Hybrid Markets, Tick Size and Investor Welfare 1

Transcription

1 Hybrid Markets, Tick Size and Inestor Welfare Egenia Portniaguina Michael F. Price College of Business Uniersity of Oklahoma Dan Bernhardt Department of Economics, Uniersity of Illinois Eric Hughson Leeds School of Business Uniersity of Colorado Draft: August 6, 2004 The first author is grateful to the Uniersity of Utah Graduate School for financial support. The second author acknowledges financial support from NSF grant SES The third author is grateful to the Guiney Research Foundation for financial support. We thank Shmuel Baruch and seminar participants at the New York Stock Exchange and at the Uniersity of Oklahoma for aluable comments and suggestions. The usual disclaimer applies.

2 Abstract This paper shows how the tick size affects equilibrium outcomes in a hybrid stock market such as the NYSE that features both a specialist and a limit order book. Reducing the tick size facilitates the specialist s ability to step ahead of the limit order book, resulting in a reduction in the cumulatie depth of the limit order book at prices aboe the minimum tick. If market demand is price-sensitie, and there are costs of limit order submission, the limit order book can be destroyed by tick sizes that are either too small or too large. We show that an intermediate tick size maximizes a market trader s welfare on a hybrid market: excessiely large ticks discourage parasitic undercutting by the specialist, but prices are bad, while if the price tick is too small, limit order depth again falls because of the parasitic undercutting by the specialist. In contrast, the specialist s profits rise as the tick size is reduced as long as the tick is not too small.

3 Introduction On January 29, 200, the NYSE completed its shift to decimalization. The SEC mandated the shift, saying that not only would it be easier for inestors to understand trading, but it would make stock prices more competitie. Today, it seems clear that the opposite has, in fact, occurred. Following decimalization, there was a massie 66% decline in the cumulatie depth in the limit order book (Research Diision of the NYSE). Reflecting that reduction in depth, Bollen and Busse (2003) find that trading costs for actiely managed mutual funds increased by a remarkable.367 percent of fund assets. Likely responding to that reduction in depth, Chakraarty, Panchapagesan and Wood (2003) find that institutional traders re-allocated order flow toward electronic networks; and Ananth Madhaan, in priate discussion, indicated that institutions hae broken their orders down into far smaller components, reducing aerage share size by more than 50%, despite the associated fixed costs of doing so. Another indication that decimalization has raised trader costs is a Charles Schwab s report of a 22% increase in cancellations or changes of limit orders in the fie days following the NYSE s completion of decimalization (AP Feb. 2, 200): traders hae to monitor their orders more carefully. Perhaps most surprisingly, a careful analysis by Chakraarty, Wood and Van Ness (2003) reeals that decimalization significantly reduced not only trading olume, but een the total number of trades. An understanding of the market design is crucial for unraeling why the moe to decimalization seems to hae backfired. The NYSE is a hybrid market in which a market order can be crossed against both a limit order book and a specialist/floor broker. On the NYSE, limit orders are submitted before a market order is realized, and accordingly hae priority at the same price oer the specialist or competing floor brokers. Gien the incoming market order and the limit order book, the specialist or a floor broker can choose whether to undercut with a slightly better price any portion of the book that they desire and take the remainder of the trade. The penny tick dramatically reduced the cost of stepping ahead of limit orders, proiding specialists and floor brokers a significant adantage at the expense of other traders. The consequences for limit orders is summarized by this complaint about the impact of decimalization by institutional traders that their efforts to buy large blocks of stock on the market are being blocked by specialists who step Chou and Lee (2003) also find that olume per trade decreased significantly after decimalization.

4 in at the last minute and bid a penny higher to buy stocks that institutional inestors would hae gotten otherwise (AP (February 7, 200) report). As long as submitting limit orders is either directly costly or indirectly costly because limit orders can become stale due to information arrial and a specialist can selectiely step in front of limit orders then to offset the reduced likelihood of execution, the optimal response of limit traders may be to submit fewer orders and set prices further from the quote mid-point. Harris (996) argues that a larger minimum price ariation (tick) makes it less profitable for front-runners to take trades away from large traders in markets that enforce time priority. Consistent with this argument, he finds that order display increases with tick size. Rock (990) was the first to model a hybrid market structure. Seppi (997) is the first to analyze formally the effect of tick size on a hybrid market such as the NYSE. Seppi assumes competitie limit order traders, price-insensitie market demand, and a monopolistic specialist. The specialist decides which portion of the book to undercut, and limit order traders break een conditional on being executed the (exogenous) cost of order submission equals their (positie) expected trading profits. The contribution of this paper is to explore how the hybrid market design of the NYSE interacts with the tick size to affect limit order depth, specialist profits and inestor welfare. To do this, we integrate rational, price-sensitie market traders into the model. If market orders are not endogenized, then as Seppi finds, a smaller tick necessarily raises specialist profits. Both the direct effect it is less costly for the specialist to undercut a gien tick and the indirect effect cumulatie depth in the limit order book falls, reducing the competition that the specialist faces make this almost immediate. But, both when decimalization was first announced and when it was implemented, the price of a seat on the NYSE fell, suggesting that the market did not beliee that decimalization would lead to greater specialist profit. For specialist profit not to rise, it must be that there is an endogenous reduction in the size and olume of market orders. When market order traders hae price-sensitie demands, this is exactly what happens they respond to the reduced depth by submitting smaller orders, as Chakraarty, Wood and Van Ness find. We find that in equilibrium, as in Seppi (997), at eery tick size sae the smallest, the cumulatie depth of the limit order book falls as tick size is reduced, because the specialist finds 2

5 undercutting more attractie. In turn, the endogenous reduction in market demand reinforces this direct effect on depth, as the reduced market demand further reduces the alue of submitting a limit order. Indeed, as in Glosten and Milgrom (95), when market demand is too price-sensitie, the limit book can become empty. We show that fixing the price-sensitiity of market orders, reducing tick size reduces depth; and then show that fixing the tick size, increasing price sensitiity reduces depth. When market demand is sufficiently price-sensitie, markets feature a non-empty limit order book only when the tick size is sufficiently large. For smaller tick sizes, the increased ability of the specialist to undercut the limit book makes it impossible for limit order traders to break een at any price. The equilibrium limit order book is empty, and consequently, there are wide quoted spreads. Parlour and Seppi (200) recognize that a competing limit order market can cause similar problems for a hybrid market. The endogenous reduction in market demand suggests that the specialist may prefer a large tick size. Howeer, we show that as long as the tick size is not so small that the limit book becomes empty, the specialist prefers a smaller tick because undercutting is so much easier with a smaller tick. This result is consistent with the eidence in Coughenour and Harris (2004) that specialist profits and participation rates increased after the decimalization for stocks in which public order precedence used to be especially costly (small stocks and actiely traded stocks with tight spreads). For large stocks, they find specialist profits to decline. The reduction in profits for the larger stocks may hae resulted from the splitting of large orders into small components. In fact, in our model the specialist s share would fall to zero if all orders were sufficiently small. Because limit order traders break een, aerage market trader losses rise as tick size falls. It is complicated to determine how the price paid by each indiidual market order is affected when tick size falls some pay less and others pay more. The reason is that although the specialist pursues a more aggressie undercutting strategy, limit depth also declines, so that a greater proportion of the market order is filled by the specialist, who charges a high clean-up price. We derie numerically how the tick size maximizing the utility of a particular market order trader aries with the inestor s willingness to pay, and hence, how it aries across the order sizes that agents trade. Inestors who submit small orders benefit from smaller tick sizes; but aggregating across all inestors, the tick size that maximizes the utility of market traders (limit traders make zero) 3

6 on a hybrid market is strictly positie. Interpreting this result in the context of decimalization, the only beneficiaries were sufficiently small retail traders, and to the extent that mutual funds aggregate small inestor trades, the moe to decimalization may hae hurt een small inestors. The contrast between the optimal tick size in a hybrid market and that in a pure limit order market is sharp: in a pure limit order market, all market order traders prefer a tick size of zero. Howeer, gien a particular positie tick size, oer a wide range of tick sizes, traders who submit relatiely smaller orders prefer a hybrid market oer one featuring an open limit order book. The paper is organized as follows. We next present and analyze the model. We characterize equilibrium outcomes in section 2. We illustrate the effect of tick size on market equilibrium in section 3. In section 4 we analyze the effects of tick size on welfare, and show ia a numerical example that the optimal tick size on a hybrid market is positie as opposed to an optimal tick size of zero on a pure limit order market. In section 5, we allow the alue of the asset to moe after limit orders are submitted. As a result, a limit order can become stale, for example, a limit sell order may be priced below the asset s alue. The specialist profits by taking the opposite side of such an order before the market order arries, thereby inflicting losses on the limit order trader. When limit orders can become stale, limit order submission costs arise endogenously. Section 6 concludes. The model Our model builds on Seppi (997). There is an asset that pays out per share. This alue is common knowledge. Agents can submit market orders to a hybrid limit order/specialist market. The limit market is made by a continuum of agents with preferences c +q, where c is consumption of money, q is the number of shares of the asset held. Market orders can also be handled by a specialist who shares the preferences c + q. We focus on the market buy orders and limit sell orders so that we consider market order submitters with a relatie preference for the asset, with preferences c + qβ, where β >. 2 We assume that β is distributed according to G( ) with density g( ). After Proposition 2, for ease of 2 The analysis of market sell orders is analogous. 4

7 exposition we let β be uniformly distributed on [β min, β max ]. Fixing β min, as we increase β max, the preference for the asset relatie to cash increases, making market order demand less price sensitie. That is, the greater is β, the less price sensitie are market orders. We normalize inestors initial endowments to zero. There is a distribution F ( ) oer the maximum number of N, that a liquidity trader can buy, where N and β are independently distributed. The liquidity trader can buy claims to the asset at prices that are positie integer multiples of a tick, d > 0. At a gien price, liquidity proiders who submit limit orders hae priority oer the specialist. For liquidity proiders other than the specialist, there is a small per-share cost c > 0 of submitting a limit order, which is less than the tick size, i.e., c < d. Later, we endogenize this assumption. The specialist can trade costlessly. Finally, as in Seppi (997), there is a trading crowd with a reseration price r that can absorb arbitrarily large orders. Alternatiely, one could assume that there is no trading crowd, but that the specialist neer quotes a price greater than X% aboe. This closely mirrors the price continuity requirement on the NYSE that preents price-gouging. This alternatie assumption leaes the qualitatie results unaffected. Finally, in Section 5, when we endogenize limit order submission costs by allowing the asset alue to moe so that limit orders can become stale, the need to exogenously specify a trading crowd anishes. The market timing is as follows:. Liquidity proiders submit limit orders. We denote the depth at price p j by s j. 2. A liquidity trader is selected and submits her market order. 3. The specialist offers a price, deciding which limit orders, if any, to undercut. 4. Trades are consummated and payoffs are realized. To simplify the analysis, we initially assume that the common asset aluation,, and the crowd s reseration price, r are multiples of the tick size, d. In our analysis of equilibrium, we discuss how results are altered slightly if we relax this assumption. Later, we proide a welfare analysis that relaxes this assumption altogether. 5

8 2 Equilibrium Let p( ) be the price schedule faced by the liquidity trader. Gien β, N, the market trader will buy M shares, M = min(n, Y ), where Y is the greatest number such that β p(y ). Liquidity proiders other than the specialist submit limit orders so that the marginal limit trader at each price earns zero expected profits. Integrating oer possible types (β, N), we compute the expected profits to each limit order. The zero-profit limit order at p j soles P r(executed) (p j ) = c. Finally, the specialist chooses a clean-up price to maximize profits: he undercuts the limit book at the price that maximizes trading profits. We assume the specialist undercuts the limit book if indifferent. The specialist s expected profit from undercutting price p j is: j E[π j ] = (p j )(M s i ). The specialist s expected profit from not undercutting is: He therefore undercuts when i= j E[π j ] = (p j )(M s i ). i= j E[π j ] = (p j )(M s i ) E[π j ] = (p j )(M i= j s i ). () i= Gien that the common aluation is on the grid, equation () simplifies: the specialist undercuts price p j if and only if j M s i + js j. (2) Let t j be the maximum market order size such that the specialist undercuts price p j : i= j t j = s i + js j. (3) In turn, this implies that a liquidity trader faces cut-off price p j if and only if i= t j < M t j+. (4) Substituting equation (3) into (4) we see that inequality (4) holds if and only if s j+ > j j + s j. (5) 6

9 If condition (5) is iolated, then it cannot be an equilibrium. To see this, obsere that if condition (5) is iolated, then it is neer optimal for the specialist to quote p j ; he does better to quote p j+. That is, the specialist s clean-up price jumps from p j to p j+. But, then execution probabilities at p j and p j+ are the same because limit orders at p j are executed only when those at p j+ are executed, and the only time that limit orders at p j are undercut by the specialist is when those at p j+ are also undercut. Because per-share reenues differ at the two prices, marginal limit order submitters cannot be indifferent between them, and hence it cannot be an equilibrium. Seppi (997) (Prop. 2) shows that when market order flow is price insensitie, then in equilibrium, s j+ is always large enough relatie to s j that (5) holds. We will show that (5) holds (i) independent of tick size only if market orders are sufficiently price-insensitie, and (ii) for more price-sensitie market demand, only if the tick size is sufficiently large. Gien the strategies of the liquidity trader and the specialist, a limit order at price p j is executed if (i) this price is not too high for the liquidity trader and (ii) the maximum number of claims, N, exceeds the corresponding threshold, t j. Because at eery price exceeding p, the threshold exceeds the cumulatie depth, and because the specialist maximizes profits, either all limit orders at a particular price are executed, or none are. The sole exception is when the asset alue is not on the grid so that p is less than a full tick. Then, the specialist cannot profitably undercut p. 3 The probability of execution at p j is ( P r(execution) = P r(n > t j )P r β > p j and the zero-profit/indifference condition for the marginal limit order at p j is ( ( )) pj ( F (t j )) G = c p j = c jd. (6) Using (6), we sole recursiely for the limit book at each price. At p, ( ( )) p ( F (s )) G = c d, 3 The zero-profit condition at p is not altered by this assumption, because for the marginal limit order trader, the probability of execution is still the probability that the market order is enough to fill the depth at p. If either p is not affordable, or the liquidity shock is too small, the marginal limit order at p is not executed, so we compute the depth at p in the same way regardless of whether the specialist finds it profitable to undercut p or not. At prices aboe p, all limit traders are marginal since either all limit orders at a particular price are executed, or none are. At p, unlike at higher prices, if is not on the grid, not all limit orders are marginal. ), 7

10 and ( ) c c H s = d( G( p : )) d( G( p )) 0 : otherwise At the next price, the threshold is t 2 = s + 2s 2. Unless this threshold is exceeded, a limit order at price p 2 is not executed. Substituting the solution for s into the corresponding zero-profit condition, we can sole for s 2 ; and continuing we can sole for the entire book. The following three propositions characterize the equilibrium and the effect of tick size and price sensitiity of market demand on the equilibrium, proided that equilibrium exists. In Section 3, we show that under certain conditions, the equilibrium limit order book is empty. (7) Proposition If the limit order book is non-empty, the equilibrium is unique and can be found using Seppi s (997) solution procedure. Proposition 2 Proided that the equilibrium limit order book is non-empty, the cumulatie depth Q j = j i= s i at or below any price p j on grids P such that p j P decreases as tick size decreases, and the specialist s expected profit increases as tick size decreases, regardless of the price sensitiity of market demand. Proposition 3 If the equilibrium limit order book is non-empty, increasing price sensitiity by reducing β max reduces depths at eery price, proided that the probability density, f( ) = F ( ) oer the maximum number of claims, N, does not increase too steeply in N, i.e., f(t j ) is not too large relatie to f(t j ) so that { } (j )f(t j ) (β max p j ) 2 (β min p j ) + (β min p j )(2d(β max p j ) + d 2 ) { +jf(t j ) (β max p j ) 2 (β min p j ) + d(β max p j ) 2} > 0. If β and N are uniformly distributed, the aboe condition holds wheneer the book is non-empty. The restriction on the distribution oer the maximum number of claims that a liquidity agent would purchase is reasonable. In fact, the most reasonable parameterization of f( ) is that it is nonincreasing, i.e., larger orders are less likely.

11 3 The effect of tick size on market liquidity So far, we hae characterized equilibrium outcomes when the book is non-empty. We next illustrate numerically how both liquidity and olume decline both with tick size and with a rise in price sensitiity. In particular, we highlight the interaction between tick size and price sensitiity. In the next subsection, we illustrate equilibrium construction when market demand is price-insensitie. We then show that liquidity declines with both increases in price sensitiity and with decreases in tick size. Last, we show how, holding price sensitiity constant, decreasing tick size can lead to an empty limit order book in equilibrium. 3. Example: Price-insensitie market demand When demand is price-insensitie, (β = ), the framework corresponds to Seppi (997). Let the maximum number of claims be uniformly distributed between 0 and 0. Finally, let the asset alue, be ; the cost of order submission, c, be 4 ; the tick size, d, be 2 ; the crowd proide infinite depth at price r = 3. Consider the limit order at price p, a tick aboe. The specialist undercuts if and only if the market order, M, is at or below the depth s. The zero-profit condition for the marginal limit order at p is therefore P r[executed] = c d = 2, so that s = 5. At the next price, p 2, the zero-profit condition is and the specialist does not undercut as long as P r[executed] = c 2d = 4, (N s s 2 )(p 2 ) > (N s )(p ). Gien the distribution of N and depth, s, we sole for depth s 2 = 5 4. Next, at p 3, three tick sizes aboe, we hae P r[executed] = 6, and s 3 = The specialist undercuts p if the order size is less than s : t = s = 5. The specialist undercuts p 2 if the order size exceeds t but is less than t 2 = s + 2s 2 = 7 2. The specialist undercuts p 3 if the order size exceeds t 2 but is less 9

12 than t 3 = s + s 2 + 3s 3 = 3. If the market order exceeds 3, the specialist maximizes profits by undercutting the trading crowd by one tick in order to take the residual market order. 3.2 Example continued: Price-sensitie market demand We begin by supposing that β is uniformly distributed on [, 5]. We let =, so that the asset alue lies on the price grid. We will demonstrate two results. First, as β max declines, so that market order flow becomes more price-sensitie on aerage, the limit order book thins out, and can een anish, because for sufficiently price-sensitie demand limit orders cannot earn zero expected profits at any price. This is akin to the market breakdown found in Glosten and Milgrom (95). Second, fixing price sensitiity, as tick size falls, the limit order book can also anish. We now show that gien a distribution for β, if market trade is sufficiently price-sensitie there exists a critical tick size below which the equilibrium limit order book is empty. At each price p j = + jd, the equilibrium probability of execution times the execution profits per share must equal the per-share cost of trading: c = pr(execution)( + jd ), so that pr(execution) = c jd, where j is the number of ticks the price is aboe. Here, the equilibrium probability of execution at p = 3 2 is c d = 2. Similarly, at p 2 = 2, it is 4, and at p 3 = 5 2, it is 6. Now, the probability of execution at p is pr(execution at p ) = pr(n > s ) pr(β > + d). This implies that Soling for s yields c d = N max s N max N min β max ( + d) β max β min. s = In equilibrium then, the specialist undercuts to if M s. 0

13 Next, we sole for s 2, the depth at p 2 = 2. At s 2, the probability of being executed is the probability that M is such that the specialist would prefer not to undercut to p. That is P r[executed] = P r[(m s s 2 )(2d) > (M s )d] = P r[m > s + 2s 2 ] P r[n > s + 2s 2 ]P r[β > p 2 ]. This implies that Soling yields c 2d = N max s 2s 2 N max N min β max p 2 β max β min. s 2 = Now, we find t 2, the minimum order so that limit orders at p 2 are executed: t 2 = s + 2s 2 = Next, we sole for s 3, the limit depth at p 3 = 5 2. c 3d = N max s s 2 3s 3 N max N min β max p 3 β max β min. This equation is alid only when the specialist, if he undercuts, undercuts to p 2 and not to p. One must erify that this is indeed the specialist s optimal strategy. If it is the optimal strategy, then The cutoff at the highest price is then s 3 = 3 2. t 3 = s + s 2 + 3s 3 = Now, we must determine whether the specialist, facing this limit order schedule, has an incentie to undercut to p instead of p 2. For such undercutting not to be optimal, it must be that s 3 > 3 s 2. In this case, s 3 3 s 2 = 2 9 > 0. Table presents the results of this example. For β uniformly distributed between and 4, we obtain the results summarized in Table 2.

14 Still, s 3 3 s 2 = 5 36 > 0. Obsere that as market demand becomes more price-sensitie, depth falls at eery price in the limit book. This follows immediately from Proposition 3. Now, suppose that β is uniformly distributed between and 7 2. Were we to follow the same solution procedure, assuming that s 3 > 3 s 2, so that for some M the specialist prefers to undercut to p 2 rather than to p, we would obtain: s = 5 4, s 2 = 25 24, s 3 = and t = 5 4, t 2 = 35 6, t 3 = Now, s 3 = 3 s 2, iolating our assumption. What happens? There is no equilibrium in which the specialist trades at p 2. We now show that there is also no equilibrium where the specialist undercuts to p. In fact, the limit order book must be empty! Essentially when β is uniformly distributed on [, 7 2 ], market order demand is too price elastic. Suppose the specialist were to undercut to p. Then, the probability that orders are executed at p 2 and p 3 must be equal, because the only time when orders at p 2 execute is when p 3 is not undercut. Therefore, marginal limit orders cannot earn zero profits at both p 2 and p 3 unless both s 2 and s 3 are zero the limit book is empty aboe p. Soling the limit order zero profit condition at p where the book aboe is empty implies that the probability of execution at p must equal the probability that not undercutting p yields specialist profits that exceed zero (which is what undercutting to yields). That is, 2 = N max s N max N min β max p 3 β max β min. The only way this equation can hold is if s < 0, which cannot be. The limit order book is therefore empty at p. Hence, the entire limit order book is empty. It is important to emphasize the fact that the entire limit order book is empty follows because is on the price grid. If is not on the price grid, then the book is empty at eery price exceeding p, but at p, the limit order book fills up until the expected profit of the marginal limit order is zero. Consequently, limit orders at p are profitable on aerage. Next, we illustrate the effects on depth of changing the tick size. First, we show that when tick size shrinks, cumulatie depths decline. Second, we show that as tick size shrinks, eentually, this can lead once again to an empty limit order book. To illustrate, let tick size in the example fall to 3. Let β be uniformly distributed between 2

15 and 5. The solution is found in Table 3. The cumulatie depth at p = 2 is then s + s 2 + s 3 = In contrast, when the tick size was 2, the cumulatie depth was s + s 2 = 5.4. Now we show that when β is uniformly distributed between and 4, with a tick of 3, the limit book is empty. Because the limit book had depth at eery price, ceteris paribus, when the tick was 2, we will hae shown that reducing the tick size alone can lead to an empty limit order book. We begin by constructing an equilibrium candidate. We obtain: s = 25 6, s 2 = , s 3 = , s 4 = , s 5 = and t = 25 6, t 2 = 05 2, t 3 = 25 4, t 4 = t 5 = 53. But now, s 5 = 3 5 s 4, iolating the assumption that the specialist will undercut p 5 to p 4. This implies that the book is empty at both p 4 and p 5. Continuing using the same argument, one can show that the entire book must be empty. 3.3 A more general case Our example shows that (), when market demand is sufficiently price sensitie, the equilibrium limit order book is empty and (2), for a gien distribution for β, decreasing the tick size causes the limit book to thin out and for small enough tick sizes, can eentually lead to an empty limit order book. We now extend these results to a general probability distribution oer the maximum number of claims N. Proposition 4 Suppose that market demand is sufficiently price-sensitie in the sense that β max < 2r. Then there exists a critical tick size, d = 3 (2(r ) (β max )), such that for all d < d, the limit book is empty. 4 Welfare 4. Hybrid market The main issue surrounding the Common Cents legislation of the 990 s was how changing tick size affected the welfare of arious types of traders. Our enironment allows us to analyze the welfare of liquidity traders (liquidity proiders who make up the limit order book essentially earn zero). We now show that different classes of liquidity traders are differentially affected by changes 3

16 in tick size: traders who submit small orders prefer small tick sizes, and those who want to trade in olume prefer larger tick sizes. In our enironment, market order size is not exogenous, and in particular, it is affected by tick size. This means that different traders who happened to trade M when the tick size was d buy many different amounts when the tick size is d, so that it is not simple to determine what happens to the welfare of a cohort who trades M in a particular regime. What we do is consider the welfare of an indiidual market order trader whose type is gien by the pair, {β, N}. Because each trader begins with a zero endowment, the trader s welfare is gien by his expected utility, c + qβ. We next show numerically that the tick size maximizing the welfare of the aerage market order trader is positie, and that traders with different {β, N} disagree about the optimal tick size that maximizes their expected utilities. Consider our example from the preious section, with β U[, 5], N U[0, 0], the true alue, = + ɛ, the crowd reseration alue, r = 3, the limit order submission cost, c = 33. We calculate the expected utilities for representatie {β, N}, for fie different tick sizes, 2, 4,, 6, and 32. We assume that the specialist does not undercut the first price in the limit order book when indifferent. This is realistic, because the probability that the true alue, is exactly on the price grid is a probability zero eent. The results are gien in Table 4. Table 4 reeals that regardless of β, traders who prefer to trade small quantities, N 5, prefer the smallest tick size of 32. Traders with N = 6 are indifferent between tick sizes of 32 and 6. Traders with larger liquidity shocks, (N = 7,, 9) and traders with N = 0 and who are price sensitie, so that β.75 prefer a tick of 6. Finally, traders with N = 0, but who are less price sensitie prefer a still larger tick of. This is not surprising. Traders who submit small orders trade only at the lowest price in the book, and thus benefit from a small tick size. Further, if such traders trade small quantities because they are price-sensitie, a larger tick will preclude them from trading. Traders who submit larger orders, either because N is large or because β is large, benefit more when depth in the book at higher prices is greater, which is the case when the tick size is larger. The table illustrates this tradeoff, and shows that for large orders, adequate depth is more important than receiing a ery 4

17 Table : Equilibrium with β U[, 5] Prices Limit Depths Specialist s Price p 0 = 0 M p = p 2 = 2 p 3 = < M < M < M Table 2: Equilibrium with β U[, 4] Prices Limit Depths Specialist s Price p 0 = 0 M 4 p = < M 25 p 2 = 2 9 p 3 = < M < M Table 3: Equilibrium with β U[, 5] with tick size 3 Prices Limit Depths Specialist s Price p 0 = 0 M 20 p = p 2 = p 3 = 2 p 4 = 7 3 p 5 = < M 2 2 < M < M < M < M 5

18 Table 4: Optimal Tick Size as a Function of Trader Type {N, β}, β/n or Notes: Optimal tick size preferred by each trader type, {N, β}oer the choice set {,,,, } Traders with N 5 prefer a tick of regardless of β. Traders with N = 6 are indifferent between tick 32 sizes and. Traders with N = 7 9 prefer a tick of, regardless of β. Finally, for traders with N = 0, those with small β.75 prefer a tick of ; the remainder prefer a larger tick of. 6 6

19 good price on a small portion of one s order. Next, we integrate oer trader types in our example and consider the effect of altering tick size on aggregate olume, aerage utility, specialist participation, and specialist profits. Our measure of aggregate olume is simply the aerage trade size. Specialist participation is measured as the fraction of the order flow that the specialist takes. Specialist profit is also a per-trade aerage. The results are gien in Table 5. First note that specialist participation rises sharply as tick size falls, because undercutting is so much easier with a smaller tick. When the tick is, specialist participation is still only 3.6%, but at a tick of 32, the specialist takes nearly half of all order flow. This is consistent with the empirical analysis of Coughenour and Harris (2004). Next, obsere that specialist profits also rise sharply as the tick size falls, again because the specialist undercuts more aggressiely. Surprisingly, in practice, reducing the tick size has not led to a large rise of the price of a seat on the NYSE. Recent reductions in trading olume oer time on the NYSE are clearly too small to explain why seat prices hae not risen. Perhaps the true explanation is that the specialist s increased ability to undercut has led traders to split their orders into eer-smaller components, reducing the specialist s adantage. This is also consistent with the finding in Coughenour and Harris (2004) that for large stocks, specialist profits and participation has fallen with the switch to decimalization. Indeed, in our model, were all orders smaller than the aggregate depth at p, when p is not on the grid, the specialist s share would fall to zero. Notice that reducing tick size first increases, and then decreases aggregate olume. For traders who wish to submit large orders, a smaller tick reduces trading olume, on aerage. To see why, obsere that from (6), at any fixed price p j, the threshold market order t j is independent of tick size. Consider a trader with highest affordable price p. If the tick is d, the next aailable price is p + d = p. The corresponding threshold amounts are t and t. Suppose N is large enough to exceed t. Under the tick size d, she can submit an order as large as t and still trade at her highest affordable price p since the specialist will undercut p unless the market order exceeds t. Consider now what happens if the tick size is reduced to d 2. The threshold t at the new intermediate price p satisfies the condition t < t < t, since (i) t and t do not change with tick size, and (ii) thresholds are monotonically increasing in price in equilibrium. Now, to ensure 7

20 that the specialist still trades at p, the trader must not submit an order larger than the new intermediate threshold, which is smaller than t. Hence, trading olume falls with tick size. What happens if N is not sufficient to exceed t? In that case, as tick size falls, the market order will either stay the same size if N t, or decrease if N > t. Finally, what happens if the highest affordable price is p? In that case, reducing tick size does not affect trading olume because the same threshold t must not be exceeded either for p (under tick size d) or for p (under tick size d 2 ). For traders who wish to submit small orders, it is clear that reducing tick size can increase order size. These two counterailing effects produce the hump-shaped relationship obsered in the table. Last, note that aerage trader utility is maximized at a tick of 6. Howeer, recall from Table 4 that different trader types prefer different tick sizes. 4.2 Pure limit market Here, we consider the welfare of the market order trader in a pure limit market. Welfare analysis for the pure limit market is analogous to that in a hybrid market, except there is no specialist to consider. The strategy of the market order trader is still to trade as many claims as possible until the highest affordable price is reached, or until the demand for shares is exhausted, whicheer happens first. The marginal limit order traders at each price still make zero (although the inframarginal limit traders expect profits), so the equilibrium depths at eery price in the limit order book are still determined from the zero-profit condition. The cumulatie depths in the pure limit market are then equal to the preiously defined threshold amounts t j at the corresponding prices in the hybrid market. This is because in the pure limit market, the probability of executing a marginal limit order is the probability that the market order exceeds the cumulatie depth at that price; and in the hybrid market, the probability of execution is the probability that the market order exceeds the threshold amount at the same price. In a pure limit order market, the optimal tick size is zero. Numerical results not reported here confirm that this is the case. 4 4 This conclusion would change in a dynamic enironment where limit order traders can adjust their orders

21 The question that we address here is for a gien tick size larger than zero, which types of liquidity trader prefer the pure limit market, and which prefer the hybrid market? To answer this question, we compute equilibrium limit order submission strategies and equilibrium utilities for all types of trader oer the fie tick sizes detailed aboe in the absence of a specialist. Not surprisingly, traders with smaller N prefer the hybrid market. Howeer, as the tick size decreases, more and more traders prefer the pure limit market to the hybrid market. But conergence to a strict preference for pure market is extremely slow. Een at a tick size of 256 submission cost), traders with N < 7 still prefer the hybrid market. (with smaller order In the case considered aboe, equilibrium in the hybrid market features actie limit order submission. Were it the case that market traders were sufficiently price-sensitie, for small tick sizes, the limit order book is empty in the hybrid market. In that case, the pure limit market is far superior for all traders except perhaps the specialist. 5 Endogenous limit order submission costs Although it may seem that our results, as well as those in the preious literature, depend critically on the tradeoff between fixed limit order submission costs and profits from limit order execution, we now show that qualitatie results are unaffected when we endogenize limit order submission costs by allowing limit orders to become stale. Until now, we ignored one of the adantages the specialist has oer traders who submit limit orders he can react much more quickly to exogenous changes in asset alue than can limit order submitters. Put another way, the specialist may know much more about current asset alue when he trades, than do limit order traders. We model this information asymmetry by assuming that the specialist knows the asset s current alue, but the limit order traders know only the distribution of realized asset alues, so that limit sell (buy) orders may become stale when asset alue rises (falls). In the case that limit orders become stale, we assume that the specialist cleans up the limit order book by profitably buying from (selling to) the limit order book when the asset alue rises (falls). intertemporally. In that case, an optimal tick size would be found by analyzing the tradeoff between monitoring costs, and the possibility of being undercut by a new limit order. 9

22 To illustrate this point, we alter the model timing slightly so that limit orders can become stale: the alue of the underlying asset changes after the limit orders are submitted, but before the market order size is realized. For simplicity, we consider a two point distribution for asset alue: the asset alue is with probability π, and +, > 0, with probability π. This corresponds to the case where there is an informational announcement that is either good news or bad news. It is immediate that because is bounded, it is costless to submit limit buy orders at any p and to submit limit sell orders at any p +, because these limit orders cannot become stale. This implies that there will be infinite depth, proided by a crowd of limit orders at the first tick, and at the first tick +. These limit orders earn (weakly) positie expected profits; profits are strictly positie wheneer either or + δ is not a feasible price. It is also immediate that limit orders in the book cannot cross, and that limit sell (buy) orders must exceed (be less than) the asset s expected alue. Sell limit orders at p < + π expect to lose at least ( π) with probability π. If these limit orders always executed against an incoming market order wheneer the asset alue were, they would make slightly less than π( π). The same argument applies on the limit buy side. The depth of the limit order book at price p j is found from the zero-profit condition, π( + p j ) = ( π)p r(m > t j )(p j ). () The left-hand side of equation () is the expected loss of a limit sell order to the specialist at price p j when the price moes against it times the probability the asset alue rises. The right-hand side is expected limit order profits when the asset alue falls times the probability that it does so. We begin by considering limit sell orders in the following motiating example. For simplicity, let β =, so that market demand is insensitie, and M = N. Let N U[0, 0], let π = 2, and consider the tick sizes d = 2, and d = 4. Let =, and = 2. Because is bounded, order submission costs are effectiely zero at a price of 3 (they cannot become stale), the book has equilibrium depth of at least 0 at price 3. Hence the crowd is endogenized it is replaced with limit orders at price 3. Limit orders that do become stale (when the asset alue is high) are assumed to be executed 20

23 with probability one at their limit prices. For example, an order at price 2 loses per share when stale. When the asset alue is low, the limit orders do not become stale, and the specialist s optimal strategy is the same as in the basic model. The resulting equilibria are gien in Table 6. As before, reducing tick size reduces cumulatie depth at prices aailable at both tick sizes. For example, at p = 5 2, cumulatie depth is 20 9 at tick size of 2, and it is < 20 9 when the tick size is 4. This example shows that both the crowd and the order submission cost can be endogenized. Essentially, in this new setting, limit orders at each price face a different effectie submission cost. Limit sell orders at higher prices face lower costs than more competitie limit orders because they lose less when stale. The example aboe shows that reducing tick size, ceteris paribus, reduces cumulatie depth at prices that are aailable on both grids. We now generalize formally these results. Proposition 5 Let the zero-profit condition be (). The equilibrium is unique. Limit sell orders are submitted at prices aboe the asset s expected alue, π +. Proposition 6 In equilibrium, the cumulatie depth Q j = j i= s i at any price p j > π + on grids P such that p j P decreases as tick size decreases, and the specialist s expected profit increases as tick size decreases. Next, we consider what happens when market demand is price-sensitie and β is uniformly distributed between β min and β max. As before, when market demand is sufficiently price-sensitie, the equilibrium limit order book is nonempty only when tick size is sufficiently large. Proposition 7 Suppose that market demand is sufficiently price-sensitie in the sense that β max +. Then there exists a critical tick size, d = 2 ( 2 (β max )), such that for all d < d, the limit book is empty. 2

24 Table 5: Effect of Tick Size on Specialist Participation, Profits, Aerage Trade Size, Aerage Trader Utility Tick Size Specialist Participation Specialist Profit Trade Size Trader Utility Notes: Specialist participation, measured as a fraction of total order flow, specialist profits, aerage trade size, and aerage trader utility as a function of tick size. The maximum number of shares an agent us willing to trade, N U[0, 0]; and β U[, 5]. Utility is gien by c + qβv, and agents hae an initial endowment of zero. Specialist profit is a per-transaction (not per-share) aerage integrating oer the uniform distributions for trader type, {N, β} Aerage trade size and aerage trader utility are also aerages integrating oer trader type. Table 6: Equilibrium with insensitie liquidity trade with tick sizes 2 and 4 P rices LimitDepths Specialist sp rice LimitDepths Specialist sp rice (d = /2) (d = /2) (d = /4) (d = /4) p = 0 neer 0 neer p = 5 4 na na 0 neer p = neer 0 neer p = 7 4 na na 0 neer p = 2 0 M M 4 p = na na 5 4 < M 20 3 p = < M < M 60 7 p = 42 4 na na 44 M > 60 7 p = neer neer 22

25 6 Conclusion This paper studies the effect of changing tick size on liquidity and on the welfare of market participants in a hybrid market such as the NYSE. The general message supported by our results is that decreasing tick size too much may hae undesirable effects on both liquidity and welfare. In the context of price-sensitie market demand, we demonstrate that cumulatie depth in a hybrid market decreases as tick size falls. We also show that for sufficiently price-sensitie market demand, when tick size is too small, equilibrium features wide quoted spreads, ery little trading actiity, and an empty limit order book. Market order sizes fall with tick size for all but the smallest orders. Next, we demonstrate ia a numerical example an intuitie result that the change in expected utility of a market order trader is maximized in a hybrid market when tick size is positie. Howeer, different types of traders disagree on the optimal tick size: traders who submit small orders prefer smaller tick sizes. Note howeer, that small inestors who pool their trades in a mutual fund prefer larger tick sizes, consistent with Bollen and Busse s (2003) empirical eidence. Specialist profits are maximized at a tick size of zero. Finally, we show that our results are not drien by fixed limit order submission costs. We find qualitatiely similar results where the cost of limit order submission is drien by the possibility that limit orders can become stale. References Bollen, N.P.B., Busse, J., Common Cents? Tick Size, Trading Costs, and Mutual Fund Performance, Vanderbilt Uniersity and Emory Uniersity, unpublished. Bollen, N.P.B., Whaley, R., 99. Are teenies better? Journal of Portfolio Management 25, Chakraarty, S., Wood, R., The effect of decimal trading on market liquidity. Working Paper, Purdue Uniersity, unpublished. Chakraarty, S., Panchapagesan, V., and Wood, R., Institutional Trading Patterns and Price Impact Around Decimalization. Working Paper, Purdue Uniersity, unpublished. Chakraarty, S., Wood, R., and Van Ness R Decimals And Liquidity: A Study Of The NYSE. Journal of Financial Research, 27, Chou, R., W. Lee, Decimalization and Market Quality, National Central Uniersity and Ching-Yun Institute of Technology. 23

26 Coughenour, J., and L. Harris, Specialist Profits and the Minimum Price Increment. Working paper. Glosten, L., Milgrom, P., 95. Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics 2, Goldstein, M., Kaajecz, K., Eighths, sixteenths, and market depth: changes in tick size and liquidity proision on the NYSE. Journal of Financial Economics 56, Harris, L.E., 996. Does a large minimum price ariation encourage order exposure? Working Paper, Marshall School of Business, Uniersity of Southern California, unpublished. Parlour, C., Seppi, D., 200. Liquidity-based competition for order flow. Working Paper, Carnegie Mellon Uniersity, unpublished. Rock, K., 990. The specialist s order book and price anomalies. Working Paper, Harard Uniersity, unpublished. Seppi, D., 997, Liquidity proision with limit orders and a strategic specialist. Reiew of Financial Studies 0, Appendix Proof to Proposition : Seppi (997) shows uniqueness when the 2nd term on LHS of (6) is equal to. Here, for each distribution of β, and gien the price grid, we get a unique alue for the 2nd term. Hence, we still get a unique alue for the st term on the LHS. Hence the solution is unique een in the price-sensitie case, proided that the equilibrium exists. Proof to Proposition 2: See Seppi (997), Proposition. The alidity of the proof does not depend on the price sensitiity of the market order. It does, howeer, depend on the condition that the threshold amount t j be independent of tick size and be monotonically increasing in price p j. We show later that, with sifficiently price-sensitie market orders, the monotonicity condition is iolated for a small enough tick size. Proof to Proposition 3: Express depth at price p j as s j = j (t j t j + (j 2)s j ). It is straightforward to show that s increases with β max. Therefore, as long as we can show that the difference t j t j increases with β max, we will hae shown that the depth at eery price in the limit book increases with β max. 24

27 To show this, differentiate t j and t j with respect to β max. Since dt j dβ max > 0 for any j, and threshold must be monotonically increasing in price for equilibrium to exist, all we need to demonstrate is that this condition becomes dt j dβ max dt j dβ max > 0. In the case of uniform distributions for both N and β, (β min p j + jd)(β max p j ) 2 (j )(β min p j )(2d(β max p j ) + d 2 ) > 0. Gien our assumption that jd = p j and β min, the aboe inequality is true. In the case of general probability distribution for N, the condition becomes { } (j )F (t j ) (β max p j ) 2 (β min p j ) + (β min p j )(2d(β max p j ) + d 2 ) { +jf (t j ) (β max p j ) 2 (β min p j ) + d(β max p j ) 2} > 0. Whether or not this inequality is true depends, in general, on the probability distribution oer the maximum number of claims, F ( ). All of the terms in the expression aboe are positie except the term jf (t j )(β max p j ) 2 (β min p j ) < 0. If the weight F (t j ) on this term is sufficiently large relatie to the weight on the first two terms, F (t j ) (which is equialent to the density function increasing too steeply in N), the condition aboe is not true. Therefore, the proof works under the condition that the probability density is not too steeply increasing in N. Proof to Proposition 4: The zero-profit condition for the marginal limit order at price p j is ( βmax p ) j ( F (t j )) = c β max β min p j. (9) We must check whether the threshold t j is monotonically increasing in price. important because if t j+ Monotonicity is t j, there does not exist a range of market orders for which the specialist optimally trades at p j. For market orders below t j, the specialist prefers to trade at p j than at p j ; and for market orders aboe t j+, the specialist s profit is greater if he trades at p j+ than at p j. In equilibrium, the zero-profit condition must hold for the marginal limit order at eery price. Howeer, if the specialist always skips price p j, the probability of execution at p j is the same as the probability of execution at the price one tick aboe, since the only time that the limit orders at p j get executed is when the limit orders at p j+ are executed also, and the only time the limit orders at p j get undercut is when the limit orders one tick aboe are undercut 25