Economics Leers 76 (00) 53 58 www.elsevier.com/ locae/ econbase Marke-makers supply and pricing of financial marke liquidiy Pu Shen a,b, *, Ross M. Sarr a Research Deparmen, Federal Reserve Bank of Kansas Ciy, 95 Grand Avenue, Kansas Ciy, MO 64198, USA b Universiy of California, San Diego, CA, USA a,b Received 7 Augus 001; acceped 3 December 001 Absrac The bid/ ask spread (inverse of liquidiy) in urbulen financial markes modeled heoreically adjuss o marke-makers average coss. Marke liquidiy declines (spread increases) wih increasing absolue value of marke-makers securiy invenories and volailiy of securiy price and order flow. 00 Elsevier Science B.V. All righs reserved. Keywords: Liquidiy; Bid/ ask spread; Order flow; Marke-maker; Invenory JEL classificaion: G1 1. Inroducion: marke-makers and marke liquidiy Liquidiy in a financial marke he abiliy o absorb smoohly he flow of buying and selling orders comes from he deph of buyers and sellers in he marke and from marke-makers (specialiss on he organized exchanges). The marke-makers business is acing as inermediaries: buying from public sellers and selling o public buyers. The marke-maker provides marke liquidiy and makes his profi from he difference beween his buying and selling (bid and ask) prices. One simple measure of he liquidiy of a financial marke a any momen is he bid/ask spread. Liquid markes will be characerized by a narrow spread, illiquid markes by a wide spread. In acual markes, he behavior of he bid/ ask spread varies wih marke condiions. Increasingly urbulen markes, such as hose around Ocober 19, 1987, lae Augus hrough Ocober 1998, and April 10 14, 000, are accompanied by significan expansion of he bid/ask spread. In his paper we develop a simple model of marke-maker pricing ha gives an elemenary explanaion of he variaion *Corresponding auhor. Research Deparmen, Federal Reserve Bank of Kansas Ciy, 95 Grand Avenue, Kansas Ciy, MO 64198, USA. Tel.: 11-816-881-543. E-mail address: pu.shen@kc.frb.org (P. Shen). 0165-1765/ 0/ $ see fron maer 00 Elsevier Science B.V. All righs reserved. PII: S0165-1765(0)0009-0
54 P. Shen, R.M. Sarr / Economics Leers 76 (00) 53 58 in marke liquidiy wih variaion in marke condiions. The bid/ ask spread is presened as a proporion of he securiy s price. The bid/ ask spread is he price of he marke-maker s services; i adjuss in equilibrium as he resul of he marke-maker s opimizing behavior. The marke-maker is characerized as a monopolisic compeior, a profi maximizer subjec o an (average) zero-profi condiion due o he hrea of enry. The marke-maker s cos is he opporuniy (or borrowing) cos of his capial; his marginal cos is (weakly) increasing as his financial posiion is exended (an increasing long or shor posiion), which is eiher financed by deb or by placing increasing equiy a risk. The principal resul of his model is ha he bid/ask spread is an increasing funcion of he size of he marke-maker s ne long or shor posiion. The bigger he invenory of he marke-maker, he higher will be his average cos of capial, and he wider will be he bid/ask spread. Hence, imbalance in he buy sell mix of order flow is priced in he bid/ask spread. In he case of a capial marke averse o he risk of possible marke-maker insolvency, he marke-maker faces increasing marginal cos in he form of an ineres rae increasing in he size of his invenory posiion. Hence, increasing asse price risk and order flow volailiy also increase he marke-maker s average cos (by Jensen s inequaliy) and he equilibrium bid/ ask spread. A disincive elemen of he reamen presened here is he simpliciy of he marke-maker s problem. He acs primarily as a reseller of a good subjec o he risk of price variabiliy. The model ignores issues of differenial informaion (informed versus uninformed raders) aking hem o be independen of he volume and volailiy consideraions (paricularly for he marke as a whole) ha he presen sudy emphasizes. The model also separaes he problem of finding equilibrium marke price from he problem of seing (percenage) bid/ask spread and focuses solely on he laer. The model provides a simple explanaion for declining marke liquidiy during periods of price volailiy 1 and of order flow imbalance (eiher for he marke as a whole or for a single securiy). In his seing, he marke-maker ypically accumulaes a large ne posiion in he securiy he specializes in; he marke-maker buys (sells) when he public sells (buys). The significanly increased securiy invenory posiion leads o increased average cos which is hen priced in he bid/ask spread.. Modeling he marke-maker We consider he marke-maker for a single securiy whose price a dae is denoed P, conceived as he midpoin of bid and ask prices. The evoluion of price over ime is exogenous. The marke-maker ses a symmeric proporional spread S (equal o half he bid/ask spread) a dae represening his price markup for he ask price and he markdown for he bid price. A dae, he faces a (long, buying) l demand volume V and a (shor, selling) supply volume V. The marke-maker s posiion (ne holding) of he securiy may be posiive or negaive. A he sar of dae, we denoe he posiion as N. The marke-maker s posiion in he securiy evolves over ime, increased by his purchases and reduced by l s 11 his sales a, i.e. N 5 N V 1V. The marke-maker passively acceps all orders o buy and sell. His only sraegic acion is o adjus s 1 For example, he Wall Sree Journal of April 17, 000, repored by he end of he day [April 14, 000], i seemed as if he only buyers were NASDAQ marke makers and Big Board floor specialiss, who have o buy wih heir own capial when no oher buyers can be found. Afer he NASDAQ s unprecedened Monday-hrough-Friday losing sreak, marke makers are holding hree imes as much sock as usual... he specialis communiy is sreched.
P. Shen, R.M. Sarr / Economics Leers 76 (00) 53 58 55 he spread S. Invenory (of he securiy he rades) eners his choices of S because he coss of carrying he invenory need o be covered by he revenues from making he marke. The marke-maker sars period wih a cash posiion M, carried over from he previous period. In conducing business a dae, he marke-maker incurs coss C. We denoe he marke-maker s ne asse value posiion (no his profi) a he sar of dae as P. The value of he marke-maker s posiion a he beginning of hen is P 5 P N 1 M. A 1 1, he value of he posiion is P 5 P N 1 M. The marke-maker s cash posiion evolves as 11 11 11 11 l s 11 11 M 5 (1 1 S )PV (1 S )PV 1 M C. There are wo componens in C 11, he cos of providing marke-making services. One is he direc cos of rading: order aking, execuion, and record keeping. The oher componen is he cos of carrying an invenory of he securiy in which he marke-maker rades. For simpliciy he firs componen is ignored here; since we are ineresed in variaion in he bid/ask spread and variaion in he marke-makers average coss, we ignore his porion of marginal cos (assumed o be consan). 3. Marke-makers pricing: a zero-profi condiion Wihou modeling explicily he compeiive srucure of marke-making, we assume a zero-profi condiion. In he case of NASDAQ, his represens he oucome of ease of enry ino marke-making. On he NYSE, his may be aken o represen he noion of he cos of mainaining an orderly marke or a normaive ceiling (no necessarily zero) on specialis profis. 3 The zero-profi condiion implies E [P ] 5 E [M 1 P N ] 5 P 5 M 1 PN. 11 11 11 11 On average, he marke-maker assumes no ne posiion on he securiies in which he is making a marke; expeced sales and purchases are on average equal. Assume he disribuion of he volume of buy and sell order o be he same, wih mean V and variance s. Then 0 v E [M ] 5 (1 1 S )PV (1 S )PV 1 M E [C ] 5 SPV 1 M E [C ]. 11 0 0 11 0 11 Now assume ha he disribuion of price and volume a 1 1 are no correlaed. Furher, we assume a maringale condiion, ha he expeced mean of he securiy price a 1 1 is equal o he price a. Then E [P N ] 5 E [P ]E [N ] 5 P (N 1V V ) 5 PN. 11 11 11 11 0 0 Therefore, he zero-profi condiion implies Conceivably, he cos funcion can differ depending on wheher he marke-maker carries a long or shor posiion, hough we do no rea his possibiliy below. 3 This expression could be presened including a ime discoun facor, wihou changing he characer of he resuls.
56 P. Shen, R.M. Sarr / Economics Leers 76 (00) 53 58 or E [M 1 P N ] 5 SPV 1 M E [C ] 1 PN 5 M 1 PN, 11 11 11 0 11 E [C ] 5 SPV. 11 0 A he marke equilibrium, fulfilling a zero-profi condiion, he markup spread (half he bid/ ask spread) is S* 5 E [C 11]/PV 0. This expression is he cornersone of his line of research. The marke-maker adjuss he bid/ ask spread a any momen o cover expeced (variable average) coss a expeced rading volume. The marke-maker pursues expeced average cos pricing in a variable sochasic environmen. 4. The quadraic cos case and he linear absolue value cos case In order o derive a predicion from he pricing model above, we mus specify he form of he cos funcion C 1 1. How does he size of he marke-maker s ne posiion, in he securiy in which he makes a marke, affec his average coss? Holding he securiy invenory N1 1 requires financing, and 4 he average cos of capial may vary wih he size of he marke-maker s invenory posiion. When a marke-maker relies on self-financing, he cos of carrying invenory is he opporuniy cos of capial; a quadraic cos funcion can be inerpreed as reflecing he marke-maker s risk aversion. Alernaively, if he marke-maker s capial is deb-financed, an increasing risk premium, represened as a quadraic cos funcion, may be added o ineres raes on lending o a marke-maker whose posiion is increasingly leveraged. Thus we sugges he specificaions: Case 1. (quadraic cos) C 5 a(p N ). Then 11 11 11 Thus l s E [C 11] 5 ae [((P1DP 11)N 11)]5 a(p 1 s p)e [(N1V 11 V 11)] 5 a(p 1 s p)(n 1 s v 1 V 0). a * ]] PV p v 0 0 S 5 (P 1 s )(N 1 s 1 V ). Hence, in he quadraic cos case, he marke-maker s bid/ ask spread varies posiively wih price risk, s p, wih rading volume risk, s v, and he size of he marke-maker s rading invenory exposure N. As markes become more volaile in price or volume, he bid/ ask spread expands. As he 4 The uni ime inerval is mos appropriaely conceived as a single rading day. A he close of rade, financing coss are incurred on he marke maker s ne posiion. The marke maker adjuss he spread during he rading day o aemp o opimize his abiliy o cover coss on he closing invenory posiion.
P. Shen, R.M. Sarr / Economics Leers 76 (00) 53 58 57 marke-maker s exposure embodied in he size of his invenory expands, so does he bid/ ask spread. For he case of risk neuraliy, we consider he case where he cos of financing he invenory is aken o be linear in is absolue value. Case. (linear absolue value cos) C 5 aup N u. Then 11 11 11 E [C ] 5 ae [P ]E [un u] 5 ap E [un u]. 11 11 11 11 However, now we need o know he disribuion funcions of buy and sell volumes o calculae he mean of un u: 11 l s 11 11 11 E [un u] 5 E [un V 1V u] l s l s l s l s 5 E (NV 11 1V 11)dv dv E (NV 11 1V 11)dv dv l s l s N V 1V $0 N V 1V,0 11 11 11 11 under he condiion ha buy and sell volumes are independen and idenically disribued. Furher, assume hey boh follow uniform disribuions. Tha is, f(v) 5 1/sv in he range of V0 s v # v # 5 V0 1 s v, and zero oherwise. Then he relaionship beween E [un11u] and unu is nonlinear when he value of unu is relaively small. However, for large enough unu (when unu $V0 1 s v), i can be demonsraed (proof available 6 from he auhors) ha E [un u] 5 un u. Tha is, S* 5 aun u/v. 11 0 In he linear absolue value cos case we find ha he bid/ask spread varies direcly wih he exen of he marke-maker s securiy invenory exposure o marke risk. The bigger he marke-maker s 7 posiion (in absolue value), he bigger is he spread. While some of he empirical implicaions of his model are indisinguishable from oher exising 8 models, such as ha he bid/ ask spread varies wih he price volailiy of he underlying securiy, oher predicions are more novel. For example, his model suggess ha even emporary imbalances in order flow are likely o lead o larger bid/ask spreads as hey end o increase boh ne securiy invenory posiions of marke-makers and he variance of order flow. 5 The mean of he disribuion is V0 and he variance is s v /3. 6 Our seing hus leads o he conclusion ha when he iniial invenory posiion is large, he ime series of invenory posiion has a uni roo. In conras, he ypical models ha allow marke-maker o se bid and ask prices separaely end o predic invenory posiion is mean revering. Empirical evidence provides indirec suppor ha, a leas for high volume socks, invenory posiions may be highly persisen (Hasbrouck and Sofianos, 1993). 7 Allowing a correlaion beween volume and price disribuions would srenghen he posiive relaion beween he absolue invenory posiion and bid/ ask spread. A sudden posiive (negaive) surge of invenory should signal negaive (posiive) price changes are more likely, which makes he exising invenory even more cosly o he marke maker. 8 See he references secion for a represenaive lis of relevan papers (Amihud and Mendelson, 1980; Copeland and Galai, 1983; Easley and O Hara, 1987; Garman, 1976; Glosen and Milgrom, 1985; Ho and Soll, 1981; Madhavan, 000; O Hara and Oldfield, 1986; Soll, 1978).
58 P. Shen, R.M. Sarr / Economics Leers 76 (00) 53 58 5. Conclusion A marke-maker s expeced average coss increase wih he size (absolue value) of his invenory posiion; he cos of financing invenory increases wih is size. Furher, in he case where he marke-maker faces increasing marginal financing cos, expeced average coss increase wih securiy price and order flow volailiy. These coss, in equilibrium, mus be covered ou of he bid/ask spread. Consequenly, he bid/ ask spread varies direcly wih securiy price and order flow volailiy and wih he size of he marke-maker s securiy invenory posiion. Since an increasingly urbulen asse marke is characerized by imbalance of public buy and sell orders and by increasing price volailiy, he model predics and explains ha such a marke is likely o be accompanied by deerioraion in marke liquidiy. Acknowledgemens The opinions in his paper are hose of he auhors and do no represen he views of he Federal Reserve Sysem or of he Federal Reserve Bank of Kansas Ciy. References Amihud, Y., Mendelson, H., 1980. Dealership marke: marke making wih invenory. The Journal of Financial Economics 8, 31 53. Copeland, T.E., Galai, D., 1983. Informaion effecs on he bid ask spread. The Journal of Finance 38, 1457 1469. Easley, D., O Hara, M., 1987. Price, rade size, and informaion in securiies markes. The Journal of Financial Economics 19 (1), 69 90. Garman, M.B., 1976. Marke microsrucure. The Journal of Finance 3 (), 57 75. Glosen, L.R., Milgrom, P.R., 1985. Bid, ask, and ransacion prices in a specialis marke wih heerogeneously informed raders. The Journal of Financial Economics 14 (1), 71 100. Hasbrouck, J., Sofianos, G., 1993. The rades of marke makers: an empirical analysis of NYSE specialiss. The Journal of Finance 48 (5), 1565 1593. Ho, T.S.Y., Soll, H.R., 1981. Opimal dealer pricing under ransacions and reurn uncerainy. The Journal of Financial Economics 9 (1), 47 73. Madhavan, A., 000. Marke microsrucure: a survey. Journal of Financial Markes 3, 05 58. O Hara, M., Oldfield, G.S., 1986. The microeconomics of marke making. Journal of Financial and Quaniaive Analysis 1 (4), 361 376. Soll, H.R., 1978. The supply of dealer services in securiies markes. The Journal of Finance 33 (4), 1133 1151.