Промышленный лизинг Промышленный лизинг  Методички 

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quantities of a sci uiiiy quickly, anonymously, and with relatively linle price imparl. To mainiain liquidity, many organized exchanges use marketmak-ers, individuals who stand ready lo buy or sell whenever the public wishes lo sell or buy. In return lor providing liquidity, marketmakers are grained monopoly rights by the exchange to post different prices for purchases and sales: They buy at the bid price , and sell al a higher ask price / . This ability to buy low and sell high is the marketmakers primary source of compensation for providing liquidity, and although the bid-ask spread /, - I\ is rarely larger than one or two ticks-the NYSE tact Hook: IWH Data reports that the spread was $(1.25 or less in 90.K% of the NYSE bid-ask quotes from 1994-over a large number of trades marketmakers can earn enough to compensate them for their services.

The diminutive size of typical spreads also belies their potential importance in determining the time-series properties of asset returns. For example, Phillips and Smith (1980) show that most of the abnormal returns associated with particular options trading strategies are eliminated when the costs associated with the bid-ask spread are included. Blunie and Slanibaugh (1983) argue that the bid-ask spread creates a significant upward bias in mean returns calculated with transaction prices. More recently, Keim (1989) shows thai a significant portion of the so-called January effect-the fact dial smaller-capitalization slocks seem to outperform larger capitalization stocks over the few flays surrounding the turn of the year- may be attributable to dosing prices recorded at the bid price at the end of December and closing prices recorded at the ask price al the beginning of January. F.ven if the bid-ask spread remains unchanged during this period, the movement from bid to ask is enough to yield large portfolio returns, especially for lower-priced stocks for which the percentage bid-ask spread is larger. Since low-priced stocks also tend to be low-capitalization slocks, Keims (1989) results do offer a partial explanation lor the Januarv effect.17

The presence of the bid-ask spread complicates matters in several ways. Instead of one price for each security, there are now three: the bid price, the ask price, and the transaction price which need not be either the bid or the ask (although in some cases it is), nor need it lie in between the two (although in most cases it does). I low should returns be calculated, from bid-lo-bid, ask-lo-bid, etc.? Moreover, as random buys and sells arrive at the market, prices can bounce back and forth between the ask and the bid prices, creating spurious volatility and serial correlation in returns, even if the economic value of the security is unchanged.

Krim (l!IHl> also do, mucins i he trial ion between other calendar anomalies (die weekend eltei l, holiday cllei is, etc.! .nut systematic moscnu-nls between the bid and ask prices.

3.2.1 Bid-Ask Bounce

To account for the impact of the bid-ask spread on the time-series properties of asset returns, Roll (1984) proposes the following simple model. Denote by P* the time-/ fundamental value of a security in a frictionless economy, and denote by s the bid-ask spread (see Glosten and Milgrom [19851, for example). Then the observed market price P, may be written as

p, = r; + iA (3-2.D

/, IID


+1 with probability 5 (buyer-initiated) -1 with probability \ (seller-initiated)

where /, is an order-type indicator variable, indicating whether the transaction at lime t is at the ask (buyer-initiated) or at the bid (seller-initiated) price. The assumption that P* is the fundamental value of the security implies that E[/(] = 0, hence Pr(/,= 1) = Pr(/,= - 1) = \. /\ssume for the moment that there are no changes in the fundamentals of the security; hence P = P is fixed through time. Then the process for price changes ДР, is given by

Л Pi = Д/?+ (/,-/, ,)£ = (3.2.3)

. and under the assumption that /, is IID the variance, covariance, and autocorrelation of AP, may be readily computed

Var[AP, ] = - (3.2.4)

Cov[ ДР, 1 , АР, ] = -- (3.2.5)

Cov[ ДР, * , AP, ) = 0, k > 1 (3.2.6)

Corr[ ДР, , , ДР, ] = -i . (3.2.7)

Despite the fact that fundamental value P*t is fixed, ДР, exhibits volatility and negative serial correlation as the result of bid-ask bounce. The intuition is clear: If P* is fixed so that prices take on only two values, the bid and the ask, and if ihe current price is the ask, then the price change between the current price and the previous price must be either 0 or s and the price change between the next price and the current price must be either 0 or -s. The same argument applies if the current price is the bid, hence the serial correlation between adjacent price changes is nonpositive. This intuition

applies more generally to cases where the order-type indicator /, is not 111),w hence the model is considerably more general than it may seem.

The larger the spread s, the higher the volatility and the (irsi-ordcr autocovariance, both increasing proportionally so that the first-order autocorrelation remains constant at - 5. Observe from (3.2.6) that the bid-ask spread does not induce any higher-order serial correlation.

Now let the fundamental value P* change through time, but suppose-that its increments are serially uncorrelated and independent of/,.1 Then (3.2.5) still applies, but the first-order autocorrelation (3.2.7) is no longer - 5 because of the additional variance of Д/- * in the denominator. Specifically ifcr2(AP*) is the variance of ДР,*, then

Corr[ Д/, 1 , l\P, } = -


(л2/2) + о-2(ДР*)

< 0.


Although (3.2.5) shows that a given spread s implies a first-order auloco-variWe of -j-2/4, the logic may be reversed so that a given autocovariance coefficient and value of /; imply a particular value for s. Solving for s in (3.2*5) yields

j s = 2/- Cov[AP, ДР,]~ , (3.2..))

hence s may be easily estimated from the sample autocovarianccs of price changes (see the discussion in Section 3.4.2 regarding the empirical implementation of (3.2.9) for further details).

tstimating the bid-ask spread may seem superfluous given the fact that bid-ask quotes are observable. However, Roll (1984) argues that the quoted spread may often differ from the effective spread, i.e., the spread between the actual market prices of a sell order and a buy order. In many instances, transactions occur at prices within the bid-ask spread, perhaps because mar-ketmakers do not always update their quotes in a timely fashion, or because they vish to rebalance their own inventory and are willing to belter their quotes momentarily to achieve this goal, or because they arc willing to provide discounts to customers that arc trading for reasons other than private information (see Eikcboom [1993], Glostcn andMilgrom [1985], Goldstein [1993], and the discussion in the next section for further details). Rolls (1984) model is one measure of this effective spread, and is also a means for

Tor example, serial correlation in (of eitltet sign) does not change the I act that bid-ask bounce induces negative serial correlation in price changes, although it tloes alien the magnitude. See Choi, Salandro, and Shastri (1988) for an explicit analysis of this case.

Roll (1984) argues that price changes must be serially uncorrelated in an informationally efficient market. However, l.eroy (197:)), Lucas (1978), and others have shown that this need not be the case. Nevertheless, for short-horizon returns, e.g., daily or intradaily returns, it is difficult to pose an empirically plausible equilibrium model of asset returns that exhibits significant serial correlation.

accounting for the effects of the bid-ask spread on the time-series properties of asset returns.

3.2.2 Components of Ihe Bid-Ash Spread

Although Rolls model of the bid-ask spread captures one important aspect of its effect on transaction prices, it is by no means a complete iheory of the economic determinants and the dynamics of die spread. In particular, Roll (1984) takes s as given, but in practice the size of the spread is die single most important quantity that markelmakei s control in their strategic interactions with other market participants, ln fact, Glostcn and Milgtom (1985) argue convincingly that л is determined endogenouslyand is unlikely to be independent of/ as we have assumed in Section 3.2.1.

Other theories of the marketmaking process have decomposed the spread into more fundamental components, and these components often behave in different ways through time and across securities. Estimating die separate components of the bid-ask spread is critical for properly implementing these theories with transactions data. In this section we shall turn to some of the econometric issues surrounding this task.

There arc three primary economic sources for the bid-ask spread: order-processing costs, inventory costs, and adverse-selection costs. The first two consist of the basic setup and operating costs of trading and recordkeeping, and the carryingof undesircd inventory subject lo risk. Although these costs have been the main focus of earlier literature,2 il is the adverse-selection component that has received much recent attention.21 Adverse selection costs arise because some investors are better informed about a securitys value than ihe marketmaker, and trading with such investors will, on average, be a losing proposition for the marketmaker. Since markctmakers have no way lo distinguish the informed from the uninformed, they are forced to engage in these losing trades and must be rewarded accordingly. Therefore, a portion of the markelmakers bid-ask spread may be viewed as compensation for taking the other side of potential information-based trades. Because this information component can have very different statistical properties from the order-processing and inventory components, il is critical lo distinguish between them in empirical applications. To do so, Glostcn (1987) provides a simple asymmetric-information model that captures the salient features of adverse selection for the components of die bid-ask spread, and we shall present an abbreviated version of his elegant analysis here (see, also, Glostcn and I larris [19S8J and Stoll 11989]).

See, lor example, Amihud and Memlelson (1980), liagehot (1971). Demsetz (HIGH), По and Stoll (19H1). Stoll (1978), and Tinic (I97Z).

See liagehot (1971), Copcland and Calai (198:)), l-Vasley and Oll.na (1987), Closlcn (1987), Glostcn and Harris (1988), Closten and Milgrom (19КГ.), and Stoll (1989).

(Hasten s Decomposition

Denote I)) , and P i ht- hid and ask prices, res рос lively, and lei P he the true or rommim-hifimiwthn market price, the price that all investors without private inhumation (uninformed investors) agree upon. Under risk-neulrality, the common-information price is given hy P = E[/ £2] where Й denotes the coiniuon or public information set and P* denotes the price that would result il everyone had access to all information. The bid and ask juices may then be expressed as the following sums:

/ = /-Л -< ), (3.2.10)

l = /,-fA +f;, (3.2.11)

s = /> - / = (A + Ab) + (C + С ), (3.2.12)

where Л + Л/, is the adverse-selection component of the spread, to be determined below, and C -f (. includes the order-processing and inventory components which Gloslon calls the grass profit component and lakes as exogenous. If uninformed investors observe a purchase at the ask, then they will revise their valuation of the asset from P to l+Aa to account for the possibility that the trade was information-motivated, and similarly, if a sale at the bid is observed, then / will be revised to P-/!<,. But how are A and Ai, determined?

C.losleu assumes that all potential marketmakers have access to common information only, and he defines their updating rule in response to transactions al various possible bid and ask prices as

fi(.v) = l;j /* I Q U ( investor buys at v j (3.2.13)

Ну) = f /* I QU ( investor sells al yl ]. (3.2.M) A and Ai, are then given by the following relations:

A = (/ )-/, Л = P-h(Ph). (3.2.15)

Under suitable restrictions for a() and />(), an e(]iiilibrium among competing iiiarkctiuakcrs will determine bid and ask prices so that the expected profits from mai kelinaking activities will cover all costs, including ( + (/, and /l H-/h,; hence

l = (/ )+(;, = /+( (/ )-/) + <;, = P + A + C (3.2.10)

i\ = hp ) - t: = / - (/> - hp )) - <: = p - л - с: . (3.2.17)

Sec Ainibllll .Hid MciicIcIvmi ( HMD); (.nlicil. Mliirr, Scliw.nl/. unit YVliilMinih (1 ),41 ); tin aiKlSlnll (IIHI);.iinI Nloll (Ц17Н) In, nii.clcKnl these costs.

An immediate implication of (3.2.16) and (3.2.17) is that only a portion of the total spread, С +Сь, covers the basic costs of marketmaking, so that the quoted spread А +Аь+Са+Сь can be larger than Stolls (1985) effective spread-the spread between purchase and sale prices that occur strictly within the quoted bid-ask spread-the difference being the adverse-selection component A +Ai,. This accords well with the common practice of marketmakers giving certain customers a better price than the quoted bid or ask on certain occasions, presumably because these customers are perceived to be trading for reasons other than private information] e.g., liquidity needs, index-portfolio rebalancing, etc.

Implications for Transaction Prices To derive the impact of these two components on transaction prices, denote by P the price at which the nth transaction is consummated, and let

P, = PJa + Pbh, (3.2.18)

where / (Д) is an indicator function that takes on the value one if the transaction occurs at the ask (bid) and zero otherwise. Substituting (3.2J16)-(3.2.17) into (3.2.18) then yields

P = Ъ[Г\ПиА]1а + Е[Г\аиВ]1ь+Са1а-Ск1ь (3.il9) = P. + CoG, (3-2.20)

P = Е[Р*ПиА]/ 4-Е[Р*ПиЛ]/4 (3.2.21)

1С if buyer-initiated trade

(3.2.22) C(, if seller-initiated trade

!4-l if buyer-initiated trade , .

7 (3.2.23) - 1 if seller-initiated trade

where A is the event in which the transaction occurs at the ask and В is the event in which the transaction occurs at the bid. Observe that P is the common information price after the nth transaction.

Although (3.2.20) is a decomposition that is frequently used in this literature, Glostens model adds an important new feature: correlation between P and Q . If P is the common information price before the nth transaction and P is the common information price afterwards, Glosten shows that

\Aa if a = -t-i

Covin. Qj,\P) = EMP] where As I (3.2.24)

[Ль if Qj, = -1.

That P and Q must be correlated follows from the existence of adverse selection. If Qj,= + 1, the possibility that the buyer-initiated trade is information-based will cause an upward revision in P, and for the same reason,

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