A quantization tree method for pricing and hedging multi-dimensional American options

A quazao ree mehod for prcg ad hedgg mul-dmesoal Amerca opos Vlad BALLY Glles PAGÈS Jacques PRINTEMS Absrac We prese here he quazao mehod whch s well-adaped for he prcg ad hedgg of Amerca opos o a baske of asses. Is purpose s o compue a large umber of codoal expecaos by projeco of he dffuso o opmal grd desged o mmze he square mea projeco error [24]. A algorhm o compue such grds s descrbed. We provde resuls cocerg he orders of he approxmao wh respec o he regulary of he payoff fuco ad he global sze of he grds. Numercal ess are performed dmesos 2, 4, 6, wh Amerca syle exchage opos. They show ha heorecal orders are probably pessmsc. Key words: Amerca opo prcg, Opmal Soppg, Sell evelope, Opmal quazao, local volaly model. 2 AMS classfcao: 9B28, 6G4, 65C5, 65C2, 93E23, 65 N 5. Iroduco ad referece model The am of hs paper s o prese, o sudy ad o es a probablsc mehod for prcg ad hedgg Amerca syle opos o muldmesoal baskes of raded asses. The asse dyamcs follow a d-dmesoal dffuso model bewee me ad a maury me T. We especally focus a classcal exeso of he Black & Scholes model: he local volaly model. Neverheless, a large par of he algorhmc aspecs of hs paper ca be appled o more geeral models. Prcg a Amerca opo a couous me Markov process S [,T ] cosss solvg he couous me opmal soppg problem relaed o a obsacle process. I hs paper we are eresed Markova obsacles of he form h = h, S whch are he mos commoly cosdered facal markes. Roughly speakg, here are wo ypes of umercal mehods for hs purpose: Frs, some purely deermsc approaches comg from Numercal Aalyss: he soluo of he opmal soppg problem adms a represeao v, S where v sasfes a parabolc varaoal equaly. So, he varous dscrezg echques lke fe dfferece or fe eleme mehods yeld a approxmao of he fuco v a dscree pos of a Laboraore d aalyse e de mahémaques applquées & Proje MahF INRIA, UMR 85, Uversé de Mare-la-Vallée, Cé Descares, 5, Bld Descares, Champs-sur-Mare, F-77454 Mare-la-Vallée Cedex 2, Frace. bally@mah.uv-mlv.fr Labo. de Probablés e Modèles aléaores, CNRS UMR 7599, Uversé Pars 6, case 88, 4, pl. Jusseu, F-75252 Pars Cedex 5. gpa@ccr.jusseu.fr Cere de Mahémaques & CNRS UMR 85, Uversé Pars 2, 6, aveue du Gééral de Gaulle, F-94 Créel. prems@uv-pars2.fr

me-space grd see e.g. [33] for a applcao o a valla pu opo or [8] for a more comprehesve sudy. Secodly, some probablsc mehods based o he dyamc programmg formula or o he approxmao of he lowes opmal soppg me. I -dmeso, he mos popular approach o Amerca opo prcg ad hedgg remas he mplemeao of he dyamc programmg formula o a bomal ree, orgally aed by Cox-Ross & Rubse as a elemeary alerave o couous me Black & Scholes model. However, le us meo he poeerg work by Kusher 977 see [28] ad also [29] whch Markov cha approxmao was frs roduced, cludg s lks wh he fe dfferece mehod. Ths ook place before he massve developme of Mahemacal Face. Cocerg he cossecy of me dscrezao, see [34]. These mehods are que effce o hadle valla Amerca opos o a sgle asse bu hey quckly become racable as he umber of uderlyg asses creases. Usually, umercal mehods become effce because he space grds are bul regardless of he dsrbuos of he asse prces. The same problem occurs for fe sae Markov cha approxmao à la Kusher. Cocerg he exeso from bomal o mulomal rees, s seems dffcul o desg some rees ha are boh compable wh he dmeso/correlao cosras ad he probablsc srucure of he dyamcs. More recely, he problem gave brh o a exesve leraure order o overcome he dmesoaly problem. All of hem fally lead o some fe sae dyamc programmg algorhm eher s usual form or based o he backward dyamc approxmao of he lowes opmal soppg me. I Barraqua & Mareau [7], a sub-opmal -dmesoal problem s solved: everyhg s desged as hough he obsacle process self had he Markov propery. I [36], he algorhm devsed by Logsaff & Schwarz s based o codoal expecao approxmao by regresso o a fe sub-famly ϕ S I of a bass ϕ k S k of L 2 σs, P. The Moe Carlo rae of covergece of hs mehod s deeply aalyzed by Cléme e al. [6]. I [4], Tsskls & Va Roy use a smlar dea bu for a modfed Markov raso. I [], Braode & Glasserma geerae some radom grds a each me sep ad compue some compao weghs usg some sascal deas based o he mporace samplg heorem. I [2] ad [22] Fouré e al. aed a Moe Carlo approach based o Mallav calculus o compue codoal expecaos ad her dervaves. Ths leads o a purely probablsc mehod. I [35], Los ad Réger exed hs approach o Amerca opo prcg ad Greek compuao. The crucal sep of hs mehod s he varace reduco by localzao. Opmal localzao s vesgaed [27] ad [9]. I hs paper, we develop a probablsc mehod based o grds lke he orgal fe sae Markov cha approxmao mehod orgally descrbed [5]. Frs, we dscreze he asse prce process a mes := kt/, k =,..., f ecessary, we roduce he Euler scheme of he prce dffuso process, sll deoed by S k for coveece hroughou he roduco. The key po s he followg: raher ha selg hese grds a pror, we wll use our ably o smulae large samples of S k k o produce a each me a grd Γ k of sze N k whch s opmally fed o S k amog all grds wh sze N k he followg sese: he closes eghbor rule projeco q Γ S k of S k oo he grd Γ k k s he bes leas square approxmao of S k amog all radom vecors Z such ha ZΩ N k. Namely } S k q Γ S k 2 = m { S k Z 2, Z : Ω R d, ZΩ N k. k I ha sese we wll produce ad he use a each me sep he bes possble grd of sze N k o approxmae he d-dmesoal radom vecor S k. For hsorcal reasos comg 2

from Iformao Theory, boh he fuco q Γ ad he se q Γ Ω are ofe called opmal k k quazer of S k. The resulg error boud S k q Γ S k 2 s called he lowes quadrac k mea quazao error. I has bee exesvely vesgaed Sgal Processg ad Iformao Theory for more ha 5 years see [25] or more recely [24]. Thus, oe kows ha goes o a a ON d k rae as N k. Excep some specfc -dmesoal cases of lle umercal eres, o closed form s avalable eher for he opmal grd Γ k, or for he duced lowes mea quazao error. I fac lle s kow o he geomerc srucure of hese grds hgher dmeso. However, sarg from he egral represeao vald for ay grd Γ S k q Γ S k m 2 = E S 2 x 2 x Γ ad usg s regulary properes as a almos everywhere dffereable symmerc fuco of Γ, oe may mpleme a sochasc grade algorhm ha coverges o some locally opmal grd. Furhermore, he algorhm yelds as by-producs he dsrbuo of q Γ S k,.e. he weghs PS k =x k, k, x k, Γ k ad he duced quazao error. Boh are volved he Amerca opo prcg algorhm see Seco 2.2. Thus, Fgure llusraes o he bvarae ormal dsrbuo how a opmal grd ges coceraed o heavly weghed areas hs grd was obaed by he CLV Q algorhm descrbed Seco 2.4. The paper s orgazed as follows. Seco 2 of he paper s devoed o he descrpo of he quazao ree algorhm for prcg Amerca opos ad o s heorecal rae of covergece. The, he ree opmzao, cludg he algorhmc aspecs, s developed. Ths seco s parally adaped from a geeral dscrezao mehod devsed for Refleced Backward Sochasc Dffereal Equaos RBSDE [3]. Tme dscrezao Seco 2. amous o approxmag a couously exercsable Amerca opo by s Bermuda couerpar o be exercsed oly a dscree mes, k =,...,. The heorecal premum of he Bermuda opo sasfes a backward dyamc programmg formula. The quazao ree algorhm s defed Seco 2.2: smply cosss pluggg he opmal quazer Ŝ := q Γ S k of S k hs formula. k Some weghs appear ha are obaed by he sochasc grd opmzao procedure meoed above. I Seco 2.3, he rae of covergece of hs algorhm s derved for Lpschz couous payoffs as a fuco of he me dscrezao sep T/ ad of he L p -mea quazao errors S k q Γ S k p, k =,.... The a shor backgroud o k opmal quazao s provded Seco 2.4. I Seco 2.5, he grd opmzao of he quazao ree s addressed, usg a sochasc approxmao recursve procedure. The las subseco proposes a effce aalyc mehod o desg a pror he sze N k of he grd a every me s proposed, gve ha N := N +N + +N elemeary quazers are avalable. I ha case, we oba some error bouds of he form C /2 +N/ d. Whe he payoff s sem-covex he same holds rue wh sead of /2. I Seco 3, we desg a approxmag quazed hedgg sraegy followg some deas by Föllmer & Soderma o complee markes. We are a poso o esmae some bouds for he duced hedgg defaul, called local resdual rsks of he quazao ree. Ths s he am of Seco 4. To hs ed, we combe some mehods borrowed from RBSDE Theory, aalycal echques for p.d.e. ad quazao heory. We oba dffereds of raes of covergece for he hedgg sraegy far from ad close o he maury. 3

Seco 5 s devoed o he expermeal valdao of he mehod. We prese exesve umercal resuls whch ed o show ha whe he grds are opmal he quadrac quazao sese, he spaal order of covergece s beer ha ha obaed wh usual grd mehods. The ess are carred ou usg mul-dmesoal Amerca exchage opos o geomerc dex a sadard d-dmesoal decorrelaed Black & Scholes model. Ths rae, acually beer ha forecas by heory, compesaes for he drawback of a rregular approxmao see below. Two segs have bee seleced for smulao: oe -he-moey ad oe ou-of-he-moey, boh several dmesos d = 2, 4, 6,. I he wors case d= case, he compued prema rema wh 3, 5% of he referece prce. The ma feaures of he quazao approach. Before gog o echcales, oe may meo a obvous mehodologcal dfferece bewee he quazao ree algorhm ad he regresso mehod [36]. The Logsaff-Schwarz approach makes he choce of a smooh bu global approxmao whereas we prvlege a rregular pecewse cosa bu local approxmao. Amog he expeced advaages of he local feaure of quazao approxmao, a prome oe s ha may lead o hgher order approxmaos of he prce, volvg he spaal dervaves.e. he hedgg see e.g. [6] for a frs approach ha dreco. A secod asse, probably he mos mpora for operag applcaos, s ha, oce he asse prce process has bee appropraely quazed, ca almos saly prce all possble Amerca valla payoffs whou ay furher Moe Carlo smulaos. Fally, whe he dffuso process S s a fuco of he Browa moo a me.e. S = ϕ, B lke he Black & Scholes model, he quazao ree algorhm may become compleely parameer free: suffces o cosder a quazao of he Browa moo self whch cosss of some opmal quazao grds of mul-varae ormal dsrbuos wh he approprae szes. Such opmal grds ca be compued sysemacally a very accurae way ad he kep off le see [39]. Quadrac opmal N-quazao of he N ; I d dsrbuos has bee carred ou sysemacally for varous szes N {,..., 4} ad dmesos d {,..., }. Some fles of hese opmal grds cludg her weghs ca be dowloaded a he URLs: www.proba.jusseu.fr/pageperso/pages.hml or www.uv-pars2.fr/www/labos/cmup/homepages/prems. Fally, oe ha hs mehod of quazao has bee mplemeed he sofware prema see hp://www-rocq.ra.fr/mahf/prema/dex.hml. The referece model. We cosder a marke o whch are raded d rsky asses S,..., S d ad a deermsc rskless asse S := e r, r R bewee me := ad he maury me T >. Oe ypcal model for he prce process S := S,..., S d of he rsky asses s he followg dffuso model ds = Sr d + σ j e r S dw j, S := s >, d,. jq where W := W,..., W q s a sadard q-dmesoal Browa Moo defed o a probably space Ω, A, P ad σ : R d Md q := R d q s bouded ad Lpschz couous..2 The flrao of eres wll be he aural compleed flrao F := F S [,T ] of S whch cocdes wh ha of W as soo as σσ ξ > for every ξ R d. For oaoal coveece, we roduce cξ := Dagξσξ, ξ := ξ,..., ξ d R d. 4

where Dagξ deoes he dagoal marx wh dagoal ery ξ a row. The fucos cξ ad he drf bξ := r ξ are Lpschz couous so ha a uque srog soluo exss for. o Ω, A, P. Furhermore, s classcal backgroud ha, for every p, here exss a cosa C p,t > such ha E s sup S p < C p,t + s p. [,T ] The dscoued prce process S := e r S s he a posve P-margale sasfyg d S = c S dw, S := s,.3 Here P s he so-called rsk eural probably Mahemacal Face ermology. As log as q d, he usual compleeess of he marke ecessarly fals. However, from umercal po of vew, hs has o fluece o he mplemeao of he quazao mehod o compue he prce of he dervaves: we jus compue a P-prce. Whe comg o he problem of hedgg hese dervaves, he he compleeess assumpo becomes crucal ad wll lead us o assume ha q = d ad ha he dffuso coeffce cx s verble everywhere o R + d. Whe q = d ad σx σ Md d,. s he usual d-dmesoal Black & Scholes model: he rsky asses are geomerc Browa moos gve by S = s exp r 2 σ. 2 + σ j W j, d. jd A Amerca opo relaed o a payoff process h [,T ] s a corac ha gves he rgh o receve oce ad oly oce he payoff h a some me [, T ] where h [,T ] s a F-adaped oegave process. I hs paper we wll always cosder he sub-class of payoffs h ha oly depeds o, S.e. sasfyg h := h, S, [, T ] where h : [, T ] R + s a Lpschz couous..4 Such payoffs are somemes called valla. Uder Assumpos. ad.4, oe has E sup h p < + for every p. [,T ] Oe shows a complee marke ha he far prce V a me for hs corac s V := e r ess sup { Ee rτ h τ F, τ T }.5 where T := {τ : Ω [, T ], F-soppg me}. Ths smply meas ha he dscoued prce Ṽ := e r V of he opo s he Sell evelope of he dscoued Amerca payoff h := h, S wh h, x := e r h, e r x..6 Ths resul s based o a hedgg argume o whch we wll come back furher o. Noe ha sup V sup h L p, p. [,T ] [,T ] Oe shows see [8] usg he Markov propery of he dffuso process S [,T ] ha V := ν, S where ν solves he varaoal equaly ν max + L r,σν, ν h =, νt,. = ht,...7 5

where L r,σ deoes he fesmal geeraor of he dffuso.. The, s clear ha he approxmao problem for V appears as a specal case of he approxmae compuao of he Sell evelope of a d-dmesoal dffuso wh Lpschz coeffces. To solve hs problem -dmeso, may mehods are avalable. These mehods ca be classfed wo famles: he probablsc oes based o a weak approxmao of he dffuso process S by purely dscree dyamcs e.g. bomal rees, [33] ad he aalyc oes based o umercal mehods for solvg he varaoal equaly.7 e.g. fe dfferece or fe eleme mehods. Whe he dmeso d of he marke creases, hese mehods become effce. A hs sage, oe may assume whou loss of geeraly ha he eres rae r. s : hs amous o assumg ha we are a dscoued world wh S gve by.3 ad h gve by.6 sead of S ad h respecvely. Noaos: C b Rd deoes he se of fucos fely dffereable wh bouded dffereals so ha hey have a mos lear growh. The leers C ad K deoe posve real cosas ha may vary from le o le.. wll deoe he Eucldea orm ad. he er produc o R d = R d. M := sup x Mx wll deoe he operaor orm of he marx M R d q d rows, q colums ad M s raspose. I parcular x.y = x y. 2 Prcg a Amerca opo usg a quazao ree I hs seco, he specfcy of he margale dffuso dyamcs proposed for he rsky asses.3 wh r = has lle fluece o he resuls, so s cosless o cosder a geeral drfed Browa dffuso S = S + bs s ds + cs s dw s, 2. where b : R d R d ad c : R d Md q are Lpschz couous vecor felds ad W [,T ] s q-dmesoal Browa moo. 2. Tme dscrezao: he Bermuda opos The exac smulao of a dffuso a me s usually ou of reach e.g. whe σ s o cosa he specfed model.. So oe uses a Markova dscrezao scheme, easy o smulae, e.g. he Euler scheme: se = kt/ ad S k+ = S k + bs k T + cs.w k W k, S = s. 2.2 The, he Sell evelope o be approxmaed by quazao s ha of he Euler scheme. Somemes, he dffuso ca be smulaed smply, esseally because appears as a closed form S := ϕ, W. Ths s he case of he regular mul-dmesoal Black & Scholes model se σx := σ.. The, s possble o cosder drecly he he Sell evelope of he homogeeous Markov cha S k k for quazao purpose. Ths me dscrezao correspods, he dervave ermology, o approxmag he orgal couous me Amerca opo by a Bermuda opo, eher o S or o S self. By Bermuda opo, oe meas ha he se of possble exercse mes s fe. Error bouds are avalable a hese exercse mes see Theorem below. 6

We wa o quaze he Sell evelope of S k or S k or of ay famly of homogeeous dscree me F k -Markov chas X k k whose rasos, deoed P x, dy, preserves Lpschz couy he followg sese: for every Lpschz couous fuco f : R d R [P fx fy f] Lp + C b,σ,t T/[f] Lp where [f] Lp := sup x y x y 2.3 see, e.g., [3] for a proof. I fac hs geeral dscree me markova seg s he aural framework for he mehod. To allevae oaos, we drop he depedecy ad keep he oao X k k. The F k -Sell evelope of h, X k, deoed by V k k, s defed by: V k := ess sup {E hθ, X θ F k, θ Θ k } where Θ k deoes he se of {,..., }-valued F l -soppg mes. I sasfes he socalled backward dyamc programmg formula see [37]: { V := h, X, 2.4 V k := max h, X k, EV k+ F k, k. Oe derves usg he Markov propery a dyamc programmg formula dsrbuo: V k = v k X k, k {,..., }, where he fucos v k are recursvely defed by { v := h,., v k := max h,., P v k+ 2.5, k. Ths formula remas racable for umercal compuao sce hey requre o compue a each me sep a codoal expecao. Theorem below gves some L p -error bouds ha hold for V k V k our orgal dffuso framework. Frs we eed o roduce some defo abou he regulary of h. Defo A fuco h : [, T ] R d R s sem-covex f ξ, ξ R d, R +, h, ξ h, ξ δ h, ξ ξ ξ ρ ξ ξ 2 2.6 where δ h s a bouded fuco o [, T ] R d ad ρ. Remarks: Noe ha 2.6 appears as a covex assumpo relaxed by ρ ξ ξ 2. I mos suaos, s used he reverse sese.e. h, ξ h, ξ δ h, ξ ξ ξ + ρ ξ ξ 2. The sem-covexy assumpo s fulflled by a wde class of fucos: If h,. s C for every [, T ] ad h ξ, ξ s bouded, ρ-lpschz ξ, uformly he h s sem-covex wh δ h, ξ := h x, ξ. If h,. s covex for every [, T ] wh a dervave δ h,. he dsrbuo sese whch s bouded, ξ, he h s sem-covex wh ρ =. Thus, embodes mos usual payoff fucos used for prcg valla ad exoc Amerca syle opos lke h, ξ := e r K ϕe r ξ + wh ϕ Lpschz couous o ses {ϕ L}, L >. The oo of sem-covex fuco seems o appear [4] for prcg oe-dmesoal Amerca opos. See also [32] for rece developmes a smlar seg. 7

Theorem Le h : [, T ] R d R be a Lpschz couous fuco ad le p [, +. Le X k = S k or S k ad le V k k deoe he Sell evelope of h, X k k. a There s some posve real cosa C depedg o [b] Lp, [c] Lp, [h] Lp ad p such ha, k {,..., }, V k V k p ect + s. 2.7 b If furhermore X k = S k, k =,..., ad f he obsacle h s sem-covex, he, k {,..., }, V k V k p ect + s 2.8 2.2 Spaal dscrezao: he quazao ree The sarg po of he mehod s o dscreze he radom varables X k by some σx k - radom varables X k akg fely may values R d. Such a radom vecor X k s called a quazao of X k. Equvalely, oe may defe a quazao of X k by seg X k = q k X k where q k : R d R d s a Borel quazg fuco such ha q k R d = X k Ω = N k < +. The elemes of he se X k Ω are called elemeary quazers. Le N = N + N + + N deoe he oal umber of elemeary quazers used o quaze he whole Markov cha X k k. We am o approxmae he dyamc programmg formula 2.4 by a smlar dyamc programmg formula volvg he sequece X k k. 2.2. Quazao ree ad quazed pseudo-sell evelope We assume ha seco ha for every k {,,..., }, we have access o a sequece of quazaos X k = q k X k, k =,..., of he Markov cha X k k. We deoe by {x k,, xk N k } = q k R d he grd of N k pos used o quaze X k ad by x k = x k,..., xk N k he duced N k -uple. The quesos relaed o he opmal choce of x k ad q k wll be addressed Seco 2.4 below. Noe ha our orgal seg X = s, so ha X = s s he bes possble L p -mea quazao of X ad N =. The quazed dyamc programmg formula below s devsed by aalogy wh he orgal oe 2.4: oe smply replaces X k by s quazed radom vecor X k V := h, X, V k := max h, X k, E V k+ X k, k. Noao: for he sake of smplcy, from ow o, we wll deoe Êk. := E. X k. 2.9 The ma reaso for cosderg codoal expecao wh respec o X k s ha he he sequece X k k N s o Markova. O he oher had, eve f he N k -uple x k has bee se up a pror for every X k, hs does o make he umercal processg of hs algorhm possble. As a maer of fac, oe eeds o kow he jo dsrbuos of X k, X k+, k =,...,. Ths s elgheed by he proposo below whose easy proof s lef o he reader. From ow o, for coveece, we wll gve he prory o he N-uple oao. 8

Proposo Quazao ree algorhm For every k {,..., }, le x k := x k,..., xk N k, q k : R d {x k,..., xk N k } ad X k = q k X k be a quazao of X k. Se, for every k {,..., } ad every {,..., N k }, p k := P X k = x k = PX k C x k, 2. ad, for every k {,..., }, {,..., N k }, j {,..., N k+ } π k j := P X k+ = x k+ j = pk j p k wh X k = x k = P X k+ C j x k+ X k C x k p k j := P X k+ C j x k+, X k C x k. 2. Oe defes by a backward duco he fuco v k by v x := h x, {,..., N } N k+ v k x k := max h, x k, πj k v k+ x k+, N k, k,.2.2 j= The, V k = v k X k sasfes he above dyamc programmg 2.9 of he pseudo-sell evelope. Remark: If X k = S k or S k, he X = X = s ad v X = v s s deermsc. I more geeral segs oe approxmaes E v X by j E v X N = p vx. Implemeg he quazao ree algorhm 2.2 o a compuer rases wo quesos: How s possble o esmae he parameers p k ad p k j volved 2.2? Is possble o hadle he complexy of such a ree srucured algorhm? Parameer esmao A frs Moe Carlo approach: he racably of he above algorhm reles o he parameers πj k := pk /pk j. So, he ably o compue hem a a reasoable cos s he key of he mehod. The mos elemeary soluo s o process a wde scale Moe Carlo smulao of he Markov cha X k k o esmae he parameers p k ad p k j as defed by 2. ad 2.. A esmae of he ph power of he L p -mea quazao error X k X k p = E m p N X k x k p ca also be p compued. Whe X k k s a Euler scheme or Black & Scholes dffuso hs makes o problem. More geerally, hs depeds upo he ably o smulae some sample pahs of he cha sarg from ay x R d. We wll see furher o paragraph 2.4 how o choose he sze ad he geomerc locao of he N k -uples x k a opmal way. Complexy of he quazao ree: heory ad pracce A quck look a he srucure of he algorhm 2.2 shows ha gog from layer k + dow o layer k eeds κ N k N k+ elemeary compuaos κ s he complexy duced by a coeco j. Hece, he cos of a quazao ree desce s approxmaely Complexy = κ N N + N N 2 + + N k N k+ + + N N. = 9

The a elemeary opmzao uder cosra shows ha κ N 2 Complexy κn2 + 4. Lower boud s for N k = N/+, upper boud for he urealsc values N k = N 2 {,}. Ths purely combaoral lower boud eeds o be ued. I fac, mos examples he raso of he Markov cha behaves such a way ha, a each layer k, may erms of he raso marx [πj k ] are eglgble because xk ad xk+ j are remoe from each oher R d : he Moe Carlo esmaes of hese coeffces wll be. Hece, he complexy of he algorhm s ν κn raher ha lower boud κ N 2 / +, where ν deoes he average umber of acve coecos above a regular ode of he ree. Thus, he cos of such a desce s smlar o ha of a oe dmesoal bomal ree wh ν 2 N me seps such a ree approxmaely coas νn pos. 2.3 Covergece ad rae usg L p -mea quazao error I hs paragraph we provde some a pror L p -error bouds for V k V k p, k =,...,, based o he L p -mea quazao errors X k X k p, k =,...,, where quazer X k s a Voroo quazer ha akes N k values x k,..., xk N k. Ths error modulus ca be obaed as a by-produc of a Moe Carlo smulao of X k k : oly requres o compue, for every P Xk -dsrbued smulaed radom vecor, s dsace o s closes eghbor he se {x k,..., xk N k }. The esmaes Theorem 2 below holds for ay homogeeous Markov cha X k k havg a K-Lpschz raso P x, dy x R d sasfyg, for every Lpschz fuco g, [P g] Lp K[g] Lp. 2.3 Ths s he case of a dffuso ad of s he Euler scheme wh Lpschz drf ad dffuso coeffce as meoed before, see 2.3. Noe ha K may be lower ha : hs s, e.g., he case f X k s he Euler scheme of a Orse-Uhlebeck process wh drf bx := ax, a > ad sep T/ < /a. Theorem 2 Assume ha he raso P x, dy of he cha X k k s K-Lpschz, ha h s Lpschz couous x, uformly me ad se [h] Lp := max k [h,.] Lp. Le V k k ad V k k be lke 2.4 ad 2.9 respecvely. For every k {,..., }, le X k deoe a quazao of X k. The, for every p, V k V k p =k d X X p wh d := + 2 δ p,2 K [h] Lp,, d := [h] Lp δ u,v sads for he Kroecker symbol. Proof: Sep : We frs show ha he fucos v k recursvely defed by 2.5 are Lpschz couous wh [v k ] Lp K k [h] Lp. 2.4 Clearly, [v ] Lp [h] Lp ad oe cocludes by duco, usg he equaly maxa, b maxa, b max a a, b b.

Sep 2: Se Φ k := P v k+ k =,...,, Φ ad h k := h,., k =,...,. The fuco Φ k sasfes Ev k+ X k+ F k = Ev k+ X k+ X k = Φ k X k. Oe defes smlarly Φ k by he equaly Êk v k+ X k+ X k := Φ k X k, k =,..., ad Φ. The V k V k h k X k h k X k + Φ k X k Φ k X k [h] Lp X k X k + Φ k X k ÊkΦ k X k + ÊkΦ k X k Φ k X k.2.5 Now Φ k X k ÊkΦ k X k Φ k X k Φ k X k + Êk Φ k X k Φ k X k [Φ k ] Lp X k X k + Êk X k X k. Hece, Φ k X k ÊΦ kx k p 2[Φ k ] Lp X k X k p. Whe p = 2, he very defo of he codoal expecao as a projeco a Hlber space mples ha oe may remove he facor 2 he equaly. Now ÊkΦ k X k Φ k X k = Êk Ev k+ X k+ X k Êk v k+ X k+ v k+ X k+ v k+ X k+ = Êk sce X k s σx k -measurable. Codoal expecao beg a L p -coraco, follows ÊkΦ k X k Φ k X k p V k+ V k+ p. Fally, follows from he above equales ad 2.5 ha V k V k p [h] Lp + c[φ k ] Lp X k X k p + V k+ V k+ p, k {,..., }. O he oher had, V V p [h] Lp X X p, so ha V k V k p [h] Lp + 2 δ p,2 [Φ ] Lp X X p =k The defo of Φ ad he K-Lpschz propery of P x, dy complee he proof sce [Φ ] Lp = [P v + ] Lp K[v + ] Lp. 2.4 Opmzao of he quazao We beg by a bref roduco o opmal quazao of radom vecors see [24] for o overvew, he we address he problem of opmal quazao of Markov chas. 2.4. Opmal quazao of a radom vecor X Le X L p Ω, A, P. From a probablsc vewpo, opmal L R p -mea quazao p d cosss sudyg he bes L p -approxmao of X by some radom vecors X = qx akg a mos N values. Mmzg he L p -mea quazao error X qx p ca be decomposed o wo successve phases:

Opmzao phase. A N-uple x = x,..., x N R d N beg se, fd a quazer q x : R d {x,..., x N } f ay such ha { } X q x X p = f X qx p, q : R d {x,..., x N }, Borel fuco. Opmzao phase 2. Fd a N-uple x R d N f ay ha acheves he fmum of X q x X p over R d N,.e. { X q x X p = f X q x X p, x R d N}. The soluo o he frs opmzao problem s purely geomerc: s he closes eghbor projecos, deoed q x, duced by he Voroo essellaos of x as defed below. Defo 2 a Le x := x,, x N R d N. A Borel paro 2 C x, =,..., N of R d s a Voroo essellao of he x f, for every {,..., N}, C x sasfes C x {y R d x y = m y x j }. jn b The closes eghbor projeco or Voroo quazer fuco q x duced by he Voroo essellao C x s defed for every ξ R d, by q x ξ = N x C xξ. c The radom vecor X x = q x X = x C xx N s called a Voroo quazao of X. The N-uple x s ofe called a N-quazer. Noao: From ow o, he oao X x wll always deoe a Voroo quazao of X. Whe here s o ambguy, he expoe x wll ofe be dropped ad we wll deoe X sead of X x. Noe ha, he closure ad he boudary of he h cell C x are he same for ay Voroo essellao. Ths boudary s cluded o a mos N hyperplaes. If he dsrbuo P X of X weghs o hyperplae ha s P X H = for every hyperplae H of R d he all he Voroo essellaos are P X - equal ad all he Voroo quazaos X x have he same dsrbuo. The secod opmzao problem cosss mmzg o R d N he symmerc fuco x X X x p. Frs, oe ha he L p -mea quazao error sasfes X X x p p = N = E C x X x p =E m X x p N = m x ξ p P X dξ. 2.6 R d N I follows ha he L p -mea quazao error depeds o X hrough s dsrbuo P X. The secod cosequece of 2.6 s a mpora ad aracve feaure of he L p -mea quazao error compared o oher usual error bouds: s a Lpschz couous fuco of he N-quazer x := x,..., x N. Hece, as soo as P X has a compac suppor, x X X x p reaches a mmum a some L p -opmal N-quazer x. Whe P X o loger has a compac suppor, hs s sll rue: oe shows by duco o N see [24] or [38], ha x X X x p reaches a absolue mmum o R d N a some x R d N. 2 I wha follows, we wll assume ha a paro may coa he empy se: hs wll happe whe x = x j for some j. 2

Proposo 2 A L p -opmal N-quazer x for X L p Ω, P sasfes X X { } x p = m X Z p, Z : Ω R d, radom vecor, ZΩ N. 2.7 Proof: Le ZΩ = {z,..., z N }. Se z := z,..., z N wh possbly z = z j. The X X x p X X z p = m Xω z p p p Pdω m Xω Zω p Pdω = X Z p. p Ω Moreover, he followg smple facs hold rue see [24] or [38] ad he refereces here: If supp P X has a fe suppor, ay opmal N-quazer x has parwse dsc elemes, ha s q x R d = X x Ω = N. The closed covex hull H X of supp P X coas a leas a opmal quazer obaed as he projeco of ay opmal quazer o H X. Furhermore, f supp P X s covex.e. equal o H X, he he N dsc compoes of ay opmal N-quazer x all le H X. Ths also holds rue for H X -valued locally opmal N-quazers. Rae of covergece: The ma fuco of he L p -mea quazao error beg o be a error boud, s mpora o elucdae he behavor of X X x p as he sze N of he opmal N-quazer x go o fy. The frs easy fac s ha goes o as N.e. lm m X X x p =. N x R d N Ideed, le z k k N deoe a everywhere dese sequece of R d -valued vecors ad se x N := {z,..., z N }. The X X x N p goes o zero by he Lebesgue domaed covergece heorem. Furhermore m x R d N X X x p X X x N p. The rae of hs covergece urs ou o be a much more challegg problem. Is soluo, ofe referred o as Zador s Theorem, was compleed by several auhors Zador, see [25], Bucklew & Wse, see [3] ad fally Graf & Luschgy see [24]. Ω Theorem 3 Asympocs Assume ha E X p+η < + for some η >. The + p lm N p d m X X x p = J N x R d N p p,d ϕu d d d+p du R d 2.8 where P X du = ϕu λ d du + νdu, ν λ d λ d Lebesgue measure o R d. The cosa J p,d correspods o he case of he uform dsrbuo o [, ] d. Lle s kow abou he rue value of he cosa J p,d excep dmeso where J p, = 2 p p+. Some geomerc cosderaos lead o J 2,2 = 5 8 see [25] or [24]. Neverheless, some upper ad lower bouds were esablshed, based o ball packg echques 3 ad o he roduco of radom quazers see e.g. [7] ad [24]. I follows ha J p,d d 2πe p/2 as d + see [24]. Ths heorem says ha m x R d N X X x p = C X,p,d N d + on d : hs s accordace wh he raes obaed wh uform produc lace grds of sze N = m d for umercal egrao wh respec o he uform dsrbuo over [, ] d. Eve ha very case, o such lace grd s a opmal quazer excep whe d =. The cocluso s ha, for ay dsrbuo P X, opmal quazao produces for every N he bes machg N-grd for P X. Asympocally, a sequece of opmal quazers yelds he lowes possble cosa C X,p,d, wh a obvous umercal eres. 3

2.5 How o ge opmal quazao usg smulao Opmal quazao of a sgle radom vecor: how o ge? I fac he L p -mea quazao error fuco s eve smooher ha Lpschz couous. Ths s a he org of a mpora a sochasc opmzao mehod based o smulao. Frs, we cosder for coveece s p h power, deoed D p, defed for every x = x N,..., x N R d N by D p x = X X x p = E m X x N p p = d p x, ξp dξ N R d N X where d p x, ξ := m x N ξ p. N The leer D refers o he word dsoro used Iformao Theory. The fuco d p x, ξ s ofe called local Lp -dsoro. N Oe shows see, e.g., [24] or [38] ha, f p >, D p s couously dffereable a N every x R d N sasfyg he admssbly codo j, x x j ad P X N = C x =. 2.9 The, s grade D p x s obaed by formal dffereao, ha s N where D E p x := dp Rd d p N N x, X = x, ξp N x x X dξ d p N x x, ξ := p x ξ x ξ x ξ p C xξ,, wh he coveo =. The above dffereably resul sll holds whe p = f P X s couous.e. P X {ξ} =, ξ R d. Oe oes ha D p has a egral represeao wh respec o he dsrbuo N of X. Whe he dsrbuo P X s smulaable, hs srogly suggess o mpleme a sochasc grade desce derved from hs represeao o approxmae some local mmum of D p : whe d 2, he mplemeao of deermsc grade desce N becomes urealsc sce would rely o he compuao of may egrals wh respec... o P X. Ths sochasc grade desce s defed as follows: le ξ N be a sequece of..d. P X -dsrbued radom varables ad le γ N be a sequece of, -valued seps sasfyg γ = + ad γ 2 < +. 2.2 Se, for every admssble x R d N he sese of 2.9, ad every ξ R d d x d p p x, ξ := N x, ξ. N x N The, sarg from a deermsc al N-uple X = x wh N parwse dsc compoes, oe defes recursvely for every, X = X γ p xd p N X, ξ 2.2 hs formula a.s. gras by duco ha x has parwse dsc compoes. From a heorecal vewpo, he ma dffculy s ha he assumpos usually made ha esure he a.s. covergece of such a procedure are o fulflled by D p see, e.g. [8] N 4

or [3] for a overvew o Sochasc approxmao. To be more specfc, le us sress ha D p x N,..., x N does o go o fy as max N x goes o fy ad D p s N clearly o a Lpschz fuco. So s o a approprae Lyapuov fuco. However some weaker codoal a.s. covergece resuls he Kusher & Clark sese have bee obaed [38] for compacly suppored absoluely couous dsrbuos P X he case p = 2. I dmeso, regular a.s. covergece holds f furhermore he desy fuco of P X s bouded. The quadrac case p = 2 s he mos commoly mplemeed for applcaos. I s kow Iformao Theory leraure as as he Compeve Learg Vecor Quazao CLV Q algorhm. The syhec formula 2.2 ca be dealed as follows: se X := X,..., X N, Compeve phase: selec + argm X ξ + 2.22 X + Learg phase: + := X+ γ X+ ξ+ + X+ ξ+ X + ξ+ p 2.23 [.6em]X + := X, +. Compao parameer procedure: Assume ha X L p+η for some η, ] ad le γ be a sequece of, -valued seps sasfyg γ = + ad The, oe defes recursvely he followg sequeces γ +η < +., p + := p γ + + γ + {=+}, N,, p :=, N, D r,+ N := D r, N + + γ + X+ ξ+ r, D r, N := where r [, p]. The, o he eve {X x }, {,..., N}, r [, p], p D r, N a.s. P X C x, as, 2.24 a.s. D r N x as. 2.25 Two aural choces for γ are γ = γ ad γ = / for some umercal expermes see [39]. The proof of 2.24 ad 2.25 reles o some usual margale echques comg from Sochasc Approxmao see [38] or [3] for a dealed proof he secod seg. Whe γ = /, oe has a smple syhec expresso for 2.24 ad 2.24 whch ca be aracve for umercal purpose, amely p = {s {,..., } ξ s C X s } ad D r, = N s= X s s ξs r. 2.26 These compao procedures are cosless sce hey use some by-producs of he compeve ad learg phases of he procedure. They yeld he parameers P X -weghs of he Voroo cells C x, L p -mea quazao error X X x p eeded for a umercal use of he quazer x. The fac ha hese compao procedures work o he eve {X x } whaever he lmg N-uple x s shows her cossecy. 5

Cocerg he praccal mplemeao of he algorhm, s o be oced ha, he quadrac case p = 2 CLV Q algorhm, a each sep, he N-uple X + remas he covex hull of X ad ξ +. Ths duces a sablzg effec o he procedure whch s observed o smulaos whch explas why he regular CLV Q algorhm s more ofe mplemeed ha s o-quadrac couerpars. See [39] for a exesve umercal sudy of he CLV Q algorhm for Gaussa radom vecors. Ths lead o a large scale quazao of he mulvarae ormal dsrbuos dmesos d = up o d = wh a wde rage of values of N. Opmzao of he quazao ree: he exeded CLV Q algorhm The prcple s o modfy a Moe Carlo smulao of he cha X k k by processg a CLV Q algorhm a each me sep k. Oe sars from a large scale Moe Carlo smulao of he Markov cha X k k.e. depede copes ξ := ξ,,..., ξ,, ξ := ξ,,..., ξ,,..., ξ := ξ,..., ξ,... of X k k. Our am s ow o produce for every k {,..., } a quadrac opmal quazer X k, := x k,,..., xk, N k wh sze N k, wh s raso kerel [π,k j ], he dsrbuo p,k Nk of X x k k ad he duced mea L p -quazao errors p 2. Noe ha, f oe ses he π,k j = p,k j p,k p,k j ad p,k := P of he jo dsrbuo marces [p,k j ]. {X k+ C j x,k+ } {X k C x,k } = j p,k j, k =,...,. So oe may focus o he esmao I he preseao below of he exeded CLVQ algorhm, we assume ha he Markov cha sars X = x R d, bu oher choces are possble. We also assume ha k {,..., }, P Xk s couous ad E X k 2+η < + 2.27 for some η >. Ths s o a very demadg assumpo whe dealg wh a dffuso process sampled a dscree mes or a Euler scheme. We adop here he seg whch he compao sep sequece s γ = / ad we rely o he o-recursve expressos lke 2.26. We propose o compue he L r -mea quazao error for a fxed r [, 2] usually r = or 2 applcaos. The he algorhm reads as follows.. Ialzao phase = : Ialze he sarg N k -uples X k, := {x,k,..., x,k N k }, of he CLV Q algorhms ha wll quaze he dsrbuos P Xk, k =,..., [se N = ad X, = {x }]. Ialze he jo dsrbuo couers β k, j :=, {,..., N k }, j {,..., N k }, k =,...,. Ialze he margal dsrbuo couer α k, :=, N k, k =,...,. Ialze he L r -mea quazao couer d k, :=, N k, k =,...,. 2. Updag + : A sep, he N k -uples X k,, k, have bee obaed. We use ξ + := ξ,+,..., ξ k,+,..., ξ,+ o carry o he opmzao process a every me sep.e. updag he grds X k, o X k,+ as follows. For every k =,..., : Smulae ξ k,+ usg ξ k,+ f k 2 or x f k =. Selec he wer he k h CLVQ algorhm.e. he dex k,+ {,..., N k } sasfyg ξ k,+ C k,+x k,. 6

Updae he k h CLV Q algorhm: X k,+ = X k, γ + {= k,+ } Xk, ξ k,+, N k. Updae of he L r -mea quazao error couer d k, : d k,+ := d k, + X k, k,+ ξ k,+ p. Updae he dsrbuo couers β k, := β k, j Nk,jN k ad α k, Nk, k =,..., se α,+ = + ad,+ := : β k,+ j := β k, j + { = k,+, j = k,+ }, N k, j N k α k,+ := α k, + { = k,+ }, N k. Oe { shows, lke for 2.24, ha for every k {,..., }, o he eve X k, x k, } { X k, x k, }, π k, j β k, j α k, := βk, j α k, d k, a.s. a.s. a.s. p,k j = PX k C x k,, X k C j x k,, 2.28 N k, j N k, p,k = PX k C x k,, N k, 2.29 π,k j = PX k C j x k, X k C x k,, 2.3 N k, j N k, a.s. D X k,2 N k x k, as +. 2.3 From a praccal vewpo, hs exeded verso has he same feaures as he regular CLV Q algorhm as far as covergece s cocered. Oe mpora fac s ha he opmzaos of he quazers a he successve me seps are processed smulaeously bu depedely: he quazao opmzao a me sep k does o affec ha of me sep k +. 2.6 A pror error bouds me ad space Proposo 3 below s he applcao of Theorem 2 o he geeral dffuso model 2. a mes = kt/ ad s Euler scheme. The error srucure s he same excep ha he real cosa does o deped o opmaly of he quazers X k s o requred. The ma resul of hs seco s Theorem 4 whch addresses he las opmzao problem: assumg ha every quazao X k s opmal, wha s he opmal dspachg of he elemeary quazers amog he me dscrezao seps. Proposo 3 Assume ha he coeffces b ad c of he dffuso 2. ad he obsacle fuco h are Lpschz couous. Le v k X k k be he pseudo-sell evelope of h, X k k defed by 2.9. For every p [, +, here exss a posve real cosa C [b]lp,[σ] Lp,[h] Lp,T,p > such ha, k {,..., }, V k v k X k p C [b]lp,[σ] Lp,[h] Lp,T,p X l X l p. 2.32 l=k 7

Oe ges rd of sce he Lpschz coeffce K of boh chas S k ad S k sasfy lm sup K < + see [3] for deals. To go furher we eed a ew kd of assumpo o he margal dsrbuos of X k : we wll assume ha he L p -mea quazao errors of he X k are ϕ-domaed of he followg sese: here exss a radom vecor R L p+η P η > ad a sequece ϕ k, k< such ha, for every, every k {,..., } ad every N, m x R d N X k X x k p ϕ k, m R R x p. 2.33 x R d N The po s ha he dsrbuo of R may deped o p bu o o N, k or. I s show [3] ha uformly ellpc dffusos cc x ε I d, ε > sasfyg eher b, c C b Rd hece wh possbly lear growh followg [3] or b ad c are bouded, b s dffereable, c s wce dffereable ad Db, Dc ad D 2 c are bouded ad Lpschz followg [23], Theorem 5.4, p.48-49, fulfll he domao propery 2.33 wh ϕ k, := c b,σ,t k/. We show here ha he local volaly model.3 also sasfes hs domao propery. Proposo 4 Local volaly model Assume ha q d ad ha σ :, + d R d q s uformly ellpc σσ ξ ε I d, ε >, bouded, hree mes dffereable ad sasfes l,..., l k {,..., d}, k σ j ξ l ξ,..., ξ d = O ξ l k ξ l ξ l k as ξ + 2.34 for every k =, 2, 3. The S k k sasfes he ϕ-domao propery 2.33 wh ϕ k, := c σ,t s k/ c σ,t > ad R := Z l + e Zl ld, Z N ; I d, 2.35 Remark: Assumpo 2.34 ca be weakeed o ξ σσ e ξ,..., e ξd s bouded, wce dffereable wh bouded Lpschz frs wo dffereals. Proof: Oe sars from he elemeary equaly, vald for every ξ, ξ R ad every ρ >, e ρξ e ρξ ρ ξ + e ξ ξ + e ξ. 2.36 Le Y := ls /s,..., ls d /s where S deoes a soluo of.3 wh r =. The Y s a dffuso process soluo of he SDE dy = δy d + ϑy dw, Y =,...,, wh δy := σ l. e y,..., e yd 2 ad ϑy := σe y,..., e yd. 2 ld I follows from Assumpo 2.34 o σ ha δ ad ϑ are wce dffereable ad ha δ, Dδ ad D k ϑϑ, k =,, 2 are Lpschz couous ad bouded. Ths mples see [23], Theorem 5.4, p.48-49 ha, for every, T ], Y has a absoluely couous dsrbuo P Y = p yλ d dy sasfyg p y α π βz y α, β > where π βz deoes he desy fuco of β Z, Z N ; I d. 8

Now le N ad le r := r N be a L p -opmal N-quazer of he radom vecor R. Oe defes for every k =,...,, a N-quazer x k, := x k, N by x k, l := s l exp β r l, l =,..., d. Now, comg back o S whch sars ow a S := s, oe has for every k =,...,, f S k Ŝx x R d p S + N p k Ŝxk, p p = E m N sl e Y l k ld x k, p α E m N s l β k Z e l k e β r ld l p α β p/2 max ld sl p E m N Z + ez Z + e Z r p. The las equaly follows from 2.36. Ths complees he proof. Assume ha every quazao X k s L p -opmal wh sze N k. The, combg he bouds obaed Theorem me dscrezao error ad Proposo 3 spaal dscrezao error wh Zador Theorem Theorem 3, asympocs of opmal quazao yelds he followg error srucure C θ + C 2 k= k N d k wh N + + N = N 2.37 me s excluded sce X = s perfecly quazes S = s. Mmzg he rgh had of he sum s a easy opmzao problem wh cosra. The, order o mmze 2.37, oe has o make a balace bewee he me ad spaal dscrezao errors. The resuls are dealed Theorem 4 below. Theorem 4 Opmzed quazao ree ad resulg error bouds Assume ha b, σ ad h are Lpschz couous, ha S k k s ϕ-domaed he sese of 2.33 by ϕ k, := c k/ Le, N +. Se X = S = s ad assume ha, for every k {,..., }, Xk s a L p -opmal Voroo quazao of X k wh sze N k = X 2d+ k Ω := k N d d 2d+ 2d+ + + + + d k d 2d+, 2.38 where x := m{k N k x} he N = ad N N + + N N +. Le v k X k k be he quazed pseudo-sell evelope of h, X k k. a Dffuso: If X k := S k, k =,...,, he max V v k X k p C p e C pt k wh θ = f h s sem-covex ad θ = /2 oherwse. b Euler scheme: If X k := S k, k =,...,, he max V v k X k p C p e CpT k + s θ + + d N d + s + + d N d.. Remark: If, N + wh = on he N k 3d+2 d k 2d+ 2d+ N 2.38. 9

3 Hedgg Tacklg he queso of hedgg Amerca opos eeds o go deeper facal modelg, a leas from a heursc po of vew. So, we wll shorly recall he prcples ha gover he prcg ad hedgg of Amerca opos o jusfy our approach. Frs, we come back o he orgal dffuso model.3 whch drves he asse prce process S wh r =. We assume ha so ha q = d ad ξ R d, σσ ξ ε I d 3. ε Dagξ 2,..., ξ d 2 cc ξ σσ ξ 2 I d where σσ ξ := sup ξ R d σσ ξ. Noao: For oaoal coveece we wll make he coveo hroughou hs seco ha f X s a couous me process ad = kt/, X k+ := X k+ X k, k =,...,. 3. Hedgg couous me Amerca opos Frs we eed o come back shorly o classcal Europea opo prcg heory. Le h T be a Europea coge clam ha s a oegave F T -measurable varable. Assume for he sake of smplcy ha les L 2 P, F T. The represeao heorem for Browa margale shows see [4] ha h T = Eh T + H s.dw s = Eh T + Z s.ds s 3.2 where H s a dp d-square egrable F-predcable process ad Z s := [cs s ] H s. Hece M := Eh T F sasfes M = M + Z s.ds s. A aalogy wh dscree me model shows ha he egral Z s.ds s represes he algebrac ga from me up o me T provded by he sraegy Zs l ld,s [,T ] a every me s [, T ] he porfolo coas exacly Zs l us of asse l. So, a me T, he value of he porfolo vesed rsky asses S,..., S d s exacly h T moeary us: pu some way roud, he porfolo Z replcaes he payoff h T ; so s aural o defe he heorecal premum as Premum := Eh T F = Eh T + Z s.ds s. 3.3 If h T := ht, S T, he Markov propery of S mples ha Premum := p, S. If h s regular eough, he p solves he parabolc P.D.E. p + L r,σp =, pt,. := ht,. ad a sraghforward applcao of Iô formula shows ha Z = x p, S. Le us come back o Amerca opo prcg. If oe defes he premum process V [,T ] of a Amerca opo by he P-Sell evelope of s payoff process, he hs premum process s a supermargale ha ca be decomposed as he dfferece of a margale M ad a odecreasg pah-couous process K.e., usg he represeao propery of Browa margales, V = M K = V + Z s.ds s K K :=. 2

So, f a rader replcaes he Europea opo relaed o he ukow Europea payoff M T usg Z, he s poso o be he couerpar a every me of he ower of he opo case of early exercse sce M = V + K V h. I case of a opmal exercse of hs couerpar he wll acually have exacly he payoff a me sce all opmal exercse mes occur before he process K leaves. If he varaoal equaly.7 adms a regular eough soluo ν, x, he Z = x ν, S. I mos deermsc umercal mehods, he approxmao of such a dervave s usually less accurae ha ha of he fuco ν self. So, s hopeless o mpleme such mehods for hs purpose as soo as he dmeso d 3. 3.2 Hedgg Bermuda opos Le V k deoe he heorecal premum process of he Bermuda opo relaed o h, S k k. I s a F k k -supermargale defed as a Sell evelope by V := ess sup {E k hτ, S τ, τ Θ k } where Θ k deoes he se of {,..., }-valued F-soppg mes. The, he F k -Doob decomposo of V as a he F k -supermargale yeld: V = M k A k, where M k s a F k -L 2 -margale ad A s a o-decreasg egrable F k -predcable process A :=. I fac, he creme of A k ca easly be specfed sce A k := A k A k = V E k V = h, S k E k V +. 3.4 The represeao heorem appled o each me erval [, + ], k =,..., he yelds a F-progressvely measurable process Z s s [,T ] sasfyg M k := k Zs.dS s, k, wh E c S s Zs 2 ds < + 3.5 keep md ha < U s.ds s > = c S s U s 2 ds. Now, such a seg, couous me hedgg of a Bermuda opo s urealsc sce he approxmao of a Amerca by a Bermuda opo s drecly movaed by dscree me hedgg a mes. So, seems aural o look for wha a rader ca do bes whe hedgg oly a mes. Ths leads o roduce he closed subspace P of L 2 c S. dp d := {Z s [,T ] progressvely measurable, c S s Z s 2 ds < + } defed by P = { ζ s s [,T ], ζ s := ζ k, s [, +, ζ k F k -measurable, E } c S s ζ s 2 ds <+. 3.6 ad he duced orhogoal projeco proj oo P for oaoal smplcy a process ζ P wll be ofe referred as ζ k k. I parcular, for every U L 2 c S. dp d c S. proj U. L 2 dp d c S. U. L 2 dp d. 2

Dog so, we follow classcal deas roduced by by Föllmer & Soderma [2] for hedgg purpose complee markes see also []. Oe checks ha P s somerc wh he se of square egrable sochasc egrals wh respec o S k k, amely { } ζ k. S k+, ζ k k P. k= Compug proj Z ṇ amous o mmzg E k= k+ c S s Z s ζ k 2 ds ζ k k P. Seg ζ := proj Z ṇ ad sadard compuaos yeld ζ := E k k+ k+ cc S s ds E k cc S s Zs ds over = E k S k+ S k+ E k M k+ S + 3.7 = E k S k+ S k+ E k V k+ S k+. 3.8 The las equaly follows from he fac ha A k s F -measurable ad from he margale propery of S k. The creme R + := k+ Z s ζ.ds s = M k+ ζ. S k+ 3.9 represes he hedgg defaul duced by usg ζ k sead of Z ṇ. The sequece R k k s a F k -margale creme process, sgular wh respec o S k k sce E k R k+ S k+ =. I s possble o defe he local resdual rsk by k+ E k R k+ 2 = E k c S s Zs ζ k 2 ds, k {,..., }. 3. A lle algebra yelds he followg, whch s more approprae for quazao purpose: E k R + 2 =E k V + E k V + 2 E k S k+ S + E k V + S k+ 2. 3. Formulae 3.8 or 3., based o S k ad V k have aural approxmaos by quazao. O he oher had, 3.7 ad 3. are more approprae o produce some a pror error bouds whe smulao of he dffuso s possble. 3.3 Hedgg Bermuda opo o he Euler scheme Whe he dffuso cao be easly smulaed, we cosder he couous me Euler scheme defed by [, +, S = S k + cs k W W k, S := s >. Ths process s P-a.s. defed sce s a.s. ozero bu may become egave adverse o he orgal dffuso. The, mmckg he above subseco, leads o defe some processes Z, M ad A by V := M k A k Doob decomposo M k := k Z s cs s dw s = k Z s.ds s wh s = f s [, + A k := A k A k = V E k V = h, S k E k V +. 22

ad A :=. The smpler formulae for he hedgg process hold ζ := E k S k+ S + E k V + S k+ = + E k k+ Z s ds. 3.2 The relaed hedgg defaul ad local resdual rsk are defed by mmckg 3. ad 3.: R + := k+ Z s ζ.ds s = M k+ ζ. S k+ 3.3 E k R + 2 := E k V + E k V + 2 E k S k+ S + E k V 2 + S k+ 3.4. 3.4 Quazed hedgg ad local resdual rsks The quazed formulae for sraeges ad resdual rsks are smply derved from formulae 3.8 or 3.2 by replacg S k S k respecvely by her quazao Ŝ Ŝ respecvely ad Vk := v k S by V k seco 2 ha V ζ k := v k Ŝ V k := v k Ŝ respecvely. I follows from := v k S k s approxmaed by v k Ŝ. So, oe ses for he dffuso := E Ŝ+ Ŝ + Ê k v k+ Ŝ+ v k Ŝ Ŝ+ Ŝ. 3.5 R + 2 := E k V + E k V + 2 E k Ŝ+ Ŝ + E k V + Ŝ+ 2.3.6 Oe derves her couerpars ζ k, R + 2 for he Euler scheme by aalogy. The po o be oced s ha compug ζ k or ζ k a a gve elemeary quazer x k of he k h layer requres o ver oly oe marx whch does o cos much. 4 Covergece of he hedgg sraeges ad raes Ths seco s devoed o he evaluao of he dffere errors quazao, resdual rsks duced by me ad spaal dscrezaos. 4. From Bermuda o Amerca me dscrezao Frs, oe exeds he defo of V a ay me [, T ] by seg V := V + Z s.ds s = V + k+ Z s.ds s + A k+, [, +. 4. Ths defo mples ha, for every k {,..., }, he lef-lm of V sasfes V = V + A k+. 4.2 Proposo 5 Assume ha he payoff process h = h, S where h s a sem-covex fuco. Assume ha he dffuso coeffce c s Lpschz couous. a For every k {,..., }, V k V k ad for every, +, Furhermore P-a.s., for every [, T ], V V + A k+. { V V C h,c T + E max st S s 2, V V [h] Lp E max k S k S k. b The followg boud holds for he hedgg sraeges he cc merc E c S s Z s Zs 2 ds + E c S s Zs c S s Z s 2 T ds C h,c. 4.3 23

Proof: a The equaly bewee V ad V a mes s obvous sce V s defed as a supremum over a larger se of soppg mes ha V k. The, usg he supermargale propery of V, equaly 4. ad Jese equaly yeld V V + E V + + A k+ E V k+ + E V + V k+ + A k+ + A k+. Now, usg he expresso 3.4 for A k+ ad V h+, S k+ mply A k+ = h, S k E k V + + h, S k E k h+, S k+ + We eed a hs sage o use he regulary of h sem-covex Lpschz couous h, S k h+, S k+ = h, S k+ h+, S k+ + h, S k h, S k+ [h] Lp + δ h, S k.s k+ S k + ρ h S k+ S k 2. Hece h, S k E k h+, S k+ [h] Lp + + ρ h E k S k+ S k 2 for some cosa C h,c >. Fally, yelds k+ [h] Lp + + ρ h E k Trcc S s ds [h] Lp + + Cρ h + + E max A k+ C T c,h T C c,h + E k max S s 2, s [,T ] + E k max s [,T ] S s 2 S s 2 s [,T ]. 4.4 To complee he equaly for V V, we frs oce ha, f [, + k+ k+ V = V k+ Zs.dS s + A k+ h+, S k+ Zs.dS s 4.5 so ha V = E V E h+, S k+ = h, S + E h+, S k+ h, S. Usg aga he sem-covexy propery of h a, S fally yelds ha V T + C c,h + E max S s 2 h, S. s [,T ] As s a supermargale as well, ecessarly sasfes P-a.s. V T + C c,h + E max S s 2 Sellh, S = V s whch yelds he expeced resul. The secod equaly s obvous oce oced V V max h, S k h, S k [h] Lp max S S k. b Oe cosders he càdlàg sem-margale V V = V V + Z s Zs.dS s K A where := k o [, +. I follows from Iô formula for jump processes ha c S s Z s Z s 2 ds + T A 2 + V V 2 24

Now V s V s dk s A s = = 2 V s Vs Z s Zs.dS s + 2 V s V s dk s + V s Vs dk s + T V s V s da s A k 2 V s V s dk s A s. sce V k = V k + A k V + A k. Ths yelds, usg he equaly obaed a ad 4.4, V s V s dk s A s C h,c T C h,c T + E s sup S u 2 dk s + A max sut < T A k K T +sup E s sup S u 2 s [,T ] sut Oe checks ha V s Vs Z s Zs.dS s s a rue margale so ha E c S s Z s Zs 2 ds C h,c T K T 2 + + max s [,T ] S s 2 2. Now K T L 2 sce K T V + Z s.ds s whch yelds he expeced resul. + +sup E s sup S u 2 s [,T ] sut The equaly volvg he Euler scheme s obaed followg he same approach usg ow V V. E c S s Z s c S s Z s 2 ds 2 E Now K T 2 V 2 + 2[h] Lp E C E sup K T 2 < +. Cocerg K T oe has V s V s dk s K s + EhT, S T ht, S T 2 E s max S S k dks + K s + [h] 2 S Lp T S T 2 2 s sup E max S S k KT + K T + C S T S T 2 2 [,T ] C sup E max S S k 2 K T 2 + K T 2 + C ST S T 2 2 [,T ] T C h,c K T 2 + K T 2 +. 4.6 Z s Zs.dS s 2 C + sup S s 2 + O/, hece s [,T ] KT K T 2 V 2 + V 2 + Zs.dS s Z s.ds s 2 C + O/ by 4.6 2. so ha sup K T 2 < +. Pluggg hs back 4.6 complees he proof. We are ow poso o ge a frs resul abou he corol of resdual rsks duced by he use of dscree me hedgg sraeges. I shows ha hs corol s esseally ruled by he pah-regulary of he process Z. 25

Theorem 5 If h ad c are Lpschz couous ad h s sem-covex, he, c S. Z. ζ ṇ L 2 dp d c S. Z. proj Z. L 2 dp d + C 4.7 proj Z s he projeco of Z o P. Furhermore c S. Z. proj Z. L 2 dp d goes o as. Remark: The erm c S. Z. proj Z. L 2 dp d whch rules he rae of covergece of c S. Z. ζ ṇ L 2 dp d clearly depeds o he pah-regulary of Z s. Theorem 6c below provdes some elemes abou s ow rae of covergece. Proof: Se for coveece ζ := proj Z. Mkowsk equaly yelds c S. Z s ζ ṇ L 2 dp d c S. Z. ζ. L 2 dp d + c S. ζ. ζ ṇ L 2 dp d. Now ζ. ζ ṇ = proj Z. Z ṇ so ha by Iequaly 4.3 Proposo 5b, c S. ζ. ζ ṇ L 2 dp d c S. Z. Z ṇ L 2 dp d C. Now, le F be a bouded adaped couous-pah process. Se Φ s := T [, +. Usg he properes of proj, oe ges k+ F u du, s c S. Z. ζ. L 2 dp d 2 c S. Z. F. L 2 dp d + c S. F. proj F. L 2 dp d 2 c S. Z. F. L 2 dp d + c S. F. Φ. L 2 dp d T 2 c S. Z. F. L 2 dp d + cs s 2 dswf, T L 2 F 2 2 P where wf, δ deoes he uform couy modulus of F. Oe cocludes usg ha L c S dp d s everywhere dese L 2 c S dp d. 4.2 Hedgg error duced by he quadrac quazao We wll focus o he error a me =. Proposo 6 Assume ha σ s Lpschz couous, bouded ad uformly ellpc ad ha h s Lpschz couous. Assume ha he dspachg rule 2.38 of he N k apples ad ha he quadrac quazao of he S k are opmal. Assume ha N ad go o + so ha lm N/ d 2d+ + = +. The, for every ζ ζ C + s ε m ld s l 2 3 2. N/ d Proof: The hedgg vecors ζ ad ˆζ sasfy respecvely E S S ζ = E V V S 4.8 E Ŝ Ŝ ζ = E V V Ŝ 4.9 where V = v S ad V = v s, ec. The quadrac quazao Ŝ of S beg opmal ad S =Ŝ =s beg deermsc, oe has E S Ŝ = Ŝ. I parcular E S =E Ŝ ad Ŝ 2 S 2 = S s 2 C T/ + s. 26

The E S S E Ŝ Ŝ = E S Ŝ S Ŝ so ha E S S E Ŝ Ŝ S Ŝ 2 2 CN 2 d. Now E V V S E V V Ŝ Oe derves from 4.8 ad 4.9 ha E S S ζ ζ Ŝ 2 V V 2 + V V + V 2 S Ŝ 2 C + s N/ d + C N d E V V S E V V Ŝ + E S S E Ŝ Ŝ ζ C + s N/ d + C N 2 d ζ. Now, follows from cc ξ ε Dagξ 2 ha E S S ε m ld ESl s 2 ds I d ε m ld ESl s 2 ds I d = m ld sl 2 ε T I d so ha E S S ε m ld sl 2 /T. ζ C ε. Hece ζ ζ C ε m ld s l + s + 2 N/ d C ε m ld s l + s + 2 N/ d Frs, oe derves from 4.8 ha N d N d + + + ζ N 2 d N 2 d + ζ ζ + The dspachg rule 2.38 mples ha N = C d N 2d+ + on, so ha, gve N he above assumpo, lm = +.e. d 2 N d goes o. Cosequely ζ ζ C ε m ld s l + s + 2 N/ d d N d + ε N 2 d.. ε Ispecg he hree erms o he rghhad sde of he equaly complees he proof. Remark: The above proof pos ous he fac ha a quazao ree opmzed for he premum compuao s o opmal a all for he hedgg. So, he above error boud could be mproved f oe adops aoher dspachg polcy, opmzed for he hedgg, alhough wll ever reach he performaces devoed o he premum compuao. 27

4.3 Approxmao of he sraegy: rae of covergece I hs seco we evaluae he global resdual rsk o me ervals [, T ], T < T, duced by he use of he me dscrezao of he dffuso wh sep T/, amely E δ c S s Z s ζ s 2 ds, 4. where Z s defed by 3.2 ad ζ := proj Z s he projeco o he se P of elemeary predcable sraeges. Our basc assumpo hs seco s Σ c C b, + d, σσ x ε I d. Assumpo Σ s fulflled whe σ C b, + d C b, + d ad k σ. x = O/ x as x +, k, =,..., d. Theorem 6 Assume ha Σ holds, ha h s Lpschz couous ad ha s, + d. a For every T [, T here exss some real cosas K ad θ ad a eger q 2 oly depedg o T ad o he bouds of σ ad s frs wo dervaves such ha E c S s Z s ζ s 2 ds + T T 3/2 + s q K e θ l. 4. b Le δ := ρ 3 ρ >. There exss some real cosas K ad θ ad a eger q 2 depedg o ρ, T ad o he bouds of σ ad s frs wo dervaves such ha E δ c S s Z s ζ s 2 ds K ε 5/2 ε 5/2 2 + s q eθ l. 4.2 6 c If furhermore h s sem-covex. The raes obaed ems a ad b rule he rae of covergece of c S. Z. ζ ṇ L 2 Ω [,T ]dp d Theorem 5 whe T = T < T or T = T δ respecvely. Remarks: The erm e θ l s due o he o-uform ellpcy of S: hs s he cos of rucao aroud zero. Oe may look a ha some way roud: f we had worked wh he uformly ellpc dffuso X = ls sead of S, he he obsacle fuco would have become h, exp x, wh a expoeal growh. So a rucao would have bee ecessary wh a smlar cos. I mos facal applcaos he obsacle h s a mos Lpschz couous for example h, x = e r K e r x + for a pu of srke K. However, f he obsacle s more regular, amely h C,2, he o regularzao s eeded ad he resulg error s Oe θ l / ad Oe θ l / /3 clams a ad b of Theorem 6 respecvely. Fally, case of a uformly ellpc dffuso our mehod of proof would lead o O/ ad O/ /3 raes respecvely. Some echcal dffcules arse whe evaluag he erm 4. drecly, so we frs reduce he problem o a smpler oe. Ths s doe wo seps. Lemma Sep Se H s := c S s.z s ad η s := T E k+ The, uder he assumpos of Theorem 6, E H u du, s [, +. c S s Z s ζ s 2 ds C + 2 E H s η s 2 ds. 4.3 28

k+ Proof: We emporarly defe z s := E k Z r dr, s < +. Noe ha z + s a adaped process whch s pecewse cosa. Sce ζ s he L 2 projeco of Z o he subspace of hese ype of processes, we have E c S s Z s ζ s 2 ds E 2E c S s Z s z s 2 ds H s η s 2 ds + 2E η s c S s z s 2 ds. I remas o prove ha he secod erm he rgh had of he above equaly s domaed by C/. We wre hs erm as E k= k+ c S s E k + wh I := E J := E k+ Z u du + E k k+ k= k+ k= k+ c S s c S k k+ E k + E k + k+ c S u Z u du 2 2 Z u du ds, ds 2I + J 2 c S u c S k.z u du ds. Le us evaluae J. Se s := f s [, +. Codoal Schwarz s equaly mples ha E k+ c S u c S k Z u du 2 k+ E k c S u c S k 2 du E k [c ] 2 Lp k+ E k S u S k 2 du E k k+ k+ Z u 2 du Z u 2 du. Now, classcal resuls abou dffusos wh Lpschz couous coeffces yeld ha, for every u [, +, E k S u S k 2 C + E k + sup S 2. [,T ] for some posve real cosa C. Cosequely J C T E k+ E k + sup S 2 E k Z u 2 du k= [,T ] = C T E k+ + sup S 2 E k Z u 2 du C + sup [,T ] where λ k+ := k+ k= [,T ] S 2 E k λ k+ 2 k= 2 Z u 2 du for every k {,..., }. Sce he λ k s are oegave, λ 2 k+ k= 2 λ k+ k= 29

so ha Fally E 2 E k λ k+ 2 E k= J C 2 λ k+ E k λ k+ + 2 E k= = 2 E λ k+ E k λ k+ 2 + 2 E 4 E k= k= T + sup S 2 2 [,T ] 2 λ k+ k= 2 λ k+ k= 2 2 λ k+ = 4 E Z u du 2. Z u 2 du. I s a sadard resul o dffusos ha + sup [,T ] S 2 2 s fe. I remas o prove ha he erm volvg Z s fe. Sce cc S s ε DagSs 2,..., Ss d 2, follows ha Z s 2 ε max d Ss 2 H s 2 so ha, by Schwarz s Iequaly, 2 /2 E Z s ds 2 E sup S T 4 /2 4 /2 8 E H s ds 2 C E H s ds 2 <+ T where S := /S,..., /S d sasfes a SDE wh bouded coeffces, so ha s supremum has fe polyomal momes. Fally, he las equaly s a sadard fac from RBSDE heory see [9] or [2]. So we have proved ha J C/. The erm I ca be reaed he same way roud. Sep 2. The secod ype of dffculy whch appears s due o he followg wo facs: The obsacle h, S s o suffcely smooh ad so we do o have a ce corol o he creasg process K. The dffuso process S s o uformly ellpc because c = ad so we do o have ce evaluaos of he desy of S. I order o overcome hese dffcules we wll replace S by a ellpc dffuso deoed S ad, whe ecessary, he obsacle h by a smooher obsacle h. Namely, le ε, ] ad λ >. We cosder: A fuco h C,2 R + R d, R usg a regularzao by covoluo of order ε of h. I parcular, sce h s Lpschz couous, we have h h C h ε ad + L c h C h ε 4.4 where L c s he fesmal geeraor of he dffuso S. ad A fuco ϕ λ C b R, R sasfyg ϕ λ ξ := ξ f ξ e λ, ϕ λ ξ := e λ 2 m, ϕ m λ 2 f ξ 2 e λ 4.5 C m e Cmλ 4.6 where C m s a real cosa o depedg upo λ. The he approxmag dffuso coeffce c λ defed for every x = x,..., x d R d by c λ x := cϕ λ x,..., ϕ λ x d 3

sasfes c λ Cb Rd ad c λ c x ε λ 4 e 2λ I d. 4.7 We cosder ow he soluo S x of he SDE ds x = r S x d + c λ S x dw, S x = x. Le P x, dy deoe s markov sem-group defed by P fx = EfS x. We wll deoe by S he soluo S s sarg a s, + d. The relaed Sell evelope sasfes he RBSDE Y = ess sup τ T E hτ, S τ Y = ht, S T + K T K H s.dw s for some o decreasg process K ad some progressvely measurable dp d-square egrable process H see [9] ad [2] for hs opc. We also cosder he approxmao η s = T E k+ Lemma 2 Assume ha Σ holds. The H s ds, s < +. E H s η s 2 ds C ε 2 + + s 2 e Cλ2 /T 2 + E H s η s ds. 4.8 Proof: We rely o he sably propery of RBSDE see [9] ad [2]. E H s H s 2 ds C E sup C st hs, S s hs, S s 2 h h 2 + E sup hs, S s hs, S s 2. st Le τ := f{ > S e λ }. Oe checks drecly o model. ha Pτ T = P f S s e λ = P sup l S s λ Ce Cλ2 /T. st st Sce S = S o he eve { τ}, we oba T E H s H s 2 ds C h h 2 + E C ε 2 + + s 2 Pτ T Cε 2 + + s 2 e Cλ2 /T. sup hs, S s 2 + hs, S s 2 {τt } st O he oher had sce η ad η are he L 2 dp d-projecos of H ad H respecvely o he space P of elemeary predcable processes, we complee he proof by og ha 2 T E η s η s ds E H s H s 2 ds. 3

We eed ow some aalycal facs ha we brefly recall here see [9] ad [2]. Frs of all we have he represeao Y = u, S, H = c λ xu, S 4.9 where u s he uque soluo a varaoal sese see [2] of he P DE + L cλ u, x + F, x, u, x =, ut, x = ht, x, 4.2 wh F, x, u, x = ϑ, x {u,x=h,x} + L cλ h, x + where ϑ s a measurable fuco such ha ϑ. Se F x := F, x, u, x. I follows from 4.4 ha sup sup F x C h /ε where C h s he real cosa roduced T x R d 4.4. Wh hs oao 4.2 becomes + L c u, x + F x =, ut, x = ht, x, a varaoal sese. The, s a sadard fac ha u sasfes he mld form of he above P DE u, x = P T h T x + P s F s xds. 4.2 We focus ow o he sem-group P. I s well kow see [3] ha uder assumpo 4.7, P x, dy = p x, ydy ad for every k N ad every mul-dex α = α,..., α m N m we have x, y R d, [, T ], Dx α p x, y K α,k + x q α,k e K α,kλ ε +k+ α /2 x y 2 C e α +d k+ 2 4.22 where Dx α := α + +α m x α x d, α = α d α + + α d ad K α,k ad q α,k are real cosas depedg o α, k ad C α bu o o λ. Oe derves some sraghforward cosequeces from hs evaluao. Frs, usg ha h T y C + y, follows from 4.22 ha here exss some cosas K ad q such ha, for every, T ], x = x,..., x d R d, P h T x k x C 2 P h T x k x l x R d p x, y x k + y dy K K ε 2 + x q ekλ, 2 P h T x k ε 3/2 x + x q ekλ 4.23 K ε 5/2 + x q ekλ. 4.24 3/2 Smlarly, usg ha F C h /ε ad chagg K for KC h, oe ges for F s x, P F s x k x K ε 3/2 + x q ekλ ε, 2 P F s x k x l x K ε 2 + x q ekλ ε 4.25 ad 2 P F s x k x K ε 5/2 + x q ekλ ε. 4.26 3/2 32

Lemma 3 Se for every [, T ad every x R d, v k, x := u x k, x, k =,..., d. Assume ha 4.7 holds. For every, T, v k, x K ε 3/2 For every, [, T, v k, x v k, x K ε 5/2 Le δ, T, for every, T δ, + x q e Kλ ε +. 4.27 T + x q ekλ ε v k, x v k, x K ε 2 + x + x q ekλ ε + Proof: We ake dervaves he mld equao 4.2 for u ad we oba v k, x = P T h T x k x + T 3/2. 4.28 ε T T + l x x + δ. 4.29 δ P s F s x k xds. Le us beg by. For every [, T ad every x, x R d, v k, x v k, x = P T h T x k x P T h T T P x k x + s F s x k x P s F s x k x ds. Hece, f [, T δ, oe derves usg 4.24 ad 4.25 ha v k, x v k, x P T h T x k x P T h T x k x T + P s F s +δ x k x P s F s x k x ds +δ + P s F s +δ x k x ds + P s F s x k x ds K ε 2 + x + x q e Kλ T + T δ ε l x x +. δ ε whch yelds he secod equaly. Clam follows smlarly. Le us come o clam. Assume whou loss of geeraly ha <. v k, x v k, x = P T h T x k x P T h T T P x k x + s F s x k x P s F s x k x ds so ha v k, x v k, x P s F s x k xds Hece, oe derves usg 4.24 ad 4.26 ha v k, x v k, x K P T h T x k x P T h T x k x + P s F s x k x P s F s x k x ds + P s F s x k x ds. ε 5/2 + x q ekλ ε 33 T 3/2 + T +

whch complees he proof. The above lemma ad he represeao H x = c λ xu, S x yeld Lemma 4 a Le T [, T ] ad δ, T T ]. For every s, [, T ], E H x s H x 2 /2 K+ x q+ e Kλ ε 5/2 2 + T T + l s + δ. 4.3 T T 3 2 ε δ b Le δ, T. For every s, [, T δ, s δ, E H x s H x 2 /2 K q+ ekλ T + 2 + x ε δ ε 5/2 s + δ. 4.3 Proof: a The fucos c λ are Lpschz couous wh [c λ ] Lp Ce Cλ ad sasfy c λ x C + x where he real cosa C does o deped o λ, cosequely c λ x xu, x c λ x x u, x Ce Cλ x u, x x x +C+ x x u, x x u, x. Combg he bouds obaed Lemma 3 for he fucos v k, x leads o c λ x xu, x c λ x x u, x K ε 5/2 + x + x q ekλ ε K T T 3/2 ε 5/2 ε T T + T T + l T δ + x + x q ekλ ε x x + δ + T T 3/2 T 2 + l x x + δ + δ Cosequely, usg Holder Iequaly ad he /2-Holder regulary of S x from [, T ] o L 4 P uformly wh respec o λ, oe has for every s, [, T ], H x s H x 2 K T T 3/2 + S x s + S x q e Kλ 4 ε ε 5/2 K T T 3/2 ε 5/2 + x q ekλ ε 2 + T + l T 2 + l δ T δ b Sll usg he esmaes Lemma 3 ad, hs me, lu u ad yelds c λ x xu, x c λ x x u, x K ε 5/2 Oe cocludes he same way roud. <T + x + x q ekλ ε. S x s S x 4 + δ + s + x s + δ. T 3/2 δ T + 2 x x + 2 δ. δ Proof of Theorem 6: a Usg 4.3 sll usg he oao q sead of q + 2 k+ E H s η s ds = E k+ 2 H <T s H r dr k+ ds k+ k+ E H s H r 2 drds k+ K 2 ε 5 T T 3 + s 2q e2kλ ε 2 34 T ε + l + 2 δ. δ

Moreover, as a cosequece of he frs wo lemmas, E c S s.z s ζ s 2 ds C + C + s 2 e Cλ2 /T + C ε 2 K 2 + s 2q ε 5 T T 3 e 2Kλ ε 2 2 + T T + l + 2 δ. δ A hs sage, we choose our parameers λ, ε ad δ, depedg o. We se λ := T 2C l, δ := 4/ so ha, The, se AT, T := such ha Cosequely E 2 + T T + l + δ 4 + T + l + lt/4. δ K ε 5/2 T T 3/2 + s q ad ake he regularzao parameer ε := ε ε 2 := AT, T 4 + T + l + lt/4 e 2Kλ. C c S s Z s ζ s 2 ds C + C + s 2 e Cλ2 /T + 2 CAT, T C + C + s 2 + CAT, T l e 2K T C l 4 + l + lt/4 e 2Kλ C + s q 2 K ε 5/2 l + T T 3 e 2K T C 2. b Oe carres ou a smlar opmzao process, based hs me o 4.3. Oe ses, for large eough, δ := ρ /3, ε 2 := K ε 5 + s q e 2Kλ T + 2/ρ + ρ /6, λ := T l. 6C 5 Numercal resuls o Amerca syle opos I hs seco, we prese some umercal expermes cocerg he prcg ad he hedgg of Amerca syle opos dmesos d = 2 up o. Ths sudy wll be dvded wo pars. Frs, we wll show how o umercally esmae he spaal accuracy each dmeso order o be able o produce a good choce of me ad spaal dscrezao. Secodly, we wll compue some prces ad hedges followg hs choces. 5. The model We specfy he uderlyg asse model. o a d-dmesoal Black & Scholes B&S model,.e. cosa volales σ l wh cosa dvded raes µ l, l =,..., d: ds l = r µ l S l d + σ l S l dw l, [, T ], l =,..., d, 5.32 where W [,T ] deoes a d-dmesoal sadard Browa moo. The raded asses vecor are e µ l S l, l =,..., d, so ha he dscoued prce sasfes 5.32 wh r =. 35

The asses are assumed o be depede for echcal reasos: urs ou o be he wors seg for quazao, so he mos approprae o carry ou covcg umercal expermes. Beyod s mporace for applcaos, he of B&S model S s a closed fuco of, W sce S l = s l exp r µ l + σ 2 l /2 + σ lw l. Therefore, oe ca eher mpleme a quazao ree for S [,T ] or for W [,T ]. Alhough he payoffs fucos are, srco sesu, o loger Lpschz couous as fucos of W, we chose he secod approach because of s uversaly: a opmal quazao of he Browa moo ca be acheved very accuraely oce for all ad he sored off le. Ideed, he Browa quazao s made of opmal quazaos of he d-dm sadard Normal dsrbuos by approprae dlaaos see Fgure whch are acually sored wh all her compao parameers for a wde rage of szes see [39]. We focus o Amerca syle geomerc exchage opos whch payoffs read hξ = max ξ ξ p ξ p+ ξ 2p, wh d := 2p. 5.33 I follows from he prcg formula.5 ha he Europea ad Amerca prema for exchage opos do o deped upo he eres rae r so we ca se r = w.l.g. A mpora remark s ha here exss a closed form for he Black & Scholes premum of a Europea exchage opo wh maury T a me gve by Ex BS θ, ξ, ξ, σ, µ := erfd expµθ ξ erfd σ θ ξ, d ξ, ξ, σ, θ, µ := lξ/ξ + σ 2 /2 + µθ σ θ ad erfξ := ξ e u2 2 du 2π wh θ := T, σ := d l= σ 2 l /2, µ := p µ l l= d l=p+ µ l, ξ := We wll also use some Amerca geomerc pu payoffs: hξ,..., ξ d := K ξ ξ d /d. + p S l, ξ = l= d S l. 5.34 l=p+ I hs case, he explc formulæ for he Europea Pu wh srke K ad maury T a me wh µ = ad σ = σ, =,..., d reads P BS θ, K, ξ, σ, r := erf d 2 + σ θ/d exp rθk erf d 2 ξ, 5.35 where θ = T ad ξ = d 2 K, ξ, σ, θ, r := lξ/k + r + σ2 /2dθ σ, θ/d d = /d S exp σ2 d. 2d 5.2 Specfcao of he umercal scheme Le us specfy ow he mplemeed umercal scheme. As meoed above, our approach o prcg cosss frs quazg he d-dm Browa moo W. More precsely, le T > ad, N wo egers; se := T ad := k. Spaal dscrezao depeds o he me. We use he opmzed dspachg rule 2.38 sze o he N k -quazer of me so ha N =, N + N + N 2 + + N N +. Frs, we compue 36

for every k {,..., } a opmal quadrac N k -quazer of N ; I d by processg a CLV Q algorhm 2.2 he fal covergg phase s refed usg a radomzed verso of he so-called Lloyd I fxed po procedure, see e.g. [26]. For furher deals abou he mplemeao, see [39]. As a secod sep, we ge he opmal N k -quazer x k =,...,N k of by a -dlaao. All he compao parameers weghs p k, pk j, L2 -quazao W k errors are he esmaed by a sadard Moe Carlo smulao. Noe ha all hese quaes are uversal objecs ha ca be kep off le, oce compued accuraely eough. I hs very parcular bu mpora case, we oly eed he orgal CLVQ algorhm defed by 2.22 ad 2.23, o s exeded verso developed for geeral dffusos. Fally, he quazao ree algorhm 2.2 reads v := h, =,..., N, v k := max where he obsacle s gve by h k := hs k, h k, jn k+ π k j vk+,..., s k,d wh s k,l j, =,..., N k, k =,..., 5.36 := s l exp µ l + σ2 l k + σ l x k, l =,..., d, 2 ad he weghs π k j are Moe-Carlo proxes of he heorecal weghs.e. πj k := PW + C j x k+, W k C x k PW k C x k. Abou he error duced by he Moe Carlo approxmao, see [4] ad []. Followg 3.5 he hedgg δ k a x k s he compued by δ k,l := N k+ j= π k jv k+ j v k e µ l+ s k+ j,l e µ l s k N k+ j= π k je µ l+ s k+,l j e µ l s k,l 2, l =,..., d. 5.37 I pracce, we ofe eed o roduce he quazao ree algorhm a sequece of corol varae varables. Ths s usually acheved by cosderg a F S -margale M k := m, S k where he fuco m s explcly kow. The oe ses Mk := m, s k so ha he explc sequece M k N k,k,.e. approxmaely sasfes: N k+ j= π k j M k+ j M k. 5.38 The approxmao comes from he spaal dscrezao by quazao fac f he equaly dd hold would be of o umercal eres. Here, a effce choce s o ake M k = Ex BS T, p l= s k,l, d l=p+ s k,l, σ, µ. 5.39 The, we use he followg proxy for he premum of he Amerca payoff h, S k k Premum h, s k := m, s k + v h m,k 5.4 37

where v h m,k k s obaed by he scheme 5.36 wh he obsacle h k m, s k k. Le us emphasze ha corol varae varables M k such ha 5.38 holds exacly s useless pracce sce hs case s o dffcul o see ha, k, v h m,k = v h,k M k. 5.3 Numercal accuracy, sably We wll ow esmae umercally he rae of covergece a me = of he umercal premum p, N := Premum h, s gve by 5.36 usg 5.4 owards a referece p h as a fuco of, N where N := N/ average umber of pos per layer. The referece premum p h s obaed by a fe dfferece mehod for valla Amerca pu opos -dmeso ad derved from a 2-dmesoal dfferece mehod due o Vlleeuve & Zaee hgher dmesos see [42]. The error erms boh me ad space gve by Theorem 4 are E, N = p, N p h c + c 2 N α wh α = /d 5.4 for sem-covex payoffs. Two quesos are rased by hs error boud: are hese raes opmal? Is possble o compue a opmal umber op of me seps o mmze he global error? We are able o aswer o he frs oe: we compue by c ad C 2 := c 2 N α by olear regresso of he fuco E, N for several fxed values of N ad. We beg by he ad 2-dmeso segs. The specfcaos of he referece model 5.32 are d =, valla pu, r =.6, σ =.2, S = 36, K = 4 ad exchage, d = 2, σ =.2, µ =.5, S = 4, S 2 = 36. I Table are dsplayed umercal approxmaos of c, C 2 ad α := lc 2 N + /C 2 N l N / N, =, 2, 3. + Noe frs ha c does o deped upo N: hs cofrms he above global error srucure 5.4. These emprcal values for α are closer o 2/d ha he heorecal /d ad srogly suggess ha α = 2/d s he rue order. Ths ca be explaed by he followg heurscs: he lear case e.g. a Europea opo compued by a desce of he quazao ree algorhm, he sem-group of he dffuso quckly regularzes he premum. The, he secod order umercal egrao formula by quazao apples: le X be a square egrable radom varable, x a opmal quadrac N-quazer; f f adms a Lpschz couous dffereal Df, he see [38] EfX P X x =x fx Dfx. EX x C x [Df] Lp X X x 2 } {{ }, 5.42 2 N N = sce x s opmal where X X x 2 2 = c X N 2/d + on 2/d as N. The opmaly of x makes he erm EX x C xx = X X bx 2 2 2 x vash. Applyg rgorously hs dea o Amerca opo prcg remas a ope queso however see [6]. Whasoever hs beer rae of covergece s a srog argume favor of opmal quazao. From dmeso 4 o, he sorage of he marx [πj k ] for creasg values of N ad large s cosly ad make he compuaos racable. The above compuaos sugges 38

a spaal order of 2/d whe he grds are opmal. I fac, rue opmal quazers become harder ad harder o oba hgher dmesos, ha s why we verfy ha spaal order becomes closer ad closer o /d raher ha 2/d. Several aswers o he secod queso are possble accordg o he varables used he error boud. Here, we chose o compue op as a fuco of N ad raher ha N ad. For a gve value of N, oe proceeds as above a olear regresso ha yelds umercal values for c ad C 2 := c 2 N /d. Fally se op d, N c :=. C 2 I lower dmeso d 3, he order α ca be esmaed ad oe may se drecly for every N, op d, N = c c 2 N /d. I Table 2 are dsplayed he umercal values. 5.4 Numercal resuls for Amerca syle opos We ow prese umercal compuaos for Amerca geomerc exchage fucos based o he model descrbed Seco 5.. Namely, we prese he prema of - ad ou-of-he moey opos as fucos of he maury T expressed year, T { k, k }. Ths dsco gves a sgh abou he umercal fluece of he free boudary. We frs sele he value of N ad he read o Table 2 he opmal umber = op d, N of me seps. Space dscrezao s he oe used for he above umercal expermes. The model parameers ad al daa are seled so ha µ ad σ rema cosa, equal o 5% ad 2% respecvely 5.34: µ := 5, µ :=, = 2,..., d, σ := 2/ d, =,..., d,. s := 42/d, =,..., d/2, s := 36 2/d, = d/2 +,..., d -he-moey, s := 362/d, =,..., d/2, s := 4 2/d, = d/2 +,..., d ou-of-he-moey. I Fgure 2 are dsplayed he compued prema a ad hedges b 2-dmeso a me = ogeher wh he referece oes as a fuco of he maury T [, T max ] for T max =. Fgure 2 emphaszes ha boh prema ad hedges 2-dmeso are very well fed wh he referece premum. I also holds rue he Ou-of-he-moey case o depced here. I geeral, he I-he-moey case, we ca see o Fgure 3a ad Table 3 ha he compued premum eds o overesmae he referece oe whe he maury grows. Ths pheomeo grows also whe he dmeso d creases. However, he maxmal error remas wh 3,5 % all he cases as dsplayed Table 3. The same pheomeo occurs for he compued hedges, wh a smlar rage hedges are o depced here. I he Ou-of-he-moey seg, we ca see o Fgure 3b ha very dffere behavors are observed o he prema. Ideed whaever he dmeso s from 4 o, he prema seem o be well compued dmeso oher ha 4 are o depced here. Fgure 4 depcs he quazed resdual rsk a = as a fuco of he maury. I suggess ha umercal compleeess of he marke has a bgger mpac -he-moey ha ou-of-he-moey. We wll ow es he fluece of he Europea premum whe used as a corol varae varable he smulaos. To hs am, we wll prce Amerca pus o a geomercal dex dmeso d = 5. The model parameers ad al daa are µ =, s =, σ = 2%, =,..., d, 39

ad r = l., K =. Ths choce s movaed by he fac ha he he Europea premum s sgfcaly lower he Amerca premum. The referece prces ad hedges are compued usg a BBSR algorhm see [2] wh me seps dmeso wh s,eq =, σ eq = σ / d, δ eq = σ2 d, 2d where s,eq, σ eq ad δ eq are he d-equvale s spo, volaly ad dvded rae. The quazed prces are sll compued usg 5.4 ad algorhm 5.36 where he corol varae varable s kow by 5.35 ad he hedges are compued usg 5.37. Table 4 shows he prce ad hedges compued for, N max =, 28. We ca see ha he prce error s.5% ad he sum of he hedge errors of each compoes s.8%. Now, Fgure 5 shows he fluece of he Europea corol varae varable 5.35. We have ploed he Amerca premum compued followg 5.35, 5.36 ad 5.4 for a opmal me ad space dscrezao foud Table 4, amely, N max =, 28. We ca see ha he Europea premum cous for a lle par he Amerca oe. Here we ca see ha he quazao s able o capure by self a sgfca par of he prce as he maury T vares [, ]. Refereces [] V. Bally, The ceral lm heorem for a olear algorhm based o quazao. Sochasc aalyss wh applcaos o mahemacal face., Proc. R. Soc. Lod., 46, No 24, 22-24, 24. [2] V. Bally, M.E. Caballero, B. Feradez, N. El Karou, Refleced BSDE s, PDE s ad Varaoal equales. Pre-pr RR-4455 INRIA, 22. [3] V. Bally, G. Pagès, A quazao algorhm for solvg dscree me mul-dmesoal dscree me Opmal Soppg problems, Beroull, 9, 3-49, 23. [4] V. Bally, G. Pagès, Error aalyss of a quazao algorhm for obsacle problems, Soch. Proc. ad her Appl., 6, -4, 23. [5] V. Bally, G. Pagès, J. Prems, A sochasc quazao mehod for o lear problems, Moe Carlo Meh. ad Appl., 7, -2, 2-34, 2. [6] V. Bally, G. Pagès, J. Prems, Frs order schemes he umercal quazao mehod, Mahemacal Face, 3, No, -6, 22. [7] J. Barraquad, D. Mareau, Numercal valuao of hgh dmesoal mulvarae Amerca secures, Joural of Face ad Quaave Aalyss, 3, 995. [8] A. Besoussa, J.L. Los, Applcaos of he Varaoal Iequales Sochasc Corol, Norh Hollad, 982, or Applcaos des équaos varaoelles e corôle sochasque, Duod, Pars, 978. [9] B. Bouchard, N. Touz, Dscree-me approxmao ad Moe Carlo smulao of backward sochasc dffereal equaos, Soch. Proc. ad her Appl.,, No 2, 75-26, 24. [] N. Bouleau, D. Lambero, Resdual rsks ad hedgg sraeges Markova markes, Soch. Proc. ad her Appl., 33, 3-5, 989. [] M. Broade, P. Glasserma, Prcg Amerca-Syle Secures Usg Smulao, Joural of Ecoomc Dyamcs ad Corol, 2, 8-9, 323-352, 997. [2] M. Broade, J. Deemple, Amerca opo valuao: ew bouds, approxmaos ad a comparso wh a exsg mehod, Revew of Facal Sudes, 9, 4, 2 25, 996. 4

[3] J. Bucklew, G. Wse, Muldmesoal Asympoc Quazao Theory wh r h Power dsoro Measures, IEEE Tras. o Iformao Theory, Specal ssue o Quazao, 28, 2, 239-247, 982. [4] A.P. Caverhll, N. Webber, Amerca opos: heory ad umercal aalyss, Opos: rece advaces heory ad pracce, Macheser Uversy press, 99. [5] D. Chevace, Numercal mehods for backward sochasc dffereal equaos, Numercal Mehods Face, L. Rogers ad D. Talay eds., Publcaos of he Newo Isue seres, Cambrdge Uversy Press, 997. [6] É. Cléme, P. Proer, D. Lambero, A aalyss of a leas squares regresso mehod for Amerca opo prcg, Face & Sochascs, 6, 2, 449-47, 22. [7] P. Cohor, Lm Theorems for he Radom Normalzed Dsoro, The A. of Appl. Probab., 4, No, 8-43, 24. [8] M. Duflo, Radom Ierave Sysems, Berl, Sprger, 998. [9] N. El Karou, C. Kapoudja, É. Pardoux, S. Peg, M.C. Queez, Refleced soluos of Backward Sochasc Dffereal Equaos ad relaed obsacle problems for PDE s, The A. of Probab., 25, No 2, 72-737, 997. [2] H. Föllmer, D. Soderma, Hedgg of o reduda coge clams, Corbuos o Mahemacal Ecoomcs, 25-223, Norh-Hollad, Amserdam, 986. [2] É. Fouré, J.M. Lasry, J. Lebouchoux, P.L. Los, N. Touz, Aplcaos of Mallav calculus o Moe Carlo mehods Face, Face & Sochascs, 3, 39-42, 999. [22] É. Fouré, J.M. Lasry, J. Lebouchoux, P.L. Los, Aplcaos of Mallav calculus o Moe Carlo mehods Face II, Face & Sochascs, 5, 2-236, 2. [23] A. Fredma, Sochasc Dffereal Equaos ad Applcaos, Academc Press, New York,, 975. [24] S. Graf, H. Luschgy, Foudaos of quazao for probably dsrbuos, Lecure Noes Mahemacs 73, Sprger, 2, 23p. [25] A. Gersho, R. Gray eds., IEEE Tras. o Ifor. Theory, Specal ssue o Quazao, 28, 982. [26] J. Keffer, Expoeal rae of Covergece for he Lloyd s Mehod I, IEEE Tras. o Iformao Theory, Specal ssue o Quazao, 28, 2, 25-2, 982. [27] A. Kohasu-Hga, R. Peersso, Varace reduco mehods for smulao of deses o Weer space, SIAM J. Numer. Aal., 4 No. 2, 43-45, 22. [28] H.J. Kusher, Approxmao ad weak covergece mehods for radom processes, wh applcaos o sochasc sysems heory, MIT Press Seres Sgal Processg, Opmzao, ad Corol, 6, MIT Press, Cambrdge, MA, 977, 984, 269 [29] H.J. Kusher, P. Dupus, Numercal mehods for sochasc corol problems couous me, 2 d edo, Applcaos of Mahemacs, 24, Sochasc Modelg ad Appled Probably, Sprger-Verlag, New York, 2, 475 [3] H.J. Kusher, G.G. Y, Sochasc Approxmaos Algorhms ad Applcaos, Sprger, New York, 997. [3] S. Kusuoka ad D. Sroock, Applcao of he Mallav calculus II, J. Fac. Sc. Uv. Tokyo, Sec IA Mah., 32, -76, 985. [32] D. Lambero, Browa opmal soppg ad radom walks, Appled Mahemacs ad Opmzao, 45, 283-324, 22. [33] D. Lambero, B. Lapeyre, Iroduco o sochasc calculus appled o Face, Chapma & Hall, Lodo, 996, 85 4

[34] D. Lambero, G. Pagès, Sur l approxmao des rédues, A. Is. Pocaré, 26, 2, 33-355, 99. [35] P.L. Los, H. Réger, Calcul des prx e des sesblés d ue opo amércae par ue méhode de Moe Carlo, workg paper, 22. [36] F.A. Logsaff, E.S. Schwarz, Valug Amerca opos by smulao: a smple leas-squares approach, Revew of Facal Sudes, 4, 3-48, 2. [37] J. Neveu, Margales à emps dscre, Masso, Pars, 97, 25p. [38] G. Pagès, A space vecor quazao mehod for umercal egrao, Joural of Compu. Appl. Mah., 89, -38, 997. [39] G. Pagès, J. Prems, Opmal quadrac quazao for umercs: he Gaussa case, Moe Carlo Meh. ad Appl., 9, 2, 35-68, 23. [4] D. Revuz, M. Yor, Couous Margales ad Browa Moo, Sprger-Verlag, 2 d edo, Berl-Hedelberg, 99, 56p. [4] J.N. Tsskls, B. Va Roy, Opmal soppg of Markov processes: Hlber space heory, approxmao algorhms, ad a applcao o prcg hgh-dmesoal facal dervaves, IEEE Tras. Auoma. Corol, 44,, 84-85, 999. [42] S. Vlleeuve, A. Zaee 22, Parabolc A.D.I. mehods for prcg amerca opo o wo socks, Mahemacs of Operao Research, 27, o, 2-49, 22. Table : Esmao of he spaal covergece expoe α of 5.4 dmesos d =, 2. d = d = 2 N N = 2 N2 = 3 N3 = 4 N4 = 6 N = 235 N2 = 455 c.47.45.45.46 3.54-3.4- C 2 3.77-3.82-3.3-3 4.79-4 6.6-4 3.55-4 α.87.9.9.89 Table 2: Esmao of he opmal umber of me seps for d =, 2, 4, 6,. d = d = 2 d = 4, N = 75 d = 6, N = d =, N = c.45.35 8.84-.46 2. c 2.2 2.5- C 2 2.62-3 2.57-3 8.75-4 op.63 N.3 N /2 9 24 5 42

Table 3: Amerca premum & relave error for dffere maures ad dmesos. Maury 3 mohs 6 mohs 9 mohs 2 mohs AM ref 4.4 4.8969 5.2823 5.65 Prce Error % Prce Error % Prce Error % Prce Error % d = 2 4.4.23 4.897.4 5.2826.57 5.655.7 d = 4 4.476.8 4.969.34 5.3284.82 5.7366.39 d = 6 4.456. 4.9276.63 5.355.38 5.7834 2.2 d = 4.437.47 4.9945 2. 5.435 2.89 5.8496 3.53 Table 4: Value of a Amerca pu a me = ad he hedgg sraegy o a geomercal dex dmeso 5 for maury T =, σ =.2, r = l., s = = K, =,..., 5. N max AM Qf. BBSR δ Qf. BBSR 28.576.584 -.739 -.779 -.75 -.75 -.789 -.756 Fgure : A 5-uple wh s Voroo essellao wh he lowes quadrac quazao error for he b-varae ormal dsrbuo. 43

a 5.8 QTF V&Z b QTF V&Z 5.6.95 5.4 5.2.9 5.85 4.8 4.6.8 4.4.75 4.2.7 4 3.8 5 5 2 25.65 5 5 2 25 Fgure 2: d = 2, = 25 ad N = 3. Amerca premum as a fuco of he maury: a; Hedgg sraegy o he frs asse: b. The cross + depcs he premum obaed wh he mehod of quazao ad depcs he referece premum V & Z cf. [42]. a 5.8 b 2.2 5.6 2 5.4.8 5.2.6 5.4.2 4.8 4.6.8 4.4.6 4.2.4 4.2 3.8 2 4 6 8 2 4 6 8 2 2 4 6 8 2 4 6 8 2 Fgure 3: d = 4. I-he-moey: a; Ou-of-he-moey: b. Amerca premum as a fuco of he maury. + depcs he premum obaed wh he mehod of quazao ad depcs he referece premum V & Z cf. [42]. 4.5 I he moey Ou he moey 4 3.5 3 2.5 2.5.5 2 4 6 8 2 4 6 8 2 Fgure 4: Quazed local resdual rsk R 2 as a fuco of he maury 4-dmeso wh = 2, N = 75 see Table 2 see he defo of local resdual rsk 3. compued owg o 3.6 he I-he-moey case sold le ad Ou-of-he-moey case dash le 44

.6.5 Dmeso 5 K = s_ = sgma =.2 r = l. = me layers Nmax = 28" EURO Ref. AM Ref. AM Qf..4.3 Prces.2..9.8.7.2.4.6.8 year Maury Fgure 5: Amerca Pu premum o a geomercal dex dmeso 5 as a fuco of maury. Here, s =, σ =.2, r = l. ad K =. Tme ad space dscrezao are, N max =, 28. The bold le depcs he referece prce compued by a BBSR d algorhm, he h le depcs he Europea premum.e. he corol varae varable ad he pos depc he quazed Amerca Premum a each me sep. 45