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

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1 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 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 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

2 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

3 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

4 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: or 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

5 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

6 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

7 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 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

8 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 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 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

9 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 N k N k+ + + N N. = 9

10 The a elemeary opmzao uder cosra shows ha κ N 2 Complexy κn 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.

11 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 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:

12 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

13 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

14 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 < 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

15 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 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 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 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

16 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 < 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 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

17 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 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 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 l=k 7

18 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 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 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 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 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

19 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 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

20 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

21 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 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 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 < 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 < 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

22 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 = E k S k+ S k+ E k V k+ S k 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 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

23 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 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 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+, [, Ths defo mples ha, for every k {,..., }, he lef-lm of V sasfes V = V + A k 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

24 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

25 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 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 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 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

26 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 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