Optimal VWAP Trading Strategy and Relative Volume

Transcription

1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 21 Sepember 27 Opimal VWAP Trading Sraegy and Relaive Volume James McCulloch and Vladimir Kazakov ISSN

2 Opimal VWAP Trading Sraegy and Relaive Volume James McCulloch Vladimir Kazakov Augus, 27 Absrac Volume Weighed Average Price (VWAP) for a sock is oal raded value divided by oal raded volume. I is a simple qualiy of execuion measuremen popular wih insiuional raders o measure he price impac of rading sock. This paper uses classic mean-variance opimizaion o develop VWAP sraegies ha aemp o rade a beer han he marke VWAP. These sraegies exploi expeced price drif by opimally fron-loading or back-loading raded volume away from he minimum VWAP risk sraegy. c Copyrigh James McCulloch, Vladimir Kazakov, 27. Conac [email protected] 1

3 1 Inroducion and Moivaion Volume Weighed Average Price (VWAP) rading is used by large (insiuional) raders o rade large orders in financial markes. Implici in he use of VWAP rading is he recogniion ha large orders raded in financial markes may rade a an inferior price compared o smaller orders. This is known as he liquidiy impac cos or marke impac cos of rading large orders. VWAP orders aemp o address his cos by bench-marking he price of rading he large order agains he volume weighed average price of all rades over a specific period of ime (generally 1 rading day). This allows any liquidiy impac coss associaed wih rading he large order o be quanified. VWAP rading also recognizes ha he key o minimizing hese coss is o breakup large orders up ino a number of sub-orders execued over he VWAP period in such a way as o minimize insananeous liquidiy demand. The VWAP price as a qualiy of execuion measuremen was firs developed by Berkowiz, Logue and Noser [4. They argue (page 99) ha a marke impac measuremen sysem requires a benchmark price ha is an unbiased esimae of prices ha could be achieved in any relevan rading period by any randomly seleced rader and hen define VWAP as an appropriae benchmark ha saisfies his crieria. An imporan paper in modelling VWAP was wrien by Hizuru Konishi [15 who developed a soluion o he minimum risk VWAP rading sraegy for a price process modelled as Brownian moion wihou drif (dp = σ dw ). In his paper he soluion is generalized o a price process ha is a coninuous semimaringale, P = A +M +P, where A is price drif, M is a maringale and P is he iniial price. I is proved ha price drif A does no conribue o VWAP risk. The relaive volume process X is also inroduced, defined as inra-day cumulaive volume V divided by oal final volume X = V /V T. I is shown ha VWAP is naurally defined using relaive volume X raher han cumulaive volume V. The minimum VWAP risk rading problem is generalized ino he opimal VWAP rading problem using a mean-variance framework. The opimal VWAP rading sraegy x here becomes a funcion of a rader defined risk aversion coefficien λ. This is relevan because VWAP rades are ofen large insiuional rades and he size of he VWAP rade iself may be price sensiive informaion ha he VWAP rader can exploi for he benefi of his clien. The opimal sraegy is hen obained for VWAP rading which 2

4 includes expeced price drif E[A over he VWAP rading period. This can be expressed in following mean-variance opimizaion (subjec o consrains on sraegy x ) where V(x ) is he difference beween raded VWAP and marke VWAP as a funcion of he rading sraegy x : x [ = max E [ V(x ) λ Var [ V(x ) x I is shown ha for all feasible VWAP rade sraegies x here is always residual VWAP risk. This residual risk is shown o be proporional o he price variance σ 2 of he sock and variance he relaive volume process Var[X. When he relaive volume process variance is empirically examined in secion 3 i is found o be proporional o he inverse of sock final rade coun K raised o he power.44. This is of imporance o VWAP raders because i formalizes he inuiion ha raded VWAP risk is lower for high urnover socks. min x Var[V(x ) σ 2 T Var[X d σ2 K.44 Finally, a pracical VWAP rading sraegy using rading bins is examined. The addiional bin-based VWAP risk from using discree volume bins o rade VWAP is shown o be O(n 2 ) for a n bin approximaion of he opimal coninuous VWAP rading sraegy x. 2 Modelling VWAP The sochasic VWAP model is based on he filered probabiliy space wih he observed progressive filraion F, (Ω, F, F = F, P). The model also defines a filraion G iniially enlarged by knowledge of he final raded volume of he VWAP sock G = F σ(v T ). The resulan filered probabiliy space (Ω, F, G = G, P) is used o define VWAP using he relaive volume process X. 3

5 2.1 A Sochasic Model of Price P The price process P will be assumed o be a sricly posiive, coninuous (special) semimaringale wih Doob-Meyer decomposiion: P = P + A + M P > Where A is price drif, M is a maringale and P is he iniial price. 2.2 A Sochasic Model of Relaive Volume X Cumulaive volume arrives in he marke as discree rades, his suggess ha he cumulaive volume process V should be modelled as a marked poin process. A very general model of poin process is he Cox 1 poin process (also called he doubly sochasic Poisson poin process, a simple (no co-occurring poins) poin process wih a general random inensiy. The Cox process has been used o model rade by rade marke behaviour by a number of financial marke researchers including Engle and Russell [1, Engle and Lunde [1, Gouriéroux, Jasiak and Le Fol [11 and Rydberg and Shephard [18. If rade coun N is modelled as a Cox process, hen inra-day rade coun can be scaled o a relaive rade coun by he simple expedien of dividing he inra-day coun (N = a K) by he final rade coun (N T = K). This defines he relaive rade coun process R,K = N /N T = a. The resulan poin process is no longer he Cox process as his has been ransformed ino a doubly sochasic binomial poin process by knowledge of he final rade coun enlarging he observed filraion F σ(n T ) (McCulloch [16). Bu he objec of ineres when execuing a VWAP rade is no relaive rade coun R,K bu he closely relaed relaive volume X. This can be modelled by a marked poin process where each occurrence or poin is associaed wih a random value (he mark) represening rade volume. Thus each rade is specified by a pair of values on a produc space, he ime of occurrence and a mark (ineger) value specifying he volume of he rade { i, v i } R + Z +. 1 Named a Cox process in recogniion of David Cox s 1955 [9 paper which he inroduced he doubly sochasic Poisson poin process. 4

6 V = N i=1 The relaive volume X is hen he raio of a random sum specified by he doubly sochasic binomial poin process as he ground process over he non-random sum of all rade volumes. V i X = V V T = N i=1 V i K i=1 V i The relaive volume process X is he cumulaive volume process ransformed by knowledge of final volume (and hus final rade coun) and is adaped o G = F σ(v T ). Noe X is a semimaringale wih respec o G because his filraion is enlarged by he sigma algebra generaed by a random variable, final volume V T, wih a counable number of possible values (corollary 2, page 373 Proer [17). 2.3 A Sochasic Inegral Model of VWAP One he reasons for he populariy of VWAP as a measure of order execuion qualiy is he simpliciy of i s definiion - he oal value of all 2 rades divided by he oal volume of all rades. If P i and V i are he price and volume respecively of he N rades in he VWAP period, hen VWAP is readily compued as: vwap = oal raded value oal raded volume = N i=1 P i V i N i=1 V i Alernaively he definiion of VWAP can be wrien in coninuous ime noaion. Le V be he cumulaive volume raded a ime and P be he 2 No all rades are acceped as admissible in a VWAP calculaion. Admissible rades are deermined by marke convenion and are generally on-marke rades. Off-marke rades and crossings are generally excluded from he VWAP calculaion because hese rades are ofen priced away from he curren marke and represen volume in which a randomly seleced rader [4 canno paricipae. 5

7 ime varying price on a marke ha rades on he ime inerval [, T. Then VWAP is defined by he Riemann-Sieljes inegral. vwap = oal raded value oal raded volume = 1 V T T P dv (1) Examining he inegral above, i is inuiive ha i relaes o he relaive volume process X = V /V T. Using he heory of iniial enlargemen of filraion (see Jeulin [14, Jacod [12, Yor [19 and Amendinger [2) VWAP can be expressed in erms of X : vwap = T P dx (2) Proof. The asserion ha he vwap random variable is he same in equaions 1 and 2 under filraions F and G respecively is proved under he assumpion ha he price process P is independen of he final volume random variable, σ(p ) σ(v T ) =, [, T. This implies ha P is also a G semimaringale wih he same Doob-Meyer decomposiion as F (heorem 2, page 364, Proer [17). Independence wih V T implies ha he price process P is unchanged by he enlarged filraion G. Cumulaive volume V arrives in he marke as discree rades and is modelled as a marked poin process (see secion 2.2 below). Noing ha V as a pure jump process has finie variaion under filraion F and he enlarged filraion G, i is readily shown ha he Riemann-Sieljes inegrals of inegrand Price P (unchanged by he enlarged filraion) and inegraor volume V are equivalen wih respec he filraion F and he enlarged filraion G. Le τ i, i = 1,..., N be he N jump imes for he volume process V on he inerval [, and V i be he corresponding jump magniudes. Then he Riemann-Sieljes inegrals wih respec o he filraions F and G are equivalen o he same Riemann-Sieljes sum because he volume jump imes and magniudes V i are he same in boh filraions and he price process is he same in boh filraions (by assumpion). P s dv s F = N i=1 P τi V i = P s dv s G 6

8 Noing ha he erm (1/V T ) is adaped o G. 1 V T P s dv s F = P s dv s V T G = P s dx s G This is a key insigh, VWAP is naurally defined using relaive volume X raher han acual volume V. One implicaion of using relaive volume is ha common relaive inraday feaures in he daily rading of socks wih differen absolue urnovers can be exploied for VWAP rading. Also, he difference beween raded VWAP and marke VWAP as a funcion of he rading sraegy V(x ) is convenienly defined using relaive volume. V(x ) = T P dx T P dx = T P d(x X ) Using inegraion by pars 3, his inegral can be ransformed ino a sochasic inegral and quadraic covariaion. V(x ) = T T P d(x X ) = P T (x X T ) (x X )dp [x X, P T Where [x X, P denoes he covariaion process beween x X and P. Since he price process P is coninuous, he relaive volume X is assumed o be a marked poin (pure jump) process and x is deerminisic, he quadraic covariaion erm is zero. Also noing ha P T (x T X T ) = he inegraion by pars equaion simplifies o: V(x ) = T (X x ) dp (3) 3 The inegrand of he sochasic inegral X is a lef coninuous (predicable) version of he relaive volume process X where for X is defined as he lef limi of X, X = lim s X s. 7

9 3 Empirical Properies of Relaive Volume X Relaive volume as self-normalized rade couns was analyzed in McCulloch [16, where deails of empirical daa collecion and analysis can be found. Briefly, New York Sock Exchange (NYSE) rade daa from he TAQ daabase was used o collec relaive rade volume daa of all socks ha raded from 1 June 21 o 31 Augus 21 (a oal of 62 rading days 4 ) for a oal of 23,158 relaive rade volume sample pahs for all socks. The relaive rade volume daa was colleced in a D hisogram wih ime in minues (39 minues + 1 end-poin) in he x-axis and relaive volume (a prime number 251 o avoid bin boundaries, plus wo end-poins) in he y-axis. 3.1 Expeced Relaive Volume E[X is S Shaped All professional equiy raders know ha markes are, on average, busy on marke open and marke close and less busy during he middle of he rading day. This is he classic U shape in rading inensiy found in all major equiy markes 5 and is, by definiion, he derivaive of he expecaion of he relaive volume de[x /d. Figure 1 plos he expeced relaive volume E[X for four groups of socks wih differen ranges of rade couns on he NYSE. The expecaion of relaive volume E[X can be approximaed wih he he following polynomial. E [ X 5 3T 22 T , [, T. (4) 3T High Turnover Socks have Lower Var[X The second feaure of empirical daa readily seen in Figure 2 is ha he low urnover sock (SUS) appears o have a higher volailiy around he mean relaive volume (shown wih red line) han he high urnover sock (TXN). 4 3 July 21 (half day rading) and 8 June 21 (NYSE compuer malfuncion delayed marke opening) were excluded from he analysis. 5 For furher discussion and explanaions of he causes of he U shaped inraday marke seasonaliy see Brock and Kleidon [5, Admai and Pfleiderer [1 and Coppejans, Domowiz and Madhavan [8. 8

10 NYSE Mean Relaive Volume wih Linear Trend Removed E [X()-/T.8 Variaion from Linear Time.6 Trade Coun Band 51-1 Trades 11-2 Trades Trades Trades.2 Analyic Approx. 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 15:3 16: Marke Time Figure 1: The mean of he relaive volume E [ X for socks wihin differen average number of daily rades. Here he consan rade line has been subraced, E [ X /T (so all means are monoonically increasing funcion of ime). The polynomial approximaion (eqn 4) is shown as he black line. This inuiion is correc and is he second imporan insigh ino VWAP rading - he volailiy of he relaive volume process X of low urnover socks is higher han high urnover socks. Figure 3 shows he empirical ime indexed variance of he relaive volume process Var[X for differen ranges of number of daily rades. I has an invered U shape where he variance is zero a = and = T, similar o he ime indexed variance of a Brownian bridge. Socks wih a lower number of daily rades have higher variance. The variances of he relaive volume process for socks wih a differen final rade coun K can be empirically scaled o fi a single curve by muliplying hem by final rade coun raised o he power.44 (K.44 ). Figure 4 plos he scaled empirical variances. 9

11 1.8 Mean SUS TXT TXN Example Sock Inraday Relaive Volume Trajecories Relaive Volume Execued : 11: 12: 13: 14: 15: 16: Figure 2: This graph shows ypical relaive volume rajecories for 3 socks represening low, medium and high urnover socks. The red line is he expeced relaive volume E[X for all socks rading more han 5 rades a day on he NYSE over he daa period. SUS is Sorage USA, TXT is Texron Incorporaed and TXN is Texas Insrumens. On 2 Jul 21 hese socks recorded 11, 946 and 2183 rades correspondingly. 1

12 NYSE Unscaled Relaive Volume Variance Var[X() 4.% 3.5% 3.% Trade Coun Band 51-1 rades 11-2 rades 21-4 rades rades 2.5% Variance 2.% 1.5% 1.%.5%.% 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 15:3 16: Marke Time Figure 3: The inverse U shaped ime-indexed variance for relaive volume Var [ X. Lower rade coun socks have a higher variance for Var [ X. NYSE Relaive Volume Variance Var[X() Scaled for Differen Final Trade Couns by K.44 Trade Coun Band 51-1 rades 11-2 rades 21-4 rades rades Scaled Variance 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 15:3 16: Marke Time Figure 4: The scaled relaive volume variances Var [ X K.44 for socks wih differen ranges of final rade couns K. 11

13 4 VWAP Trading Sraegies 4.1 Feasible Trading Sraegies Any deerminisic rading sraegy x is feasible only if i conforms o he firs consrain below. The second and hird consrains are no sricly necessary bu enforce a uni-direcional sraegy where buy VWAP raders only buy socks and sellers only sell socks. 1. Trader sars rading he VWAP sraegy a = when x = and has raded he whole sraegy a = T when x T = The relaive volume for he sraegy mus always be beween zero (nohing has been raded) and one, all order s volume was raded, x 1, [, T. 3. The sraegy mus be monoonically non-decreasing, x x +δ VWAP Trade Size I is inuiive and rue ha he greaer percenage of rading ha he VWAP rader conrols, he easier i is o rade a he marke VWAP price. In he limi, he rader conrols 1% of raded volume and exacly deermines he marke VWAP irrespecive of rading sraegy. I seems clear ha VWAP risk is proporional o he raded volume ha he VWAP rader does no conrol and his inuiion is quanified below. The relaive volume process of oher marke raders X will be assumed o be independen of he rading sraegy x adoped by he VWAP rader. Marke relaive volume process X can be wrien as a weighed sum of he relaive volume of oher marke paricipans X and he VWAP rader x. If V is he cumulaive volume process of ha does no include VWAP rader volume, hen he relaive volume of oher marke paricipans X is defined: X = V V T 12

14 Similarly he relaive volume sraegy of he VWAP rader is simply he rader final cumulaive volume v T divided by cumulaive volume a ime, v. x = v v T The proporion 6 β of he oal marke raded by he VWAP rader can be calculaed. β = v V T + v T The expeced oal relaive volume (known in G ) can be decomposed ino he relaive volume process of oher marke paricipans X and he deerminisic rading sraegy of he VWAP rader. X = (1 β) X + βx Using he definiions above, V(x ) can be rewrien as: V(x ) = T ( X x ) dp = (1 β) T (X x ) dp In he following exposiion i is assumed ha β << 1 and all O(β) erms are ignored. 4.3 The Risk of VWAP Sraegies The risk of raded VWAP wih rading sraegy x is readily expressed using equaion 3. Var [ V(x ) = Var (X x ) dp 6 Noe ha β is known under he enlarged filraion G = F σ(v T ) and a random variable under F. 13

15 Using he semimaringale generalizaion of Io s isomery his variance can be wrien as: Var (X x ) dp = E (X x ) 2 d[p, P Since he price semimaringale P is assumed coninuous, he drif erm A is coninuous and i is proved below ha he drif erm does no conribue o VWAP risk and ha he VWAP risk can be wrien jus using he maringale componen of he Doob-Meyer decomposiion. P = M + A + P Var (X x ) dp = E (X x ) 2 d[m, M (5) Proof. The inegrands of eqn 5 are idenical, so by he properies of he Riemann-Sieljes inegral, he equaliy of eqn 5 is esablished if he wo inegraing processes, he quadraic variaions, are equal (a.e) [M, M = [P, P. Using he polarizaion ideniy for quadraic covariaion. [A, M = 1 2 ( [A + M, A + M [M, M [A, A ) The drif process A is coninuous by assumpion and herefore he quadraic covariaion erm is zero (Jacod and Shiryaev [13, page 52) [A, M =. Also he drif process A is predicable, coninuous and of bounded variaion so he drif quadraic variaion erm is zero (Proer [17, heorem 22, page 66) [A, A = and he polarizaion ideniy simplifies o: [P, P = [A + M, A + M = [M, M 14

16 Since he maringale erm of he price process is coninuous he maringale represenaion heorem (Proer [17, heorem 43, page 188) can wrien as follows for a coninuous predicable process σ. M = σ s dw s Using his represenaion, he VWAP variance of equaion 5 can be furher simplified: Var [ V(x ) = E (X x ) 2 d[m, M = E (X x ) 2 σ 2 d (6) 4.4 Minimum Risk VWAP Sraegy I seems reasonable ha an opimal rading sraegy x is a sraegy ha is close o X wihou any knowledge of he acual oucome of X. Thus he opimal rading sraegy should be, by inuiion, close o he expecaion of relaive volume x = E[X. This is shown below. Following Konishi [15 he equaion can be decomposed as: x = min x 1 Var[ V(x ) = min x 1 [ (X ) 2 E 2x X + x 2 σ 2 d = min x 1 x 2 E [ σ 2 2 x E [ X σ 2 d = min x 1 = min x 1 ( E[ σ 2 x 2 E[X σ 2 2 x E[ σ 2 ( x E[X ) σ 2 2 d E[ σ E[X σ 2 2 E[ σ 2 2 ) E[X σ 2 2 E[ σ 2 d

17 This is minimized when: x = E[X σ 2 E[ σ 2 = E [ Cov [ X, σ 2 X + E[ σ 2 Thus he consrained soluion is: x = if if E [ Cov [ X, σ 2 X + 1, E[ σ 2 1 E [ Cov [ X, σ 2 X +, E[ σ 2 E [ Cov [ X, σ 2 X +, oherwise. E[ σ 2 (7) Where Cov [ X, σ 2 is he covariance beween relaive volume X and sock price variance σ 2. In financial markes lieraure he posiive relaionship beween rading volume and volailiy is a sylized fac, see Con [7, Clark [6 and Ané and Geman [3. Therefore, since he expecaion of relaive volume E[X is monoonically increasing and he covariance beween relaive volume and variance is non-negaive Cov [ X, σ 2, he minimum risk soluion (eqn 7) is feasible. Noe ha under he assumpion ha he relaive volume and sock price variance are independen or sock price variance is a deerminisic funcion hen he covariance erm is zero and he minimum risk sraegy reduces o he expecaion of he relaive volume x = E [ X. 4.5 Non-removable residual risk of VWAP rading Residual risk is he lower bound of VWAP risk ha canno be eliminaed by choosing a rading sraegy x. Subsiuion of eqn 7 ino eqn 6 gives he following bound on he residual VWAP variance: min x Var[V(x ) = T E[ X 2 σ 2 E[X σ 2 2 E[ σ 2 d 16

18 If price volailiy is assumed consan ˆσ 2 = σ 2, hen he expression above simplifies o he following: min x Var[V(x ) = ˆσ 2 T Var[X d Using he scaling propery of Var[X found above in he NYSE daa (see secion 3) hen residual VWAP risk is proporional o he esimaed sock variance divided by he final rade coun K o he power.44. min x Var[V(x ) = Cons ˆσ 2 K.44 So a sock wih 1 imes he rade coun of anoher sock wih similar price variance has approximaely one-enh he residual VWAP risk. 4.6 Opimal VWAP Sraegy wih Expeced Drif In pracise a rader may wish o bea VWAP. This is reasonable because he VWAP rader may have price sensiive informaion abou a sock. A broker can exploi his privae informaion for he benefi of his clien by adoping a VWAP rading sraegy x ha is riskier han minimum variance sraegy. This drif opimal sraegy x can be found using mean-variance approach. For definieness he VWAP order is assumed o be a buy order in his paper. Thus beaing marke is defined as a posiive expecaion E[V(x ). Expanding he expecaion and noing ha he maringale ransform has zero expecaion: E[V(x ) = E ( X x ) da + E ( X x ) dm = E ( X x ) da The quadraic covariaion beween he coninuous price drif A and he relaive volume process is zero [X, A = herefore wihou loss of generaliy 17

19 he covariance beween price drif and relaive volume can be assumed o be zero, Cov[A, X =. Denoing µ E[A, he expecaion of he VWAP reurn can be simplified o he following: E[V(x ) = T ( E[X x ) µ d (8) In general, he opimal VWAP sraegy is no he minimum VWAP risk sraegy of secion 4.4 because his sraegy does no include he expeced reurn of he VWAP rade. A sraegy ha includes expeced reurn can be specified as a classic mean-variance opimizaion using a rader specified risk aversion consan λ. [ x = max E [ V(x ) λ Var [ V(x ) x 1 Solving for his opimizaion problem: x = max E (X x ) dp x 1 λ Var (X x ) dp = min λ x 1 E [ (X x ) 2 σ 2 (X x ) µ d λ = min x 1 ( x { E[X σ 2 E[ σ 2 } ) 2 µ d 2λE[ σ 2 The above is minimized when: x = E[X σ 2 E[ σ 2 µ 2λE[ σ 2 = E [ Cov [ X, σ 2 X + E[σ 2 µ 2λE[ σ 2 (9) The consrained soluion o opimal VWAP sraegy wih drif: 18

20 if x = if E [ Cov [ X, σ 2 X + E[ σ 2 E [ Cov [ X, σ 2 X + E[ σ 2 E [ Cov [ X, σ 2 X + E[ σ 2 µ 2λE[ σ 2 µ 2λE[ σ 2 µ 2λE[ σ 2, 1, 1, oherwise. (1) An Example of Drif Opimal VWAP Trading A simple example of opimally fron-loading and back-loading he VWAP rading sraegy o exploi expeced price drif is illusraed by example opimizing sraegies wih boh posiive and negaive expeced price drif. In hese examples he VWAP period is one day T = 1. The expeced drif E[A is assumed o be a simple linear funcion of ime such ha he sock has eiher los 2% or gained 2% by he end of he rading day µ = ±.2. The sock volailiy (sd dev.) is a consan 2% (σ 2 = ˆσ 2 =.2 2 ). Risk-aversion coefficien λ = Wih hese assumpions he opimal drif rading policies of eqn 1 are: x = if E [ X ±.7 1, 1 if E [ X ±.7, E [ X ±.7, oherwise. I is clear from he example above ha he opimal sraegies for drif shif he opimal sraegy upwards ( fron-loading ) for a posiive expeced drif E[X > and downwards ( back-loading ) for a negaive expeced drif E[X <. These opimal sraegies have disconinuiies a = and = 1 where volume is insanly acquired. This is unrealisic because i assumes ha 19

21 he marke can supply insan liquidiy and eliminaes he cenral virue of VWAP rading, disribuing liquidiy demand over he VWAP period in such a way so as o minimize insananeous liquidiy demand Opimal VWAP Trading wih Consrained Trading Rae The soluion is add an addiional consrain o he opimizaion problem by seing an upper bound o he insananeous liquidiy demand ν max. This liquidiy consrain can be specified as follows: dx d v max The opimal sraegy here is consruced using he se D of feasible sraegies x as a recangular in (x, ) space wih upper lef poin a (1, ) and upper righ-poin a (1, T ), see figure 5. The lef x L and righ x R boundaries for region D are defined as inegrals of he maximum rading rae v max. x L = vs max ds x R = 1 T vs max ds All poins o he righ of x R and o he lef of x L are ouside he feasible region D. The opimal sraegy is o rade following unconsrained sraegy (9) inside D unil one of he boundaries of D is encounered and hen rade a he maximum allowable rae. if E [ Cov [ X, σ 2 X + E[ σ 2 µ 2λE[ σ 2 x L, x L x = if E [ Cov [ X, σ 2 X + E[ σ 2 µ 2λE[ σ 2 x R, x R (11) E [ Cov [ X, σ 2 X + E[ σ 2 µ 2λE[ σ 2, oherwise. 2

22 Proof ha (11) is he opimal sraegy for VWAP rading problem wih consrained liquidiy is given in appendix. The example above is re-considered now for ime-dependen consrained liquidiy, where he maximal rae of rading is assumed o be proporional o he expecaion of he rading rae of he marke (ime-derivaive of E [ X ) 1.8 x L D E[X.6 * x.4.2 unconsrained rading sraegy x R /T 1 Figure 5: The opimal back-loading VWAP sraegy for liquidiy consrained rading in example. v max = 2 d d E[ X The resulan opimal VWAP rading sraegy back-loads volume along x, shown in Figure 5. 21

23 4.7 Bins - VWAP Sraegy Implemenaion The opimal sraegies x discussed previously are coninuous. Tha is, i is assumed ha he VWAP rader has complee conrol over rading rajecory a any momen of ime during rading. This is unrealisic, raders need ime o implemen sraegy and find rading couner-paries o provide liquidiy. In order o model VWAP wih uncerain liquidiy a weaker assumpion is adoped ha rading can be divided ino number of periods where rader has conrol over he average rading rae during each period. Tha is, he rader has sufficien conrol over rading o guaranee ha he raded volume a beginning and he end of every period is equal o x. These periods are called ime bins. The acual rajecory x is generaed by a random liquidiy process and could deviae from x inside he bin bu will always coincide a is boundaries The Cos of a Subopimal VWAP Trading Sraegy The VWAP bin rajecory x is subopimal and he mean-variance cos of subopimal VWAP rading sraegies C(x ) is formulaed below. C(x ) = ( ) E[V(x ) + λvar[v(x ) ( ) E[V(x ) + λvar[v(x ) = E (x x ) µ + λ [ (X x ) 2 (X x ) 2 σ 2 d = T (x x )( µ 2λE[σ 2 X + 2λ E[σ 2 x ) + λ (x x ) 2 E[σ 2 d Noing ha he when he acual rading rajecory coincides wih unconsrained opimal soluion wih drif (eqn 9) hen he firs erm in he inegral is eliminaed and he cos of a subopimal sraegy is simplified. C(x ) = λ T (x x ) 2 E[σ 2 d (12) 22

24 4.7.2 The Bounded Cos of a Bin Trading Sraegy Bins are designed by dividing he VWAP rading period [, T ino b ime periods wih he bin boundary imes for bin i denoed as τ i 1 and τ i. = τ < τ 1 < < τ i < τ i+1 < < τ b = T By consrucion x τ i 1 = x τ i and x τ i = x τ i. Since x and x are non-decreasing funcions ha are less han or equal o 1 he deviaion beween hem is bounded. x x x τ i x τ i 1 [τ i, τ i 1 (13) Using (13) we ge from (12) he following bound of addiional cos from bins C(τ 1,..., τ b ) b (x τ i x τ i 1 ) i=1 τi τ i 1 (µ 2λ(E[σ 2 X E[σ 2 x ))d + b τi (x τ i x τ i 1 ) 2 i=1 τ i 1 λe[σ 2 d (14) Equal Volume Bins Equal volume bins are ofen used by praciioners. They are defined as x (τ i ) x (τ i 1 ) = 1 b i {1,..., b} The bin cos bound (14) for rading wih unconsrained rae hen akes he form: C(τ 1,..., τ b ) 23 1 b 2 λ T E[σ 2 d (15)

25 Thus he addiional VWAP risk from using discree volume bins o rade VWAP depends on he number of bins b as O(b 2 ) Opimal VWAP Bin Sraegy The opimal bins are obained by minimizing he bound (14) on vecor in bin boundary imes τ. The firs order condiion of opimaliy is. C(τ 1,..., τ b ) τ k = Differeniaing equaion 14 wih respec o he vecor in bin boundary imes τ gives: (2x τ i x τ i 1 x τ i+1 ) (µ τi 2λ(E[σ 2 τ i X τi E[σ 2 τ i x τ i )) + d [ τi dτ x τ i (µ 2λ(E[σ 2 X E[σ 2 x )) d τ i 1 τi+1 τ i (µ 2λ(E[σ 2 X E[σ 2 x ))d + λστ 2 i [x τ i 1 (x τ i 1 2x τ i ) x τ i+1 (x τ i+1 2x τ i ) + 2λ d [ τi τi+1 dτ x τ i (x τ i x τ i 1 ) E[σ 2 d (x τ i+1 x τ i ) τ i 1 τ i E[σ 2 d (16) = Solving his equaion for τ i can be viewed as a compuaional operaion which reduces bin-based addiional cos by varying τ i condiional on (as a funcion of fixed) τ i 1 and τ i+1. I is applied recursively o he iniial se of bins imes (eg equal-volume bins) unil convergence o he opimal bins. 24

26 The example in figure 5 plos he bin boundaries of 1 equal volume bins for he liquidiy-consrained VWAP sraegy and 1 opimal bin boundaries obained by applying recursively improving operaion are shown in Figure 6. The reducion in he addiional bin-based risk from he use of opimal insead of equal-volume bins is 4.65%..8.6 opimal bins coninuous soluion.4.2 equal-volume bins Figure 6: The opimal sraegy he example wih consrained liquidiy and is corresponding 1 equal-volume bins and 1 opimal bins. 25

27 5 Conclusion and Summary This paper builds on he paper by Hizuru Konishi [15 by developing a soluion o an opimal minimum risk VWAP rading problem. The volume process is assumed o be marked poin process and he price process o be a coninuous semimaringale. I is shown ha VWAP is naurally defined using he relaive volume process X which is inra-day cumulaive volume V divided by oal final volume X = V /V T. The novel expression for he risk of VWAP rading is derived. I is proven ha his risk does no depend on he price drif. The minimum risk sraegy of VWAP rading is generalized ino a meanvariance opimal sraegy. This is useful when VWAP raders have price sensiive informaion ha can be exploied by a VWAP sraegy. The cos of exploiing price sensiive informaion is deviaion from he minimum risk VWAP rading sraegy by fron-loading or back-loading raded volume o exploi he expeced price movemen. I is shown ha even wih a minimum risk VWAP rading sraegy is implemened here is always a residual risk. This residual risk is shown o be proporional o he price variance ˆσ 2 of he sock and he inverse of final rade coun K raised o he power.44. Higher rade coun socks have lower residual VWAP risk because he variance of he relaive volume process is lower for hese socks. A pracical VWAP rading sraegy using rading bins is consruced. The addiional VWAP risk from using discree volume bins o rade VWAP is esimaed. I is shown ha i depends on he number of bins b as O(b 2 ). 26

28 References [1 Ana Admai and Paul Pfleiderer, A Theory of Inraday Paerns: Volume and Price Variabiliy, Review of Financial Sudies 1 (1988), 3 4. [2 Jürgen Amendinger, Iniial enlargemen of filraions and addiional informaion in financial markes, Ph.D. hesis, Berlin Technical Universiy, Berlin, Germany, [3 Thierry Ané and Helyee Geman, Order flow, ransacion clock, and normaliy of asse reurns., The Journal of Finance. 55 (2), no. 5, [4 Sephen Berkowiz, Dennis Logue, and Eugene Noser, The Toal Cos of Transacions on he NYSE, Journal of Finance 43 (1988), [5 William Brock and Allan Kleidon, Periodic Marke Closure and Trading Volume: A Model of Inraday Bids and Asks, Journal of Economic Dynamics and Conrol 16 (1992), [6 Peer Clark, Subordinaed sochasic process model wih finie variance for speculaive prices, Economerica 41 (1973). [7 Rama Con, Empirical properies of asse reurns: sylized facs and saisical issues, Quaniaive Finance 1 (21), [8 Mark Coppejans, Ian Domowiz, and Ananh Madhavan, Liquidiy in an Auomaed Aucion, Working Paper. March 21 version. [9 David Cox, Some Saisical Mehods Conneced wih Series of Evens (Wih Discussion), Journal of he Royal Saisical Sociey, B 17 (1955), [1 Rober Engle and Jeff Russell, The Auoregressive Condiional Duraion Model, Economerica 66 (1998), [11 Chrisian Gouriéroux, Joanna Jasiak, and Gaëlle Le Fol, Inra-Day Marke Aciviy, Journal of Financial Markes 2 (1999), [12 Jean Jacod, Grossissemen Iniial, Hypohèse e Théorème de Girsanov, Séminaire de Calcul Sochasique 1982/83, Lecure Noes in Mahemaics 1118, Springer (1985), [13 Jean Jacod and Alber Shiryaev, Limi Theorems of Sochasic Processes, Springer, Berlin,

29 [14 Thierry Jeulin, Semi-maringales e grossissemen d une filraion, Lecure Noes in Mahemaics 92, Springer (198). [15 Hizuru Konishi, Opimal slice of a VWAP rade, Journal of Financial Markes 5 (22), [16 James McCulloch, Relaive Volume as a Doubly Sochasic Binomial Poin Process, Quaniaive Finance 7 (27), [17 Phillip Proer, Sochasic Inegraion and Differenial Equaions, Springer, 25. [18 Tina Rydberg and Neil Shephard, BIN Models for Trade-by-Trade Daa. Modelling he Number of Trades in a Fixed Inerval of Time, Unpublished Paper. Available from he Nuffield College, Oxford Websie; hp:// [19 Marc Yor, Grossissemen de filraions e absolue coninuié de noyaux, Lecure Noes in Mahemaics 1118, Springer (1985),

30 A Opimal VWAP Trading Sraegy wih Consrained Trading Rae Proof. Tha eqn 11 is he soluion he he opimal VWAP rading problem wih liquidiy consrained rading rae v v max. min x,v (µ x + λσ 2 (x 2 2x E[X )) d (17) Subjec o dx d = v, v v max, [, T, x =, x T = 1. The case in Figure 7 is considered where he unconsrained rading sraegy of eqn 9 passes hrough he origin and inersecs wih he maximal rading line x R a R < T. The proof for oher cases when he unconsrained sraegy ξ inersecs wih oher he boundaries of D is idenical. x x * x L D x R unconsrained rading sraegy R Figure 7: The feasible se D defined by consrains on he rae of rading and boundary condiions. The adjoin variable Ψ, [, T is calculaed by solving following he equaion: dψ d = µ 2λσ 2 (x 2λσ 2 E[X ), Ψ R =. (18) 29

31 Using inegraion by pars: Ψ T x T + Ψ x + T [ Ψ v + dψ d x d =. Afer adding his ideniy s lef side o VWAP mean-variance cos and dropping erms ha depend on fixed x and x T he problem of eqn 17 is ransformed o he following: min x,v Where: (µ x +λσ 2 (x 2 2x E[X )) d = min R(Ψ, x, v ) d x,v (19) R(Ψ, x, v ) = µ x + λσ 2 (x 2 2x E[X ) + Ψ v + dψ d x Consider he lef arc in x, when v = dx /d < v max, and (, R ). Here he rhs of equaion in eqn 18 is zero and herefore Ψ =. I is easy o check ha: R x (Ψ, x = x, v = v ) =, R v (Ψ, x = x, v = v ) =, (, R ). Thus R has a minimum on x D a x = x v = v < v max everywhere along lef arc of x. and on v [, v max a Consider he righ arc of x, when v = v max and ( R, T ). Here x is higher han he unconsrained rading sraegy ξ defined by eqn 9. Afer decomposing x = ξ + (x ξ ) eqn 18 becomes: dψ d = [ µ 2λσ 2 (ξ E[X ) 2λσ 2 (x ξ ) = 2λσ 2 (x ξ ) < Since Ψ R =, Ψ <, ( R, T ). I is easy o check ha: 3

32 R x (Ψ, x = x, v = v ) =, R v (Ψ, x = x, v = v ) = Ψ <, ( R, T ) Thus R has minimum on x D a x = x. By inspecion he funcion R is a linear funcion of v, so on v [, v max i has minimum on v a v = v = v max everywhere along righ arc of x. Therefore x defined by eqn 11 and v = dx /d obey consrains in eqn 17 and minimize he inegral of he equivalen mean-variance cos crierion R on x and v a every momen of ime [, T and so is he opimal soluion of eqn