b ỹ d = log(y d/x d ) (µ 0.5σ 2 )(T t ) σ T t. For monomals η p (x) := x p 1 1 xp D D E[η p (X T ) X t = x] = D d=1 one has d=1 x p d d exp{(p dµ + 0.5p d (p d 1)σ 2 )(T t )}. For the payoff functons of a max-call opton η K (x) = (max d=1,...,d x d K) +, t can be derved from the results by Johnson (1987) that E[η K (X T ) X t = x] = D e µ(t t) x d N 0,Σ (a d,+ ) d=1 K(1 D d=1 N( log(k/x d) (µ 0.5σ 2 )(T t ) σ T t )), 13

14 where N 0,Σ s the dstrbuton functon of a D-varate normal wth mean vector 0 and covarance matrx Σ. Moreover, log(x d /K) + (µ + 0.5σ 2 )(T t ) 1 (log(x 1 a d,+ = σ d /x 2 d) + σ 2 (T t )) T t, d = 1,..., D, d d,. 1 (log(x d /x D ) + σ 2 (T t )) 2 and Σ = 1 1/ 2 1/ 2 1/ 2 1/ 2 1 1/2 1/2 1/ 2 1/2 1 1/ / 2 1/2 1/2 1 Hence, for such functons the condtonal expectatons requred n (13) are avalable. Concernng the condtonal expectatons of the form (14), we assume that η(x) s a functon such that η 0 (, x) := E[η(X t ) X t = x] can be computed. Under approprate growth condtons (whch allow to ntroduce the dervatves below under the ntegral sgn), we get for d = 1,..., D and < N, η d (, x) := E ] η(xt π N ) Xπ t = x [ Wd, = σx d x d η 0 (, x). (18) Indeed, for the one-dmensonal case (D = 1) one easly computes σx d dx η 0(, x) = σx d dx E [ η 0 ( + 1, X t+1 ) X t = x ] 1 = σx e u 2 d 2 2π dx η 0( + 1, xe σu+(µ 0.5σ2 ) )du = = 1 2π 1 2π e u π d du η 0( + 1, xe σu+(µ 0.5σ2 ) )du η 0 ( + 1, xe σu+(µ 0.5σ2 ) ) d ( ) e u 2 2 du du η 0 ( + 1, xe σu+(µ 0.5σ2 ) ) u e u 2 2 du ] = [ W = E η 0 ( + 1, X t+1 ) X t = x [ W = E ] η(xt π N ) Xπ t = x. The mult-dmensonal case can be treated analogously. Usng formula (18) we can then calculate the condtonal expectatons (14) for e.g. the ndcator functons, monomals, and the call payoff. 14

15 Remark 3.3. The above example s, admttedly, somewhat smplstc. We note, however, that for more sophstcated models, good closed-form approxmatons for many European opton prces and ther deltas are often avalable. These can be appled to bult bass functons n the sprt of the prevous example, whch at least approxmately ft nto the martngale bass settng. We now study the projecton error,.e. the error nduced by choosng the bass functons, n the settng of martngale bass functons. In order to separate ths error from the smulaton error, we now assume that the orthogonal projectons on the bass can be computed n closed form. Hence, we defne β π,k N and for = N 1,..., 0, [ 2, = arg mn E η 0 (N, X π β R K t N ) β g(xt π N )] ŷ π,k +1 (x) = η 0( + 1, x) β π,k +1 ẑ π,k d, (x) = η d (, x) β π,k +1, d = 1,..., D, [ β π,k = arg mn E η 0 (, X π β R K t ) β At tme t = 0 we set +f(t, X π t, ŷ π,k +1 (Xπ t +1 ), ẑ π,k (X π t )) ] 2 β π,k = β π,k π,k +1 + β. (19) ŷ π,k 0 (x) = η 0 (0, x) β π,k 0. Theorem 3.4. Under Assumptons 2.1 and 3.1, there s a constant C such that N 1 max E[ Y t π 0 N (Xt π ) 2 ] + E[ Zt π ẑ π,k (Xt π ) 2 ] =0 ( C nf E[ η 0(N, X π β R K t N ) β g(xt π N ) 2 ] N 1 + =0 nf E[ η 0(, X π β R K t ) β E[f(t, Xt π, Yt π +1, Zt π ) Xt π ] 2 ] The proof wll be postponed to the Appendx. ).(20) Remark 3.5. Recall that the frst term on the rght hand sde of (20) vanshes, when the termnal condton g can be added to the martngale bass. The remanng error term averages over tme the squared projecton errors between E[f(t, Xt π, Yt π +1, Zt π ) Xt π ] and ts best projecton on the bass. So here we do not observe the unfavorable error propagaton over tme, whch we found n the upper bound for the projecton error of the orgnal scheme n (9). 15

16 Remark 3.6. We notce that, by a straghtforward applcaton of the law of large numbers, the smulaton error n the martngale bass settng converges to zero, as the number of smulated paths L tends to nfnty. A prelmnary error analyss for a sutably truncated scheme suggests, that the smulaton error converges at N ρ/2 for ρ [0, 1] (N the number of tme steps n an equdstant partton), f the number of bass functons K ncreases proportonal to N δ, δ 0, and the number of smulated paths L ncreases proportonal to N 2+δ+ρ (compared to N 2+2δ+ρ n the orgnal scheme). A detaled analyss s, however, beyond the scope of ths paper. 4 Numercal experments 4.1 The test example We now ntroduce the test example for our numercal experment, whch s the prcng problem of a call spread opton under dfferent nterest rates. Actually, ths example s taken from Lemor et al. (2006) and hence allows for a comparson wth ther results. We shall also consder some varatons of ths example n order to study the nfluence of larger Lpschtz constants and mult-dmensonal stuatons. Suppose we are gven a market wth D rsky assets X t, whch are modeled by Black-Scholes stocks. For smplcty we assume that the D stocks are ndependent and dentcally dstrbuted,.e. X d,t = x 0 exp{(µ σ 2 /2)t + σw d,t }, d = 1,..., D, where W t = (W 1,t,..., W D,t ) s a D-dmensonal Brownan moton and x 0, σ, µ > 0. The trader can also nvest nto a rskless bond wth rate r 0 for nvestng and rate R r for borrowng from the bond. Our am s to prce a call spread opton on the maxmum of the stocks, whch here s assumed to be of the form ( ) ( ) ξ = max X d,t K 1 2 max X d,t K 2 d=1,...,d d=1,...,d + for constants K 1, K 2 > 0. Followng Lemor et al. (2006) we choose the constants x 0 = 100, µ = 0.05, σ = 0.2, T = 0.25, r = 0.01, K 1 = 95, K 2 = 105. As nterest rate for borrowng we choose R = 0.06 for the economcally sensble case wth a small Lpschtz constant. In order to test the algorthms n a stuaton wth larger Lpschtz constant we shall also consder the case R = We run ths problem for the one-dmensonal case (D = 1), where the opton reduces to a call spread opton on a sngle stock, and for the three-dmensonal problem (D = 3). + 16

17 It follows from results by Bergman (1995) that ths opton prcng problem under dfferent nterest rates can be formulated n terms of a BSDE by Y t = ξ T D d=1 t T t ( (µ r) ry s + σ Z d,s dw d,s. D Z d,s (R r) ( Y s σ 1 d=1 D Z d,s ) d=1 ) ds Note that n the case of a vanlla call opton, the nvestor s bound to perpetually borrow money n order to hedge the opton. Hence the closed-form soluton for such opton s gven by the hedgng problem n a standard Black- Scholes settng wth a bank account gven by e Rt. Contrarly, for the call spread opton case the problem s truly nonlnear and the soluton (Y, Z) of the BSDE s not avalable n closed form. Therefore we requre a tool to measure the performance of the numercal algorthm. We here stck to an error crteron suggested and studed n Bender and Stener (2010). We now explan the dea n the general settng of the present paper. Let us suppose that some approxmaton (ŷ π(x), ẑπ (x)) of (yπ (x), zπ (x)) for every t π was computed by some numercal scheme. In the examples we consder the approxmatons obtaned by the least-squares Monte Carlo scheme (ỹ π,k,l (x), z π,k,l (x)) n (8) and by the martngale bass scheme (ŷ π,k,l (x), ŷ π,k,l (x)) n (17). Gven a generc approxmaton (ŷ π(x), ẑπ (x)), we set (Ŷ π t, Ẑπ t ) = (ŷ π (X π t ), ẑ π (X π t )) and defne (Ŷ π t, Ẑπ t ), t [0, T ], by pecewse constant nterpolaton. Then we consder as an error crteron E π (ŷ π, ẑ π ) := E[ g(x π t N ) Ŷ π t N 2 ] + max E[ Ŷ t π 0 N Ŷ π 1 t 0 j=0 f(t j, Xt π j, Ŷ 1 t π j, Ẑπ t j ) j Ẑt π j W j 2 ]. We emphasze that ths crteron does only depend on the numercal soluton (ŷ π t (x), ẑ π t (x)) and, thus, can be consstently estmated by a plan Monte Carlo approach. The second term on the rght hand sde measures, whether the approxmatve soluton s close to solvng the SDE (run as a forward SDE). The frst term on the rght hand sde measures how well t fts to the termnal condton. So, n a sense, we check how close the approxmatve soluton s to solvng the BSDE, whle we are actually nterested n how close t s to the true soluton of the BSDE. On the one hand, the error crteron s of some nterest quanttatvely due to ts smple and meanngful nterpretaton. Moreover, t s ntutvely j=0 17

18 clear that beng close to solvng the BSDE s necessary for beng close to the soluton of the BSDE. On the other hand, the crteron s also of nterest qualtatvely, because there are constants c 1, c 2, C 0 such that for suffcently fne parttons π c 1 E π (ŷ π, ẑ π ) c 2 π (21) sup E[ Y t Ŷ t π 2 ] + t [0,T ] T 0 E[ Z t Ẑπ t 2 ]dt C (E π (ŷ π, ẑ π ) + π ), (22) see Bender and Stener (2010). Ths means that the square root of the error crteron s up to terms of order 1/2 n the mesh sze of the partton equvalent to the L 2 -error between approxmaton and true soluton. We also emphasze that the constant c 2 can be taken as 0, when the drver f(t, x, y, z) does not depend on (t, x) whch s the case n our opton prcng example. Thus, n such stuaton, we arrve at the mproved lower bound T c 1 E π (ŷ π, ẑ π ) sup E[ Y t Ŷ t π 2 ] + E[ Z t Ẑπ t 2 ]dt. (23) t [0,T ] 0 Remark 4.1. Note that we cannot expect that the squared L 2 -error T sup E[ Y t Ŷ t π 2 ] + E[ Z t Ẑπ t 2 ]dt t [0,T ] 0 converges to zero faster than at the order π, because ths error typcally corresponds to the L 2 -regularty n t of the soluton Y t and so perssts, even f Ŷ t π concdes wth Y t on the grd π. So, by lookng at the error crteron, we are manly amng to judge whether the way, n whch the estmator for the condtonal expectaton s desgned n dependence of the mesh of the partton, retans the convergence rate of order 1/2 n the mesh or not. The error crteron decreases more slowly than π n the latter case. 4.2 Numercal results Case 1: Small Lpschtz constant We frst consder the one-dmensonal case (D = 1) and set R = In ths case, the nonlnearty has a rather small Lpschtz constant of (R r)/σ = Concernng the tme dscretzaton we apply an equdstant partton wth N tme steps. For the orgnal least-squares Monte Carlo scheme we choose as bass functons the payoff functon of the call spread and, followng Lemor et al. (2006), ndcator functons of K equdstant ntervals whch form a partton of the doman [40, 180]. For the scheme based on martngale bass functons, we also use the payoff functon and the same number of ndcator functons at termnal tme t N = T, and then the 18

19 bass functons at the other tme steps are computed by formulas (13) and (14). However, the ntervals for the ndcator functon are not chosen n an equdstant way, but such that X T hts each nterval wth equal probablty. For dfferent values β, γ > 0, we choose n dependence of ν N N = [ 2 2 (ν 1)], K = [ 14 (β+1)(ν 1)/2 2 5 ] + 2, L = [ 2 2 γ(ν 1)]. Table 1 shows the numercal approxmatons for the prce Y 0 of the call spread opton under dfferent nterest rates for borrowng and nvestng. Here, LSM stands for the orgnal least-squares Monte Carlo scheme by Lemor et al. (2006) and MBF stands for the use of martngale bass functons. Y 0 N β γ type 3 LSM LSM LSM MBF ,96 2,96 2,96 2,96 3 LSM LSM MBF MBF Table 1: Numercal prce Y 0 of the call spread opton For all varatons of the two algorthms the numercal prces converge to values around Overall, the convergence of the MBF-algorthm appears to be faster than for the LSM-algorthm. Moreover, n ths example n the MBF-algorthm a faster ncrease of the number of bass functons (β = 1 vs. β = 0.5) and a faster ncrease of the number of sample paths (γ = 3 vs. γ = 2) does not sgnfcantly change the numercal results. Contrarly, for the LSM-algorthm, the values for Y 0 are mproved by ncreasng β and γ. We emphasze that the choce of the parameters β and γ may drastcally change the computatonal effort. For nstance, for N = 45 and γ = 5, about 12 mllon paths must be smulated, whle for N = 45 and γ = 2 only 1024 paths are requred. In order to derve nformaton about the qualty of the whole numercal soluton (Y -part and Z-part at all tme ponts) and not only about the Y 0 -value, we plot the error crteron, whch we motvated n the prevous subsecton. Fgure 1 llustrates the error crteron (on a logarthmc scale) for β = 1, whch s estmated usng a new sample of L ndependent paths. In ths case, the projecton error n the LSM-scheme theoretcally converges at order 1/2 n the number of tme steps N. In order to get the same theoretcal convergence rate (up to logarthmc factors) for the smulaton error, γ = 5 s requred. γ = 4 s the theoretcal threshold for convergence, whle for γ = 3 convergence of the smulaton error s not supported by the theoretcal analyss n Lemor et al. (2006). The error crteron s smaller for a larger number of sample paths (.e. larger values of γ), whch ndcates that 19

20 γ = 3, LSM γ = 4, LSM γ = 5, LSM γ = 3, MBF Number of tmesteps N per partton, N = 3,..., 181 Fgure 1: Error crteron for β = 1 the larger computatonal effort mproves on how close the numercal soluton s to solvng the BSDE. Somewhat surprsngly, the dfference between the cases γ = 5 and γ = 4 s rather small and for both values of γ a convergence of the LSM-scheme at order 1/2 n the number of tme steps s ndcated by the error crteron. For γ = 3 the error crteron s sgnfcantly larger. Here t s less obvous, whether the LSM-scheme wth β = 3 converges at all, but defntely t does not seem to converge at the same speed as γ = 4, 5. For the MBF-algorthm we observe, that the error crteron s sgnfcantly lower wth γ = 3 than t s for the LSM-scheme wth γ = 5. The slope of the lne of about suggests that the MBF-algorthm wth γ = 3 converges almost at rate of 1/2. We recall that t s hardly possble to run the LSM-algorthm wth γ 4 for larger values than N = 64 (and hence to further decrease the error) n an acceptable tme due to the tremendous smulaton costs. Fgure 2 shows the error crteron for the case β = 0.5. Here, for the LSM-algorthm, the projecton error theoretcally decreases as N 1/4, and so does the smulaton error (up to logarthmc factors) for γ = 4. The theoretcal convergence threshold for the smulaton error s γ = 3. A look at the error crteron ndcates that the LSM-algorthm for γ = 2 does not seem to converge n accordance wth the theoretcal error bounds. For γ = 3 and γ = 4, the error crteron only slghtly dffers. The slope of the lnes s about -0.9 n both cases, whch corresponds to a rate of about Ths suggests that, n practce, the worst case error propagaton backwards n tme, whch s reflected n theoretcal rate 1/4, s not present. Agan, for the 20

21 γ = 2, LSM γ = 3, LSM γ = 4, LSM γ = 2, MBF γ = 3, MBF Number of tmesteps N per partton, N = 2,..., 181 Fgure 2: Error crteron for β = 0.5 MBF-scheme the error crteron s overall smaller and the scheme converges wth lower smulaton costs at γ = 2. Indeed, the addtonal smulaton effort for γ = 3 does not mprove the convergence behavor of the MBF-scheme. The slope s at dentcal to the case β = 1. In summary, n ths example we fnd that usng martngale bass functons leads to sgnfcant mprovements of the numercal approxmatons of the whole soluton of the BSDE. Moreover, the mproved numercal solutons are computed wth drastcally less smulaton effort. Case 2: Large Lpschtz constant We now test the algorthms n a stuaton wth a larger Lpschtz constant, but stll n the one-dmensonal case. As the Lpschtz constant of f enters exponentally n some of the error estmates, we expect that the numercal algorthms may run nto dffcultes. We set R = Hence, the nonlnearty n f has as Lpschtz constant (R r)/σ = 15. Of course, from the pont of vew of the fnancal applcaton an nterest rate of 301% s not relevant, but we beleve that t s mportant to test the algorthms n some extreme stuatons as well. Moreover, as R tends to nfnty, the prce of the call spread opton under dfferent nterest rates converges to the superhedgng prce under the no-borrowng constrant, see e.g. Bender and Kohlmann (2008). So the case of a large rate R for borrowng may stll be of some nterest from a fnancal pont of vew. We note that the superhedgng prce under the no-borrowng constrant can be computed analytcally for 21

22 the call spread opton by applyng the technques developed by Broade et al. (1998). It s 7.18 and serves as an upper bound for our test BSDE, n whch we use the same specfcaton for the number of tme steps, the bass functons, and the number of sample paths as n the prevous example. Y 0 N β γ type 4 LSM LSM MBF MBF Table 2: Numercal value of Y 0 for the case wth hgher Lpschtz constant Table 2 dsplays the numercal approxmatons for Y 0 calculated wth the LSM-algorthm and the MBF-algorthm. On the one hand, for the LSMalgorthm no convergence pattern can be observed for γ = 4 and N up to 64 and γ = 5 and N up to 45. As n the latter case (γ = 5) the algorthm theoretcally converges at a rate of 1/2 n the number of tme steps N, we conclude that larger values of N are requred. As the number of sample paths also ncreases as N γ, large values of N become, however, numercally untractable. Recall that N = 45 and γ = 5 already leads to 12 mllon sample paths. Nonetheless, the somewhat wld fluctuatons n the estmated Y 0 -values suggest that even larger number of sample paths cannot be avoded n the LSM-algorthm for ths example. On the other hand, for the MBFalgorthm the pattern of the estmated Y 0 -values apparently converges for γ = 2 and γ = 3. Convergence s not yet acheved for N = 181, but t seems plausble that Y 0 s about A look at the error crteron, whch s plotted n Fgure 3, confrms these observatons. The LSM-algorthm s seen not to be n the range of convergence for the gven values of N. For the MBF-algorthm we frst note that the observed convergence behavor does not really dffer for the cases γ = 2 and γ = 3. So, agan, the use of more sample paths than for γ = 2 does not appear to be necessary for ths scheme. It s nterestng that the error crteron for the MBF-algorthm s comparable n absolute values to the case of the small Lpschtz constant for N 16. Ths example demonstrates that calculatng some of the condtonal expectatons n closed form by usng martngale bass functons stablzes the algorthm. Hence the new algorthm based on martngale bass functons can compute reasonable approxmatons for the soluton of the BSDE n stuatons, where the orgnal algorthm already breaks down due to the large Lpschtz constant of the nonlnearty. Case 3: Three-dmensonal case We fnally return to the case of the small Lpschtz constant,.e. the rate for borrowng R s agan set to 6%, but we now prce a call spread opton on the maxmum of three stocks (D = 3). In the prevous examples 22

23 γ = 4, LSM γ = 5, LSM γ = 2, MBF γ = 3, MBF Number of tmesteps N per partton, N = 2,..., 181 Fgure 3: Error crteron for β = 1 and the case of a larger Lpschtz constant the number of bass functons was ncreased wth the number of tme steps N, n order to make the projecton error converge as N tends to nfnty. In ths example we test the use of a small number of bass functons. Here we take as bass functons the constant 1, the three frst-order monomals, and the payoff functon of the max-call-spread for the orgnal least-squares approach. For the MBF-algorthm, the bass functons are only specfed ths way at termnal tme and are the computed by formulas (13) and (14) at the other tme ponts. Fxng a fnte number of bass functons automatcally ntroduces a bas to the numercal scheme whch cannot be removed, but ths procedure corresponds to what s usually done n Bermudan opton prcng by the Longstaff-Schwartz algorthm. For the number of tme steps and the number of sample paths we use the same specfcatons as before. Y 0 N β γ type 4 LSM LSM MBF MBF Table 3: Prce Y 0 of the 3-dmensonal max-call-spread The numercal prces for the max-call-spread on three stocks under dfferent nterest rates are shown n Table 3. Here the values of the LSM-algorthm and the MBF-algorthm converge to smlar but slghtly dfferent values. In both cases the number of smulated paths (γ = 4 vs. γ = 5 for the LSM- 23

24 10 4 γ = 4, LSM γ = 5, LSM γ = 2, MBF γ = 3, MBF Number of tmesteps N per partton, N = 2,..., 45 Fgure 4: Error crteron for the 3-dmensonal max-call-spread algorthm, γ = 2 vs. γ = 3 for the MBF-algorthm), does not sgnfcantly change the convergence pattern. We now look at the error crteron for ths example (Fgure 4). It shows that the smple bass consstng of the payoff functon and some monomals s clearly napproprate to recover the whole soluton of the BSDE numercally. Indeed, for the LSM-scheme the error crteron stays roughly constant for N 11 at a level larger than 10. Ths clearly ndcates that the error arsng from the choce of the small bass domnates the tme dscretzaton error and the smulaton error, whch both converge lke N 1/2. In the MBFscheme the bass functons computed from the payoff functon correspond to the prce of the European opton (wthout dfferent nterest rates) and to the deltas and are therefore automatcally constructed n a more problemspecfc way. We observe that for the MBF-algorthm and N 45 the error crteron corresponds to a decrease of the error at order 1/2. Ths ndcates that the projecton error s stll domnated by the tme dscretzaton error and the smulaton error for ths range of N. We dd not try larger values for N, but of course the projecton error wll be domnant for suffcently large N. The key observaton, whch we make here, s that also for multdmensonal problems a reasonable approxmaton of the whole soluton of the BSDE may stll be possble wth only a few relevant bass functons, n partcular when one can addtonally explot the fact that some of the condtonal expectatons can be computed n closed form by usng martngale bass functons. 24

25 To conclude, n our numercal examples we fnd that the use of martngale bass functons yelds sgnfcantly better numercal approxmatons at a much lower computatonal cost compared to the orgnal least-squares Monte Carlo scheme. However, the new algorthm s less generc, because the constructon of martngale bass functons depends on the law of X and restrcts the choce of bass functons. So, we fnally recommend to explot the advantages of martngale bass functons when a good set of such functons s avalable. A Proof of Theorem 3.4 Throughout the proof, C denotes a generc constant, whch may vary from lne to lne. In order to smplfy the notaton, and wthout any real loss of generalty, we restrct ourselves to the case D = 1. We also make use of the followng abbrevatons: f π := f(t, X π t, Y π t +1, Z π t ), f π,k := f(t, X π t, ŷ π,k t +1 (X π t ), ẑ π,k t (X π t )). Furthermore, P K, = 0,... N, denotes the orthogonal projecton on the lnear span of {η 0,1 (, Xt π ),..., η 0,K (, Xt π )} as a subspace of L 2 (P ). Then we obtan by the defntons n (5) and (19) and Young s nequalty fo every γ > 0 E[ Y π t (Xt π ) 2 ] (1 + γ )E[ E[Yt π (Xπ t +1 ) Xt π ] 2 ] + (1 + γ ) γ E[ P K (f π,k ) E[f π Xt π ] 2 ] = (I) + (II). as well as the Lps- The orthogonalty and the contracton property of P K chtz condton of f and the defnton of P K yeld (II) = (1 + γ ) γ E[ P K (1 + γ ) 2 κ 2 + (1 + γ ) γ γ (f π,k ) P K (f π ) 2 + P K (f π ) E[f π Xt π ] 2 ] E[ Y π t (Xπ t +1 ) 2 + Z π t ẑ π,k (X π t ) 2 ] nf E[ η 0(, X π β R K t )β E[f π Xt π ] 2 ]. (24) 25