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Quanttatve Fnance, 2013 Vol. 13, No. 10, 1575 1586, http://dx.do.org/10.1080/14697688.2013.779013 Effcent portfolo valuaton ncorporatng lqudty rk YU TIAN, RON ROOD and CORNELIS W. OOSTERLEE School of Matheatcal Scence, Monah Unverty, Melbourne, VIC 3800, Autrala RBS The Royal Bank of Scotland, 280 Bhopgate, London EC2M 4RB, UK CWI Centru Wkunde & Inforatca, P.O. Box 94079, 1090 GB Aterda, The Netherland 1. Introducton (Receved 21 Deceber 2010; n fnal for 14 February 2013) Accordng to the theory propoed by Acerb and Scandolo (2008) [Quant. Fnance, 2008, 8, 681 692], an aet decrbed by the o-called Margnal Supply Deand Curve (MSDC), whch a collecton of bd and ak prce accordng to t tradng volue, and the value of a portfolo defned n ter of coonly avalable arket data and doyncratc portfolo contrant poed by an nvetor holdng the portfolo. Dependng on the contrant, one and the ae portfolo could have dfferent value for dfferent nvetor. A t turn out, wthn the Acerb Scandolo theory, portfolo valuaton can be fraed a a convex optzaton proble. We provde ueful MSDC odel and how that portfolo valuaton can be olved wth rearkable accuracy and effcency. Keyword: Lqudty rk; Portfolo valuaton; Ladder MSDC; Lqudaton equence; Exponental MSDC; Approxaton JEL Clafcaton: C60, G11, G12 Accordng to the theory developed by Acerb and Scandolo (2008) the value of a portfolo deterned by arket data and a et of portfolo contrant. The arket data aued to be publcly avalable and the ae for all nvetor. The arket data cont of prce quote correpondng to dfferent tradng volue. Thee quote for an aet are repreented n ter of a atheatcal functon referred to a a Margnal Supply Deand Curve (MSDC). The portfolo contrant ay vary acro dfferent player. Thee doyncratc contrant collectvely referred to a a lqudty polcy refer to retrcton that any portfolo held by the nvetor hould be prepared to atfy. Exaple of uch portfolo contrant are: nu cah aount to eet hort-ter lqudty need; arket or credt rk anageent lt; captal lt. We ntroduce the fundaental concept of Acerb Scandolo theory n ecton 2. To value her portfolo, the nvetor wll ark all the poton he could pobly unwnd to atfy the lqudty polcy to the bet prce accordng to an MSDC functon. A t turn out, wthn Acerb and Scandolo theory, the valuaton of a portfolo of aet can be fraed a a convex optzaton proble. The aocated contrant et repreented by a lqudty polcy. Although th wa already ponted out by Acerb and Scandolo theelve, the practcal plcaton of the theory have a yet not been well nvetgated. Such the a of the preent paper. We wll tudy portfolo valuaton under theacerb Scandolo theory extenvely, aung dfferent for of the MSDC functon. We frt conder a very general ettng where the MSDC haped a a non-ncreang tep functon (referred to a a ladder MSDC) n ecton 3. Th correpond to noral arket tuaton for relatvely actvely traded product uch a lted equte. We wll preent an algorth for portfolo valuaton aung ladder MSDC and a cah portfolo contrant. In ecton 4, we wll look at MSDC that are haped a decreang exponental functon, whch can be ued to decrbe le-lqud over-the-counter (OTC) traded product. We wll alo ee how the exponental functon can be ued a approxaton of ladder MSDC. All nuercal reult are collected n ecton 5. We wll fnd that, n a wde range of cae, the approxaton of ladder MSDC by exponental MSDC appear to be accurate, uggetng that not all arket prce nforaton repreented n ladder MSDC neceary for accurate portfolo valuaton. We preent our concluon n ecton 6. Correpondng author. Eal: ocar.tan@onah.edu 2013 Taylor & Franc

1576 Y. Tan et al. 2. The portfolo theory Th ecton preent relevant concept and reult fro Acerb and Scandolo (2008) that wll be ued throughout th paper. 2.1. Aet An aet an object traded n a arket and wll be characterzed by a Margnal Supply Deand Curve (MSDC). Th codfe avalable bd and ak prce correpondng to dfferent tradng volue. Defnton 2.1: An MSDC a ap : R\{0} R atfyng the followng two condton: (1) () non-ncreang,.e. ( 1 ) ( 2 ) f 1 < 2 ; (2) () càdlàg (.e. rght-contnuou wth left lt) for < 0 and làdcàg (.e. left-contnuou wth rght lt) for > 0. The varable repreent the tradng volue of the aet. Condton 1 repreent a no-arbtrage aupton. Condton 2 enure that MSDC have elegant atheatcal properte. We wll not ue th condton heavly and we only enton t for the ake of copletene. Intead, what we wll need ot of the te that an MSDC (Reann) ntegrable on t doan. We call the lt + := l h 0 (h) the bet bd and := l h 0 (h) the bet ak.thebd ak pread, denoted by δ, defned a δ := +. Defnton 2.2: Cah the aet repreentng the currency pad or receved when tradng any aet. It characterzed by a contant MSDC, 0 () = 1 (.e. one unt) for every R \{0}. Cah referred to a a perfectly lqud aet f the aocated MSDC contant. We call a ecurty any aet whoe MSDC a potve functon (e.g., a tock, a bond, a coodty) and a wap any aet whoe MSDC can take both potve and negatve value (e.g., an nteret rate wap, a CDS, a repo tranacton). Anegatve MSDC can be converted nto a ecurty by defnng a new MSDC a () := ( ). We preuppoe one currency a the cah aet. For exaple, f we chooe the euro a the cah aet, relatve to the euro, the US dollar wll be condered a an llqud aet. 2.2. Portfolo A portfolo characterzed by ltng the holdng volue of dfferent aet n the portfolo. Gven are N + 1 aet labeled 0, 1,...,N. We let aet 0 label the cah aet. Defnton 2.3: A portfolo a vector of real nuber, p = (p 0, p 1,...,p N ) R N+1, where p repreent the holdng volue of aet. In partcular, p 0 denote the aount of cah n the portfolo. When we pecfcally want to hghlght the portfolo cah we tend to wrte a portfolo a p = (p 0, p). We henceforth preuppoe a et of portfolo referred to a the portfolo pace P. We wll aue that P a vector pace o that t becoe eanngful to add portfolo together and to ultply portfolo by calar nuber. Let p = (p 0, p) P and uppoe we have an addtonal aount a of cah. We wrte p+a = (p 0 +a, p ). Defnton 2.4: The lqudaton Mark-to-Market (MtM) value L(p) of a portfolo p defned a L(p) := =0 p 0 (x) dx = p 0 + p 0 (x) dx. (1) The lqudaton MtM value can be vewed a the value of a portfolo p for an nvetor who hould be able to lqudate all her poton n exchange for cah. Defnton 2.5: The upperot Mark-to-Market (MtM) value U(p) of p gven by where U(p) := ± p = p 0 + =0 ± p, (2) { + ± =, f p > 0,, f p < 0. The upperot MtM value can be vewed a the value of a portfolo for an nvetor who ha no cah deand. In th ene, the portfolo uncontraned. A MSDC are non-ncreang, U(p) L(p). The dfference between U(p) and L(p) tered the upperot lqudaton cot and defned a C(p) := U(p) L(p). 2.3. Lqudty polcy The defnton of the lqudaton MtM value L(p) and the upperot MtM valueu(p) ugget that the value of a portfolo p ubject to certan cah contrant an nvetor hould be able to eet by wholly or partly lqudatng poton he ha taken. Thee contrant are repreented a a lqudty polcy. There could be other type of contrant bede. For exaple, an nvetor ght want to poe arket rk VaR lt on her poton, or credt lt, or captal contrant. All the contrant that an nvetor poe can be repreented a a ubet of the underlyng portfolo pace P. Thee contrant are collectvely referred to a a lqudty polcy. We refer to Acerb and Fnger (2010) and Weber et al. (2013). Defnton 2.6: A lqudty polcy L a cloed and convex ubet of P atfyng the followng condton: (1) f p = (p 0, p) L and a 0, then p + a = (p 0 + a, p) L; (2) f p L, then (p 0, 0) L. Exaple 2.7: A lqudty polcy ettng a nu cah requreent, c, acah lqudty polcy: L(c) := {p P p 0 c 0}. (4) An nvetor endorng a cah lqudty polcy hould be prepared to lqudate her poton to uch an extent that nu cah level c obtaned. We wll extenvely ue cah (3)

Effcent portfolo valuaton ncorporatng lqudty rk 1577 lqudty polce n ecton 3 and 4. We refer to Acerb (2008) and Weber et al. (2013) for addtonal exaple of lqudty polce. 2.4. Portfolo value Th ecton preent Acerb and Scandolo defnton of the portfolo value functon. We frt need the followng defnton. Defnton 2.8: Let p, q P be portfolo. We ay that q attanable fro p f q = p r + L(r) for oe r P. The et of all portfolo attanable fro p wrtten a Att(p). The followng defnton key. Defnton 2.9: The Mark-to-Market (MtM) value (or the value, for hort) of a portfolo p ubject to a lqudty polcy L the value of the functon V L : P R { }, defned by V L (p) := up{u(q) q Att(p) L}. (5) If Att(p) L =, eanng that no portfolo attanable fro p atfe L, then we tpulate the portfolo value to be. Acerb and Scandolo (2008) gve the followng propoton of the new portfolo value. Propoton 2.10: The portfolo value functon V L fro defnton 2.9 can be alternatvely defned a V L (p) = up{u(p r) + L(r) r P, p r + L(r) L}. (6) To prove th not very dffcult; we refer to Acerb and Scandolo (2008). Propoton 2.10 allow u to frae the deternaton of the value of a portfolo a an optzaton proble wth explct contrant, naely axze U(p r) + L(r), ubject to p r + L(r) L, r P. (We gnore the cae V L (p) =.) Th optzaton proble convex a L a convex et. Snce L alo cloed, th proble ha a unque optal value (whch could be ). 3. Portfolo valuaton ung ladder MSDC In the prevou ecton we have outlned the an concept of Acerb and Scandolo portfolo theory. We dcued that portfolo valuaton could be fraed a a convex optzaton proble (7). Convex optzaton proble can often be olved nuercally (Boyd and Vandenberghe, 2004). In the preent ecton we wll provde an algorth provdng an exact global oluton for proble (7) under the aupton that the MSDC for llqud aet pecewe contant; a uch we wll nae the ladder MSDC. Wthn theacerb Scandolo theory, ladder MSDC wll play a key role n odelng the lqudty of the aet. Equpped wth the fat and accurate algorth dcued n th ecton, one could olve the convex optzaton proble ncurred n portfolo valuaton ore effcently than ung conventonal optzaton technque. (7) 3.1. The optzaton proble Generally we aue a arket wheren we can quote a prce for each volue we wh to trade,.e. a arket of unlted depth. However, n a real-world arket context, we wll typcally only be able to trade volue wthn certan bound. The upper and lower bound of th doan repreent the arket depth: the upper bound repreent the axu volue we wll be able to ell agant prce we can quote fro the arket and the lower bound repreent the axu we wll be able to buy agant prce we wll be able to quote fro the arket. Weber et al. (2013) refer to th et of contrant on the portfolo pace a a portfolo contrant. In the context of lted arket depth, we wll need to retrct the doan to a ubet of the portfolo pace to olve the optzaton proble of portfolo valuaton. In what follow, we tll aue unlted arket depth o that we can earch for the optal oluton n the whole portfolo pace for plcty, wherea the ethod we tate below alo work wth lted arket depth. Reconder proble (7). Ung a cah lqudty polcy L(c) th becoe axze U(p r) + L(r), ubject to p 0 r 0 + L(r) c, (8) r P. The nequalty contrant can be replaced by the equalty contrant p 0 r 0 + L(r) = c wthout affectng the optal value of the orgnal proble. Furtherore, we ay aue that the cah coponent r 0 equal 0 a t doe not play a role n the optzaton proble. To fnd the optal oluton we hence ght a well olve axze U(p r) + L(r), ubject to L(r) = c p 0, r P. Note that, wthout lo of generalty, we ay aue that p 0 = 0; otherwe ue the cah lqudty polcy L(c p 0 ). 3.2. A calculaton chee for portfolo valuaton wth ladder MSDC In the cae of portfolo valuaton baed on ladder MSDC we can olve the aocated optzaton proble (9) nuercally, for exaple by an nteror pont algorth (Boyd and Vandenberghe, 2004). However, the pleentaton of the algorth could be coputatonally neffcent n the ene that everal teraton ght be requred to brng the oluton wthn reaonable bound n hgh denon. In addton, the non-oothne of the ladder MSDC ncreae the dffculty of pleentng conventonal convex optzaton algorth. Hence, the a of th ecton to provde an algorth for proble (9) yeldng an exact global optal oluton r. Unle otherwe tated, throughout the reander of th ecton we ue the followng aupton. For exaple, the optalty condton n the nteror pont algorth wll not apply at non-ooth pont of the ladder MSDC. See Boyd and Vandenberghe (2004) for ore nforaton. (9)

1578 Y. Tan et al. Aupton 3.1: Any nvetor hold a portfolo p contng of long poton only and ue a cah lqudty polcy L(c) (c > 0). Propoton 3.2: Under aupton 3.1, the axzaton proble (9) ha the ae optal oluton a the followng nzaton proble: nze C(r), ubject to L(r) = c p 0, r P. (10) Proof: Snce we are prepared to lqudate our portfolo for cah under a cah polcy, the portfolo p and r hould have the ae gn coponentwe. It follow that U(p r) = U(p) U(r) by the defnton of the upperot MtM value. Conequently, the objectve functon of proble (9) can be rewrtten a U(p) U(r) + L(r). Snce, gven p, we can alway deterne U(p), axzng th functon under the gven contrant wll yeld the ae optal oluton r a axzng the followng functon under the ae contrant: Obvouly, nzng U(r) + L(r). U(r) L(r) agan yeld the ae optal oluton r. Notng that C(r) = U(r) L(r) prove the reult. Reark 1: The followng nequalty hold n general: U(p r) U(p) U(r). For exaple, we ay be prepared to ncreae our hare n everal rky aet or reduce the purchae of rky aet. In tuaton lke thee, coponent of the orgnal portfolo p and correpondng coponent of to-be-lqudated portfolo r ay have dfferent gn. Equalty hold under aupton 3.1. Inforally, propoton 3.2 ple that to deterne the value of a portfolo under a cah lqudty polcy to deterne a portfolo r uch that lqudatng r n exchange for cah nze the upperot lqudaton cot C(r ). Th reult wll prove ueful at a later tage. Gven that all aet are aued to be characterzed by ladder MSDC, we can convenently break up each and every poton n our portfolo nto a fnte nuber of volue. To each of thee volue there correpond a defnte arket quote a repreented by the MSDC. The dea of the algorth to conder all of thee portfolo bt together and to lqudate the n a yteatc and orderly anner, tartng wth the porton that wll be lqudated wth the allet cot relatve to the bet bd, and ubequently to thoe that can be lqudated wth the econd allet cot, and o on, untl the cah contrant et. If the nu cah requreent that the portfolo hould be prepared to atfy exceed the lqudaton MtM value of the entre portfolo, then we wll never be able to eet the cah contrant; by defnton, we et the portfolo value to be. Alternatvely, uppoe that we ell off a fracton of each poton agant the bet bd prce and that the total cah we ubequently receve n return exceed the cah contrant. Then the value of the portfolo equal the upperot MtM value and there ext nfntely any optal oluton. We wll now ake th foral, tartng wth the followng defnton. Defnton 3.3: Gven an aet, characterzed by MSDC.Thelqudty devaton of a volue of aet defned a S () := + (), for > 0. (11) + The lqudty devaton the relatve dfference between the bet bd prce and the lat arket quote () ht for a volue. In th ene, t eaure the lqudty of aet at unt traded relatve to the bet bd. Gven any aet, the lqudty devaton a non-decreang functon, a the MSDC correpondng to that aet nonncreang. For a ecurty, the value of the lqudty devaton are n [0, 1], a the lower bound of the correpondng MSDC 0. For a wap, the value are n [0, + ). Snce the MSDC of an aet aued to be pecewe contant, each value of the lqudty devaton correpond to a axu bd ze. Ung the prevouly defned lqudty devaton, poton are lqudated n a defnte order, a follow. Gven a portfolo r = (r 0, r 1,...,r N ), aue that we want to lqudate all the r, > 0. Each non-cah poton r can be wrtten a a u J r = r j, = 1,...,N, (12) j=1 where r j called a lqudaton ze. To defne the lqudaton ze r j, conder the bd part of a ladder MSDC, whch contructed by a fnte nuber of bd prce wth axu bd ze. For each r n aet, we can dentfy a fnte nuber J of bd prce j wth lqudaton ze r j, j = 1,...,J. For the frt J 1 lqudaton ze r j ( j = 1,...,J 1), they are equal to the frt J 1 axu bd ze recognzed fro the arket. For the J th lqudaton ze r j, t le than or equal to the J th axu bd ze. Moreover, each lqudaton ze r j correpond to each bd prce j. In partcular, the frt lqudaton ze of each aet r 1 correpond to the bet bd + = 1. The lqudty devaton for each lqudaton ze can then be wrtten a S j = + j + = 1 j. (13) 1 Now we put the lqudty devaton S j n acendng order ndexed by k, and we genercally refer to any ter of th equence a S k (r) (the addton of r a an extra paraeter wll prove convenent later on). Note that the length of the lqudaton equence equal K = J 1 + + J N. In addton, we oberve that there ext a natural one-to-one correpondence between the equence {S k (r)}, the equence of lqudaton ze {r j } and the equence of bd prce { j }. Hence, whle preervng thee one-to-one correpondence, we relabel the equence {r j } and { j } a {r k } and { k }, repectvely. We call the orted ndex k the lqudaton equence, whch a perutaton of the ndex (, j).

Effcent portfolo valuaton ncorporatng lqudty rk 1579 Table 1. Bd prce nforaton of aet 1 and 2. Maxu bd ze Bd prce Maxu bd ze Bd prce (a) Aet 1 (b) Aet 2 200 11.65 200 19.58 200 11.55 600 19.5 200 11.45 200 19.2 Table 2. Lqudaton ze of our portfolo r = (0, 600, 900). Lqudaton ze Bd prce (a) Aet 1 r 11 200 11 11.65 r 12 200 12 11.55 r 13 200 13 11.45 (b) Aet 2 r 21 200 21 19.58 r 22 600 22 19.5 r 23 100 23 19.2 Table 3. Lqudty devaton and lqudaton equence. Lqudaton equence Index (, j) Aet Lqudaton ze Bd prce Lqudty devaton 1 (1, 1) 1 200 11.65 0 2 (2, 1) 2 200 19.58 0 3 (2, 2) 2 600 19.5 0.004085802 4 (1, 2) 1 200 11.55 0.008583691 5 (1, 3) 1 200 11.45 0.017167382 6 (2, 3) 2 100 19.2 0.019407559 Note that the frt N ter of the equence { k } are the bet bd +, = 1,...,N. To llutrate the above concept, conder an exaple a follow. Gven two llqud aet, the bd part that can be read fro the arket are hown n table 1. Aue that we hold a portfolo that contan 600 unt n aet 1 and 900 n aet 2. The lqudaton ze for the two aet are hown n table 2 and the orted lqudty devaton a well a the lqudaton equence are preented n table 3. To eet the cah contrant eboded n the cah lqudty polcy we tart lqudatng the portfolo fro S 1 (r), then S 2 (r), and o on, untl we have et the cah requreent. The lqudaton equence effectvely drect the earch proce throughout the contrant et toward the global oluton, and exactly o. Th uarzed n the followng theore, whch we wll prove ubequently. Propoton 3.4: Gven a portfolo p uch that each aet characterzed by a ladder MSDC. Under aupton 3.1, the optzaton proble (9) ha the ae optal oluton a the followng: nze Kk=1 S k (r), ubject to L(r) = c p 0, (14) r P. Looely put, the optal oluton the one yeldng the nu total u of lqudty devaton. Intutvely, the propoton ple that, to eet cah deand, we hould lqudate the ot lqud aet a they are eaer to ell off and ther lqudaton wll ncur le loe copared wth ore llqud aet. Proof: Let a portfolo p = (p 0, p 1,...,p N ) be gven and uppoe we lqudate a portfolo r = (r 0, r 1,...,r N ) to eet a lqudty polcy L. Aet ha a correpondng MSDC, = 0, 1,...,N. For plcty, r 0 et to be 0. Fro propoton 3.2, the optal oluton of (9) nze the upperot lqudaton cot. Ung that all aet are characterzed by ladder MSDC, the objectve functon C(r) can be rewrtten a follow: C(r) = U(r) L(r) = N J ( + r j j r j ). j=1 Note that, for each aet, + j for all j. It follow that the nu of the u of the abolute dfference between + r j and j r j the ae a the nu of the u of the relatve dfference. Hence, to fnd the optal oluton we ght a well nze

1580 Y. Tan et al. J j=1 + r j j r j + r j = N J j=1 J + j + = N S j (r) j=1 = K S k (r). k=1 On the lat lne, note that K = J 1 + + J N. Baed on th reult, we now tate the algorth for portfolo valuaton aung only ladder MSDC under aupton 3.1. For the ake of clarty we recall that the optal oluton r of proble (9) hould atfy L(r ) = c p 0. Alo, we aue that p 0 = 0 and r 0 = 0. (Otherwe, we can et the cah requreent c = c p 0.) The peudocode uarzed n algorth 1. Algorth 1: Algorth for portfolo valuaton aung ladder MSDC anda cah lqudty polcy L(c) Calculate: U(p) = N + p ; L(p) = N J j=1 j p j ; V 1 (p) = N + p 1 ; S j = ( 1 j )/ 1 ; Sort the S j a an acendng equence wth ndex k.{wth k runnng fro 1 to J 1 + + J N } f c > L(p) then return V L(c) (p) = ;{There no optal oluton atfyng the cah contrant.} ele f c = L(p) then return V L(c) (p) = L(p); {The optal oluton r = p.} ele f c V 1 (p) then {Lqudatng p 1 to the repectve bet bd eet the cah contrant.} return V L(c) (p) = U(p);{There are nfntely any optal oluton.} ele U(r) = V 1 (p); c = c V 1 (p); k = N +1; {Start loop fro the frt part wth non-zero lqudty devaton untl c becoe 0.} whle c > 0 do f c/ k > p k then U(r) = U(r) + + k p k; c = c k p k ; k = k + 1; ele U(r) = U(r) + + k (c/ k); c = 0; end f end whle return V L(c) (p) = U(p) U(r) + c {Here we have L(r) = c.} end f end f end f There are generally four cae tated n algorth 1: (1) f the cah requreent c hgher than the lqudaton MtM value L(p) uch that the cah lqudty polcy cannot be et, then we agn to the portfolo value V L(c) (p) and conclude that there no optal oluton; (2) f the cah requreent c equal to the lqudaton MtM value L(p) uch that we have to lqudate all part of the portfolo, then the portfolo value V L(c) (p) equal the lqudaton MtM value L(p) and the unque optal oluton r = p; (3) f the cah requreent c le than or equal to V 1 (p), the lqudaton value of all part of the portfolo correpondng to the bet bd, then the portfolo value V L(c) (p) equal V 1 (p) and there ext nfnte any optal oluton; (4) f the cah requreent c hgher than V 1 (p) but le than L(p), we have to lqudate the portfolo along the lqudaton equence untl the cah requreent et and the unque optal oluton r can be found by recordng the lqudaton part of correpondng aet n the calculaton procedure of the algorth. The pecewe contant MSDC n the convex optzaton proble generally ncreae the dffculty of the earch for the global optal oluton wth tandard oftware. Wth the aforeentoned calculaton chee lted n algorth 1, ntead, we can olve the optzaton proble accurately and effcently va the lqudaton equence. 4. Portfolo valuaton ung contnuou MSDC There typcally no analytc oluton to the convex optzaton proble (9). However, t can be hown that f we odel the MSDC a a contnuou functon we can obtan ple analytc oluton fro the ethod of Lagrange ultpler. In ecton 4.1 we wll frt look at contnuou MSDC wthout pong any pecfc for for the. We wll ubequently look at MSDC haped a exponental functon n ecton 4.2 and 4.3. Eprcally, we fnd that exponental MSDC can be ued to odel MSDC for ecurty-type equty aet wth dfferent cap. We then propoe to ue exponental MSDC to approxate ladder MSDC n order to prove the effcency of portfolo valuaton n ecton 4.4. We wll aue the cah lqudty polcy n th ecton. 4.1. The general cae Aue N llqud aet labeled 1,...,N wth MSDC, = 1, 2,...,N. Each aued to be contnuou on R. Th ple that (0) ext. We wll exclude the pont (0) later n th ecton. In addton, each aued to be trctly decreang. Adoptng a cah lqudty polcy, valung a portfolo contng of poton n thee aet coe down to olvng the optzaton proble (9). The oluton to th optzaton proble can be derved analytcally, a hown by the followng propoton propoed by Acerb and Scandolo (2008).

Propoton 4.1: Aung contnuou trctly decreang MS- DC and the cah lqudty polcy L(c), the optal oluton r = (0, r ) to optzaton proble (9) unque and gven by r = { 1 [ (0)/(1 + λ)], f p 0 < c, 0, f p 0 c, (15) where 1 ( ) denote the nvere of the MSDC functon ( ), and the Lagrange ultpler λ, repreentng the argnal lqudaton cot, can be deterned fro the equaton L(r ) = c p 0. We refer to Acerb and Scandolo (2008) for a proof. Reark 2: Note that we can extend the above to the cae where the MSDC are not contnuou at the pont 0,.e. the cae where there a potve bd ak pread. We have to change the defnton of the value at (0) to the lt + n the cae of long poton or to n the cae of hort poton. Obvouly, by ung the Lagrange ultpler ethod, we can generalze the cae to any lqudty polcy gvng re to equalty contrant. When ung a general lqudty polcy whch reult n nequalty contrant, we can olve the optzaton proble (7) by checkng the Karuh Kuhn Tucker (KKT) condton. In addton, the Lagrange dual ethod ay alo be ueful. 4.2. Exponental MSDC for large- and edu-cap equte We contnue the dcuon by lookng at a partcular exaple of a MSDC,.e. the exponental MSDC. A t turn out, the exponental MSDC for an effectve odel to characterze a ecurty-type aet and to deterne the portfolo value by convex optzaton. We wll dcu th n ecton 4.4. Many reearche have hown that there a relaton between the prce change and the tradng volue n the arket durng a hort te perod. Cont et al. (2011) propoe that there a quare-root relaton between the prce change and the tradng volue for S&P 500 equte. Algren et al. (2005) found a lar reult and propoed a 3/5 relaton between the teporary prce pact and the trade ze for large-cap US equte. Thee paraeter correpond to a edu-ze prce pact. In our paper, we nterpret the prce change a log(()/ + ),.e. the relatve change between the bd (or ak) prce () and the bet bd + (or bet ak ) over a hort te perod, durng whch an MSDC can be fored and denote the tradng volue to be. Large- and edu-cap equte lted on tock exchange uch a the London Stock Exchange and Euronext are actvely traded and thu relatvely lqud. Fro avalable data we oberve a quare-root relaton between the bd prce change and the volue over a hort te, a follow: ( ) () log + = k + ɛ, (16) where +, k > 0 and ɛ the noe ter. For the ak prce part, where < 0, we have the followng odel: Effcent portfolo valuaton ncorporatng lqudty rk 1581 ( ) () log = k +ɛ, (17) where, k > 0 and ɛ the noe ter. When kppng the noe ter, we ue the followng exponental MSDC odel to approxate the bd part of the ladder MSDC for a large- or edu-cap equty: () = + e k, (18) and we approxate the ak part a () = e k. (19) We gve two exaple of the ladder MSDC and the above approxated exponental MSDC by leat-quare regreon for large- and edu-cap equte n fgure 1. Suppoe there are N (large- or edu-cap) ecurty-type aet 1, 2,...,N, the bd part of whch are characterzed by () = + e k. (20) We call k the lqudty rk factor for the correpondng aet ( = 1,...,N), whch eaure the general lqudty condton of aet. Fro propoton 4.1 and reark 2, we can approxate portfolo value under dfferent lqudty polce. A an exaple, aung a portfolo wth only long poton, then we have the lqudaton MtM value p L(p) = p 0 + (x) dx = p 0 + 0 2 + k 2 (1 k p e k p e k p ), (21) and under a cah lqudty polcy L(c) wth p 0 < c, fro propoton 4.1, we have ( ) log(1 + λ) 2 r =, = 1,...,N, k wth λ ( = e x 1, x > 0, ) (22) c p 0 and 1 N (2 + /k 2) e x x 1 = 0. The lat equaton can be olved nuercally by ung the Newton Raphon teraton ethod or the Taylor expanon. Hence, the portfolo value under the cah lqudty L(c) read V L(c) (p) = U(p r ) + L(r ) ( ( ) ) log(1 + λ) 2 = + p + c. (23) For large- and edu-cap aet, nce they are generally very lqud to trade, the upperot lqudaton cot uually very all. 4.3. Exponental MSDC for all-cap equte On the other hand, for all-cap equte, we fnd there a quare relaton between the bd prce change and the volue over a hort te, whch ple a large prce pact: ( ) () log + = k 2 + ɛ, (24) where +, k > 0 and ɛ the noe ter. k

1582 Y. Tan et al. 26.02 ladder MSDC exp MSDC 4.22 4.215 4.21 ladder MSDC exp MSDC 26 4.205 () 25.98 25.96 25.94 25.92 2 1 0 1 2 3 x 10 5 () 4.2 4.195 4.19 4.185 4.18 4.175 1.5 1 0.5 0 0.5 1 x 10 5 Fgure 1. Exponental MSDC veru ladder MSDC for large- and edu-cap equte. Slarly, for the ak prce change we have ( ) () log = k 2 + ɛ, (25) where, k > 0 and ɛ the noe ter. When kppng the noe ter, we have the followng exponental MSDC odel to approxate the bd part of a ladder MSDC of a all-cap equty: () = + e k2, (26) and for the ak part we have the exponental MSDC () = e k 2. (27) We gve an exaple of a ladder MSDC and the approxated exponental MSDC for a all-cap equty n fgure 2. Suppoe that there are N (all-cap) ecurty-type aet 1, 2,...,N, whoe bd part are characterzed by the followng exponental MSDC: () = + e k 2, (28) wth +, k > 0 for all = 1,...,N. By ung a leat-quare approxaton, we can ft the value of + and k fro real data. See ecton 4.4. To llutrate th type of exponental MSDC functon, we aue a portfolo wth only long poton. Then the lqudaton MtM value read L(p) = p 0 + π = p 0 + 2 p 0 (x) dx + k erf( k p ), (29) where erf( ) the Gau error functon, whch can be obtaned nuercally. () 1.6 1.55 1.5 1.45 1.4 Fgure 2. equty. ladder MSDC exp MSDC 2 1 0 1 2 x 10 6 Exponental MSDC veru ladder MSDC for a all-cap For a cah lqudty polcy L(c) wth p 0 < c, fro propoton 4.1 we have r log(1 + λ) =, = 1,...,N, k wth λ = e z2 1, (30) ( ) z = erf 1 c p 0 ( π/2) N ( + /, k ) where erf 1 ( ) the nvere error functon, whch can alo be obtaned nuercally. Hence, V L(c) (p) = U(p r ) + L(r ) ( ) = + log(1 + λ) p + c. (31) k

Effcent portfolo valuaton ncorporatng lqudty rk 1583 Table 4. Bd of aet 1 4. Lqudaton ze Bd prce Lqudaton ze Bd prce (a) Aet 1 (c) Aet 3 200 11.65 400 29.3 200 11.55 200 29.16 200 11.45 400 29.15 200 11.1 400 28.9 200 11.05 200 28 200 11 600 27.8 200 10.3 200 27.15 500 9.3 200 27 500 6.5 400 26 1000 6.46 200 22 (b) Aet 2 (d) Aet 4 200 19.58 200 43.1 600 19.5 400 42.65 200 19.2 200 41.9 200 19.15 400 41 200 19.1 200 40.86 200 18.6 200 40.4 200 18.5 200 39 200 16.85 400 37 200 16.1 400 36 200 16.05 200 35.1 portfolo value x 10 5 3.05 3 2.95 2.9 2.85 2.8 2.75 2.7 0 0.5 1 1.5 cah needed 2 2.5 3 x 10 5 Fgure 3. Portfolo value wth dfferent cah requreent. 4.4. Approxatng ladder MSDC by exponental MSDC In ecton 3 we have defned a fat calculaton chee for portfolo valuaton wth ladder MSDC. In the real world, however, we ay face a tuaton where to collect the prce nforaton to for a ladder MSDC too cotly, or where the nforaton ncoplete or not avalable, e.g. n an over-thecounter (OTC) arket. A an order book record the tradng volue, whch for the ba of MSDC, one could odel ladder MSDC fro the odelng of order book dynac. For exaple, Bouchaud et al. (2002) found that the tradng volue at each bd (or ak) prce n the tock order book follow a Gaa dtrbuton. Cont et al. (2010) ued a contnuou Markov chan to odel the evoluton of the order book dynac. In our paper, we a to ue the bac contnuou MSDC odel to approxate ladder MSDC drectly, a we can then apply the Lagrange ultpler ethod and other convex optzaton technque to obtan analytc oluton and thu prove the effcency. For actvely traded large- or edu-cap ecurty-type aet, a portfolo valuaton baed on exponental MSDC (20) wth ther analytc oluton gnfcantly fater than wth ladder MSDC. For le actvely traded all-cap aet, we can ue the exponental MSDC odel (26) to obtan portfolo value. For OTC-traded aet, lackng prce nforaton, the exponental MSDC (28) for all-cap ecurty-type aet wth a large lqudty rk factor could be a frt odelng attept. Generally, when ung exponental MSDC odel (28) for all-cap ecurty-type aet, we need to etate or odel the paraeter + and k. The dynac of the bet bd + can be read fro arket data, or odeled by aet prce odel (e.g., geoetrc Brownan oton). If we aue that the lqudty rk factor k ndependent of +, we can eploy te ere or tochatc procee to odel k.ifk aued to be correlated wth +, we alo need to odel the correlaton. Furtherore, for ecurty-type aet traded n an OTC arket, we ay ue the ere prce nforaton of the aet to etate lqudty rk factor n the MSDC odel (28). In partcular, the lqudty rk factor ay be et at a hgh level to repreent the llqudty of the aet. For the approxaton of ladder MSDC of llqud ecurtytype aet ung all-cap exponental MSDC (28), we aue that the portfolo cont of only long poton n N llqud ecurty-type aet. If we aue that the lqudty rk factor of aet, k, ndependent of the bet bd +, then paraeter k can be etated fro the ladder MSDC of

1584 Y. Tan et al. Table 5. Lqudty devaton and lqudaton equence. Lqudaton equence Index (, j) Aet Lqudaton ze Bd prce Bet bd Lqudty devaton 1 (1, 1) 1 200 11.65 11.65 0 2 (2, 1) 2 200 19.58 19.58 0 3 (3, 1) 3 400 29.3 29.3 0 4 (4, 1) 4 200 43.1 43.1 0 5 (2, 2) 2 600 19.5 19.58 0.004085802 6 (3, 2) 3 200 29.16 29.3 0.004778157 7 (3, 3) 3 400 29.15 29.3 0.005119454 8 (1, 2) 1 200 11.55 11.65 0.008583691 9 (4, 2) 4 400 42.65 43.1 0.010440835 10 (3, 4) 3 400 28.9 29.3 0.013651877 11 (1, 3) 1 200 11.45 11.65 0.017167382 12 (2, 3) 2 200 19.2 19.58 0.019407559 13 (2, 4) 2 200 19.15 19.58 0.021961185 14 (2, 5) 2 200 19.1 19.58 0.024514811 15 (4, 3) 4 200 41.9 43.1 0.027842227 16 (3, 5) 3 200 28 29.3 0.044368601 17 (1, 4) 1 200 11.1 11.65 0.0472103 18 (4, 4) 4 400 41 43.1 0.048723898 19 (2, 6) 2 200 18.6 19.58 0.050051073 20 (3, 6) 3 600 27.8 29.3 0.051194539 21 (1, 5) 1 200 11.05 11.65 0.051502146 22 (4, 5) 4 200 40.86 43.1 0.051972158 23 (2, 7) 2 200 18.5 19.58 0.055158325 24 (1, 6) 1 200 11 11.65 0.055793991 25 (4, 6) 4 200 40.4 43.1 0.062645012 26 (3, 7) 3 200 27.15 29.3 0.07337884 27 (3, 8) 3 200 27 29.3 0.078498294 28 (4, 7) 4 200 39 43.1 0.09512761 29 (3, 9) 3 400 26 29.3 0.112627986 30 (1, 7) 1 200 10.3 11.65 0.115879828 31 (2, 8) 2 200 16.85 19.58 0.139427988 32 (4, 8) 4 400 37 43.1 0.141531323 33 (4, 9) 4 400 36 43.1 0.164733179 34 (2, 9) 2 200 16.1 19.58 0.17773238 35 (2, 10) 2 200 16.05 19.58 0.180286006 36 (4, 10) 4 200 35.1 43.1 0.185614849 37 (1, 8) 1 500 9.3 11.65 0.201716738 38 (3, 10) 3 200 22 29.3 0.249146758 39 (1, 9) 1 500 6.5 11.65 0.442060086 40 (1, 10) 1 1000 6.46 11.65 0.445493562 aet by the ethod of leat quare a follow. Provded that + ha already been deterned, we tranfor the exponental functon a log( ()/ + ) = 2 k, and etate k by n dcrete par ( n, log( ( n )/ + )) to nze the ert functon: ( ( ) 2 n ( j ) log 2 j ) k. (32) j=1 + The leat-quare etate of paraeter k then read ˆk = n j=1 2 j log( ( j )/ + ) nj=1 4. (33) j 5. Nuercal reult In th ecton we gve exaple for the varou concept dcued n th paper. In partcular, we explan the calculaton chee for effcent portfolo valuaton by ean of an exaple. Snce, for large- or edu-cap ecurty aet, the upperot lqudaton cot uually qute all, we wll focu on relatvely llqud all-cap ecurty aet and valuate portfolo ung ladder MSDC and exponental MSDC. 5.1. Portfolo wth four llqud aet The exaple here baed on four llqud all-cap ecurtytype aet. We deal wth a portfolo p = (0, 3400, 2400, 3200, 2800) wth zero cah aet and long poton n all four llqud aet. The bd prce wth lqudaton ze for the portfolo are choen at a gven te a preented n table 4. It eay to calculate the upperot MtM value U(p) and the lqudaton MtM value L(p) fro the table, that U(p) = 3.01042 10 5 and L(p) = 2.73720 10 5. Hence, the upperot lqudaton cot equal C(p) = 0.27322 10 5.Ifthe true portfolo value equal to the lqudaton MtM value, but, however, f we would ue the upperot MtM value ntead, we would overetate the portfolo value by a uch a 10%.

Effcent portfolo valuaton ncorporatng lqudty rk 1585 12 11 exp MSDC v ladder MSDC for aet A 1 ladder exp 20 19.5 exp MSDC v ladder MSDC for aet A 2 ladder exp 10 19 () 9 8 7 () 18.5 18 17.5 17 6 16.5 5 16 4 0 500 1000 1500 2000 2500 3000 3500 15.5 0 500 1000 1500 2000 2500 () portfolo value 30 29 28 27 26 25 24 23 22 exp MSDC v ladder MSDC for aet A 3 0 500 1000 1500 2000 2500 3000 3500 3.05 x 105 3 2.95 2.9 2.85 2.8 2.75 ladder exp () 44 42 40 38 36 34 exp MSDC v ladder MSDC for aet A 4 ladder exp 32 0 500 1000 1500 2000 2500 3000 Fgure 4. Exponental MSDC veru ladder MSDC for the bd prce of aet 1 4. ladder exp relatve dfference 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 2.7 0 0.5 1 1.5 2 2.5 3 0 0 0.5 1 1.5 2 2.5 3 cah needed x 10 5 cah needed x 10 5 Fgure 5. Modelng ladder MSDC by exponental MSDC.

1586 Y. Tan et al. For dfferent cah requreent, we ue the orted lqudty devaton (ee table 5) to fnd the lqudaton equence and then calculate the portfolo value (ee fgure 3). Fro the lat row of table 5, we can ee that the lqudty devaton can be a large a 44.5% for the ot llqud part of the MSDC for aet 1, whch ndcate a hgh level of lqudty rk. Fro fgure 3, we nfer that the portfolo value decreae at a fater rate a we have to lqudate the poton of an ncreang nuber of llqud aet to eet the cah requreent, whch wll defntely caue ore gnfcant loe durng lqudaton. The calculaton chee n algorth 1 provde an effcent earch drecton to the optal value guded by the lqudaton equence. For th four-aet exaple, we copare our calculaton chee wth the fncon functon wth an nteror pont algorth n MATLAB for portfolo valuaton. The optzaton repeated for around 2.5 10 5 dfferent cah requreent and the total coputaton te recorded. The averaged te for each cah lqudty polcy equal 0.568 llecond for our chee, wherea fncon take 202.7 llecond, whch ple that the te dfference a factor of 300. Snce the acendng equence of lqudty devaton how the llqudty of dfferent part of the correpondng aet, lqudatng a portfolo along the lqudaton equence wll caue nu lo of value copared wth the other knd of lqudaton. 5.2. Ung exponental MSDC to approxate ladder MSDC For the four-aet exaple wth the ladder MSDC fro ecton 5.1, we ue the exponental MSDC (28) for all-cap equte. Fgure 4 llutrate the ladder MSDC and the correpondng exponental approxatng MSDC. The latter MS- DC are etated by leat quare (ee ecton 4.4). The lqudty rk factor n the exponental MSDC are found a k 1 = 7.4193 10 8, k 2 = 3.5499 10 8, k 3 = 1.8691 10 8 and k 4 = 3.1634 10 8. Hence, we nfer that aet 1 the ot llqud and aet 3 the ot lqud, n general. In fgure 5(a), we copare the portfolo value obtaned ung the exponental MSDC wth the reference portfolo value by the ladder MSDC under dfferent cah requreent. The relatve dfference n the portfolo value preented n fgure 5(b). The relatve dfference found at ot 1.91%, o that, n th exaple, the exponental MSDC are accurate approxaton. The large approxaton error le n the tal part of the fgure and caued by the llqudty of the tal part of aet 1 and 3. Th ean that the exponental MSDC ay fal to approxate the tal part of aet 1 and 3 f there are huge drop n prce. 6. Concluon Wthn the theory propoed by Acerb and Scandolo (2008)the valuaton of a portfolo can be fraed a a convex optzaton The coputer ued for all experent ha an Intel Core2 Duo CPU, E8600 @3.33 GHz wth 3.49 GB of RAM and the code wrtten n MATLAB R2009b. proble. We have propoed a ueful and effcent algorth ung a pecfc for of the arket data functon,.e. all prce nforaton repreented n ter of a ladder MSDC. We have alo condered approxaton of ladder MSDC by exponental functon. A long a the portfolo valuated ung the new odel ncorporatng lqudty rk, one can calculate Value-at-Rk and other rk eaure for rk anageent. Another applcaton n portfolo electon. Under the new portfolo theory, the procedure of portfolo electon wll becoe a convex optzaton of the allocaton baed on the convex optzaton of portfolo valuaton. By way of future reearch, ethod to etate the lqudty rk factor n the exponental functon ay be proved and ore ophtcated odel ay be condered to replace the exponental functon. Wherea n regulated arket uch a tock exchange prce nforaton relatvely ealy avalable, bd and ak prce for aet traded n the OTC arket ay not be readly obtaned. Hence, t ee non-trval to apply th portfolo theory to thee type of arket. Extractng all relevant prce nforaton fro OTC arket, however, a challenge. Acknowledgeent The author would lke to thank Dr CarloAcerb (MSCI) for h knd help and frutful dcuon on the theory and the MSDC odel. We alo thank the anonyou referee for provdng u wth nghtful coent on our frt veron. The vew expreed n th paper do not necearly reflect the vew or practe of RBS. Reference Acerb, C., Portfolo theory n llqud arket. In Pllar II n the New Bael Accord: The Challenge of Econoc Captal, 2008 (Rk Book: London). Acerb, C. and Fnger, C., The value of lqudty: Can t be eaured?, 2010 [onlne]. Avalable onlne at: http://www.nvetentrevew.co/fle/2010/07/ The-value-of-lqudty1.pdf (acceed June 2010) Acerb, C. and Scandolo, G., Lqudty rk theory and coherent eaure of rk. Quant. Fnance, 2008, 8, 681 692. Algren, R., Thu, C., Hauptann, E. and L, H., Equty arket pact. Rk, 2005, July, 57 62. Bouchaud, J., Mézard, M. and Potter, M., Stattcal properte of tock order book: Eprcal reult and odel. Quant. Fnance, 2002, 2, 251 256. Boyd, S. and Vandenberghe, L., Convex Optzaton, 2004 (Cabrdge Unverty Pre: Cabrdge). Cont, R., Kukanov, A. and Stokov, S., The prce pact of order book event, 2011 [onlne]. Avalable onlne at: http://arxv.org/ab/1011.6402 (acceed Noveber 2011) Cont, R., Stokov, S. and Talreja, R., A tochatc odel for order book dynac. Oper. Re., 2010, 58, 549 563. Weber, S., Anderon, W., Ha, A.M., Knpel, T., Lee, M. and Salfeld, T., Lqudty-adjuted rk eaure. Math. Fnanc. Econ., 2013, 7, 69 91.