HIGH FREQUENCY MARKET MAKING

HIGH FREQUENCY MARKET MAKING RENÉ CARMONA AND KEVIN WEBSTER Absrac. Sce hey were auhorzed by he U.S. Secury ad Exchage Commsso 1998, elecroc exchages have boomed, ad by 21 hgh frequecy radg accoued for over 7% of equy rades he US. Such markes are hough o crease lqudy because of he presece of marke makers, who are wllg o rade as couerpares a ay me, exchage for a fee, he bd-ask spread. I hs paper, we propose a equlbrum model showg how such marke makers provde lqudy. The model reles o a codebook for cle rades, he mpled alpha. Afer solvg he dvdual cles opmzao problems ad defyg her mpled alphas, we frame he marke maker sochasc opmzao problem as a sochasc corol problem wh a fe dmesoal corol varable. Assumg eher decal me horzos for all he cles, or a sochasc paral dffereal equao model for her belefs, we solve he marke maker problem ad derve racable formulas for he opmal sraegy ad he resulg lm-order book dyamcs. 1. Iroduco Elecroc exchages play a creasgly mpora role facal markes ad marke mcrosrucure became he key o udersadg hem. Here marke mcrosrucure s udersood as he sudy of he radg mechasms used for facal secures [11]. Hgh frequecy radg sraeges deped very srogly o hese mechasms whch ur, vary from marke o marke. Three ma hemes were proposed o ue he commo feaures of mos of hese markes [11]. 1) The frs heme s he lm-order book, where ages ca pos radg eos a prces above ask) or below bd) he md-prce, whch s regarded as he curre far prce for he secury. These poss, called lm-orders, ca he lead o execuos f hey are pared wh a machg order. I hs case, he al age, commoly kow as he lqudy provder, gas a small amou of moey, havg sold a he ask or bough a he bd. I s sad ha case ha he lqudy aker has pad a lqudy fee for he rgh o rade mmedaely. 2) The secod heme s adverse seleco due o asymmerc sources of formao. I s based o he prcple ha lqudy akers choose whch lm-orders are acually execued, ad whe. Moreover, sce lqudy provders publcly aouce her eos by posg her lm-orders, ad hese wo marke rules creae a formao bas favor of lqudy akers ha compesaes for he lqudy fee. 3) The las heme s sascal predcos. All ages o elecroc markes aemp o predc prces o a shor me scale. Such predcos are possble boh because of he exsece of prvae formao ad he acual marke mechasms ha creae shor-rage prce correlaos. Whle sascal predcos are used by all raders o elecroc markes, he objecves of hese ages vary wdely. Some make her profs from hese predcos, whle ohers jus reduce her rasaco coss by radg smarly. Dae: Ocober 12, 212. Key words ad phrases. Lm-order book, Trasaco coss, Sochasc paral dffereal equaos, Poryag maxmum prcple, Marke makg. 1

2 RENÉ CARMONA AND KEVIN WEBSTER Afer he lqudy crss of 28, lqudy became he ceral objec of eres for marke mcrosrucure. Lqudy s rarely defed precsely, alhough uvely, should quafy how dffcul s o egage a rade. Commoly acceped measures of lqudy are he bd-ask spread 1 ad he volume prese a lm-order book. We e hese wo quaes ogeher by modelg wha we wll call he lqudy cos curve, or smply cos curve. We defe hs o be he oal fee pad by he lqudy aker as a fuco of he volume asked for. If he lqudy aker oly execues lm-orders a he bes ask or bd, he fee wll be equal o he bd-ask spread mes he volume. Some cles, however, mgh wsh o go beyod he bes bd or ask ad ake more volume from he marke. The margal fee he becomes a fuco of he volume. Trade volumes are dffcul o model, par because hey are so depede o lqudy akers decsos ad her execuo sraeges. Emprcally, rade volumes prese very heavy als ad are srogly auocorrelaed [6]): here are freque oulers, buys ed o follow buys, ad sells ed o follow sells, whch s wha makes hem hard o evaluae sascally. Lm-orders, o he oher had, are mea-reverg ad have bee wdely suded he leraure see, for sace, [6, 7, 19]). The erplay bewee red-followg lqudy akers ad mea-reverg lqudy provders s wha makes he marke reach he crcal sae of dffuso, as see [5]. From a modelg sad po, all hese arbues are wha makes a drec sascal descrpo of rade volumes so dffcul, whch s why he roduco of a codebook s desrable. 1.1. Marke makg. Marke makers are a specal class of lqudy provders. They esseally ac as a scaled dow verso of he marke self, always provdg lm-orders o boh sdes of he md-prce. How hese lm-orders are placed erms of volume, dsrbuo, ad dsace from he md-prce, deermes he prcg sraegy. We ca herefore defe he lqudy curve assocaed wh a marke maker s prcg sraegy. The ages radg wh a marke maker are cosdered as hs cles. Marke makg s a purely passve sraegy ha esseally correspods o he servce of provdg lqudy o he marke agas a fee. Ths sraegy makes moey as log as he prcg accuraely acpaes fuure prce varaos. The frs sep owards ha objecve s o fd a model for cle volumes ha s cosse wh prce dyamcs. There are wo schools of houghs o marke makg models. The frs focuses o veory rsk. There, he marke maker has a preferred veory poso ad prces accordg o hs rsk averso o dvergg veory levels. Some of he frs models of hs ype are [1, 4], whch ca be foud [11]. The secod, aed by [14], focuses o adverse seleco, usually dsgushg bewee formed ad ose rades. Ths paper belogs o he secod le of hough. Accordg o our sylzed defo of a marke maker, he laer does o possess a vew o he marke. Cles, o he oher had, have vews o he marke ad hs leads o radg eeds. Whe hs vew s shor erm, he he cle has a sascal edge o he res of he marke ad res o push ha edge o make a shor erm prof. Bu eve whe he vew s log erm, he cle wll sll aemp o mmze rasaco coss by predcg prces o a shor me scale. Log-erm sraeges ad lqudy cosras ca herefore be modeled as ose aroud opmal shor erm execuo sraeges. The uo behd hs s ha log erm cosras, whle he ma moves behd mos rades, wll o ecessarly dcae her execuo. Ths s he case because here are creasgly more layers bewee he ey ha formulaes he log erm sraegy for example vesme baks or hedge fuds) ad he oe ha acually execues for example brokers or execuo eges). Bu oly he laer acually has a shor erm vew o he marke, whch s wha book. 1 The bd-ask spread s defed as he prce dfferece bewee he lowes ask ad he hghes bd he lm-order

HIGH FREQUENCY MARKET MAKING 3 he marke maker s eresed. Opmal execuos based o shor erm belefs ad assocaed rade volumes form herefore he corersoe of he upcomg aalyss ad wll lk prces o rades. 1.2. Alpha. Alpha s he erm ofe used o sgfy ha a cle has a drecoal vew o he marke. I a adverse seleco model he am of ay marke maker s o dscover hs vew ad hedge agas. Mos papers see for example [2, 5, 18, 11]) roduce a oo of marke mpac or respose fuco, o sudy he relaoshp bewee rades ad prces. Trade sze s ofe hough o always cluded, ad a lo of work has bee doe o f varous curves o he respose fuco. A large umber of papers o execuo sraeges [1, 2, 3, 16]) rely heavly o hs oo of marke mpac. So do mos pracoers. Esseally, alpha ad marke mpac are he same quay, eve hough he sores old o jusfy hese coceps are que dffere. I he marke mpac leraure, he premse s ha a rade causes a prce moveme, whle alpha s vewed as a aemp a predcg he moveme. Respose fuco s a more eural erm ha allows for boh erpreaos. I hs paper, we prese a smple model for cle decsos ha proves, uder very geeral assumpos, a sysemac relaoshp bewee rade szes ad shor erm expeced prce varaos. I saes ha margal coss should equal expeced prce varaos. Ths s a uve resul gve ha prce varaos are he margal gas of a rsk-eural cle. I hs model, each cle has a shor erm vew o he marke ad rades opmally accordg o hs vew: hs opmal execuo sraegy depeds upo hs belef. The heorecal oo of mpled alpha appears que aurally from he sochasc opmal corol problem ha defes he cle s model. O he marke maker sde, our model res o capure he fac ha marke makers do o ac o a persoal vew of he marke. The marke maker uses he oo of cle mpled alpha o fer a approxmae prce process from hs cles behavors, esseally aggregag he vews of hs cles o form a probably dsrbuo o he prce. The opmal order book sraegy he replcaes hs dsrbuo, wh a correcve erm ha akes o accou he profably of rades, ha s, he rade-off bewee spread ad volume. 1.3. Resuls. The hrus of hs paper s o propose a framework whch a marke makg sraegy appears edogeously. Ths framework s based o 1) a resul o mply a marke vew from cle rades uder a smple ye robus model of cle behavor; 2) a procedure for he marke maker o fer hs ow vew o he marke from ha of hs cles; 3) a profably fuco ha measures how profable a posed volume o he order book s lkely o be; 4) a pealzg erm akg o accou possble feedback effecs of he marke maker s decso. The las par wll be movaed by he dea ha, jus as he marke maker mples formao from hs cles rades, cles ca fer formao from hs order book. Our hypoheses are chose, ad our resuls are derved, order o make he marke makg problem racable. Oce he heorecal framework s pu place, he marke maker opmal corol problem s formulaed ad solved, leadg o a explc marke makg sraegy. The sory old by he model s closes o ha of he formed rade leraure [14], alhough oe major dfferece s ha here s o clear cu dsco bewee formed ad ose raders. Our coservave marke maker assumes ha all of hs cles mpled alphas carry formao. Moreover, he exsece of a leas oe formed rader uderps he esmao formula used by he marke maker. The paper also relaes srogly o boh he marke mpac [2, 5, 18, 11] ad opmal execuo [1, 3, 2, 16] leraures. A cos fuco c ad a order book γ are derved

4 RENÉ CARMONA AND KEVIN WEBSTER edogeously from he premses of he model. We hope ha he proposed mpled alpha codebook wll fd applcaos oher lm-order book models. Fally, whle we gore veory rsk o make our marke maker rsk-eural, our approach sll relaes o uly fuco ad veory models such as [4, 1]. The resuls of he paper are orgazed four secos. Frs, he seup for he model s gve, whch cludes a mehodology for modelg heerogeeous belefs ad rasaco coss. Secod, a smple cle model s preseed ad solved, leadg o a relaoshp bewee rades ad alphas whch movaes he marke maker s choce for a codebook. Thrd, he marke maker model s bul sep by sep, sarg from he cle model ad workg hrough a seres of resuls ad approxmaos o lead o a reasoable corol problem. Las, he marke maker s problem s solved ad he dyamcs of he order book are deermed aalycally, For he sake of defeess, we focus o a racable example for whch we compue he wo-humped lm order book shape. A appedx s added a he ed of he paper o provde he proofs of wo echcal resuls whch, had hey bee cluded he ex, could have dsraced from he ma objecve of he modelg challege. 2. Seup of he model I hs seco, all he elemes of he model are preseed. 2.1. Heerogeeous belefs o he prce. Cosder cles ad oe marke maker who erac o a elecroc exchage, wh very large. We wll deoe by {1...} a cle dex, ad k {...} a geerc dex, wh k = correspodg o he marke maker. We frs roduce he followg seg for he model: 1) a flered probably space Ω, F, F ), P) represeg he real lfe flrao ad probably measure. The flrao s geeraed by a d-dmesoal P Weer process W. 2) a dffere flrao ad measure F k ), P k ) for each of he ages. Assume furhermore ha F k F, ha P k F k ad P F k are equvale ad ha P F = P F. 3) a d k -dmesoal P Weer process W k ha geeraes he flrao F k ). F ) ) k 4) a prce process p whch s a Iô process adaped o all he flraos. k=... 5) he drf ad volaly of p grow a mos polyomally uder all probably measures. where he las hypohess mus be udersood he a.s. ad L 2 sese. Le dp = a d + σ dw 2.1) ) be he Iô decomposo of p uder F ), P. Hypoheses 2 ad 3 mply ha here exss a ) F k adaped process rk such ha W k = B k + r k s ds 2.2) for some P k F k Weer process B k ad wh r =. Because W k s d k dmesoal, so are r k ad B k. Furhermore, by he margale represeao heorem, gve ha W k s a P margale, here exss a F ) adaped, d k d dmesoal marx Σ k such ha dw k = Σ k dw 2.3) F ) k Fally, age k has he followg vew o he marke uder ), Pk : dp = a k d + σ k db k 2.4)

HIGH FREQUENCY MARKET MAKING 5 wh σ = σ k Σ k ad a k = a + σ k r k. I parcular, σ eeds o lve he erseco of he mages of all he ) Σ k T for p o be adaped o all he flraos. Noe ha he d k s allowed o dffer from oe age o aoher, whch case he σ k mus be of dffere dmesos ad hece dffer. I cocluso, all he ages have vews o he prce process ha poeally coflc wh each oher s probably measure ad eve flrao, bu ca be compared coherely wh he larger probably space Ω, F, F ), P). Example 2.1. Cosder he case where you have hree Weer processes W 1, W 2 ad W 3. Le W ) p = W 1 + W 2 + W 3, F = σ s )s, p s) s ad r = fw ). You he have ha: dp = dw 1 + dw 2 + dw 3 uder P = fw 1 )d + db 1,1 + db 1,2 uder P 1 = fw 2 )d + db 2,1 + db 2,2 uder P 2 where he wo las decomposos are o adaped wh respec o he oher flrao, bu he prce process s everheless adaped o boh flraos. 2.2. Trasaco coss. We ow allow rades amog ages. All cles have a cumulave poso L he asse, whch sars off a a he begg of he radg perod. For he marke maker, we rescale all he quaes accordg o he umber of cles, hece L s hs average cumulave poso per cle. Cles corol her poso hrough s frs dervave, l bu cur rasaco coss. The marke maker, o he oher had, has o drec corol over hs poso, bu receves he lqudy fee c. To be precse, he marke maker aouces o hs cles a cos fuco l c l), whch deoes he prce of radg a speed l a mome. A cle he chooses her preferred radg volume l, ad pays c l) oal rasaco fees. We make he followg hypoheses o hese wo processes: 1) The rade volume process l for each cle s adaped o ) F ad ) F. 2) The cos fuco process c s adaped o all flraos. 3) Margal coss are defed: c s almos everywhere couous. 4) Cles may choose o o rade, c ) = ad he md-prce s well defed a p, c ) =. 5) Margal coss crease wh volume: c s covex. 6) There s a fxed amou of lqudy he marke maker offers. Hypoheses 1 ad 2 descrbe he formao he dffere ages have access o. 3-5 are uve properes ha he cos fuco mus verfy o make sese erms of rasaco coss. Fally, he hypohess 6 s lef purposefully vague, because s bes expressed wh oao we roduce ow. 2.2.1. Dualy ad lk o he order book. Ths seco roduces a chage of varable ha s boh mahemacally covee ad wll carry deeper facal meag whe he model s solved. The ew varable γ s defed as follows: γ α) = sup αl c l)) 2.5) l suppc ) Uder he assumpos c C 1, covex ad c ) =, hs s equvale o defg γ = c ) 1 ad γ ) =. I ha case, γ s smply he Legedre rasform of c. The followg properes ca be derved:

6 RENÉ CARMONA AND KEVIN WEBSTER 1) The Legedre rasform maps he space of covex fucos oo self. I s s ow verse. I parcular, γ, where he secod dervave has o be udersood he sese of dsrbuos, s a posve, fe measure. 2) α R, l R, γ α) + c l) αl Fechel s equaly), wh equaly for α = c l) or l = γ α). 3) γ s a descrpo of he lm-order book of he marke maker. Ideed, represes margal volumes as a fuco of margal coss. The propery γ beg a posve, fe measure correspods o he fac ha he marke maker ca oly pos posve, fe volumes o he order book. Ths meas we ca acually replace c by he secod dervave of s dual γ as he corol varable of he marke maker. Smlarly, α s he dual o he cle s corol varable l. Call γ he marke maker s lqudy offer ad α he cle s mpled alpha. The secod ermology wll be explaed laer. We herefore recas all he hypoheses wh hese ew dual varables: 1) The mpled alpha process α for each cle s adaped oly o ) F ad ) F. 2) The lqudy process ) γ for each marke maker s adaped o all flraos. 3) We have l = γ α ad α = c l). 4) Cles may choose o o rade ad he order book s ceered aroud he md-prce: γ ) = ad γ ) =. 5) Oly posve, fe volumes are posed o a marke maker s lqudy offer: γ s a posve, fe measure. 6) The oal mass of γ s fxed. For coveece sake we wll reormalze o oe, makg γ a probably measure. As a cosequece of 3-5, l s bouded. Furhermore, γ herefore lves o he space of fe measures, whch s a complee, separable merc space uder he Lévy-Prokhorov merc ad a covex se. 2.3. Pug all ogeher. We summarze he model. Frs a publc prce process p s gve. Assume s exogeously gve o every oe, ha s, all he ages cosder ha hey have o mpac o he prce uder her probably measure, ad oly ry o esmae s fuure movemes. The, a publc lqudy offer, whch ca boh be followed eher by he cos fuco c or s dual, s aouced. Cles pck her rade volumes l, or equvalely, her mpled alphas α. We furhermore have he followg sae varable equaos: = ld γ dl dl ) = γ α d = 1 ) 2.6) γ α d sce γ s bouded, L k has a mos lear growh. Fally, we wre a fe-horzo objecve fuco for each age, usg dsc me scales β k for each of hem. Cumulave posos apprecae or deprecae a every mome by dp ad cle pays he lqudy fee c l) o he marke maker whe readjusg her porfolo. We also rescale hs objecve fuco accordg o he umber of cles he has. [ J = E P e ) ) ] β L dp c l d [ = E P e ) ) ) ] β L dp αγ α d + γ α d [ J = E P e β L dp + 1 ) )) ) ] 2.7) α γ α γ α d

HIGH FREQUENCY MARKET MAKING 7 The frs erm of each objecve fuco s bouded by he lear growh of L k ad he polyomal growh of he drf ad volaly of p uder P k. 3. The Cle Corol Problem We frs solve he problem from he cles perspecve, order o kow wha drves rades. The resuls of hs smple model wll be used he ex seco as a codebook for a more powerful marke makg model. The seco cocludes wh a small robusess aalyss o he uderlyg hypohess of he model. 3.1. Solvg he corol problem. I hs seco, we solve he corol problem for oe geerc cle. Because her decsos have o mpac o eher p or c, wll o affec ay of he oher cles decsos. Her corol varable s o eve adaped o her flrao. Smlarly, her ow corol problem s o affeced by he oher cles decso. Ths allows us o drop he dex ) ) hs subseco. Le P, F deoe ha cle s probably measure ad flrao ad W her Weer process. Remember ha he cle res o solve he followg corol problem: ) A admssble corol l s a sochasc process adaped o F ha lves he suppor of c. Because of he hypohess γ, 1 = 1, we have ha he suppor of c s cluded [ 1, 1], whch meas ha he corol se A s bouded. The sae varable L verfes he dyamcs: The objecve fuco s sup E P l [ dl = l d 3.1) ] e β L dp c l ) d) We assume β large eough for he problem o be well defed. Usg he oao roduced he prevous seco, we have he ceral resul: Theorem 3.1. Impled alpha) A cle who rades opmally wll follow he relaoshp [ ] α = E P e βs ) dp s F where α s defed by he codebook c l ) = α. Proof. The am ow s o apply he Poryag maxmum prcple o he above corol problem. The ga fuco s o sadard form, bu a smple egrao by pars solves hs ssue. e β L dp = [ e β L p ] e β l p d + 3.2) 3.3) e β L βp d. 3.4) The frs erm s equal o because we assume L = ad lm e β L p = by he lear growh of L ad he polyomal growh of p. The cle herefore maxmzes: [ ] e β βl l ) p c l )) d 3.5) We wre ou he geeralzed Hamloa of he sysem: E P H, ω, L, l, Y ) = ly + e β βl l)p c l)) 3.6)

8 RENÉ CARMONA AND KEVIN WEBSTER The geeralzed Hamloa s lear L ad cocave l by covexy of c, ad herefore overall cocave L, l). p ad c are exogeously defed Iô process. The, he backward equao has a uque soluo dy = βp e β d Z d W 3.7) [ ] Y = E P βp s e βs ds F By polyomal growh of he volaly of p, he Z erm of he Backward Sochasc Dffereal Equao BSDE from ow o) sasfes he growh codo A.6). Therefore, he caddae opmal corol ) s l verfyg e β Y p = c l ). Ths deermes l = γ e β Y p. Therefore he forward equao of L has a uque soluo. Fally, he quay [ ] c l ) = e β Y p = E P βp s e βs ) ds p F 3.9) = E P [ ] e βs ) dp s F 3.8) 3.1) ca be see as he mpled alpha of he deal, ha s, he dfferece bewee he projeced value o he fuure ad he curre value of p. Whle makes uve sese 2, hs a o-rval resul. Ideed, c ad hece l deped o he marke maker s decso, ad ye α = c l ) becomes a quay ha s rsc o he cle. I s depede of he marke maker s prcg ad oly depeds o he cle s vew o he marke. I uvely represes he cle s prce esmaor ad summarzes her belefs o he prce dyamcs. Noe ha he dscou facor ow becomes he me-scale of her predco. We defe he quay he cle res o predc as he realzed alpha over he me scale 1 β : α r = Ths cocdes wh wha pracoers refer o as he alpha of a cle. e βs ) dp s. 3.11) 3.2. Dyamcs of he mpled alpha. 3.3) ca be rewre erms of Iô dyamcs: dα = βα d dp + e β Z d W 3.12) = βα d dp + θ d W 3.13) wh θ = e β Z herefore beg he volaly of he esmao. All hese equaos summarze he lk bewee opmal cle volumes ad prce dyamcs uder he cle s probably measure ad flrao. Three hgs ca be oed. 1) The drf of he mpled alpha s he resul of wo opposg forces. O he oe had, he self-correlao erm guaraees a cera coherece he cle s decsos over he mescale β 1. O he oher had, he mpled alpha, hrough he dp erm, auomacally akes o accou he las prce varao o receer he esmao. 2 All he formula says s ha margal coss equal expeced margal gas.

HIGH FREQUENCY MARKET MAKING 9 2) θ s a measure of ellgece of a cle over he prce process p, gve ha he lm where θ =, a cle has a perfec vew o he marke. Coversely, cles wh a bg θ wll have a hgher varace o her prce esmaor, ad ca a he lm be cosdered as ose raders. 3) We ca wre he dyamcs of α uder P o oba: dα = βα d dp + θ Σ dw + θ r d 3.14) whch provdes he lk bewee rade ad prce dyamcs. Noe ha, whle uder P, α s rsc o he cle, uder P, α may deped o he marke maker s decsos. I words, hs meas ha whle he marke maker cao fluece he cle s decso uder her ow vew of he marke, he ca affec ha vew self. 3.2.1. The cos of formao. Gve he above resul, s clear ha a marke maker has a prvleged poso o he marke: he caches a glmpse of everyoe s belef o he prce. We ca ow gve a erpreao of γ beyod he fac ha γ represes he order book: Uder a margale measure of p, c l ) represes he cos he cle pays o he marke maker for he lqudy l, ad hs s he coservave erpreao of rasaco coss. However, γ α ) represes he cos he marke maker pays uder he cle s vew of he marke. Uder P, s as f he marke maker pays he cle for formao o he prce process. Ths gves us a good uo abou he marke maker s sraegy: he collecs formao from each cle, prcg hem accordg o he curre belefs ad how much he ew formao brgs o hm: γα) s esseally how much he marke maker s wllg o pay for a predco of sregh α, assumg ha he cle s correc. 3.3. Robusess aalyss. I hs seco we defe he oo of a cash-sesve age ad geeralze he mpled alpha relaoshp o he uly fuco case. Ths llusraes he robusess ad lms of he proposed codebook. Defe he sae varables { dl = l d 3.15) dk = p l + c l )) d The secod varable represes he cle s cash poso a me. If we defe her wealh process as X = p L + K he verfes he sadard dyamcs dx = L dp 3.16) The proposed decomposo has a clear ecoomc meag: he age calculaes her wealh by addg her cash K ad he marked-o-he-md value of her asse poso L. Ths meas ha a cash-sesve objecve fuco would be of he sadard form J = E [UX τ, p τ )] = E [Up τ L τ + K τ, p τ )] 3.17) wh τ a soppg me adaped o he cle s flrao ad U her uly fuco. The specal form guaraees ha he cle does o dffereae bewee wealh cash ad wealh he asse. A age cocered wh he lqudy of he asse would o be cash-sesve. Oe could geeralze he uly framework o cash-sesve ages by cosderg a uly fuco of he form UL τ, K τ, p τ ). I hs case, he below resul would o hold. The ew Hamloa becomes: H, ω, L, K, l, Y L, Y K ) = ly L p l + c l))y K 3.18)

1 RENÉ CARMONA AND KEVIN WEBSTER where he dual varables sasfy he cash-sesve case he BSDEs Y L, = E [ X UX τ, p τ )p τ F ] Y K, = E [ X UX τ, p τ ) F ] The opmal execuo sraegy herefore verfes: [ ] c l YK,τ ) = E p τ Y K, F p 3.19) whch ca be rewre a maer very smlar o 3.3): α = E Q [p τ F ] p 3.2) where dq dp = Y K,τ E[Y X,τ ] s a legmae chage of measure f X UX τ, p τ ) s posve ad egrable. I he case where τ s expoeal ad depede of he prce process, we smply recover 3.3) uder a dffere probably measure. Gve ha we assume all he cles o have dfferg probably measures ayway, we ca whou loss of geeraly sck o 3.3). The above compuao s somewha formal, bu ca be made rgorous by gvg explc egrably assumpos o τ such ha he growh codo A.2) s verfed. Ths s parcular he case whe τ s depede of p ad has expoeal momes, or f τ s bouded. 4. Reworkg he Marke Maker s <odel I hs seco, we work uder he measure P ad ofe drop he superscrp referrg o he marke maker. Our goal s o provde a opmal marke makg sraegy. Because of s complexy, he problem cao be solved full geeraly, ad we propose a se of approxmaos whch we jusfy o facal grouds. The ex four subsecos defy a successo of smplfcaos guded by uo based o he behavor of a ypcal marke maker: 1) Frs, he should o hold a vew o he marke. Mahemacally speakg, hs meas ha hs model for he prce, he ma explaaory varables are hs cle s belefs. The error assocaed o hs frs smplfcao s proved o be small, hough a fuco of he marke maker s corol. 2) Secod, he should model how hs cles vews mgh evolve o he fuure. A sraghforward sysem of correlaed Orse-Uhlebeck processes s proposed o serve hs purpose. Ths wll be used o defe a approxmae objecve fuco for he marke maker. 3) The hrd approxmao s made for mahemacal coveece: we assume ha he marke maker has a fe umber of cles. Ths leads o he prevous models becomg SPDEs. 4) Fally, a placeholder fuco s proposed o model he source of error defed he frs subseco. 4.1. Approxmae Prce Process. As he marke maker should o hold a vew o he marke, we refra from drecly modelg p uder he marke maker s measure ad flrao. Isead, he marke maker cosrucs hs model from he cle s mpled alphas. Ths s doe he followg fasho: Equao 3.14) ca be ured aroud o descrbe prce dyamcs usg he mpled alpha of cle uder P: dp = β α d dα + θ Σ dw + θ r d 4.1) ad hs equao holds rue for all. Noce ha here s o coradco wh he uqueess of he Iô decomposo: hese represeaos correspod o he same Iô process rewre erms

HIGH FREQUENCY MARKET MAKING 11 of he varable α. Usg a sequece of posve ad cosa weghs λ averagg o 1.e. such ha 1/) =1 λ = 1), we oba: dp = 1 λ ) ) β α d dα ) 1 + λ θ Σ 1 dw + λ θ r d 4.2) =1 =1 Aga, hs equao s jus a reformulao of he Iô decomposo of p. The advaage of ha rparcular epreseao s ha he frs erm s adaped o ) F. Hece, f he also kows or raher, chooses) he weghs λ ), he he marke maker ca follow he frs erm of hs =1... decomposo real me. We roduce he specal oao p λ for hs erm ad we refer o as hs prce esmaor: So: dp λ = 1 λ β α d dα). 4.3) =1 The remader whch we deoe ɛ λ cludes quaes ha are ukow o hm sce ) ) dɛ λ 1 = λ θ Σ 1 dw + λ θ r d 4.4) =1 If we replace p by p λ he marke maker s problem, he oly dfferece appears he objecve fuco, whch ow coas a exra erm. We refer o s as he error erm: [ ) ] err = E e β 1 L λ θ r d 4.5) o he marke maker s objecve fuco. Nex we defe σ ) [ ] 2 = E e β θ 2 d. 4.6) Ths quay s a measure of how osy cle s, ad ca be esmaed by fg he expresso for he mpled alpha o hsorcal daa. To be more specfc, recall ha uder he cle measure ad flrao, [ ] α = E P e β s ) dp s F 4.7) ad ha θ s he error o hs esmao. Hece, he smaller θ, he closer he mpled alpha s o he realzed oe, whch meas ha he cle s parcularly well formed. Noe ha he face leraure, cles are ofe called formed f her marke mpac fuco her average alpha) s uusually large. Our oo of ellgece of he prce process does o cocde wh hs pracce, as oe ca have a he same me a sysemacally small hough correc alpha. I hs paper, a formed rader s a rader for whom he mpled ad realzed alphas early cocde, whereas for a ose rader, relaoshp 3.3) has much hgher varace due, for example, o lqudy cocers, a srogly o-lear uly fuco, or a poor flrao). The β ad σ ca be esmaed from hsorcal daa by regressg he mpled alpha agas he realzed oe, ha s for example by solvg he leas squares regresso problem f β> 1 N N k=1 α k k =1 =1 e βs k) dp s where he k are he mes he pas a whch cle raded. =1 ) 2 4.8)

12 RENÉ CARMONA AND KEVIN WEBSTER A sklled marke maker ca herefore cosruc hs approxmae prce process by choosg a λ whch pus mos of s wegh o such ellge cles ad lle wegh o ose raders wh large σ s. Usg Cauchy-Schwarz s equaly ad he parcular choce of weghs λ = ad assumg L s uformly bouded by a cosa L, we oba: err 2 β 1 L) 2 E βe β 1 2 λ θ rd =1 βl) 2 1 e β E λ θ 2 1 e β =1 βi 1 L) 2 e β 1 σ ) 2 j σj 2, 4.9) ) E r 2 d =1 =1 E r 2 d where I 1 1 = m λ =1 λ σ ) 2 = 1 =1 σ ) ) 1 2. 4.1) I ca herefore be see as a measure of he aggregae ellgece he cles have over he prce process. Noe ha suffces for oe σ o be of order ɛ for I o be of order ɛ 1. By Grsaov s heorem, we he have ha E r 2 d = 2 d E log dp F, 4.11) dp F ad fally, err 2 2β 2 I 1 L) 2 e β 1 =1 E log dp F dp F d 4.12) Two mpora remarks are order a hs po. Frs, as log as a leas oe age s well formed, I 1 s small. Secod, he depedece of p ad α upo he marke maker s corol γ s hdde he Rado Nykodym dervave dp F dp. To udersad why s uforuaely) reasoable o assume F ha hs erm may be srogly depede upo γ s because a cle ca use he formao o he order book γ publcly avalable as oe of he sources of formao he uses o form hs probably measure o he prce. Ths problem wll be addressed he las subseco. 4.2. Approxmae Objecve Fuco. I hs subseco, we ake a crucal mehodologcal sep. Isead of maxmzg [ )] J = E e β L dp + 1 α γ ) α 1 ) d γ α d 4.13) we assume ha he marke maker maxmzes he approxmae objecve fuco J λ = e β 1 E L λ β β)α + α 1 λ j α j γ α) γ α) d 4.14) =1 j=1

HIGH FREQUENCY MARKET MAKING 13 subjec o a cosra of he form err 2 C for some cosa C >. The ew objecve fuco was obaed by replacg dp by dp λ ad egrag by pars. Because of our choce of he form of he approxmae prce 4.3), he marke maker s objecve fuco does o deped upo p aymore, ad he oly eeds o model he cle belef dsrbuo. Ths s cosse wh he uo ha a marke maker should o hold a vew o he marke. Raher, he should model he behavor of hs cles wh respec o each oher, ad prce accordg o he formao hey provde. Ths s exacly he approach used wha follows. However, we sll eed o propose a model for he α. For reasos of racably we choose hem as correlaed Orse-Uhlebeck processes: dα = ρα d + σdm + νdw 4.15) wh ρ > ad he M Weer processes whch are depede of each oher ad of W. σ 2 + ν 2 s he overall volaly level of a cle, ad ν he volaly ha s due o some commo formao amogs cle sfor example, he moveme of he mdprce). The ex sep of our sraegy s o roduce a pealzao erm o accou for he possble feedback effecs hdde sde he error erm roduced whe replacg he objecve fuco by s approxmao. Bu frs, we ake he lm o defy effecve equaos provdg formave approxmao o he properes of he orgal sysem comprsg fely may cles. 4.3. Ifely May Cles. Assume ha he umber of cles of he marke maker s large eough o jusfy a approxmao he asympoc regme large. Ths wll grealy mprove he racably of he model by allowg us o work fuco spaces ad rely o sochasc calculus ools o solve he model. The masay of hs subseco s he Sochasc Paral Dffereal Equao SPDE for shor): dv = ) 1 2 σ2 + ν 2 ) v + ρ d v ) d ν v dw 4.16) descrbg he dyamcs of a fe dmesoal measure valued Orse-Uhlebeck process. The followg lemma lks hs macroscopc SPDE o our mcroscopc Orse-Uhlebeck model for he mpled alphas. Proposo 4.1. If ɛ ) 1 s a sequece of radom varables such ha α, ɛ ) 1 s a d sequece depede of W ad he sequece M ) 1, ad such ha he jo dsrbuo, say m, of all he couples α, ɛ ) sasfes: α p + ɛ 2 )mdα, dɛ) <, 4.17) for all p >, he for each, he lm 1 ν = lm ɛ δ α 4.18) exss almos surely he sese of weak covergece of measures, almos surely for every holds: α p dν α) < 4.19) for every p >, ad he measure valued process ν ) s a weak soluo of he SPDE 4.16) he sese ha for ay wce couously dffereable fuco f.e. f C 2 ) such as f ad s wo dervaves have a mos polyomal growh, we have ha 1 d f, v = 2 σ2 + ν 2 ) f ρd f, v d + ν f, v dw 4.2) =1

14 RENÉ CARMONA AND KEVIN WEBSTER Furhermore, for each >, he measure v possesses a L 2 desy almos surely. Proof. See appedx. The papers [13, 12] provde smlar resuls for a more geeral class of mcroscopc models, cludg exsece ad uqueess of he soluo of he SPDE 4.16). However, gve he smple form of he dyamcs chose our parcular model, we ca provde a explc form for he soluo ad dealed properes o he aure of s als. Noe ha he correlao bewee cle belefs s crucal havg a fully sochasc model, gve ha for ν =, ν oly sasfes a deermsc paral dffereal equao. We shall use four measure valued soluos of he above SPDE. I each case, he weghs ɛ are explc fucos of he parameers σ ad β roduced earler. Because he values of hese parameers appear as oucomes of sascal esmao procedures pracce, assumg ha hey are radom ad sasfy some form of ergodcy s o resrcve 3. To cosruc hese measures we assume ha α) 1 s a d sequece of radom varables whose commo dsrbuo has fe momes of all orders. The frs of our four measures s obaed by choosg ɛ 1 for all 1. The: 1 µ = lm δ α. 4.21) Nex we assume ha {σ ) 2 } 1 s a sequece of posve radom varables of order 1 such ha {α, σ ) 2 )} 1 s a d sequece depede of W ad he sequece M ) 1, so by choosg ɛ = σ ) 2 for all 1 we ca defe: 1 I = lm =1 =1 σ ) 2 δα. 4.22) We shall also assume ha he umber I = E[σ ) 2 ] s fe ad srcly posve, ad for each we defe he o-egave measure λ by: λ = I 1 I. 4.23) Fally, we assume ha β ) 1 s a sequece of bouded radom varables such ha {α, σ ) 2, β )} 1 s a d sequece depede of W ad he sequece M ) 1, so by choosg ɛ = β σ ) 2 for all 1 we ca defe: β = I 1 1 lm β σ ) 2 δα. 4.24) =1 We ow make a few remarks o he properes shared by esseally all he soluos v ) of he SPDE 4.16). For he sake of defeess, we shall assume ha v ) s a o-egave measure valued process solvg hs SPDE. 1) The oal mass 1, v of he measure v s cosa over me. Ideed, usg he cosa es fuco f 1 4.16) we see ha d 1, v = 4.25) I parcular, he ellgece assumpo I = 1, I ɛ 1 s coserved over me. 2) If we use he es fuco f = d 4.16) where he dey fuco d s defed by dα) = α, he we see ha he frs mome of v s self a Orse-Uhlebeck process mea reverg aroud sce: d d, v = ρ d, v d + ν 1, v dw. 4.26) 3 The fac ha we have o elarge he Browa flrao a me = o radomze he coeffces does o mpac he margale represeao heorem.

HIGH FREQUENCY MARKET MAKING 15 3) Usg f = d 2 we see ha he secod mome mea revers aroud σ 2 + ν 2) 1, v sce: d d 2, v = σ 2 + ν 2, v 2ρ d 2, v ) d + 2ν d, v dw. 4.27) These Sochasc Dffereal Equaos SDEs for shor) guaraee he exsece of a cosa C 1 such ha: E d 2, v e C 1 E d, v E d 2, v e 1 2 C1. We shall use hese esmaes for he measures µ, λ ad β. I fac smlar esmaes hold for momes of all order as ca be proved by duco from 4.16). We do o gve he deals as hey wll o be used wha follows. Comg back o he opmal corol problem of he marke maker, sce γ belogs o a space of probably measures wheever he corol γ ) s admssble, he followg esmaes hold: γ 1 γ α) α γ, v 1, v = O1) E γ, v E d, v = o e β) E d γ, v E d, v = o e β) As a mmedae cosequece of he above remarks we have: s a probably mea- Corollary 4.2. For ay progressvely measurable process γ ) such ha γ sure, he sae dyamc equao of he marke maker: dl = γ, µ d 4.28) makes sese, ad for suffcely large β, he approxmae objecve fuco J λ = where ᾱ = d, λ, s well defed. e β E [L d, β + L βd + d ᾱ ) γ γ, µ ] d 4.29) 4.4. Modelg he Error Term. As argued he frs subseco, he approxmao echque hges o oe hypohess: he exsece of a leas oe ɛ-ellge cle. Furhermore, he cles probably measure by whch we mea he dsrbuo of he cles mpled alphas) s poeally a fuco of he marke maker s corol. Ths causes a udesrable olear feedback effec whch eeds o be reed o avod exploso of he approxmao error. I hs subseco we propose a drec descrpo of he error erm, leavg ope he queso of how o derve from a specfc model of he cles probably measures. Iuvely, he feedback effec correspods o how much ew formao he order book shape reveals o he cles. The cles belefs a me ca be summarzed by he probably measure µ ad he order book by γ. A reasoable model for he error erm s gve by he expresso Hν µ) = E = ɛe dν dµ e β Hγ µ )d 4.3) where Hν µ) deoes he Kullback-Lebler dsace also kow as relave eropy) defed by { ) f dµ, wheever ν << µ; oherwse 4.31)

16 RENÉ CARMONA AND KEVIN WEBSTER wh fx) = x log x. Noe ha Hν µ) s mmal for γ = µ by covexy of f. As explaed he roduco, we choose hs parcular dsace for s uve erpreao ad he fac ha leads o a explc expresso for he wo hump order book shape edogeous o he model. However, he resuls hold for geeral srcly covex fucos f, whch case he pseudo-dsace H defed 4.31) s kow as he f-dvergece bewee he measures µ ad ν. See [8]. 5. The Marke Maker s Corol Problem Wh all he peces of our model place, we ow solve he marke maker s corol problem. 5.1. Model Summary. We cosder a sequece β, σ ) 2) of radom varables such ha he 1 assumpos of Proposo 4.1 are sasfed wh ɛ 1, ɛ = σ ) 2 ad ɛ 1 = β σ ) 2 respecvely, so ha he measure-valued processes µ ), λ ) ad β ) cosruced 4.21), 4.23) ad 4.24) are well defed. As explaed earler, he marke maker s corol a me s he covex fuco γ, ad gve our choce of pealzg erms, we expec s secod dervave γ o be a probably measure absoluely couous wh respec o µ order o avod fe peales. So we refe he defo of he se A of admssble corols for he marke maker as he se of radom felds g x)),x R such ha g x)) s progressvely measurable for each x R fxed, ad g, µ = 1 for each. Gve a admssble corol g A, we defe he fuco γ as he a-dervave of he fuco γ α) = g 1 [,α], µ sasfyg γ ) =. The objecve of he marke maker s o choose a admssble corol order o maxmze hs modfed objecve fuco defed as: e β E [L d, β + L βd + d ᾱ ) γ γ ɛf g, µ ] d 5.1) where he corolled dyamcs of he sae L of he sysem are gve by: dl = γ, µ d. 5.2) We used he fucos γ ad γ for cossecy wh earler dscussos. See below for forms of he sae dyamcs ad marke maker objecve fuco wre exclusvely erms of he desy g ). The above quaes are well-defed because of he esmaes prove for he measures µ, λ ad β solvg he SPDE. 5.2. Solvg he Marke Maker Corol Problem. We deoe by Y he adjo varable of L so ha he geeralzed radom) Hamloa of he problem ca be defed as H, L, g, Y ) = γ, µ Y + e β L d, β + βld + d ᾱ )γ γ ɛf g, µ ) 5.3) for ay deermsc fuco g as log as we defe γ ad γ by γ α) = g 1 [,α], µ ad γα) = α γ α)d α. Noe ha he above expresso of he Hamloa, g s deerms ad does o deped upo, bu γ whch s he cumulave dsrbuo fuco of he probably measure γ wh desy g wh respec o µ s radom ad depeds upo. Clearly, so s γ. Ths Hamloa ca be rewre as H, L, g, Y ) = e β L d, β βµ + e β Hg) 5.4) where he modfed Hamloa H s defed by Hg) = γ, µ e β Y + d ᾱ )γ γ ɛf g, µ, ad he oher erms do o deped upo he corol. Sce he sochasc maxmum prcple prove appedx says ha we ca look for he opmal corol by maxmzg he Hamloa, we shall

HIGH FREQUENCY MARKET MAKING 17 maxmze he modfed Hamloa. For each choce of he admssble corol g ), we cosder he correspodg sae process L ) gve by 5.2) ad he adjo equao: dy = e β d, β βµ d Z dw. 5.5) Sce he dervave of he Hamloa wh respec o he sae varable L does o deped upo he corol or he sae L, he adjo process ca be deermed depedely of he choce of he admssble corol g ) ad he assocaed sae L ) solvg 5.2). Gve he explc formulas we have derved for µ, we oba: Lemma 5.1. The soluo o he adjo Backward Sochasc Dffereal Equao 5.5) s gve by: Y = e β β + ρ d, β βµ, 5.6) ad verfes he growh codo A.6). Proof. The exac form 5.6) of he soluo ca be guessed by gog back o he explc sysem of fely may Orse-Uhlebeck processes ad akg he lm. However, for he proof, we show ha he process Y ) gve by 5.6) s he soluo by drec speco, compug he Iô dffereal of Y defed by 5.6) ad usg he fac ha he radom measures β ad βµ also solve he SPDE 4.16). Usg 4.26), we ge: dy = βy d ρy d + e β ν β + ρ dw = e β d, β βµ d + e β ν β + ρ dw ad by he growh properes of he frs mome, lm Y =. Moreover, Z = e β ν/β + ρ) clearly verfes he growh codo A.6). The form of he modfed Hamloa jusfes he roduco of he quay: α = ᾱ + e β Y = d, λ + β βµ β + ρ so ha, f we compue he modfed Hamloa alog he pah of he adjo process we ge: Hg) = α γ γ, µ ɛ f g, µ. α ca be vewed as he shadow alpha of he marke maker. 4 The frs erm he defo of α s he average belef for alpha uder he weghed cle measure λ, whle he secod erm akes o accou msmaches he me-horzos of he cles. For each, we defe he profably fuco m by: α m α) = α α )[µ [α, )) 1,] α)]. 5.8) For each α R, m α)) s a progressvely measurable sochasc process ad for each fxed, he fuco α m α) s almos surely couous α, excep for a possble jump m + ) m ) a α =. m s bouded ad vashes a he fves because of he egrably propery 4.19) of he soluos of he SPDE. The profably fuco quafes he expeced prof for a order placed a me a he prce level α. Ideed, he absolue value α α s equal o he spread he marke maker expecs o ga per flled order, ad up o a possble chage of sg, he erm µ [α, )) 1,] α) s equal o he proporo of cles ha wll fll he order. If he arrvals 4 Ths ermology s jusfed by he fac ha, whe he marke maker does have hs ow vew o he marke ha s, hs ow model for dp ), we would oba he same opmzao problem replacg α by E [ e βs ) dp s F ]. 5.7)

18 RENÉ CARMONA AND KEVIN WEBSTER of he ages of our model occurred accordg o a Posso process sead of smulaeously, hs would be he fllg probably of he order. The respecve corbuos of hese wo erms are commoly parsed by pracoers. Noce also ha, he degeerae case ɛ =, he profably fuco s he dervave of he Hamloa he dreco of he corol. We ow defy he modfed Hamloa erms of he corol g whou volvg he adervaves γ a γ. Lemma 5.2. For each, we have he dey d α )γ γ, µ = m, γ. 5.9) Proof. Successve egraos by pars ad smplfcaos yeld: γ, d α ) µ [, )) 1,] ) ) = γ, µ [, )) 1,] ) ) d α )µ α δ = γ, µ δ + γ d α ), µ + α γ ) = d α )γ γ, µ where he frs egrao by pars s jusfed by he fac ha γ s bouded, he Drac dsrbuo has compac suppor ad he smple facs: lm µ [α, )) =, ad lm µ [α, )) 1 α = lm µ, α]) =, α α α ad αµα) vashes a he fes. The lear growh of γ jusfes he secod egrao by pars. The las le uses he al codos of γ ad γ. We ca ow sae ad prove he ma resul of hs seco: Theorem 5.3. The soluo o he marke maker s corol problem s gve by g α) = e mα)/ɛ e m α)ɛ dµ α) 5.1) The quay he rgh had sde of ca be see as he chage of measure from he dsrbuo of he cles alphas o he order book of he marke maker. Proof. Usg Lemma 5.1 ad equao 5.9) eables us o rewre he modfed Hamloa as: Hg) = m g, µ ɛ f g, µ 5.11) whch we eed o maxmze for each fxed, over g L 1 µ ) wh g, µ = 1 ad g. Aga, for each fxed ad almos surely, by covexy of f, H s cocave g ad bouded from above, so ha M = sup A Hg) exss ad s fe. Le g ) 1 be a maxmzg sequece A,.e. a sequece of admssble g such ha Hg ) M. By Komlos s lemma see for example [9]), here exss a subsequece g φ) ad a eleme g L 1 µ ) such ha 1 =1 g φψ)) g µ -a.e. as for ay subsequece of he orgal subsequece. Clearly, g µ almos everywhere. To prove g, µ = 1 we show uform egrably of {g } usg De la Vallée-Pouss s heorem see [15]). Ideed, by defo of he maxmzg sequece, Hg ) M 1 for large eough. Hece ɛ f g, µ M + 1 + m. 1 f s o-egave, covex, creasg o [1, ) ad verfes lm x xfx) =. Ths cocludes he proof of he admssbly of g as a corol. Fally, by cocavy of he modfed Hamloa, he supremum s aaed a g. We ow characerze he maxmal eleme we jus cosruced. We roduce a Lagrage mulpler η R o relax he problem o he se of g L 1 µ ) such ha g, so ha he Lagraga reads: Lg) = m η)g ɛf g, µ 5.12)

HIGH FREQUENCY MARKET MAKING 19 Classcal varaoal calculus yelds he opmaly codo g = f ) 1 ɛ 1 m η) ), 5.13) for some η. The exsece ad uqueess of a Lagrage mulpler reormalzg g s obvous gve he explc formula for f ) 1. We coclude ha g s he desred opmum. Fgure 1. Graph of a smulaed profably curve m wh α.33. The maxmum profably s oly 5% of he volaly used o smulae prces. The average spread beg comparable o he volaly see [18]), hs meas ha he marke maker s margs are very h hs smulao. 5.3. Ierpreao. Fgure 1 ad Fgure 2 llusrae wha he marke maker does: 1) He res o sck o a shape o oo far away from µ, hs cle alpha dsrbuo. Ths s o avod feedback effecs ad assocaed errors o he prce esmao. 2) He also akes o accou he profably fuco m, whch leads hm o make a bg hole he ceer of he dsrbuo µ. The combed effecs lead o he famlar double hump shape of he order book, as see [18]. Oher cosequeces of he lqudy formula are 1) I he lm where oe cle s perfecly ellge of he prce ɛ ), he marke maker places a Drac mass o he order book. I parcular, he oly rades o profable secos of he book. 2) I he ose rade lm ɛ ), he marke maker smply reproduces he cle alpha dsrbuo. 3) If he reur dsrbuo of he asse s mea-reverg whch s o he case for opos wh maures, for example), he so wll µ ad hece γ ad c. I he case of a Europea opo, γ coverges o a Drac a he payoff a maury.

2 RENÉ CARMONA AND KEVIN WEBSTER Fgure 2. Bmodal dsrbuo: Opmal order book γ for ɛ =.1 usg he smulaed alpha dsrbuo ad he eropc pealzg facor. Umodal dsrbuo: Cle alpha dsrbuo. The profably used s he same as fgure 1. 6. Cocluso I hs paper, we propose a equlbrum model for hgh frequecy marke makg. We show ha each cle chooses hs level of rade by a expeced dscou of fuure prces accordg o hs belefs. We call hs he mpled alpha of he cle. Ths relaoshp ca be used as a codebook o lk rade ad prce dyamcs. The dffere cles ca be dffereaed accordg o he me scales of her mpled alphas ad he varaces of he errors hey make esmag he prce. Ths leads o a expresso for he laer ha he marke maker ca use o avod havg hs ow vew o he marke. He ca he cosruc a profably curve whch dcaes whch secos of he order book are he mos profable. I solvg he marke maker opmzao problem, we use a pealzg erm o smooh he objecve fuco ad capure possble feedback effecs. All hese resul a racable framework whch we ca solve he marke maker s corol problem ad defy he equlbrum order book dyamcs. Appedx A. A covee form of he sochasc maxmum prcple We prese a form of he Poryag sochasc maxmum prcple alored o he eeds of he aalyss of Seco 5. The se-up s que geeral, followg loosely [17] Chaper 3. We assume ha Ω, F, F ), P) s a flered probably space, F ) s geeraed by a Weer process W ), ad we le A ω) be such ha for all, ω), A s a Borel covex subse of a Polsh opologcal vecor space E, adaped o F ). The admssble corol processes α ) are he progressvely measurable processes E such ha α A for all. We also assume ha he dyamcs of he corolled sae X are govered by a Ordary Dffereal EquaoODE) wh radom

HIGH FREQUENCY MARKET MAKING 21 coeffces valued R : dx = f X, α )d A.1) where he coeffce f : ω,, X, a) f X, a) s Lpschz X uformly all he oher varables. Assume a esmae of he form X X e C A.2) where C s a cosa. We also assume ha f s couously dffereable.e. C 1 ) X. The coroller s objecve fuco s gve by [ τ ] Jα) = E j X, α )d + g τ X τ ) A.3) where τ s a soppg me adaped o F ) ad j ad g are radom real-valued cocave fucos whch are C 1 X. Furhermore, we assume ha j, g ad τ are such ha E [ g τ X τ ) ] ad E [ τ j X, α ) d ] are fe ad uformly bouded from above. Nex, we defe he Hamloa H ω, X, a, Y ) = f ω, X, a)y + j ω, X, a) A.4) for Y R ad for each admssble corol α ), he adjo equao: dy = X H X, α, Y )d Z dw A.5) wh Y τ = X gx τ ) ad Z such ha E τ Z 2 d <. Uder hese codos we have he followg resul. Theorem A.1 Poryag s maxmum prcple). Le ˆα be a admssble corol ad ˆX be he assocaed sae varable. Suppose here exss a soluo Y, Z) o he adjo equao A.5) such ha [ τ ] E e C Z 2 d <, A.6) ad he Hamoa verfes for every H ˆX, ˆα, Y ) = max a A H ˆX, a, Y ), a.s. A.7) ad for every, almos surely, he fuco X, α) H ˆX, ˆα, Y ) s cocave, he ˆα s a opmal corol, ha s, Jα) Jˆα) for all admssble α = α ).

22 RENÉ CARMONA AND KEVIN WEBSTER Proof. If X ) s he sae assocaed o aoher admssble corol α ), we have he cha of relaoshps: [ E g τ X τ ) g τ ˆX ] τ ) E [X τ ˆX ) ] τ Y τ A.8) [ τ = E X ˆX ) X H ˆX, ˆα, Y ) + f X, α ) f ˆX ) ) ], ˆα ) Y d A.9) [ τ E H ˆX, ˆα, Y ) H X, ˆα, Y ) + f X, α ) f ˆX ) ) ], ˆα ) Y d A.1) [ τ = E H ˆX, ˆα, Y ) H X, ˆα, Y ) H ˆX ) ], ˆα, Y ) + H X, α, Y ) d [ τ E j X, α ) j ˆX ) ], ˆα ) d A.11) [ τ ] E j X, α ) j ˆX, ˆα ). A.12) A.8) sems from he cocavy of g ad ermal codo of Y. A.9) follows from he dyamcs of Y ad he relaoshp: [ τ E X ˆX ) ] 2 [ τ ] Z 2 d 2X E e C Z 2 d < A.13) whch guaraees ha he local margale par s a margale. A.1) holds by he cocavy of he Hamloa. A.11) s jus he defo of H. Fally, A.12) s a cosequece of he fac ha ˆα maxmzes H ˆX,, Y ). Appedx B. Dervao of he SPDE 4.16) We gve he ma seps of he proof of Proposo 4.1. By depedece of he commo radomess W ad he dosycrac radomess M, α, ɛ ) 1, we ca freeze he radomess of W ad work afer codog wh respec o F W, he σ-algebra geeraed by W, whou affecg he depedece properes of he dosycrac radom varables. By depedece of he M, he α ad ɛ, we have ha, codoal o F W, he radom varables α = α e ρ + e ρ s) νdw s + σdms ) B.1) are d ad Gaussa. So f f C 2 s such ha f ad s wo dervaves have a mos polyomal growh, gve he assumpo 4.17) o he jo dsrbuo m of all he couples α, ɛ ), we ca apply he law of large umbers ad ge: 1 lm ɛ fα ) = E [ ɛ 1 fα 1 ) F W ] =1 = ɛf α e ρ + e ρ s) νdw s + σx e 2ρ s) ds 1 e x2 2 mdα, dɛ)dx R 3 2π The oher resuls ca be derved drecly from hs explc represeao.

HIGH FREQUENCY MARKET MAKING 23 Refereces [1] A. Alfos, A. Fruh, ad A. Sched. Opmal execuo sraeges lm order books wh geeral shape fucos. Quaave Face, 12):143 157, 21. [2] R. Almgre ad N. Chrss. Opmal execuo of porfolo rasacos. Joural of Rsk, 32):5 39, 2. [3] R. F. Almgre. Opmal execuo wh olear mpac fucos ad radg-ehaced rsk. Appled Mahemacal Face, 11):1 18, 23. [4] Y. Amhud ad H. Medelso. Asse prcg ad he bd-ask spread. Joural of Facal Ecoomcs, 172):223 249, 1986. [5] J.. Bouchaud, Y. Gefe, M. Poers, ad M. Wyar. Flucuaos ad respose facal markes: The suble aure of radom prce chages. Quaave Face, 42):176 19, 24. [6] J.. Bouchaud, M. Mzard, ad M. Poers. Sascal properes of sock order books: Emprcal resuls ad models. Qua.Face, 24):251 256, 22. [7] R. Co, S. Sokov, ad R. Talreja. A sochasc model for order book dyamcs. Operaos research, 583):549 563, 21. [8] I. Csszar. Iformao-ype measures of dfferece of probably dsrbuos ad drec observaos, volume 2. 1967. [9] F. Delbae ad W. Schachermayer. The Mahemacs of Arbrage. Sprger Verlag, 21. [1] M. B. Garma. Marke mcrosrucure. Joural of Facal Ecoomcs, 33):257 275, 1976. [11] J. Hasbrouck. Emprcal marke mcrosrucure. Oxford Uversy Press, 27. [12] T. G. Kurz ad J. Xog. Parcle represeaos for a class of olear spdes. Sochasc Processes ad her Applcaos, 831):13 126, 1999. [13] T. G. Kurz ad J. Xog. Numercal soluos for a class of SPDEs wh applcaos o flerg. Sochascs Fe ad Ife Dmeso, pages 233 258, 2. [14] A. S. Kyle. Couous aucos ad sder radg. Ecoomerca, 536):1315 1335, 1985. [15] P.A. Meyer. Probables ad Poeal. Norh-Hollad Pub., 1966. [16] A. Obzhaeva ad J. Wag. Opmal radg sraegy ad supply/demad dyamcs. Prepr, 25. [17] H. Pham. Couous-me sochasc corol ad opmzao wh facal applcaos. Sprger Verlag, 28. [18] M. Wyar, J.. Bouchaud, J. Kockelkore, M. Poers, ad M. Veorazzo. Relao bewee bd-ask spread, mpac ad volaly order-drve markes. Quaave Face, 81):41 57, 28. [19] I. Zovko ad J. D. Farmer. The power of paece: A behavoral regulary lm order placeme. Quaave Face, 25):387 392, 22.