An Analysis of a Least Squares Regression Method for American Option Pricing

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1 An Analyss of a Least Squares Regresson Method for Amercan Opton Prcng Emmanuelle Clément Damen Lamberton Phlp Protter Revsed verson, December 200 Abstract Recently, varous authors proposed Monte-Carlo methods for the computaton of Amercan opton prces, based on least squares regresson. The purpose of ths paper s to analyze an algorthm due to Longstaff and Schwartz. Ths algorthm nvolves two types of approxmaton. Approxmaton one: replace the condtonal expectatons n the dynamc programmng prncple by proectons on a fnte set of functons. Approxmaton two: use Monte-Carlo smulatons and least squares regresson to compute the value functon of approxmaton one. Under farly general condtons, we prove the almost sure convergence of the complete algorthm. We also determne the rate of convergence of approxmaton two and prove that ts normalzed error s asymptotcally Gaussan. KEY WORDS: Amercan optons, optmal stoppng, Monte-Carlo methods, least squares regresson. AMS Classfcaton: 90A09, 93E20, 60G40. Introducton The computaton of Amercan opton prces s a challengng problem, especally when several underlyng assets are nvolved. The mathematcal problem to solve s an optmal stoppng Équpe d Analyse et de mathématques applquées, Unversté de Marne-la-Vallée, 5 Bld Descartes, Champs-sur-marne, Marne-la-Vallée Cedex 2, France. Operatons Research and Industral Engneerng Department, Cornell Unversty, Ithaca, Y ,USA; Supported n part by SF Grant #DMS and SA Grant #MDA

2 problem. In classcal dffuson models, ths problem s assocated wth a varatonal nequalty, for whch, n hgher dmensons, classcal PDE methods are neffectve. Recently, varous authors ntroduced numercal methods based on Monte-Carlo technques see, among others, [, 2, 3, 4, 5, 9, 2]. The startng pont of these methods s to replace the tme nterval of exercse dates by a fnte subset. Ths amounts to approxmatng the Amercan opton by a so called Bermudan opton. A control of the error caused by ths restrcton to dscrete stoppng tmes s generally easy to obtan see, for nstance, [8], Remark.4. Throughout the paper, we concentrate on the dscrete tme problem. The soluton of the dscrete optmal stoppng problem reduces to an effectve mplementaton of the dynamc programmng prncple. The condtonal expectatons nvolved n the teratons of dynamc programmng cause the man dffculty for the development of Monte- Carlo technques. One way of treatng ths problem s to use least squares regresson on a fnte set of functons as a proxy for condtonal expectaton. Ths dea whch already appeared n [5] s one of the man ngredents of two recent papers by Longstaff and Schwartz [9], and by Tstskls and Van Roy [2]. The purpose of the present paper s to analyze the least squares regresson method proposed by Longstaff and Schwartz [9], whch seems to have become popular among practtoners. In fact, we wll consder a varant of ther approach see Remark 2.. In order to present our results more precsely, we wll dstngush two types of approxmaton n ther algorthm. Approxmaton one: replace condtonal expectatons n the dynamc programmng prncple by proectons on a fnte set of functons taken from a sutable bass. Approxmaton two: use Monte-Carlo smulatons and least squares regresson to compute the value functon of the frst approxmaton. Approxmaton two wll be referred to as the Monte-Carlo procedure. In practce, one chooses the number of bass functons and runs the Monte-Carlo procedure. We wll prove that the value functon of approxmaton one approaches wth probablty one the value functon of the ntal optmal stoppng problem as the number m of functons goes to nfnty. We then prove that for a fxed fnte set of functons, we have almost sure convergence of the Monte-Carlo procedure to the value functon of the frst approxmaton. We also establsh a type of central lmt theorem for the rate of convergence of the Monte- Carlo procedure, thus provdng the asymptotc normalzed error. We note that partal convergence results are stated n [9], together wth excellent emprcal results, but wth no 2

3 study of the rate of convergence. On the other hand, convergence but not the rate nor the error dstrbuton s provded n [2] for a somewhat dfferent algorthm. We also refer to [2] for a dscusson of accumulaton of errors as the number of possble exercse dates rows. We beleve that our methods could be appled to analyze the rate of convergence of the Tstskls-Van Roy approach, but we wll concentrate on the Longstaff-Schwartz method. Mathematcally, the most techncal part of our work concerns the Central Lmt Theorem for the Monte-Carlo procedure. One mght thnk that the methods developed for the analyss of asymptotc errors n statstcal estmaton based on stochastc optmzaton see, for nstance, [6, 7, ] are applcable to our problem. However, the algorthm does not seem to ft n ths settng for two reasons: the lack of regularty of the value functon as a functon of the parameters and the recursve nature of dynamc programmng. The paper s organzed as follows. In Secton 2, a precse descrpton of the least squares regresson method s gven and the notaton s establshed. In Secton 3, we prove the convergence of the algorthm. In Secton 4, we study the rate of convergence of the Monte- Carlo procedure. 2 The algorthm and notatons 2. Descrpton of the algorthm As mentoned n the ntroducton, the frst step n all probablstc approxmaton methods s to replace the orgnal optmal stoppng problem n contnuous tme by an optmal stoppng problem n dscrete tme. Therefore, we wll present the algorthm n the context of dscrete optmal stoppng. We wll consder a probablty space Ω, A, IP, equpped wth a dscrete tme fltraton F =0,...,L. Here, the postve nteger L denotes the dscrete tme horzon. Gven an adapted payoff process Z =0,...,L, where Z 0, Z,..., Z L are square ntegrable random varables, we are nterested n computng sup IEZ τ, τ T 0,L where T,L denotes the set of all stoppng tmes wth values n {,..., L}. Followng classcal optmal stoppng theory for whch we refer to [0], chapter 6, we 3

4 ntroduce the Snell envelope U =0,...,L of the payoff process Z =0,...,L, defned by U = ess- sup τ T,L IE Z τ F, = 0,..., L. The dynamc programmng prncple can be wrtten as follows: U L = Z L We also have U = IE Z τ F, wth U = max Z, IE U + F, 0 L. τ = mn{k U k = Z k }. In partcular IEU 0 = sup τ T0,L IEZ τ = IEZ τ0. The dynamc programmng prncple can be rewrtten n terms of the optmal stoppng tmes τ, as follows: τ L = L τ = {Z IEZ τ+ F } + τ + {Z <IEZ τ+ F }, 0 L, Ths formulaton n terms of stoppng rules rather than n terms of value functons plays an essental role n the least squares regresson method of Longstaff and Schwartz. The method also requres that the underlyng model be a Markov chan. Therefore, we wll assume that there s an F -Markov chan X =0,...,L wth state space E, E such that, for = 0,..., L, Z = f, X, for some Borel functon f,. We then have U = V, X for some functon V, and IE Z τ+ F = IE Z τ+ X. We wll also assume that the ntal state X 0 = x s determnstc, so that U 0 s also determnstc. The frst approxmaton conssts of approxmatng the condtonal expectaton wth respect to X by the orthogonal proecton on the space generated by a fnte number of functons of X. Let us consder a sequence e k x k of measurable real valued functons defned on E and satsfyng the followng condtons: A : For = to L, the sequence e k X k s total n L 2 σx. m A 2 : For = to L and m, f λ k e k X = 0 a.s. then λ k = 0 for k = to m. k= 4

5 For = to L, we denote by P m the orthogonal proecton from L 2 Ω onto the vector space generated by {e X,..., e m X } and we ntroduce the stoppng tmes τ [m] : τ [m] L τ [m] = L = { Z P m Z τ [m] + } + τ [m] + { Z <P mz τ [m] + }, L, From these stoppng tmes, we obtan an approxmaton of the value functon: U m 0 = max Recall that Z 0 = f0, x s determnstc. numercally IEZ τ [m] by a Monte-Carlo procedure. We assume that we can smulate ndependent paths X Z n Z n Z 0, IEZ τ [m]. 2. The second approxmaton s then to evaluate,...,x n,... X of the Markov chan X and we denote by = f, X n the assocated payoff for = 0 to L and n = to. For each path n, we then estmate recursvely the stoppng tmes τ [m] by: = L τ n,m, L τ n,m, = { Z n } + τ n,m, α m, e m X n + { Z n }, L, <α m, e m X n Here, x y denotes the usual nner product n IR m, e m s the vector valued functon e,..., e m and α m, s the least square estmator: α m, Remark that for = to L, α m, the followng approxmaton for U m 0 : 2 = arg mn Z n a e m X n a IR m τ n,m,, + U m, 0 = max Z 0, IR m. Fnally, from the varables τ n,m,, we derve Z n. 2.2 τ n,m, In the next secton, we prove that, for any fxed m, U m, 0 converges almost surely to U m 0 as goes to nfnty, and that U m 0 converges to U 0 as m goes to nfnty. Before statng these results, we devote a short secton to notaton. We also menton that the above algorthm s not exactly the Longstaff-Schwartz algorthm as ther regresson nvolves only n-the-money paths see Remark 2.. 5

6 2.2 otaton Throughout the paper we denote by x the Eucldean norm of a vector x n IR d. For m we denote by e m x the vector e x,..., e m x and for = to L we defne α m so that P m Z [m] τ = α m e m X We remark that, under A 2, the m dmensonal parameter α m has the explct expresson: α m = A m IEZ [m] τ e m X, for = to L, where A m s an m m matrx, wth coeffcents gven by A m k,l m = IEe k X e l X. 2.5 Smlarly, the estmators α m, are equal to α m, = A m, Z n τ n,m, + e m X n, 2.6 for = to L, where A m, s an m m matrx, wth coeffcents gven by A m, k,l m = e k X n e l X n. 2.7 ote that lm A m, = A m almost surely. Therefore, under A 2, the matrx A m, s nvertble for large enough. We also defne α m = α m,..., αm L and αm, = α m,,..., α m, L. Gven a parameter a m = a m,..., am L n IRm... IR m and determnstc vectors z = z,..., z L IR L and x = x,..., x L E L, we defne a vector feld F = F,..., F L by: F L a m, z, x = z L F a m, z, x = z {z a m em x } + F + a m, z, x {z <a m em x }, for =,..., L. We have F a m, z, x = z B c + L =+ z B...B B c + z L B...B L, 6

7 wth B = {z < a m e m x }. We remark that F a m, Z, X does not depend on a m,..., am and that we have F α m, Z, X = Z τ [m] F α m,, Z n, X n = Z n For = 2 to L, we denote by G the vector valued functon and we defne the functons φ and ψ by Observe that wth ths notaton, we have and smlarly, for = to L, α m, τ n,m, G a m, z, x = F a m, z, xe m x, φ a m = IEF a m, Z, X 2.8 ψ a m = IEG a m, Z, X. 2.9 α m = A m ψ + α m, 2.0 = A m,. G + α m,, Z n, X n. 2. Remark 2. In [9], the regresson nvolves only n the money paths,.e. samples for whch Z n > 0. Ths seems to be more effcent numercally. In order to stck to ths type of regresson, the above descrpton of the algorthm should be modfed as follows. Use nstead of τ [m] ˆτ [m] We analogously defne ˆτ n,m, = {Z ˆα m ex } {Z >0} + ˆτ [m] + {Z <ˆα m ex } {Z =0}, for L, wth 2 ˆα m = arg mn IE a IR m {Z >0} [m] a ex Zˆτ. +, ˆα m,, ˆF and Ĝ. The convergence results stll hold for ths verson of the algorthm wth smlar proofs, provded assumptons A and A 2 are replaced by  : For = to L, the sequence e k X k s total n L 2 σx, {Z >0}dIP. m  2 : For L and m, f {Z >0} λ k e k X = 0 a.s. then λ k = 0 for k m. k= 7

8 3 Convergence The convergence of U m 0 to U 0 s a drect consequence of the followng result. Theorem 3. Assume that A s satsfed, then, for = to L, we have lm IEZ m + τ [m] F = IEZ τ F, n L 2. Proof: We proceed by nducton on. The result s true for = L. Let us prove that f t holds for +, t s true for L. Snce Z τ [m] = Z {Z α m em X } + Z [m] τ {Z <α m + em X }, for L, we have IEZ τ [m] Z τ F = Z IEZ τ+ F {Z α m em X } {Z IEZ τ+ F } +IEZ [m] τ Z τ+ F {Z <α m + em X }. By assumpton, the second term of the rght sde of the equalty converges to 0 and we ust have to prove that B m defned by B m = Z IEZ τ+ F {Z α m em X } {Z IEZ τ+ F }, converges to 0 n L 2. Observe that B m = Z IEZ τ+ F {IEZτ+ F >Z α m em X } {α m e m X >Z IEZ τ+ F } But snce P m Z IEZ τ+ F { Z IEZ τ+ F α m em X IEZ τ+ F } α m e m X IEZ τ+ F α m e m X P m IEZ τ + F + P m IEZ τ + F IEZ τ+ F. α m e m X = P m Z [m] τ = P m IEZ [m] + τ F, + s the orthogonal proecton on a subspace of the space of F -measurable random varables. Consequently B m 2 IEZ [m] τ F IEZ τ+ F 2 + P miez τ + F IEZ τ+ F

9 The frst term of the rght sde of ths nequalty tends to 0 by the nducton hypothess and the second one by A. In what follows, we fx the value m and we study the propertes of U m, 0 as the number of Monte-Carlo smulatons, goes to nfnty. For notatonal smplcty, we drop the superscrpt m throughout the rest of the paper. Theorem 3.2 Assume that for = to L, IP α ex = Z = 0. converges almost surely to U0 m of towards IEZ [m] τ Z n τ n,m, Then U m, 0 as goes to nfnty. We also have almost sure convergence as goes to nfnty, for =,..., L. ote that wth the notaton of the precedng secton, we have to prove that lm The proof s based on the followng lemmas. F α, Z n, X n = φ α, L. 3. Lemma 3. For = to L, we have : L L F a, Z, X F b, Z, X Z = { Z b ex a b ex }. = Proof : Let B = {Z < a ex } and B = {Z < b ex }. We have : But F a, Z, X F b, Z, X = Z B B + L =+ Z B...B B c B... B Bc +Z L B c...b c L Bc... B c L. B B = {a ex Z <b ex } + {b ex Z <a ex } { Z b ex a b ex } Moreover B...B B c B... B Bc = k= k= Bk Bk + B c Bc Bk Bk, 9

10 ths gves F a, Z, X F b, Z, X L = L Z B B. Combnng these nequaltes, we obtan the result of Lemma 3.. Lemma 3.2 Assume that for = to L, IP α ex = Z = 0 then α almost surely to α. = converges Proof: we proceed by nducton on. For = L, the result s a drect consequence of the law of large numbers. ow, assume that the result s true for = to L. We want to prove that t s true for. We have α = A G α, Z n, X n. By the law of large numbers, we know that A converges almost surely to A and t remans to prove that G α, Z n, X n converges to ψ α. From the law of large numbers, we have the convergence of G α, Z n, X n to ψ α and t suffces to prove that : lm G α, Z n, X n G α, Z n, X n = 0. + We note G = G G α, Z n, X n G α, Z n, X n. We have : ex n F α, Z n, X n F α, Z n, X n ex n L = Z n L = { Z n α ex n α α ex n }. Snce, for = to L, α converges almost surely to α, we have for each ɛ > 0 : lm sup G lm sup ex n L Z n L = IE ex L = n { Z = = { Z α ex ɛ ex }, = L Z α ex n ɛ ex n } where the last equalty follows from the law of large numbers. Lettng ɛ go to 0, we obtan the convergence to 0, snce for = to L, IP α ex = Z = 0. The proof of Theorem 3.2 s smlar to the proof of Lemma 3.2. Therefore, we omt t. 0

11 4 Rate of convergence of the Monte-Carlo procedure 4. Tghtness In ths secton we are nterested n the rate of convergence of Z n, for = to L. τ n, Recall that m s fxed and that Z L and ex L are square ntegrable varables. We assume that : H : =,..., L, lm sup ɛ 0 Ȳ = + L = IEȲ { Z α ex ɛ ex } ɛ L Z + ex = + < +, where L = ex. 4. ote that H mples that IP Z = α ex = 0 and, consequently, under H we know from Secton 3 that F α, Z n, X n converges almost surely to φ α. Remark too that H s satsfed f the random varable Z α ex has a bounded densty near zero and the varables Z and ex are bounded. Theorem 4. Under H, the sequences =,..., L and α F α, Z n, X n φ α α =,..., L are tght. The proof of Theorem 4. s based on the followng Lemma. Lemma 4. Let U n, V n, W n be a sequence of dentcally dstrbuted random varables wth values n [0, + 3 such that IE W {U lm sup ɛv } ɛ 0 ɛ < +, and θ a sequence of postve random varables such that θ s tght, then the sequence W n {U n θ V n } Proof: Let σ θ = s tght. W n {U n θv n }. Observe that σ s a non decreasng functon

12 of θ. Let A > 0, we have IP σ θ A IP σ θ A, θ B + IP θ > B = IP σ B A + IP θ > B A IEσ B + IP θ > B A IEW {U B V } + IP θ > B. From the assumpton on U n, V n, W n and the tghtness of θ, we deduce easly the tghtness of σ θ. Proof of Theorem 4.: We know from the classcal Central Lmt Theorem that the sequence / F α, Z n, X n φ α s tght and t remans to prove the tghtness of F α, Z n, X n F α, Z n, X n, for = to L. Smlarly, to prove the tghtness of α α, for = to L, we ust have to prove the tghtness of G α, Z n, X n G α, Z n, X n see Secton 2 for the notaton. We proceed by nducton on. The tghtness of F L α, Z n, X n F L α, Z n, X n s obvous and that of α L α L follows from the Central Lmt Theorem for the sequence Z n L exn L and the almost sure convergence of the sequence A L I. Assume that tght for = to L. We set F = F α, Z n, X n F α, Z n, X n and α α are ow from Lemma 3., we have : wth F Ȳ n = + F α, Z n, X n F α, Z n, X n. L Ȳ n = L = Z n L + { Z n = α ex n ex n L + α α ex n }, = ex n. 4.2 From Lemma 4. and by the nducton hypothess, we deduce that F s tght. In the same way, we prove that α 2 α 2 s tght. 2

13 4.2 A central lmt theorem We prove n ths secton that under some stronger assumptons than n secton 4., the vector Z n τ n, precedng notaton, we have IEZ [m] τ =,...,L converges weakly to a Gaussan vector. Wth the Z n τ n, IEZ [m] τ = F α, Z n, X n φ α. 4.3 In the followng, we wll denote by Y the par Z, X and by Y n the par Z n, X n. We wll also use Ȳ and Ȳ n as defned n 4. and 4.2. We wll need the followng hypothess: H : For = to L, there exsts a neghborhood V of α, η > 0 and k > 0 such that for a V and for ɛ [0, η ], IEȲ { Z a ex ɛ ex } ɛk. H 2 : For = to L, Z and ex are n L p for all p < +. H 3 : For = to L, φ and ψ are C n a neghborhood of α. Observe that H s stronger than H. Theorem 4.2 Under H, H 2, H 3, the vector Z n τ n, IEZ [m] τ, α α =,...,L converges n law to a Gaussan vector as goes to nfnty. For the proof of Theorem 4.2, we wll use the followng decomposton : + F α, Y n φ α = F α, Y n F α, Y n φ α φ α F α, Y n φ α + φ α φ α. From the classcal Central Lmt Theorem, we know that converges n law to a Gaussan vector. Moreover, we have 3 F α, Y n φ α

14 where A α α = A A A converges almost surely to A and A the smlar decomposton + G + α, Y n ψ + α A A ψ + α, A converges n law. We have G + α, Y n ψ + α = 4.4 G + α, Y n G + α, Y n ψ + α ψ + α G + α, Y n ψ + α + ψ + α ψ + α. Usng these decompostons and Theorem 4.3 below, together wth the dfferentablty of the functons φ and ψ, Theorem 4.2 can be proved by nducton on. Theorem 4.3 Under H, H 2, H 3, the varables and F α, Y n F α, Y n φ α φ α G + α, Y n G + α, Y n ψ + α ψ + α converge to 0 n L 2, for = to L. Remark 4. If we try to compute the covarance matrx of the lmtng dstrbuton, we see from Theorem 4.3 and the above decompostons that t depends on the dervatves of the functons φ and ψ at α. Ths means that the estmaton of ths covarance matrx may prove dffcult. Indeed, the dervaton of an estmator for the dervatve of a functon s typcally harder than for the functon tself. Remark 4.2 Theorem 4.3 can be vewed as a way of centerng the F -ncrements resp. the G + -ncrements between α and α by the φ -ncrements resp. the ψ + -ncrements. One way to get some ntuton of the proof of Theorem 4.3 s to observe that f the sequence α I were ndependent of the Y n s, the convergence n L 2 would reduce to the convergence of α I to α. Indeed, we would have to consder expectatons of the type 4

15 IE ξ n 2 wth varables ξ n whch are centered and d, condtonally on α. man dffculty n the proof of Theorem 4.3 comes from the fact that α s not ndependent of the Y n s. On the other hand, we do have dentcally dstrbuted random varables and we wll explot symmetry arguments and the ndependence of α and Y. The For the proof of Theorem 4.3, we need to control the ncrements of F and G +. Lemma 4.2 relates these ncrements to ndcator functons. Lemma 4.3 wll enable us to localze α near α. We wll then develop recursve technques adapted to dynamc programmng see Lemma 4.3 and Lemma 4.5. In the followng, we denote by IY, a, ɛ the functon IY, a, ɛ = { Z a ex ɛ ex }. ote that IY, a, ɛ IY, b, ɛ + b a. 4.5 The followng Lemma s essentally a reformulaton of Lemma 3.. Lemma 4.2 For = to L, and a, b n IR m L, we have F a, Y F b, Y Ȳ G a, Y G b, Y Ȳ L = L = IY, a, a b IY, a, a b Lemma 4.3 Assume H and H 2, then for = to L, there exsts C > 0 such that for all δ > 0, IP α α δ C δ 4 2. Proof: Let us recall that f U n n s a sequence of..d. varables such that EU 4 have δ > 0, IP U n IEU δ 5 < +, we C δ

16 Observe that α We set Ω ɛ = { A α = A From 4.6, we know that IP Ω ɛ c α G + α, Y n ψ + α A A ψ + α, A A A ɛ} and we choose ɛ such that A 2 A on Ω ɛ. α K C ɛ 4 2, for = to L and that, on Ω ɛ, G + α, Y n ψ + α + Kɛ. ow, snce G L α, Y n = Z n L exn L, we deduce from 4.6 appled wth ths choce of U n that IP α L α L δ Choosng ɛ = ρδ wth ρ small enough, we obtan: C L δ Kɛ C L ɛ 4 2. IP α L α L δ C L δ 4 2. Assume now that the result of Lemma 4.3 s true for +,..., L. We wll prove that IP α α δ C δ 4 2. We have G + α, Y n ψ + α = G + α, Y n G + α, Y n + G + α, Y n ψ + α. From Lemma 4.2, we obtan on Ω ɛ, α α Kɛ + L Ȳ n IY n, α, α α =+ + K G + α, Y n ψ + α. The last term can be treated usng 4.6. Therefore, t suffces to prove that δ > 0, IP S δ C δ 4 2, 6

17 L where S = Ȳ n =+ IP S δ IP IY n L Ȳ n =+, α, α α. But IY n, α, ɛ δ + L =+ IP α α ɛ. By assumpton, for = + to L, we have IP α α ɛ C L we know from H that δ L =+ =+ IEȲ n IY n, α, ɛ ɛk, wth K = L =+ IEȲ n IY n, α, ɛ δ ɛk and, usng 4.6 agan, we see that IP L Ȳ n =+ IY n, α, ɛ δ C δ ɛk 4 2. Choosng ɛ = ρδ, wth ρ small enough, we obtan the result of Lemma 4.3. ɛ 4 2. Moreover k, so we have Before statng other techncal results n preparaton for the proof of Theorem 4.3, we ntroduce the followng notatons. Gven k {, 2,..., L}, λ and µ n IR +, we defne a sequence of random vectors U k λ, µ, =,..., L, by the recursve relatons U k L λ, µ = λ U k λ, µ = λ + µ k L Ȳ n =+ I Y n, α k, U k λ, µ, L 2. Wth ths defnton, U k λ, µ s obvously σy,..., Y k -measurable. We also observe that t s a symmetrc functon of Y,..., Y k because α k depends symmetrcally on Y,..., Y k. The next lemma establshes a useful relaton between U k and U k. Lemma 4.4 Assume H 2. There exst postve constants C, u, v such that for each I, one can fnd an event Ω wth IP Ω c C/ 2 and, on the set Ω we have, for k {,..., L} and {,... L }, U k λ, µ + α k α k U k λ + Lµ + uȳ k, v + µ. 7

18 Proof: We have Snce A α k = A k k k G + α k, Y n. s the mean of d random varables wth moments of all orders and mean A, we can fnd Ω, wth IP Ω c = O/ 2, on whch A k 2 A, for k =,..., L, =,..., L. On ths set, we have, for some postve constant C, α k α k C Ȳ k k + Ȳ n + k k C k k G + α k, Y n G + α k, Y n. 4.7 Here we have used the nequalty ka k k A k Ȳ k. We may choose Ω n such a way that k k Ȳ n remans bounded on Ω. ote that, for = L, the last sum n 4.7 vanshes. Usng Lemma 4.2 for L 2, we have, on Ω, α k α k uȳ k for some constants u and v. I + v k Ȳ n L =+ I To complete the proof of the lemma, we observe that Y n, α k, U k λ, µ I Y n, α k Y n, α k, α k α k 4.8, U k λ, µ + α k α k. ow, for L 2, by gong back to the recursve defnton of U k λ, µ and separatng the k-th term of the sum, we obtan U k λ, µ k µ λ + LµȲ k + L I, α k Ȳ n =+ Y n, U k λ, µ + α k α k 4.9 ow let V = U k λ, µ + α k α k. By combnng 4.8 and 4.9 we get V λ + Lµ + uȳ k + µ + v k Ȳ n L =+ I Y n, α k, V 8

19 Lemma 4.5 Assume H and H 2. For all ε 0, ] and for all µ 0, there exsts a constant C ε,µ such that λ 0, {,..., L }, IEU 2 λ, µ C ε,µ + λ ε. Proof: We wll prove by nducton on k = L, L,..., 2 that sup IEU k λ, µ C ε,µ + λ k L ε 4.0 We obvously have 4.0 for k = L, snce U L L λ, µ = λ/. We now assume that 4.0 holds for k + and wll prove t for k. For k, we have, usng the symmetry of U k λ, µ wth respect to Y,..., Y k, IEU k λ, µ = λ + µ k IE Ȳ k L =+ I Y k, α k For = +,..., L, we have, usng 4.5, Lemma 4.4, and the notaton V k λ, µ = U k λ + Lµ + uȳ k, v + µ,, U k λ, µ IE Ȳ k I IEȲ k I IEȲ k I Y k Y k Y k, α k, U k λ, µ, α k, α k, U k, V k λ, µ ote that IEȲ k Ω c Ȳ L 2 IP Ω c = O/. At ths pont we would lke to use H α k λ, µ + α k + IEȲ k Ω c and the nducton hypothess. However, we have to be careful because V k λ, µ depends on Y k. For { +,..., L }, we wrte wth IEȲ k I A l = IEȲ k {l Ȳ k <l} I lip Ȳ k l Y k, α k, V k λ, µ = Y k, α k /p IEȲ k { Ȳ k <l} I, V k λ, µ Y k A l, l=, α k, V k λ, µ p, 9

20 for all p, + Hölder. ow, IEȲ k { Ȳ k <l} I Y k, α k IEȲ k I, V k λ, µ Y k, α k, U k λ + Lµ + ul, v + µ and we may condton wth respect to σy,..., Y k and use Lemma 4.3 and H to, obtan IEȲ k { Ȳ k <l} I Y k, α k, V k λ, µ CIEU k λ + Lµ + ul, v + µ + C 2. We can now apply the nducton hypothess, and we easly deduce 4.0 for k, usng the fact that IP Ȳ k l = o/l m for all m I. Proof of Theorem 4.3: We prove that lm F α, Y n F α, Y n φ α φ α = 0 n L 2. The proof s smlar for the second term of the Theorem. We ntroduce the notaton a, b, Y = F a, Y F b, Y φ a φ b. We have to prove that 2 lm IE α, α, Y n = 0. Remark that for n = to, the pars α, Y n and α, Y have the same law, and for n m, α, Y n, Y m and α, Y, Y have the same dstrbuton. So we obtan IE α, α, Y n 2 = IE 2 α, α, Y + IE α, α, Y α, α, Y. But α, α, Y 2 Ȳ + IEȲ. Snce the sequence α goes to α almost surely and IP Z = α ex = 0 for = to L by assumpton, we deduce that α, α, Y goes to 0 almost surely. Consequently, we obtan that IE 2 α, α, Y tends to 0. It remans to prove that lm IE α, α, Y α, α, Y =

21 We observe that IE α 2, α, Y Y,..., Y = 0, snce IE F α 2, Y Y,..., Y = φ α 2 almost surely. Ths gves and we ust have to prove that lm IE α 2, α, Y α 2, α, Y = 0, IE α, α, Y α, α, Y α 2, α, Y α 2, α, Y = 0. We have the equalty α, α, Y α, α, Y α 2, α, Y α 2, α, Y = α, α 2, Y α, α, Y + α 2, α, Y α, α 2, Y, We want to prove that and lm IE α, α 2, Y α, α, Y = lm IE α 2, α, Y α, α 2, Y = Both equaltes can be proved n a smlar manner. We gve the detals for 4.3. Frst, note that gven any η > 0, we have, usng H 2 and Hölder s nequalty, IE α 2, α, Y α, α 2, Y { α α η } C p IP α α η /p, for all p >. We know from Lemma 4.3 that IP α α η C/ 2 4η. Therefore, f η < /4, lm IE α 2, α, Y α, α 2, Y { α α η } = 0. On the other hand, we have, usng Lemma 4.2, L α, α 2, Y Ȳ I Y, α 2 = φ α φ α 2,, α α 2 + 2

22 and L α, α, Y Ȳ I =, α, α α Y + φ α φ α. By the same reasonng as n the proof of Lemma 4.4, we have postve constants C, s and t such that, for each I, one can fnd a set Ω wth IP Ω c C/ 2, on whch α α 2 U 2 sȳ + Ȳ, t, =,..., L. Usng Lemma 4.3 and H 3, we may also assume that, on Ω, and for some postve constant K. We now have, for η < /4, φ α φ α 2 φ α φ α K K L α = α 2 L α α =, IE α, α 2, Y α, α, Y = L l= IE α, α 2, Y Ȳ I Y l, α l, η + K η + o/. In order to prove 4.2, t now suffces to show that, for, l L, lm Ȳ IE I Y, α 2, V 2 + V 2 Ȳ I Y l, α l, η + η = 0, wth the notaton V 2 = U 2 sȳ + Ȳ, t. We wll only prove that lm Ȳ IE I Y, α 2 snce the other terms are easer to control. For m I, let, V 2 Ȳ I Y l, α l, η = 0, A m = {m Ȳ + Ȳ < m}. 22

23 We have IE Ȳ I Y, α 2, V 2 IE Ȳ I = IEȲ I Y Y Ȳ I Y, α 2, U 2 l, α l, η Am sm, t Ȳ I Y l, α l, η IEȲ I Y l, α l, η., α 2, U 2 sm, t Here we have used the fact that Y s ndependent of Y,..., Y. We now condton wth respect to σy,..., Y 2 n the frst expectaton and we use H to obtan and Lemma 4.5 IE Ȳ I Y, α 2, V 2 Ȳ I Y l, α l, η + m Am C ε ε+η. Here ε s an arbtrary postve number and, by takng ε < η < /4, summng up over m and usng Hölder s nequalty we easly complete the proof. Acknowledgement: the research on ths paper has been stmulated by the proet mathf semnar on the topc Monte-Carlo methods for Amercan optons, durng the fall The talks gven by P. Cohort, J.F. Delmas, B. Jourdan, E. Temam were especally helpful. References [] Bally, V. and G. Pagès, A quantzaton algorthm for solvng multdmensonal optmal stoppng problems, Preprnt, December [2] Broade, M. and P. Glasserman, Prcng Amercan-style securtes usng smulaton, Journal of Economc Dynamcs and Control 2, , 997. [3] Broade, M., P. Glasserman, and G. Jan, Enhanced Monte-Carlo estmates for Amercan opton prces, Journal of dervatves 5, 25-44, 997. [4] Broade, M. and P. Glasserman, A stochastc mesh method for prcng hgh-dmensonal Amercan optons, Preprnt, ovember 997. [5] Carrère, J., Valuaton of Early-Exercce Prce of optons Usng Smulatons and onparametrc Regresson, Insurance: Mathematcs and Economcs, 9, 9-30,

24 [6] Dupacova, J. and R. Wets, Asymptotc behavor of statstcal estmators and of optmal solutons of stochastc optmzaton problems, Ann. of Stat., 6, o 4, , 988. [7] Kng A.J. and T.R. Rockafellar, Asymptotc theory for solutons n statstcal estmaton and stochastc programmng, Math. of Operatons Research 8, o, 48-62, 993. [8] Lamberton, D., Brownan optmal stoppng and random walks, to appear n Appled Math. and Optmzaton, [9] Longstaff, F.A. and E.S. Schwartz, Valung Amercan optons by smulaton: a smple least-squares approach, Revew of Fnancal Studes 4, 3-48, 200. [0] eveu J. Dscrete-parameter Martngales. orth Holland, Amsterdam, 975. [] Shapro A., Asymptotc propertes of statstcal estmators n stochastc programmng Ann. of Stat., 7, o 2, , 989. [2] Tstskls, J.. and B. Van Roy, Regresson methods for prcng complex Amercan-style optons, preprnt, August To appear n IEEE Transactons on eural etworks. 24