FinMarketsTrading

If the best bid price, B, is lower than C s (v, –L, t), the trader with type yl submits a sell limit order with price A (v, –L, t) ¼ Cb (v, L, t þ 1) rather than a sell market order.

If the trading time, T, was deterministic, the functions Cb (v, þL, t) and Cs (v, –L, t) could be calculated using the backward induction. However, the trading process in the Foucault model has a non-zero probability to continue at any time. Hence, there is no terminal point in time from which one can start recursive calculations. Foucault (1998) offers a stationary solution for the system of equations (5.9) and (5.10), in which the functions Cb (v, þL, t) and Cs (v, –L, t) do not depend on time. Namely,

V o l a t i l i t y E f f e c t

In the generic case with non-zero volatility s, the assumption made in the former section that traders with type yh (yl) submit only long (short) orders is dropped. The attention is now focused on the order placement strategy when arriving traders face an empty order book and submit both bid and ask limit orders. Foucault shows that in equilibrium only traders with type yh purchase the asset and only traders with type yl sell the asset. The bid/ask spread in equilibrium equals

Hence, the spread increases with volatility. The first terms in the righthand sides of (5.14) and (5.15) being proportional to s account for the winner’s curse. Namely, these costs relate to the case when a trader submits a bid (offer) with the price higher (lower) than the asset future value—and this order is filled (unfortunately for the trader). As a result, the spread and hence the cost of market orders increases, which motivates traders to submit limit orders rather than market orders. This, in turn, decreases the probability of execution at higher volatility. The second terms in the right-hand sides of (5.14) and (5.15) reflect the execution costs. Obviously, increasing r leads to a smaller spread and to a higher probability of execution.

N E W D E V E L O P M E N T S

The Parlour (1998) and Foucault (1999) models are attractive in that they are simple enough to have analytic solutions; yet, they offer rich and testable implications. Therefore, these models serve as the starting points for further analysis of the limit-order markets.

Foucault, Kadan, and Kandel (2005, called further FKK) study how traders’ patience affects order placement strategy, bid/ask spread, and market resiliency. The latter is measured with the probability that the spread after liquidity shock returns to its initial value before the next transaction. FKK made several assumptions to obtain a tractable equilibrium solution. In particular, there is an infinite liquidity source for buying (selling) an asset at price A (B); A > B. This liquidity source defines the outside spread, K ¼ A – B. All prices here and further are given in units of ticks. Buyers and sellers arrive sequentially according to a Poisson process and trade one unit of the asset. Each trader can place one market order or limit order, which depends on the state of the order book and trader’s impatience level. The latter is a measure of trader’s perceived opportunity cost5 and can have two values: low dP and high dI; dI > dP. The proportion of patient traders, 0 < u < 1, remains constant. Limit orders can be submitted j ticks away from the current best price (j ¼ 0 for market orders) and cannot be changed or cancelled. Namely, the order bid price and order ask price equal b – jD and a þ jD, respectively, where b B and a A are the best bid and best ask prices; D is the tick size. The inside spread is therefore equal s ¼ a – b. It is expected that limit orders must improve price, that is, narrow the spread. The trader’s utility function contains a penalty proportional to the execution waiting time and impatience level. Namely, the expected profits of trader i (i ¼ I, P) with filled order j are equal:

where Vb > AD and Va < BD are the buyers’ and sellers’ asset valuations, respectively. Hence, for a given spread s, traders place orders to maximize the value:

The FKK model benefits from its strong constraints in that it has a tractable solution for the equilibrium placement of limit orders and expected execution times. It shows, in particular, that expected execution times are

higher in markets with a high proportion of patient traders. This motivates liquidity providers to submit more aggressive orders and to lower the inside spreads. Market orders are less frequent when the portion of patient traders is high, as it is impatient traders who use market orders. New limit orders may top current best prices by more than one tick, which creates holes in the order book—an effect found in real markets (Biais et al. 1995). Another finding is that the tick size may increase the average inside spread if the portion of patient traders is small, since they can place even less aggressive limit orders. Also, a lower order arrival rate leads to smaller spreads as liquidity providers strive to shorten the execution times.

Rosu (2008) has expanded the FKK model in that traders can dynamically modify their limit orders. This implies multi-unit orders and a variable (random) number of traders at any time. The model is still analytically tractable. In particular, it describes the distribution of orders in the order book and market impact from orders of different size. Rosu discerns temporary (or instantaneous) and permanent (or subsequent) market impacts. The former is the result of executing a market order; the latter results from the regrouping of remaining limit orders that are submitted by strategic makers.

The Rosu model (2008) has a number of interesting predictions. First, higher activity of impatient traders causes smaller spreads and lower price impact. Market orders lead to a temporary price impact that is larger than the permanent price impact (so-called price overshooting). Limit orders can cluster outside the market, generating a hump-shaped LOB. In other words, there is a price level inside the order book where aggregated order size has a maximum. Rosu shows also how trading may widen the bid-ask spread. For example, when a sell market order moves the bid price down, the ask price also falls but by a smaller amount. Another observation is related to the full order book, that is, when the total cost of limit orders—including the waiting cost—is higher than that of market orders. When the order book is full, patient traders may submit quick (or fleeting) limit orders with the price inside the spread. These orders illustrate traders’ attempts to simultaneously gain some cost savings and immediacy.

Goettler et al. (2005) have offered a more generic model in which traders can submit market orders and multiple limit orders at different prices. This model has no analytic solution and search of the market equilibrium represents a computational challenge. Traders in this model are characterized by their private estimate of the asset value and by their maximum trading size. The order book is assumed to be open, which allows traders to make rational decisions. Traders arrive sequentially and submit several market orders and limit orders to maximize their expected trading gains. Unexecuted limit orders are subject to stochastic cancellation over time when the asset value moves in adverse direction. The Goettler et al. model

offers a rich set of conditional order dynamics. In particular, it shows that traders with low private values often submit ask orders below the consensus value of the asset. On the other hand, overly optimistic traders submit bid orders above the consensus value. As a result, many market orders yield negative transaction costs, that is, these buy (sell) orders matched below (above) the consensus value. Another finding is that the transaction price is closer to the true value of the asset rather than the midpoint of the bid-ask spread. Hence, the former is a better proxy to the true asset value.

SUMMARY

Solutions of several stylized models of order-driven markets that are reviewed in this chapter have properties that coincide with empirical findings in continuous order-driven markets. In particular,

&A thinner LOB has a wider bid/ask spread.

&Market orders are more likely submitted after market orders on the same side of LOB.

&Buyer (seller) is more likely to submit market order in case of thick LOB on the bid (ask) side.

&Buyer (seller) is more likely to submit limit order in case of thick LOB on the ask (bid) side.

&The bid/ask spread increases with volatility.

&Winner’s curse occurs when a limit order filled due to price volatility turns out to be mispriced.

&Patient traders submit less aggressive limit orders. As a result, the bid/ask spread increases.

&Limit orders may cluster outside the market, generating a humpshaped LOB.

Ask

SP *

Bid

FIGURE 6.1 Possible transaction price paths in the Roll’s model.

In (6.1), S is the bid/ask spread assumed to be constant; qt is the directional factor that equals 1 for buy orders and 1 for sell orders; Pt is the fundamental price that is assumed to follow the random walk and is not affected by trading. Therefore,

where et ¼ N(0, s2). Another assumption is that probabilities for buying and selling are equal and independent of past transactions:

Roll focuses on estimating the covariance Cov(DPt, DPt 1). Note that if

S ¼ 0, then DPt ¼ DPt and Cov(DPt, DPt 1) ¼ Cov(et, et 1) ¼ 0. Consider now Cov(DPt, DPt 1) for the case with non-zero spread. Possible transactional

price paths between two consecutive time periods are depicted in Figure 6.1. Note that P within the Roll’s model is fixed (unless new information is

available). Hence, if Pt 1 is at the bid, DPt can be either 0 or þS; if Pt 1 is at the ask, DPt can be either 0 or S. The corresponding probabilities are listed in Table 6.1. Since transaction at t 1 can be bid or ask with the same probability, the joint probability has the following form (see Table 6.2).

TABLE 6.1 Probability Distribution for Successive Price Changes DPt and DPtþ1 in the Roll’s Model

Obviously, the relation (6.5) makes sense only if the empirical covariance is negative. While this is generally the case, several assumptions made in the Roll’s model may look simplistic. In particular, empirical data in both equity (Hasbrouck 2007) and FX (Hashimoto et al. 2008) markets show that qt can be correlated (so-called runs), that is, buys (sells) follow buys (sells). Fortunately, if

then the Roll’s formula for the spread can be easily modified (de Jong & Rindi 2009):

Trading within the Roll’s model does not affect bid and ask prices.

Hence, within this model, the realized spread (1.4) equals the effective spread (1.3): SR ¼ SE.

T H E G L O S T E N - H A R R I S M O D E L

The Glosten-Harris model (1998) expands the Roll’s model in that it treats the bid/ask spread as a dynamic variable and splits it into the transitory, Ct, and the adverse selection, Zt, components

This implies that the fundamental price is now affected by the adverse selection component

Glosten & Harris (1998) assume in the spirit of Kyle’s model (see Chapter 4) that both spread components are linear upon the trading size Vt:

Let’s calculate the price change for the round-trip transaction of a sale immediately following a purchase of the same asset size. Namely, let’s put qt ¼ 1 and then qt 1 ¼ 1 into (6.12). Then (neglecting et),

In fact, DPt is a measure of the effective spread that is conditioned on the round-trip transaction. It differs from the unconditional quoted spread 2Ct þ 2Zt that is observable to uninformed traders.

The model coefficients c0, c1, z0, and z1 can be estimated using empirical market data. Glosten & Harris (1991) made such estimates with transactional data for 20 NYSE stocks traded in 1981. Their results confirm the prediction made using the information-based microstructure models (see Chapter 4) that the adverse-selection spread component grows with increasing trade size.2

S T R U C T U R A L M O D E L S

Both Roll’s and Glosten-Harris models depart from the efficient market canon in that the observable (transactional) prices are not treated as martingales anymore. One can argue that bid/ask bounces described in the Roll’s model are small short-lived frictions in otherwise efficient market, and the mid-price equated with the asset fundamental value still follows the random walk. Glosten and Harris go further by offering a price discovery equation (6.10) in which trading volume of informed investors can result in a (possibly long-lived) market impact.

The information-based market impact may have lagged components related to quote revisions and past trades. Hasbrouck (1991) suggests that information-based price impact can be separated from the inventory-based price impact since the former remains persistent at intermediate time intervals while the latter is transient.3 Hasbrouck (1991) offers a structural

model in which it is assumed that the expectation of the quoted mid-price conditioned on public information It available at time t approaches the true asset value PT at some future time T:

In (6.14), pat and pbt are ask and bid prices at time t, respectively. Let’s introduce the revision of the mid-price:

In (6.16), ai and bi are the coefficients; e1,t is a disturbance caused by new public information; xt is the signed order flow (positive for buy orders and negative for sell orders) that, in turn, has the general form

It should be noted that the term order flow employed in empirical market microstructure research generally refers to trading (transactional) volumes. However, equating trading volumes and order flows neglects the limit orders that are not filled. This distinction is particularly important in limit-order markets where market orders are not permitted. Empirical data available for academic research often do not specify what side of the market initiated transactions. It is usually assumed that a buy (sell) order initiated a transaction if its price is higher (lower) than the bid/ask mid-price.

The specific of the Hasbrouck’s model (1991) is that the equation for rt(6.16) has the contemporaneous term, b0xt, while (6.17) has only lagging terms. This implies that the quote revision follows trade impact immediately but the latter cannot instantly reflect the former. Hasbrouck assumes also that disturbances have zero means and are not serially correlated:

The system (6.16)–(6.18) represents a bivariate vector auto-regression (VAR).4 Assuming that x0 ¼ e2,0 and e1,t ¼ 0, the long-term price impact can be estimated using the cumulative impulse response:

t¼0