Approximate hedging for non linear transaction costs on the volume of traded assets

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1 Noame mauscrp No. wll be sered by he edor Approxmae hedgg for o lear rasaco coss o he volume of raded asses Romuald Ele, Emmauel Lépee Absrac Ths paper s dedcaed o he replcao of a covex coge clam hs a facal marke wh frcos, due o deermsc order books or regulaory cosras. The correspodg rasaco coss rewre as a o lear fuco G of he volume of raded asses, wh G >. For a sock wh Black-Scholes md-prce dyamcs, we exhb a asympocally coverge replcag porfolo, defed o a regular me grd wh radg daes. Up o a well chose regularzao h of he payoff fuco, we frs roduce he frcoless replcag porfolo of h S, where S s a fcve sock wh elarged local volaly dyamcs. I he marke wh frcos, a proper modfcao of hs porfolo sraegy provdes a ermal wealh, whch coverges probably o he clam of eres hs, as goes o fy. I erms of order book shapes, he exhbed replcag sraegy oly depeds o he sze G of he bd-ask spread. The ma ovao of he paper s he roduco of a Lelad ype sraegy for o-vashg o-lear rasaco coss o he volume of raded shares, sead of he commoly cosdered raded amou of moey. Ths duces los of echcales, ha we pass hrough usg a ovave approach based o he Mallav calculus represeao of he Greeks. Key words Lelad Lo sraegy, Dela hedgg, Mallav Calculus, rasaco coss, order book. Mahemacs Subjec Classfcao 9G ; 6G44 ; 6H7 JEL Classfcao G G3 Iroduco The curre hgh frequecy of radg o he facal markes does o allow o eglec he frcos duced by marke orders for buyg or sellg a gve umber of shares. Depedg o he lqudy of he sock of eres, he margal prce of CEREMADE, CNRS, UMR 7534, Uversé Pars-Dauphe E-mal: [email protected] [email protected]

2 Romuald Ele, Emmauel Lépee ay exra u of sock ca be sgfcaly dffere. The shape of he order book ad he sze of he bd-ask spread deerme he uderlyg cos duced by passg a order o he marke. Modelg order book dyamcs ad more mporaly quafyg he mpac of he rades o he uderlyg prce have brough a lo of aeo he rece leraure. Our cocer hs paper s o look owards effce aleraves order o replcae opos he presece of rasaco coss, relaed o he presece of order books. Ths kd of duced cos rewres as a fuco of he raded amou of shares sead of he more classcal ad less realsc raded amou of moey. For smplcy here, he order book shape s supposed o be deermsc ad has a saoary asympoc behavor whe he umber of raded shares goes o zero. More precsely, radg γ shares of sock a me duces a cos G, γ where he possbly o-lear fuco G sasfes G, γ G γ + O γ, for γ small eough. We cosder a facal marke wh oe bod ormalzed o ad oe sock S wh Black Scholes md-prce dyamcs. Observe ha G erpres as he half sze of he bd-ask spread. The order book duces frcos o ay poso ake o he sock ad we vesgae he replcao of a Europea opo wh payoff hs, where h s a covex fuco. I he classcal framework of proporoal rasaco coss o he amou of raded moey, Lelad [8 roduced a geous mehod order o hedge effcely call opos o a dscree me grd. Hs dea reles o he use of he frcoless hedgg sraegy assocaed o a Black Scholes sock wh a suably elarged volaly, relaed o he chose frequecy of radg. As he umber of radg daes goes o fy, Lo [ or Kabaov ad Safara [6 verfed ha he ermal value of he correspodg porfolo coverges o he clam hs of eres, uder he addoal codo ha he rasaco coss coeffce vashes suffcely fas as well. Ths urealsc assumpo has recely bee releved by Lépee [9 va a proper modfcao of he replcag sraegy. The ma movao of he paper s he roduco of Lelad-Lo approxmae hedgg sraeges he realsc framework descrbed above, where he amou of rasaco coss s a o lear fuco of he umber of raded shares of asse. Ths parcular feaure mples ha he aural Lelad-ype elarged volaly s assocaed o a local volaly model sead of a Black Scholes oe. Ideed, we cosder he prcg fuco Ĉ ad assocaed dela hedgg sraegy Ĉx duced by a fcve asse wh local volaly 8 ˆσ :, x σx + σg π x,. where σ s he Black Scholes volaly of he sock ad / s he mesh sze of he regular revso grd. I he mperfec marke of eres, we exhb a porfolo sarg wh al wealh Ĉ, S ad duced by a proper modfcao of he dela hedgg sraegy Ĉ x, S T, he spr of [9. The ma resul of he paper s he covergece probably of he ermal value of hs porfolo o he clam of

3 Asympoc Hedgg 3 eres hs, as he umber of revso daes eds o fy. Ths covergece requres o cosder payoff fucos h wh bouded secod dervaves. For dervaves wh less regular payoff fucos such as he classcal call opo, oe smply eeds o replace h by a well chose more regular payoff fuco h, characerzed erms of umber of radg daes of he hedgg sraegy. The approxmae hedgg sraegy roduced hs paper allows herefore o replcae asympocally a covex coge clam hs a marke wh o vashg rasaco coss coeffce relaed o deermsc order books. The ehaced sraegy oly reles o he sze G of he bd-ask spread ad o o he global shape of he order book. The cosderao of a fcve asse wh local volaly dyamcs of he form. duces los of echcales sce he Lo Kabaov mehodology requres precse esmaes o he sesves of he prcg fuco Ĉ erms of he umber of radg daes. The raher compuaoal obeo of hese esmaes reles o a ovave approach based o he Mallav represeao of he Greeks roduced [4. The paper s orgazed as follows: The ex seco preses he facal marke wh frcos ad he replcao problem of eres. Seco 3 s dedcaed o he ma resuls of he paper: he cosruco of he modfed volaly ad correspodg fcve prcg ad hedgg fucos, he Dela correco for he cosderao of o-vashg rasaco coss coeffce, he payoff regularzao ad he covergece of he ehaced replcag sraegy. Seco 4 deals he proof of he covergece, whereas echcal esmaes o he dervaves of he fcve prcg fuco Ĉ are repored Seco 5. Noaos. For a fuco f from [, R o R, we deoe by f, f x, f x, f xx,... he me ad space paral dervaves. For a fuco f from R o R, he frs ad secod dervaves are smply deoed f ad f. We deoe by C a geerc cosa, whch may vary from le o le. For possbly radom cosas, we use he oao C ω. Hedgg uder rasaco coss o he raded volume of shares I hs seco, we roduce he marke model ad formulae he facal dervave replcao problem uder rasaco coss duced by order book frcos.. The marke model We cosder a facal marke defed o a probably space Ω, F, Q, edowed wh a -dmesoal Browa moo W. We deoe by F = F he compleo of he flrao geeraed by W. Our model s he sadard wo-asse model wh he me horzo T = assumg ha s specfed uder he uque margale measure Q. The o-rsky asse s he umérare S =, ad he dyamcs of he rsky asse s gve by he

4 4 Romuald Ele, Emmauel Lépee sochasc equao S = S + σs udw u,, where σ > s a cosa. Up o cosderg dscoued processes, all he resuls of he paper exed as usual o facal markes wh o zero deermsc eres raes. I a frcoless complee marke of hs form, he prce a me of a facal dervave hs s gve by C, S where C s he uque soluo of he PDE { C, x + e = σ x C xx, x =,, x [,, C, x = hx, x,. I presece of realsc rasaco coss, where couous hedgg s o adequae aymore, hs paper develops a asympoc hedgg sraegy for he facal dervave hs.. The order book frcos We ed o ake o accou he frcos duced by he use of marke orders he facal marke. Whe a porfolo maager buys or sells a gve quay γ of sock S, he presece of order books mples a addoal cos, whch s relaed o he volume γ of he order. We model hese order book relaed coss va he roduco of a o lear couous deermsc cos fuco G. Wheever a age rades a possbly egave quay γ of socks S o he facal marke a me, he shall pay a mmedae cos G, γ >. We make he followg saoary assumpo o he asympoc behavor of he cos fuco G o he eghborhood of γ =. Codo G: There exss a cosa G > such ha G, γ = G γ + O γ,. Remark. Whe S represes he md-prce dyamcs of he rsky facal asse, G erpres smply as he bd-ask spread of he asse he order book of eres. We shall see he followg ha for asympoc replcao purpose, oly he sze G of he bd-ask spread s releva our approach. Remark. Of course, assumg ha he order book s deermsc ad ha he bd-ask spread remas cosa s urealsc ad hece resrcve. Neverheless, we oule hs paper ha hs smple framework already rases eresg mahemacal problems ad leads o promsg coclusos. The cosderao of dyamc radom order books, for whch o uamous model has emerged he leraure, shall be lef for furher research.

5 Asympoc Hedgg 5.3 Porfolo dyamcs ad replcao Due o he presece of frcos o he marke, ducg drec or drec rasaco coss, we oly cosder porfolo sraeges, where he maager chages hs marke poso o a fe umber of revso daes. For smplcy, we assume he paper ha he revso daes defe a uform deermsc me grd,.e. := /, for. Remark.3 As observed [ or [3, he use of o uform me grd, where he umber of radg daes creases as he maury s geg closer, allows o mprove he covergece of he Lelad ype approxmae hedgg sraegy. Oe ca expec hs propery o rema sasfed our coex. A rgorous proof of hs resul requres very compuaoal fer esmaes, whch go beyod he scope of hs already echcal paper. For he cosderao of radom me es, we refer o he ce resuls of [5, whch produces a robus asympoc hedgg sraegy for vashg lear rasaco coss wre erms of he raded amou of moey. A porfolo o he me erval [, s gve by a al capal x R ad a F-adaped pecewse-cosa process H N, where H L Ω represes he umber of shares of sock hold he porfolo o he me erval [, +, for ay <. Due o he order book frcos, he value of he porfolo process V assocaed o he pecewse-cosa vesme sraegy H s gve by V = V + Hu ds u G, H H,, N.. We am a hedgg he coge clam wh payoff hs, where h s a covex fuco, for whch precse regulary requremes are gve Seco 3.3 below. We look owards a porfolo V, wh ermal value covergg o hs as he umber of radg daes eds o fy. 3 Asympoc hedgg va volaly modfcao ad payoff regularzao I order o exhb a porfolo sraegy, whose asympoc ermal value aas he clam of eres hs despe he frcos, we formally expla Seco 3. he Lelad mehodology ad cosder a fcve asse wh upgraded volaly. Sce rasaco coss rewre our framework as a fuco of he volume of raded asse, he fcve asse has o Lpschz local volaly dyamcs. Afer verfyg Seco 3. ha hs sochasc dffereal equao has a uque soluo, we roduce he correspodg prcg ad hedgg fucos of he clam hs for a frcoless marke. Up o a proper sraegy modfcao, we exhb Seco 3.4 a asympoc hedgg sraegy for he covex clam hs. For payoff fucos wh few regulary such as call opo, a well chose addoal regularzao mehod s exposed Seco Cosruco of he elarged volaly fuco I he frcoless Black Scholes model, he prce fuco of he covex clam hs s he uque soluo C.,. of he PDE e ad he exac self-facg

6 6 Romuald Ele, Emmauel Lépee replcao porfolo s gve by C, S = EhS + C xu, S uds u,. I exacly replcaes he coge clam hs ad s self-facg. I he presece of rasaco coss, Lelad suggesed hs famous paper [8 o subsue he volaly σ by a arfcally elarged oe σ, relaed o he mesh / of he radg replcao grd. We brefly recall he ma deas behd hs volaly elargeme ad deal formally how adaps o he framework of frcos cosdered here. For a sequece of volaly fucos σ o be deermed below, cosder he followg PDEs { u, x + σ xx u xx, x =,, x [,,, u, x = hx, x, for N. The soluo C of hs equao f exss s he frcoless prcg fuco of a facal dervave wh payoff fuco h, wheever he sock has σ local volaly dyamcs. We look owards a volaly fuco σ allowg o ake o accou he rasaco coss duced o he radg daes. More precsely, Io s formula mples ha he formally supposed smooh fuco C verfes C, S = C, S + C x u, S uds u + [ σ σ S u SuC xxu, S udu, for ad N. Hece, he process C, S ca be approxmaely defed as a porfolo process wh dyamcs of he form. wheever he las erm o he rgh had sde above correspods o he rasaco coss cumulave sum,.e. equalzg he varaos: [ σ σ S u SuC xxu, S u u G u, Cx u + u, S u+ u Cx u, S u, for N. A formal Taylor approxmao gves C x u + u, S u+ u C x u, S u = C xu, S u u + C xxu, S u S u+ u S u, C xxu, S u S u+ u S u, for N. Sce h s a covex fuco, we expec Cxx ad follows formally from Codo G ogeher wh he relao S u+ u S u σs u W u+ u W u ha [ σ σ S u u G σ W u+ u W u S u, N. Takg he codoal expecao gve F u ad pluggg he classcal esmae E W u+ u W u = u/π, hs leads o [ σ σ S u u G σ u S u π, N.

7 Asympoc Hedgg 7 For he regular radg grd cosdered here, u = / provdes he followg caddae for he upgraded volaly fuco: σ : x, σ + G / 8 π σ, N. 3.3 x Observe ha hs caddae upgraded local volaly fuco s degeerae a ad we prove he ex paragraph he well posed-ess of he correspodg local volaly fcve asse ad assocaed prcg fuco. 3. The fcve asse dyamcs Le us cosder a sequece of fcve asses, whose dyamcs are gve by he caddae upgraded volaly σ defed 3.3. We expec he fcve asses Ŝ o solve he followg sochasc dffereal equao Ŝ = S + γ Ŝu dw u, T, N, 3.4 where we roduced he oao γ : x σ xx = σ x + σγ x, wh γ := G / 8, N. 3.5 π Sce he dffuso coeffces γ are o Lpschz, he exsece of a uque process wh such dyamcs does o follow from he classcal heorems. We puzzle ou hs dffculy usg he Egelber & Schmd crero as dealed he followg lemma. Lemma 3. Whaever al codo, x [,,, he sochasc dffereal equao 3.4 adms a uque srog soluo Ŝ s s, sarg from x a me. Furhermore, hs soluo remas o-egave. Proof. We fx N ad, x [,,. For ay z R, observe ha he dffuso coeffce γ defed 3.5 sasfes: f ε ε dy =, for ay ε >, he γz =. 3.6 γ z + y Ideed, for z, akg ε = z /, we ge ε ε dy < γ z+y, so ha he lef had sde codo of 3.6 mples z =, leadg o γ z =. Hece, he dffuso coeffce γ sasfes he Egelber & Schmd crero, ad, here exss a weak soluo o 3.4 wh al codo, x, see Theorem 5.4 Seco 5 of [7. We ow observe ha he dffuso coeffce γ also sasfes γ z γ y = σ z + σγ z σ y + σγ y σ z y + σ y + σγ z σ y + σγ y, z, y R,

8 8 Romuald Ele, Emmauel Lépee sce he dervave of y σ y + σγ z s upper bouded by σ. We deduce γ z γ y σ z y + σγ z y l z y, z, y R, wh l : u σu + σγ u. Sce ε du lu =, for ay ε >, we deduce from Proposo.3 Seco 5 of [7 ha pahwse uqueess holds for he sochasc dffereal equao 3.. Togeher wh he exsece of a weak soluo verfed above, hs mples he exsece of a uque srog soluo o 3. for ay al codo, x, see Corollary 3.3 Seco 5 of [7. Fally, Ŝ remas o-egave, sce s couous ad Markova, ad he uque srog soluo sarg a s he ull oe. 3.3 Payoff regularzao ad relaed prcg fuco We ow qure he properes of he prcg fucos assocaed o he fcve asses Ŝ ad frs dscuss he regulary of he payoff fuco of eres. We am a hedgg he coge clam wh payoff hs, where he payoff fuco h s supposed o sasfy he followg: Codo P: The covex fuco h : [, R s affe ousde he erval Codo P: [/K, K, wh K >. Observe ha mos of he classcal covex payoffs sasfy hs codo. I parcular, uder Codo P, he map h s Lpschz ad we deoe by L > s smalles Lpschz cosa. I he followg, we shall somemes requre he payoff fuco o be couously dffereable. Besdes, order o cosder o-vashg rasaco coss, we eed a corol o he secod order varaos of he payoff fuco. I order o do so, we regularze he covex map h, as dealed he followg lemma. Lemma 3. There exss a sequece of covex maps h valued C [,, R such ha, for large eough, h h L l γ /6, h L, h 3L γ/6 l [/K,K. 3.7 Proof. We observe ha h s affe o [, /K ad roduce he exeso of h o R, whch remas affe wh he same slope o,. For smplcy, hs exeded map s also deoed h. For N, we roduce he covoluo bewee h ad he square kerel wh suppor [ l/γ /6, l/γ /6 : h : x [, 4 3 h x + y l γ /6 y dy. Sce h s L-Lpschz ad y dy = 3/4, we compue h h 4 L ly 3 γ /6 y dy = L l L l, N. 3 γ /6 γ /6

9 Asympoc Hedgg 9 Fx N. Observe ha h C [,, R ad, deog abusvely h he rgh dervave of h, we have h x = 4 h x + y l y dy 3 γ /6 = 4 x+ γ hz /6 z x γ/6 3 l l dz, x. x Sce h L, we deduce ha h L. Dffereag he secod expresso of h above, we deduce ha h x = 4 x+ hz γ/6 γ/6 x z 3 l l dz = 8 h x + y l γ /6 3 γ /6 l ydy, x for x. Usg oce aga ha h L, hs yelds h 8L 3 y γ/6 l dy = 8L 3 γ /6 l γ/6 3L l. 3.8 Besdes, sce h s affe o [K,, we deduce ha h x = 4 hk x K + y l y dy = hkx K = hx, 3 γ /6 for ay x K + l/γ /6. The exac same reasog apples for x /K l/γ /6. Hece, for large eough such ha γ /6 / l K, h s affe ad herefore h = ousde he erval [/K, K. Combed wh 3.8, hs complees he proof. Remark 3.4 Wheever h s valued C [,, R, he regularzao procedure s o ecessary sce 3.7 s sasfed as soo as s large eough. Hece oe ca smply use h sead of h. The sequece of regularzed approxmag payoff fucos h had, we ca ow roduce he assocaed valuao PDEs, gve by: { Ĉ e =, x + σ xx Ĉxx, x =,, x [,,, Ĉ, x = h x, x,., for N. The exsece of a uque srog soluo for hs PDE s gve Proposo 3.3 below. For sake of compleeess ad sce he correspodg dffereal operaor s o uformly parabolc o [,,, he proof of hs proposo s repored Appedx. As expeced, he soluo of he PDE erpres as he valuao fuco of he opo wh payoff h o he ermal value of he fcve asse Ŝ, roduced he prevous seco. Proposo 3.3 For ay N, he PDE e has a uque soluo deoed Ĉ, whch moreover sasfes Ĉ, x = E,x [h Ŝ,, x [,,, N. 3.9

10 Romuald Ele, Emmauel Lépee 3.4 Dela correco ad asympoc hedgg for o vashg rasaco coss coeffce Eve a frcoless complee seg, a coge clam ca ever be perfecly replcaed pracce, sce couous me hedgg s o feasble. As dealed Seco.3, we cosder porfolos where he poso he asses chages o he regular dscree me grd. I hs framework, we clam ha he upgrade σ of volaly ad he regularzao h of he payoff dealed Seco 3. ad Seco 3.3 allows o couerbalace asympocally he frcos due o order book relaed rasaco coss. Ths clam s he coe of he ex heorem, whch s he ma resul of he paper. Theorem 3.4 Cosder he sequece of porfolos V assocaed o he al codos Ĉ, S ad he vesme sraeges H defed by H := Ĉx, S Ĉ x j, S Ĉ j x j, S j, j for [, + ad <. The, he sequece of porfolo values rewre V = Ĉ, S + Hu ds u G, H H,, N, ad V coverges probably o he payoff hs as goes o. 3. The proof of hs heorem s preseed Seco 4 below, ad requres sharp esmaes o he dervaves of Ĉ, whose proofs are pospoed o Seco 5. Remark 3.5 Observe ha he hedgg sraegy does o smply coss cosderg he Dela assocaed o he fcve asse Ŝ. Ideed, as observed [6, for he classcal framework of rasaco coss proporoal o he amou of moey, hs orgal Lelad replcag sraegy does o coverge o he clam of eres, uless he rasaco coss vash fas eough as he umber of radg daes creases. As [9, he exra erm he defo of H allows o cosder o vashg rasaco coss. I parcular, observe ha he chage of poso a me, for, he porfolo V s gve by Ĉ x, S Ĉ x, S. Remark 3.6 Our ma resul also allows o quafy he effecs of a volume based radg axao, o he cos of hedgg sraeges for covex dervaves. Ideed, order o reder mos of he hgh frequecy radg arbrage opporues rreleva, he regulaor s sll lookg owards he bes way o creae a ax o radg orders. Neverheless, he exac cosequeces of such a regulao o asse maageme sraeges or more geerally rsk maageme sraeges s o ye compleely udersood. Smple quesos o hs subjec sll lack fully sasfyg aswers: Should he regulaor creae a ax o he volume of raded asse or he quay of raded moey? Should he use a lear ax? Wha are he cosequeces of usg a dffere shape of ax fuco? I our smplfyg Black Scholes framework, our coclusos are ha he global shape of he axao does o really maers from a hedgg perspecve sce oly he asympoc behavor aroud s releva. Besdes, Theorem 3.4 exhbs he volaly chage relaed o a volume based axao sead of a more classcal amou based oe.

11 Asympoc Hedgg 4 Proof of he ma resul Due o he cosderao of volume relaed o lear rasaco coss, he exhbed radg sraegy s based o a prcg fuco of a sock model wh o lear dyamcs. Hece, classcal esmaes are o avalable for he sesves of he prce fuco erms of he volaly parameer. Bu, we requre o udersad precsely he depedece of he prce sesves wh respec o he umber of radg daes whch affecs he modfed volaly parameer. We overcome hs dffculy, usg Mallav dervave ype represeao of he Greeks, as dealed he ex subseco. Ths leads o sharp esmaes, whch allow o derve he covergece of he approxmag replcag porfolo o he clam of eres a maury. 4. Represeao ad esmaes for he modfed prce fuco sesves Recall ha he prce fuco Ĉ s gve by Ĉ :, x E,x [h Ŝ. 4. A well chose probably chage leads classcally o a ce represeao of he Dela of he opo preseed below. Lemma 4. For N ad ay al codo, x [,,, he s.d.e. d S u = γ S u dw u + γ γ S u du 4. has a uque soluo S, whch moreover remas srcly posve. Besdes, we have Ĉx, x = E,x [ h S,, x [,,, N. 4.3 Proof. Fx N. The exsece of a uque soluo o 4. follows from smlar argumes as he oe preseed Lemma 3.. Besdes, sce ρudu = where { σ } y + σγ ρ : u exp u σ y + σγ dy = σ + σγ y σ u + σγ, u Theorem.6 ad.7, [, esure ha S remas srcly posve for a gve posve al codo. The mappgs y σ e y ad y σ e y adm locally Lpschz frs dervaves because her secod dervaves are locally bouded. Le deoe S := l Ŝ. By vrue of Theorem 39 V.7 ad Theorem 38 V.7[, we deduce ha here exss a verso of he mappg y S,y, whch s couously dffereable ad so s x Ŝ,x o,, for ay,. Precsely, for a gve al codo, x [,,, he age process Ŝ s gve by Ŝ u = + γ Ŝs Ŝs dw s, s T.

12 Romuald Ele, Emmauel Lépee Besdes, dffereag expresso 4. provdes Ĉx, x = E,x [ h Ŝ Ŝ. Assume for he mome ha Ŝ s a posve margale ad roduce he ew equvale probably P defed by dp = Ŝ dq, so ha Ĉ x, x = E P,x [ h Ŝ. 4.4 Grsaov heorem assers ha he process W gve by dw u = dw u γ Ŝ u du s a sadard Browa moo uder P. Hece, he dyamcs of Ŝ uder P are gve by dŝ u = γ Ŝ u dw u + γ γ Ŝ u du. Therefore, he law of Ŝ uder P s decal o he oe of S uder Q ad 4.4 rewres as 4.3. The res of he proof s dedcaed o he verfcao ha Ŝ s deed a posve margale. For ay p N, le us roduce he soppg me τ p := f{s : Ŝ s x/ + p}, wh he coveo ha f =. Applyg Growall s lemma, we verfy ha sup s Ŝs τ s square egrable, hece Ŝṇ τ s a margale. Le us defe he chage of measure dq p := Ŝ τ pdq. The, E[ Ŝ E[ Ŝ τ p τ p = = Q p τ p =, p N. 4.5 As τ p p, le us defe he sequece τ p p assocaed o he process S gve by 4.. By cosruco, observe ha τ p has he same law uder Q p ha τ p uder Q, for ay p N. I follows ha Q p τ p = = Q τ p = Q τ = where τ s he frs me whe S hs zero. Bu S remas srcly posve, so ha 4.5 mples ha E[ Ŝ. Sce Ŝ s a supermargale, we he coclude. We ow provde a expresso for he secod dervave of he prce fuco Ĉ, he spr of he Mallav represeao of he Greeks preseed [4. Lemma 4. For ay N, we have Ĉxx, x = E,x [ h S πudw u,, x [,,, 4.6 where π s defed by π u := S u γ S u, u. 4.7 Proof. Fx ay al codo, x [,, ad N. Dffereag 4.3 wh respec o x, we drecly compue Ĉ xx, x = E,x [ h S S = E,x [ h S D s S S s ˆγ S s ds.

13 Asympoc Hedgg 3 Recall ha he Mallav dervave ad he age process oly dffer by her al codos. Hece, recallg he defo 4.7 of π, he egrao by pars formula yelds [ Ĉxx, x = E,x D s[ h S πs ds = E,x [ h S πs dw s. Smlarly, he hrd dervave of he prce fuco also has such ype of represeao expecao, where we emphasze ha he sochasc egrals cosdered below are of Skorokhod ype, sce he egrad s o ecessarly F-adaped. Lemma 4.3 For ay N, we have Ĉxxx, x = E,x [ h S π udw u,, x [,,, 4.8 where π s defed by π u := xπu + πu πs dw s, u. 4.9 Proof. Fx ay al codo, x [,, ad N. Dffereag 4.6 wh respec o x ad followg a smlar reasog as above yelds Ĉxxx, x = E,x [ h S xπs dw s + h S S πudw u = E,x [ h S xπs dw s + D s[ h S πs πudw u ds. Hece, he Mallav egrao by pars formula provdes Ĉxxx, x = E,x [ h S xπs dw s + h S ad he defo 4.9 cocludes he proof. π s π udw u dw s, The exac same le of argumes provdes a smlar represeao for he fourh dervave of he prcg fuco. Lemma 4.4 For ay N, we have Ĉxxxx, x = E,x [ h S ˆπ udw u,, x [,,, 4. where ˆπ s defed by ˆπ u := x π u + πu π s dw s, u. 4. These represeaos allow o derve esmaes o he depedace of he dervaves of he prcg fuco Ĉ, erms of he parameer. The raher compuaoal obeo of hese esmaes s repored Seco 5 below.

14 4 Romuald Ele, Emmauel Lépee Proposo 4.5 There exs a cosa C ad a couous fuco f o, whch do o deped o N, such ha Ĉx, x C, 4. Ĉxx, C x x / γ 4.3 Ĉ xxx, x Ĉ xxxx, x fx γ + Ĉ x, x C C γ x + γ x 3/, 4.4 fx γ + fx 5/4 γ 5/4 + fx 3/ γ 3/, 4.5 fx 4/3 l, 4.6 for ay, x [,, ad N. Remark 4.7 Observe ha 4.3 also dcaes ha he prce fuco Ĉ s covex wh respec o he space varable. Ideed, he prcg fuco hers he covexy of he payoff. Ths observao s crucal order o esure ha a volaly upgrade allows o compesae he rasaco coss. 4. Asympocs of he hedgg error The subseco s dedcaed o he proof of Theorem 3.4, he ma resul of he paper. We verfy below ha he sequece V of ermal values for approxmae replcag porfolos coverges o hs, as he umber of radg daes eds o fy. For ay N, we rewre he hedgg sraegy H as H = Ĥ + K wh Ĥ := Ĉ x, S ad K := j Ĉ x j, S j Ĉ x j, S j, 4.7 for [, + ad. We also deoe Ĥ := Ĥ+ Ĥ ad K := K+ K. Therefore he ermal value of he caddae replcag porfolo V rewres V = Ĉ, S + Hu ds u G, Ĥ < Besdes, he dyamcs of Ĉ ad he defo 3.3 of ˆσ yelds h S = Ĉ, S + Ĉ x u, S uds u + + K, N. 4.8 σγ S u Ĉ xxu, S udu, N. Pluggg he wo expressos above ogeher drecly leads o he followg racable decomposo of he hedgg error V hs = F + F + F + F 3 + F 4,

15 Asympoc Hedgg 5 for ay N, where F := h S hs + F := F := Ĥ Ĉx, S ds, K ds F3 := G Ĥ + K F 4 := = Ĥ Ĉx, S ds G, Ĥ + K, = σγ S Ĉxx, S d = G, Ĥ + K, G Ĥ + K. We ow prove ha each sequece of radom varables F j for j =,..., 4 goes o zero probably, as goes o fy. Proposo 4.6 The sequeces F, F, F ad F 3 coverge o probably as goes o. Proof. We prove he covergece of each sequece separaely. Sep. Covergece of F. By cosruco of h, 3.7 mples ha he frs erm h S hs eds o as h h. The secod oe coverges o because Ĉx., S. s bouded accordg o 4.. As for he las erm, observe from 4.3 ha [ Ĥ = Ĉx, S Ĉx, S = h S E h S S = S h E [ S S S = S, N. As E S S C /, we deduce from 3.7 ha Ĥ C γ/6 + C γ/6 l l E S S, N. From he dyamcs 4. of S, we compue drecly E S S C γ / so ha Ĥ goes o as goes o fy. Very smlarly, we show ha K coverges also o ad Codo G provdes he covergece of F o. Sep. Covergece of F. Applyg he Io formula, we drecly compue ha Ĥ Ĉ x, S = M M + A A, < +, <, 4.9 where he sequece of processes M ad A are gve by.. [ M := σs u Ĉxxu, S udw u ad A := Ĉxu, S u + σ S uĉ xxxu, S u du,

16 6 Romuald Ele, Emmauel Lépee for ay N. Sce S has bouded momes, 4.3 ogeher wh he Cauchy Schwarz equaly yeld EM M 4 C γ + du C u γ, < +, <. Besdes, 4.4 ogeher wh 4.6 dcae ha E A A 4 C C l du u l γ + 4/3 γ / + 4/3 4 du u + γ / du u Pluggg he las wo esmaes 4.9 leads drecly o E F C γ + C l γ + 8/3 4, < +, <. d Cγ + for ay N, so ha E F goes o as goes o fy. Sep. Covergece of F. From he defo of K gve 4.7, we drecly compue C l /3 + C γ, F = Ĉ xu, S S S du Combg he Cauchy Schwarz equaly ogeher wh 4.6 yelds E F C l C l Sep 3. Covergece of F 3. For ay, observe ha E[ S S / du u 4/3 5/6 C l. Ĥ + K = Ĉ x, S Ĉ x, S = Ĉ xx, S S S, where he radom varable S s bewee S ad S. Hece, 4.3 ogeher wh Codo G yeld F 3 C ωχ 3 where χ 3 := γ S S, for ay N. Bu Eχ 3 Cγ l, hece F 3 as goes o. Proposo 4.7 The sequece F 4 coverges o probably as goes o.

17 Asympoc Hedgg 7 Proof. For ay N, we wre F 4 = 4 = L wh he summads L := σγ S Ĉxx, S d σγ S Ĉxx, S, = L := Ĉxx σγ, S G σ W = L 3 := σg S Ĉxx, S W = L 4 := G =, σs u Ĉ xxu, S udw u σs u Ĉxxu, S udw u G H + K. Observe ha he prevous decomposo uses he covexy of he prce fuco gve 4.3, see Remark 4.7. I ow suffces o show ha L for =,... 4 as dealed he seps below. Sep. Covergece of L. We have L C ω L + L where, by vrue of 4.3, L S S := γ γ d, L := γ = = = S Ĉ xx, S Ĉxx, S d. We have E L C γ /. For he secod erm, we use he Taylor expaso Ĉ xx, S Ĉ xx, S = Ĉ xxx, S S S + Ĉ xx, S, for some radom varables ad S, for <. Besdes, dffereag he dyamcs of Ĉ, we observe ha Ĉ x = σ Ĉ xx σ x + σγ xĉ xxx σ x + σγ xĉ xxxx, 4.3 for ay x, ad N. Hece, combg 4.3, 4.4 ad 4.5, we ge L C ωγ + C ω γ = S S γ d d γ + d γ + d + d, 5/4 γ 3/ 3/ for ay N. Hece he Cauchy Schwarz equaly ad a drec compuao yeld EL l C + +. /4 3/8 Sep. Covergece of L. γ 5/4,

18 8 Romuald Ele, Emmauel Lépee We use he equaly E W = /π from whch we deduce σγ E [ G σ W = V ar G σ W = σ G, for ay. The depedece of he cremes of he Browa moo ogeher wh 4.3 yeld EL C γ C l. Sep 3. Covergece of L 3. We use he equaly a b a b. Therefore, he Cauchy-Schwarz equaly ad he Io somery gve us E L 3 C = [ S E Ĉ xx, S S u Ĉxxu, S u / du. By he Io formula, we ge d[s Ĉ xx, S = f dw + g d where f := σs Ĉ xx, S + σs Ĉ xxx, S, g := S Ĉ xx, S + σ S 3 Ĉ xxxx, S + σ S Ĉ xxx, S, for ad N. Hece, we derve E L 3 C E fs ds + = Esmaes 4.3 ad 4.4 provde E f u du C γ + E g s ds /. 4.3,. γ Besdes, combg 4.3 ogeher wh 4.3, 4.4 ad 4.5, we ge E gu du Cγ γ / + γ + γ 5/4 γ 3/ + 5/4. 3/ Pluggg hese las wo esmaes 4.3, smlar compuaos as Sep yeld o he covergece of E L 3 o zero. Sep 4. Covergece of L 4. We frs verfy ha we may replace K by K where K := Ĉxu, S udu,. To do so, suffces o show ha χ where χ := Ĉ x u, S u Ĉ xu, S du.

19 Asympoc Hedgg 9 Usg a Taylor expaso, we compue Ĉ xu, S u Ĉ xu, S = Ĉ xxu, S S u S, for some radom varable S bewee S u ad S, for ay u. Hece 4.3 ogeher wh 4.3, 4.4 ad 4.5 mply ha χ C ω χ where χ Su S := γ + Su S Su S + + Su S du, = / γ γ 5/4 γ 5/4 3/ γ 3/ for N. As E S u S C / for u +, we easly coclude ha E χ. A las, replacg K by K ad usg he equaly a b a b, we deduce from Io s formula ogeher wh 4.4 ha L 4 C ω Ĉ xxxu, S udu c ω du uγ + du uγ. 5 Prce sesves esmao Ths seco s dedcaed o he obeo of he esmaes preseed Proposo 4.5 above, whch allow o upper boud he sesves of he prce fuco Ĉ erms of he umber of radg daes. The corol of each sesvy s preseed separaely. These esmaes, amely 4., 4.3, 4.4, 4.5 ad 4.6, are obaed usg he Mallav represeao of he Greeks dealed Seco 4.. Ths parcular feaure s ew he classcal scheme of proof for he obeo of Lelad ype covergece heorems. I all he seco, we fx, x [,, ad om he subscrp {, x} order o allevae he oaos. 5. Esmaes 4. ad 4.3 o he frs ad secod dervaves Frs observe ha esmae 4. drecly follows from he represeao 4.3, sce h s bouded. The res of hs subseco s dedcaed o he obeo of 4.3. We fx, x [,,. Usg 4.6 ogeher wh he Cauchy Schwarz equaly, we derve Ĉxx, x h E / π u du, N. 5.3 We ow focus more closely o he dyamcs of he processes π defed by 4.7. Frs, accordg o he dyamcs of S, he age process S sasfes d S u = γ S u S u dw u + γ S u + γ S u γ S u S u du,

20 Romuald Ele, Emmauel Lépee for N. Besdes, Io s formula mples ha / γ S has he followg dyamcs d = γ S u γ S u γ S u d S u + γ S u γ S u γ S u du γ S u = γ S u γ S u dwu γ S u du, N. A drec applcao of he egrao by pars formula hece mples dπ u = γ S u S u Therefore, we deduce ha { πu = π σ γ } exp 8 γ S s ds du = σ γ 8 γ S u π udu, N γ x, u, N Pluggg hs expresso ogeher wh γ x σγ x 5.3 provdes 4.3. Ideed 5.34 also dcaes ha π ad hece S are o-egave, so ha Ĉ xx, x = E,x [ h S S. 5. Esmae 4.4 o he hrd dervave Ths subseco s dedcaed o he obeo of 4.4 ad dvdes 3 seps. Sep. Esmae decomposo Usg 4.8, we derve Ĉ xxx, x h E Z where Z := Le us roduce he sequece of processes Z gve by Z s := s By he defo of π gve 4.9, we compue π udw u, N π udw u, N Z = xπudw u + πuz dw u = xπudw u + Z πud uz du = xπudw u + Z πu du πu D uπs dw s du u = Z πu du + xπu πs D sπuds dw u, N.

21 Asympoc Hedgg Pluggg hs expresso 5.35 ad usg Io s formula, we deduce Ĉxxx, x C A / + B, N, 5.37 where A ad B are respecvely defed by A := E π uz u dw u ad B := E xπu πs D sπuds dw u, for N. We ow fx N ad ed o corol he erms A ad B separaely. Sep. Corol of A Recall from 5.34 ha π / γ x. Hece, we ge from a drec applcao of Io s formula ha A = E πu π s dw s dw u = E π u πs dw s du. We recall from 5.34 ha π / γ x ad deduce from he prevous expresso A = E π u πs dw s du γ x E π s dsdu Usg oce aga he same relao ogeher wh ˆγ x σγ x yelds A / γ x 4 σ γx Sep 3. Corol of B We ow ur o he more rcae erm B. Le us roduce he oao. b := xπ [D sπuπ s ds, so ha B = E b udw u. 5.4 By vrue of he margale mome equales, here exss C > such ha B C E / b u du C E sup b u, N. 5.4 u I order o corol he las erm o he r. h. s., we look owards he dyamcs of b. Dffereag he dyamcs of π gve 5.33, we compue separaely d xπu = σ γ 8 γ S u xπ udu + σ γ γ S u S u πudu, γ S u 3 dd sπu = σ γ 8 γ S u Dsπ udu + σ γ γ S u D s S u πudu, s, γ S u 3

22 Romuald Ele, Emmauel Lépee Sce D s S r = S r γ S s / S s = S r /{ π s } for s r, we deduce πs D sπuds σ γ = π s σ γ s 8 γ S r Dsπ r drds + γ S r D s S r π s 4 γ S r 3 r πs drds r = πs D sπ σ γ r ds dr 8 γ S r + r σ γ γ S r S r π 4 γ S r 3 r dr, for u. Combg hs expresso wh 5.4, we ge db u = σ γ 8 γ S u b udu + σ γ γ S u πu udu γ S u Noce ha b = γ x/ γ x <. From he dyamcs of b, we observe ha b creases as log as b s egave. Oce becomes posve, mus rema o egave, sce he egave par of he drf dsappears as soo as b reaches. Ideed, b = L π /π where. L := b σ γ + γ S r π 4 γ S r r π rdr 5.45 s srcly creasg. From here, we deduce ha b ad L have he same sg. Hece b s always o egave o [τ, where τ := f{s [,, b s = }. Therefore, we ge b u b {bu } + b u {u τ } b σ γ b r {u τ } τ 8 γ S r dr + σ γ γ S r {u τ } π τ 4 γ S r r rdr, for ay u, whch drecly leads o b u b + Γ u, wh Γ := Sce γ s o-egave ad π s decreasg, we deduce ha E sup b u u. γ S r rπ r dπ r γ x γ x + EΓ We ow focus o he las erm of hs expresso ad observe from a drec applcao of he egrao by pars formula ha Γu = γ x π u γ S u πu + πr rd γ S r πr γ S r dr, u We compue γ x = σ x + σγ σ x + σγ, γ x = σ γ x 4 γ x 3, 3 γ x = 3σ γ γ x 4 γ x 4,

23 Asympoc Hedgg 3 ad deduce from he applcao of Io s formula ha d γ S u = σ γ dw u γ S u du = σ γ 4 γ S u 4 γ S u dwu γ S u dπ u πu Pluggg hs expresso 5.48 drecly leads o Γ u γ x π + N u + Γ u, u, where N :=. π r r σ γ 4 γ S dwr. Sce Γ, follows ha N r u u s a supermargale whece EN. We deduce a upper boud o EΓ whch plugged 5.47 provdes E sup b u u γ x γ x + γx π = Togeher wh 5.4 ad he expresso γ x/ γ x C/x, we ge B C x γ x C γ x 3/, whch, combed wh 5.37 ad 5.39, provdes γ x γ x Esmae 4.5 o he fourh dervave Ths subseco s dedcaed o he obeo of 4.5. Fx N. The represeao 4. drecly provdes Ĉ xxxx, x h E ˆπ udw u, 5.5 ad we ow ed o corol he erm E ˆπ udw u several seps. Sep. A racable Decomposo for E. ˆπ udw u Le roduce he oao Z u := b s + π s Z s dw s, u, so ha Z = π s dw s, where b s defed above ad gve by b := π. π r D rπ dr. The defo of ˆπ gve 4. mples ˆπ udw u = π udw u + πu Z dw u = Z + π u Z dw u. 5.5

24 4 Romuald Ele, Emmauel Lépee Usg egrao by pars formulae, observe ha π u Z dw u rewres Z Z πud u[ Z du = Z Z πub u + πuz u du πu D ub s + D u[πs Zs dw s du u s s = Zu d Z u + Z u dzu πud ub s du Zs πud uπs du dw s s s r πs πu du dw s πs πud uπr du dw r dw s. Pluggg hs expresso ogeher wh Z = b s + Zs πs + πs Zs dw s ad he defo of b 5.5, we oba ˆπ udw u = c s dw s + Zu d Z u + Z u dzu { s s } + Zs b s + πs b r dw r πs πu du dw s, where c := b. π r D rb dr. Iroducg he dyamcs of Z ad Z he prevous expresso, we ge s ˆπ udw u = c s dw s + 3 Zs b s + πs b r dw r dw s s s + Zs + πr Zr dw r πr dr dw s. π s Usg Io s formula ogeher wh he defo of Z, we deduce E ˆπ udw u 3C + 3C + C3, 5.53 where we se C := E π s s Z r dw r dw s, C := E We ow requre o corol hese hree erms separaely. Zs b s dw s, C3 := E Sep. Corol of C Usg wce he margale mome equaly, we compue c s dw s. C C π E Z u du c π E sup Z u u C π E sup C π u b u + π E sup E sup b u + c π u. u Z u

25 Asympoc Hedgg 5 Pluggg 5.5 hs expresso, follows ha C C γ x γx γ x +. γx Sce γ x σγ x ad γ x / γ x 3/x, we deduce ha C C γ x x γx Sep 3. Corol of C Applyg he margale mome equaly ogeher wh he relao 5.46, we deduce C C E sup b uzu C u b E sup Zu + E sup Γu Zu u u where Γ defed 5.46 s o egave ad creasg. Usg oce aga he margale mome equaly, we derve C C b π + C E sup Γu Zu u Observe ha he egrao by pars formula yelds dγ u Z u = γ S u uz u π udπ u + Γ u π udw u. The Jese equaly appled o he cocave fuco x x yelds he equaly fuudu fudu fuu du. Sce π s decreasg, we deduce ha sup Γu Zu u sup Γr π rdw r + Γ Γ /, 5.56 u u where, usg 5.49, we have Γ u := = γ S r r Zr πr dπr πr 4 γ S r rdr πr Zr γ S r dr πr Zr r γ S r dπ r πr + πr 3 rd Z, γ S + N u, r for u, wh N a local margale. Hece, we deduce ha Γ u Nu + χ + χ where χ := π r 4 γ S r rdr, χ := 4 r Z r π r dπ r. Applyg Io s formula o πr 4 γ S r r r ogeher wh he relao 5.49 yelds χ = π 4 γ x + 4 r π r 4 γ S r dπ r π r r γ S r π r 3 dπ r + N,,,

26 6 Romuald Ele, Emmauel Lépee where N, s a lower bouded local margale, so ha E χ π 4 γ x. From he margale equaly ogeher wh Io s formula, we ge E χ 4E sup Zr r r πr dπr 4 π π 3, 3 where he las equaly follows from he mooocy of π ogeher wh Doob s equaly. We deduce ha E χ +χ <, so ha Γ u Nu +E[ χ +χ F u, whch mples ha N s a supermargale. Therefore EN ad EΓ E[χ + χ. Hece, he wo prevous equales ogeher wh 5.56 lead o E sup u Γ u Z u E sup u Γr πr dw r + EΓ π 4 π 4 γx / The margale mome equaly ad he mooocy of Γ ad π esure u E sup Γr πr dw r CE Γr πr dr Cπ EΓ. u Pluggg EΓ γ x π observed 5.5 ogeher wh he defos of π ad b he prevous expressos ad 5.55 leads o C C γx γ x 3 + γx γ x + γ x γ x 4 + γ x 4 3/ Sce γ x/ γ x 3/x ad γ x σγ x, we compue C for some couous fuco f. fx γ + fx 5/4 γ 5/4, 5.57 Sep 4. Corol of C3 We ow ur o he las erm C3 ad observe from he margale mome equaly ha C3 = E c s dw s CE c s ds C E sup c s s I order o corol hs las erm, we compue he dyamcs of c defed as b. π s D sb ds. We deduce from he dyamcs of b gve 5.44 ha d b γ u = 8 γ S u b udu + γ γ 4 γ 3 S u S u b udu [ + γ γ γ S u πu π u udu + γ 4 γ γ S u S u πu udu.

27 Asympoc Hedgg 7 Smlarly, we compue dd sb γ u = 8 γ S u Dsb udu + γ γ 4 γ 3 S u D s S u b udu [ + γ γ γ S u πud sπu udu + γ 4 γ γ S u D s S u πu udu, for s u. Sce D. S u = Sṇ /{ πs }, we deduce followg he same le of argumes as Sep 3 of he prevous seco ha dc γ u = 8 γ S u c udu + γ γ 4 γ 3 S u b u u S u du [ + γ γ γ S u πu ub udu + γ 4 γ γ Su Su π u u du. Therefore, Io s formula ogeher wh he defo of π leads o c u π u = c π γ 4 [ γ γ γ S r r b r π r γ 8 π r dr γ S r γ γ S r πr r dr, u. Sce π ad γ are decreasg, hs relao combed wh 5.58 mples C3 C c + EX + EY, wh X := γ S r r b r πr π r dπr,. Y := γ S r γ S r γ S r πr r dπr. We frs focus o he process Y ad, sce π s decreasg, observe ha. Y π γ S r γ S r γ S r r dπr. 5.6 Applyg Io s formula o he process u πu γ S u π γ x = 3 γ S r rπ r γ S r dr + r γ S r dπ r u π u/ γ S u u, we ge γ S r π r γ S r dr γ S r 4 γ S r 3 r πr γ S r r γ S r dr N Y u, where N Y s a local margale gve by N Y :=. r πr γ S r / γ S r dw r. Pluggg dπr = πr γ S r γ S r dr he prevous equaly provdes βu := r π γ S r r γ S r dr 5.6 γ S r γ S r = u π u γ S u π γ x + rπ r γ S r dr + N Y u, u. 5.6

28 8 Romuald Ele, Emmauel Lépee Le pck v [, ad defe for r [v,, N r := r v N Y u du. By vrue of Theorem 65, IV -6, [, N r r [v, s a local margale. Moreover, 5.6 mples ha r r βu du v v Besdes, observe ha 3 v u πu [ γ S u du + rπ r γ S r dr = 8 γ r v rπ r γ S r drdu + N r uπ u γ S u du + N r 5.63 rdπr 8 γ π Ths esmae ogeher wh 5.63 ad β mply ha N r r [v, s a supermargale, as a local margale bouded from below. Therefore, sce β s creasg, we deduce from 5.63 ad 5.64 ha Eβv v E v βu du 4 γ π, v <. As v, usg he Faou lemma sce β, we derve EY γ 4 π Eβ 6 π We ow focus o he erm X ad observe from 5.44 ha b d r πr = σ γ γ S r π 4 γ S r r dr = γ S r dπr, so ha b /π s creasg ad herefore d b r /πr γ S r dπr. Hece, Io s formula mples drecly Xu γ x b π π γ S r b r πr πr dr γ S r πr r dπr b r + πr r πr d γ S r, for u. Pluggg 5.49 hs expresso, we deduce X u γ x b π + Y + N X u, u, 5.66 where N X s a local margale. Sce EY <, we deduce ha N X s a supermargale so ha EN X. Hece, combg 5.59 ogeher wh 5.65 ad 5.66 provdes C3 C c + γ x b π + π 3 = C γ x γ x + 3 γx γ x 3 + γ x 3. Sce γ γ x Cγ /x, γ / γ x C/x ad / γ x C γ x, hs yelds C3 C γ x / γ x 3/ Pluggg 5.54, 5.57 ad ad 5.53 provdes 4.5.

29 Asympoc Hedgg Esmae 4.6 o he crossed dervave Ths subseco s dedcaed o he obeo of 4.6. Ths fer esmae s ecessary order o cosder rasaco coss coeffces whch do o vash as he umber of radg daes goes o fy. I requres he obeo of sroger esmaes o Ĉ xx ad Ĉ xxx whch are made possble va he corol 3.7 o he sequece of payoff fucos h. We recall ha he al codo, x s fxed ad E,x deoes E[. S Le us frs derve some a pror esmaes o S ad S. = x. Lemma 5. There exs a cosa C ad a couous fuco f o, whch do o deped o such ha E,x S u C, u, 5.68 E,x S u Cfx, u, 5.69 E,x S u Cfx, u, 5.7 E,x S u 3/ Cfx, u, 5.7 E,x S u C γ fx, u, 5.7 Proof. We fx N ad u [, order o verfy each esmae separaely. Proof of Recall ha S sasfes d S u = γ S u S u dw u + σ S u du. Usg he dyamc of S ad he Io formula, we verfy easly ha S has fe momes of all orders. As S u = π u γ S u, we deduce ha S u has also fe momes of all orders. We also kow ha he process S s posve ad. γ S u S u dw u s a local margale whch urs ou o be a margale oce sopped by a sequece of soppg mes τ k, a.s. as k. By he Faou Lemma, we deduce ha τ k, E S u + lm f E σ S u du + E k Usg he Growall lemma, we coclude abou σ S r dr., Proof of By vrue of 5.68, we have xe,x S u = E,x S u C. Hece, a Taylor expaso drecly leads o Proof of 5.7 E,x S u = E,x S u E, S u Cx.

30 3 Romuald Ele, Emmauel Lépee From he s.d.e. sasfed by S u, we deduce ha here s a cosa C such ha E S u Cγ gx for some couous fuco g. To do so, suffces o use equaly 5.69 ad apply he Growall lemma. Recall ha S = π γ S = π σ S + σγ S. As π π, we coclude abou 5.7. Proof of 5.7. We have xe,x S u 3/ = 3/E,x S u / S u. Usg he Cauchy-Schwarz equaly ad Iequales 5.69 ad 5.7, we deduce ha xe,x S u 3/ Cgx, for some couous fuco g. Hece 5.69 follows from a Taylor expaso. Proof of 5.7. We have xe,x S u = E,x S u S u. We he use he Cauchy Schwarz equaly wh Iequaly 5.7 ad he equaly E,x S u Cγ gx. The cocluso follows as prevously. We ow provde fer esmaes o Ĉ xx ad Ĉ xxx. Lemma 5. There exss a couous fuco f such ha Ĉ xx, x fx,, x [,,, N. 4/3 γ l Proof. Fx N. From 4.7 ad 5.34, we compue [ Ĉxx, x = E[ h S S = E h S γ S π C γ h E [π, sce h vashes ousde a compac subse of, whch does o deped of ad hece h S γ S s bouded by C γ h. We ow look owards a sharp esmae of E[π. The expresso of π gve 5.34 ogeher wh Jese equaly yeld [ E [π π γ E e σ 8 γ S u du π γ E γ γ S u e σ 6 γ S u du, 5.73 where we used he boud xe x C, x, for he las equaly. We spl he expecao of he r.h.s. he expresso above wo pars. The frs oe s bouded for large eough as follows, by vrue of 5.69 ad 5.7: [ E γ S u e σ γ 6γ S u { S u γ } γ e σ / γ 6 fx, 5.74 where f s a couous fuco whch may chage from le o le. Observe ha he Cauchy-Schwarz equaly ad 5.7 yelds E[ S u S u γ γ /6 fx. Therefore, he secod erm s bouded by [ E γ S u e σ γ 6γ S u { S u σ γ γ } + γ 5/6 fx γ 5/6 fx. 5.75

31 Asympoc Hedgg 3 Togeher wh x /3 e x C, x, pluggg 5.74 ad yelds E [π π γ γ 5/6 fx fx /3 γ 5/ /3 Togeher wh 3.7, pluggg hs esmae he frs equaly of hs proof cocludes he proof. Lemma 5.3 Fx N. There exss a couous fuco f such ha Ĉ xxx, x fx,, x [,,, N. 4/3 γ l Proof. Fx N. As observed Seco 5., we have [ Ĉxxx, x h A / + B, where B := E h S b udw u 5.77 ad A ad b are respecvely gve 5.37 ad 5.4. As already observed 5.38, we have A γ x E π s dsdu π γ x s Ee s σ γ 4 γ S r drdsdu. s Usg he boud x / e x C for x, we deduce A C π [ s γ x E γ S γ s 3/ r e s σ γ 8 γ S r drdsdu. Sce he expoeal o he r.h.s s smaller ha, we drecly deduce from 5.69 ha C π γ /4 fx γ x γ / fx 3/4 5/4 γ 5/4 A / fx 4/3 γ l We ow focus o he secod erm o he r.h.s. of 5.77 ad rewre B = E b u h S D u S du = E Observe from 5.45 ha he process b s gve by b u = π u π b + πu S b u h S πu du, u. σ γ γ S r π 4 γ r rdr. S r Moreover, recall ha S = π γ S ad h vashes ousde a compac subse depede of. Pluggg hese esmaes he expresso of B, we ge B C [ γ h b π E[π + E π σ γ γ S r π 4 γ r r dr. S r

32 3 Romuald Ele, Emmauel Lépee Recallg he process β defed 5.6, observe ha he expresso of b ogeher wh γ + γ ad 5.34 lead o B C [ γ h γx γ x E[π + E π r dπr C γ h x E[π + π E [π + E [π β + E [π β h fx + C γ h E [π 4/3 γ 7/6 β, 5.79 where he las equaly follows from The res of he proof s dedcaed o he corol of E [π β. We follow he oaos of he prevous seco ad observe from he mooocy of β ogeher wh 5.63 ha Eπ β lm Eπ v v v βu du lm Eπ rπr 3 v γ S r dr + v v N Y r dr Sce he frs erm he parehess s bouded by Cπ γ, 5.76 yelds Eπ β Cπ γ fx + lm Eπ 4/3 γ 7/6 v v Regardg he las erm, we frs observe from 5.63 ha v v N Y r dr. 5.8 Nr Y rπr dr 3 v γ S r dr C vπ γ, v u. Ths provdes a upper boud for. v N Y r dr ad he egrao by pars formula yelds π u v N Y r dr C vπ γ dπr + v v π r N Y r dr, v u. Moreover, he las erm o he r.h.s s a supermargale as a bouded from below local margale. Hece, by vrue of he Lebesgue heorem, we fally deduce ha lm v Eπ v v N Y r dr C vπ γ lm E v v π d r π =. Combg hs esmae wh 5.77, 5.78, 5.79 ad 5.8 ad 3.7 cocludes he proof. Proof of 4.6. I order o derve he upper boud 4.6, suffces o derve he expresso of Ĉ x, x from Ĉ xx, x ad Ĉ xxx, x by dffereag he p.d.e. e ad o plug he esmaes of Lemma 5. ad Lemma 5.3..

33 Asympoc Hedgg 33 6 Appedx: proof of Proposo 3.3 Noe ha we cao mmedaely coclude abou he exsece of a soluo of e because he operaor s o uformly parabolc o, [ [, [. Tha s why, we shall brg he problem back o aoher oe he doma of whch sasfes he requred uform parabolcy. Fx N. By vrue of Lemma 3., recall ha Ŝ s he uque soluo of he sochasc equao Ŝ,,x s = x + s γ Ŝ,,x u dw u, s,, x [,,, where we use he overscrp, x order o emphasze he al codo. Iroducg γ m : x σ x + σγ x + m, we deoe by Ŝ,m he soluo of Ŝ,m,,x s = x + s γ m Ŝ,m,,x u dw u, s,, x [,,, for ay m >. Sce γ m γ m /, for m >, hece Ŝ,m,,x Ŝ,,x L Ω, P as m goes o, uformly, x [,,. We deduce ha Ĉ,m :, x E h Ŝ,m,,x coverges uformly o Ĉ :, x E h Ŝ,,x. Applyg Lemma 3.3 p wh Codo A p 3 [3, mples, ogeher wh h L, ha Ĉ,m, x Ĉ,m u, y L E Ŝ,x, Ŝ,u,y K x y + u, for m >,, u ad x, y R for a gve R,, where he cosa K depeds o, m ad R. We deduce ha Ĉ,m s couous for ay m > ad hece so s Ĉ. Fx m >. We use argumes of Seco 6.3 [3 ad ry o follow her oaos. Le us cosder he followg ses Q m :=, m, m, B m := {} m, m, { } T m := {} m, m, S m := [, m, m, For each y S m, s easy o observe ha here exss a closed ball Ky m such ha Ky m Q m = ad Ky m Q m = {y}. I follows ha he fuco W y proposed p 34 [3 defes a barrer for each y S m. Besdes, Ĉ ad h are couous ad σ s Lpschz o Q m. By vrue of Theorem 3.6 p 38 [3, we deduce ha he Drchle problem u, x + σ xx u xx, x =, x Q m T m ut, x = h x x B m u, x = Ĉ, x, x S m

34 34 Romuald Ele, Emmauel Lépee adms a uque soluo u,m, couous o Q m wh couous dervaves u,m, u,m xx o Q m T m. Moreover, Theorem 5. p 47 [3 mples ha u,m has he followg sochasc represeao u,m [Ĉ, x = E τ m, Ŝ,,x τ m τ m < + h Ŝ,x τ m =,, x Q m, where τ m s he frs me where Ŝ,,x exs Q m. The defo of Ĉ mples u,m [Ĉ [, x = E τ m, Ŝ,,x τ = E m h Ŝ,x = Ĉ, x,, x Q m. As m, we deduce ha Ĉ solves he PDE e. Moreover, C :, y Ĉ, e y solves he followg uformly parabolc PDE { v, y + σ e y v yy, y σ e y v y, y =,, y [, R v, y = he y, y R. By vrue of Theorem 3.6 [3, C s also he uque soluo of he same PDE resrced o a arbrary smooh bouded doma. Moreover, Theorem 5. p 47 [3, mples ha C has a uque probablsc represeao. We deduce ha Ĉ s he uque soluo of e. Refereces. Chery A.S. ad Egelber H.J. Sgular Sochasc Dffereal Equaos. Lecure Noes Mahemacs. Sprger.. Des E., Yur Kabaov. Mea Square Error for he Lelad-Lo Hedgg Sraegy: Covex Pay-off. Face ad Sochascs, 4, 4, ,. 3. Fredma A. Sochasc Dffereal Equaos ad Applcaos. Volume. Academc Press, Fouré E., Lasry J.M., Lebuchoux J., Los P.L., Touz N. Applcaos of Mallav Calculus o Moe Carlo Mehods Face, Face ad Sochascs 3, 39-4, Fukasawa, M.. Coservave dela hedgg uder rasaco coss, Prepr. 6. Kabaov Y., Safara M. O Lelad s sraegy of Opo Prcg wh Trasaco Coss. Face ad Sochascs,, 3, 39-5, Karazas.I, Shreve.S.E. Browa Moo ad Sochasc Calculus. Sprger Verlag. 8. Lelad H. Opo prcg ad Replcao wh Trasacos Coss, Joural of Face, XL, 5, 83 3, Lépee E. Modfed Lelad s Sraegy for Cosa Trasaco Coss Rae. Mahemacal Face. do:./j x. To appear.. Lo K. E Verfahre zur Replkao vo Opoe uer Trasakokose seger Ze, Dsserao. Uversä der Budeswehr Müche. Isu für Mahemak ud Daeverarbeug, Pergamechkov S. Lm Theorem for Lelad s Sraegy. Aals of Appled Probably, 3, 99 8, 3.. Proer P.E. Sochasc Iegrao ad Dffereal Equaos. d. Ed.. Sochasc Modellg ad Appled Probably. Sprger. 3. Seke J., Yao J. Hedgg Errors of Lelad s Sraeges wh me-homogeeous Rebalacg. Prepr.