Dealers learn about asset values as they set prices and absorb informed order flow. These flows cause inventory imbalances. This study models price setting in markets such as foreign exchange, U.S. treasury bonds, European sovereign bonds, and the London Stock Exchange, where market makers have multiple instruments to smooth inventory imbalances and update priors about asset values. Estimating a dealer pricing model with multiple instruments for inventory control and information-gathering yields support for what at times have been elusive inventory and asymmetric information effects. The model presented yields direct measures of the structural-liquidity cost parameters faced by market makers, akin to Kyle’s Lambda. For example, the estimates presented suggest that a $10 million incoming purchase pushes price up by roughly one basis point, and dealers expect to immediately lay off one-third of every incoming order. Compared with estimates of price setting in single dealer markets, price shading is found to have a smaller role in inventory management and information effects are shown to be stronger. Hence, estimating traditional microstructure models (based on only one market maker per asset) on data from asset markets where market makers have multiple instruments misses information from sources other than incoming order flows, and overemphasizes price shading in managing inventories.


Dealers learn about asset values as they set prices and absorb informed order flow. These flows cause inventory imbalances. This study models price setting in markets such as foreign exchange, U.S. treasury bonds, European sovereign bonds, and the London Stock Exchange, where market makers have multiple instruments to smooth inventory imbalances and update priors about asset values. Estimating a dealer pricing model with multiple instruments for inventory control and information-gathering yields support for what at times have been elusive inventory and asymmetric information effects. The model presented yields direct measures of the structural-liquidity cost parameters faced by market makers, akin to Kyle’s Lambda. For example, the estimates presented suggest that a $10 million incoming purchase pushes price up by roughly one basis point, and dealers expect to immediately lay off one-third of every incoming order. Compared with estimates of price setting in single dealer markets, price shading is found to have a smaller role in inventory management and information effects are shown to be stronger. Hence, estimating traditional microstructure models (based on only one market maker per asset) on data from asset markets where market makers have multiple instruments misses information from sources other than incoming order flows, and overemphasizes price shading in managing inventories.

This paper presents and tests a model of dealer inventory management in a market with active interbank trading. It is widely understood that such “two-tier” markets, which include the foreign exchange market, the U.S. Treasury market, and the London Stock Exchange, operate differently from markets without interdealer trading like the New York Stock Exchange (NYSE).1 Hansch and Neuberger (1996), for example, show that dealers will not protect themselves from informed customers with wide spreads, but will instead seek to attract them with narrow spreads. Similarly, Osler, Menkhoff, and Schmeling (2008) show differences arising in price discovery between the two types of markets. The paper attempts to extend existing models of market maker price setting to capture the essential features of markets with interdealer trading in an empirically tractable way.

The model presented focuses on dealer pricing and inventory management practices in the two-tier foreign exchange market. Models of market making, largely based on Madhavan and Smidt (1991); and Madhavan and Smidt (1993), assume that the market has only one tier: dealers only trade with customers. This paper extends such models to include a second tier, specifically an active interdealer market. Market maker price setting is grounded in the two microstructure-pricing effects. The first is the inventory effect, in which the dealer must manage a finite stock of the asset against a demand that responds to a random-walk fundamental value.2 In this situation, if the dealer passively fills orders, the probability of a stock-out is unity. Hence, inventory models argue that dealers shade prices away from the expected asset value to induce trades that unwind undesired positions. The second effect is the asymmetric information effect, in which, for example, the dealer faces a market where some insiders have information about the asset’s liquidation value.3 Recognizing that incoming order flow partially reflects this information, the dealer changes her price accordingly. The model presented attempts to extend these two general effects in an empirically tractable way to markets where dealers initiate trades as well as set prices for incoming order.

At the general-equilibrium level, the “hot potato” model of Lyons (1997) favors dealer pricing with multiple instruments, as high trading volume in the foreign exchange market results from dealers passing on inventory imbalances. At the dealer level, the Ho and Stoll (1983) framework permits interdealer trading (although it does not arise in the model solution), which is the basis of the approach presented here, and recent works by Osler, Menkhoff, and Schmeling (2008); Ramadorai (2008); and Taylor and Reitz (2008) have also focused on two-tier markets. Other features of the model presented are found in Lyons (1995); Mello (1996); and Romeu (2005), which speculate that nonlinearities in dealer pricing models related to inter-transaction time or multiple inventory control instruments may be present in estimations of foreign exchange dealer behavior.

The model presented nests existing market maker pricing models and could contribute to explaining empirical difficulties in previous studies. While there is ample evidence supporting asymmetric information,4 finding evidence supporting predicted inventory effects has proved more challenging (in both one- and two-tier markets). For example, Madhavan and Smidt (1991); and Hasbrouck and Sofianos (1993) reject expected inventory effects in equity and futures markets, respectively. Madhavan and Smidt (1993) only find evidence of unexpectedly long-lived effects after modeling inventory mean reversion and shifts in the desired inventory level. Manaster and Mann (1996) find robust effects opposing theoretical predictions. Romeu (2005) shows that the inventory and information effects found in Lyons (1995) are not simultaneously present in subsamples, as the model would suggest. In foreign exchange markets, Yao (1998) and Bjonnes and Rime (2000) also find little evidence of inventory effects.5 Two-tier markets, however, underscore the impact of dealers employing every alternative when rebalancing portfolios, rather than relying solely on price-induced order flow. Hence, as dealers face increasing marginal losses for inducing flows through price shading, they turn to other methods of unloading unwanted positions. Furthermore, communication with others while making outgoing trades is as informative as communication through incoming order flow. A dealer may use this information to update prior beliefs about asset values and adjust inventory levels. The model presented empirically identifies these parts of observed inventory and price changes correlated with innovations in information, but unrelated to either inventory carrying costs or informative incoming order flow. Hence, the results suggest that ability to make outgoing trades lowers inventory-driven price changes, increases learning about asset values, and may be one reason why prior estimations have had difficulties identifying inventory effects.

The empirical results presented support the model and offer direct estimates of the model primitives of liquidity costs, asymmetric information, and inventory effects. The model offers novel results, for example asymmetric information effects driving price changes are likely twice as large as estimated in other studies (using the same data)—not only is the price response to order flow effect larger, but there are more instruments. One can graphically compare prices with the new information signals that the dealer sees.

Inventory pressure on prices is found to be one-fourth previous estimates, which is reasonable if multiple instruments keep inventory management costs at the lower end of an increasing marginal cost curve. After controlling for inventory and information effects, the base bid-ask spread is wider than previously estimated, and statistically indistinguishable from the market spread convention (3 pips).6 When setting prices, the dealer plans to trade out about one-third of the difference between her current and the optimal inventory positions. A standard ($10 million) incoming trade moves the dealer’s price less than 2 pips or $1,000, and the expected cost of executing an outgoing trade is about double that amount.

A Federal Reserve intervention of $300 million recorded in the data temporarily moves prices about 6.7 pips per $100 million.7 This increases the asymmetric information impact of trades on price changes by 15 percent, which suggests that order flow becomes more informative as the market learns of the intervention. Moreover, the estimate of how much our dealer shades her price in response to inventory imbalances is fairly robust to intervention. This, taken with the result on asymmetric information, suggests that the central bank intervention was transmitting information rather than inducing portfolio balance effects. Finally, the base spread tightens by 5 percent when the intervention is included in the estimation.

The data employed—one week of trading by an active foreign exchange dealer—suggest making limited generalizations without other wider samples. Nevertheless, estimates of asymmetric information effects and the impact of Central bank intervention are in line with other studies.8 Price setting in all types of markets provides an incentive to minimize guaranteed losses from inducing trades via price changes, not just in two-tier markets. While laying off inventory on other dealers is not an alternative outside of two-tier markets, there is evidence that similar phenomenon exist in centralized markets, such as the NYSE.9 The model presented here suggests that market participants share intraday inventory efficiently and exhaust the gains from sharing a large inventory position quickly and with low price impact. As a result, the transitory effects of inventory imbalances, while present, are less important in determining intraday price changes. Hence, a more efficient aggregation of the dispersed information embedded in order flow suggests microstructure models are an important component to understanding permanent price movements.

I. Intraday Price Discovery in Markets with Multiple Dealers

The model begins with features familiar from Madhavan and Smidt (1991); and Madhavan and Smidt (1993) and other models. Consider a foreign exchange dealer in the context of trade time (each period t represents a new incoming trade). She enters every period t with inventory It of currency I (for example, euro) whose value is measured in terms of currency K (for example, dollar), the numeraire. At the beginning of period t, the dealer receives incoming order flow from customers’ denoted qjt. Incoming order flow is measured from the perspective of the customer: positive if the customer buys from the dealer (and the dealer sells to the customer), negative otherwise.

Although the dealer sees only the total of this incoming order flow, it comprises two separate components: qjt = Xt + Qt. The first component, Xt, represents the net demand of uninformed traders. This group includes nonfinancial firms that import and export, as part of their normal commercial business, index funds that purchase and sell assets solely in response to their own inflows of funds. We assume XN(0,σx2).

The rest of incoming order flow, Qt, is informed. These counterparties most likely represent hedge funds and other members of the active trading community, according to dealers as well as current research (Osler, Menkhoff, and Schmeling, 2008). They are assumed to receive a noise-free signal of the currency’s true value, vt. Under the assumption of constant relative risk aversion, which we leave implicit, these traders’ demand is proportional to the gap between the dealer’s quoted price, pt, and the asset’s true value.

Qt = δ(vtpt),δ>0.(1)

Figure 1 depicts the timing of the model, which shows incoming order flow,

Figure 1.
Figure 1.

The Timing of the Model

Citation: IMF Staff Papers 2010, 002; 10.5089/9781589069121.024.A007

Note: The figure describes the timing of the model. At every event:1. if t ≠ T, the dealer knows her current inventory (denoted It), and a new incoming trade (one source of information for updating priors) occurs. The incoming quantity is qjt.2. The dealer decides her price (denoted by Pt) and plans her outgoing trade (denoted by qtout). These are the alternate methods available for offsetting inventory disturbances caused by the incoming trade.3. Between events, the dealer executes the planned outgoing trade (qtout) and faces a quantity shock, (denoted by γt). This is another source of information for updating priors.4. In addition, the dealer observes time elapsed between trades (denoted by Δτ).5. At the next event (t + 1), the dealer uses the new incoming trade qji + 1 as well as the quantity shock between trades and the time elapsed between trades to update priors on the evolution of the asset value, and set prices.
qjt = Qt + Xt = δ(vtpt) + Xt.(2)

Standard models assume that to eliminate inventory associated with qjt, the dealer will have to wait until she has an opportunity to provide quotes to other incoming callers. In the foreign exchange market, however, dealers can trade in the interdealer market, an alternative that can influence the choice of pt. We assume that dealers with suboptimal inventory levels always trade aggressively, meaning they call another dealer directly or they place a market order on an electronic interdealer exchange (these now dominate interdealer trading in the major markets).10 In reality, dealers can supply liquidity in the interdealer market as well as demand it, a complication it would be appropriate to address in future research.

At the time she quotes price pt to the customers behind incoming order flow qjt, the dealer anticipates liquidating the amount qout of the associated inventory in the interbank market. Unlike qjt, the dealer’s chosen trades are measured from her own perspective: positive if the dealer buys currency I, negative otherwise. In choosing qout the dealer anticipates that when the next incoming trade arrives, which we define as period t + 1, her inventory will be:

E[It+1|Ωtj] = Itqjt + qtout.(3)

The dealer assumes that the amount paid or received from qout will be (μt + αqout)qout. Here μt represents the dealer’s expectation of the traded asset’s true value, E[vt|Ωtj] = μt, and α represents “slippage,” meaning the expected adverse price movement associated with trading the amount qout.

Between one incoming customer trade and the next, the dealer could trade more in the interbank market than just qout. Market-relevant information—which of course arrives continuously over real—time news services and from associates in the market—could change the dealer’s incentives and trigger further interbank trading. We denote by Zt the total amount of interbank trading between incoming trades, and γt the component of that interbank trading that was not planned when the dealer quoted price pt: Zt = qout + γt. The dealer’s realized inventory when the next incoming trade arrives is thus:

It+1 = Itqjt + ZtItXtδ(vtpt) + qout + γt.(4)

An example using actual dealer transactions helps motivate the key assumptions regarding qtout and γt. Table 1 shows the first five incoming trades received by the New York-based foreign exchange dealer used in this study (these data are discussed in detail below). The first column indexes the trades according to their order of arrival; the second column shows the price set by the dealer at each incoming trades. The next columns show incoming order flow, followed by the inventory at the beginning of the trade. The last column shows qtout + γt, which are observed jointly. Consider, for example, the third incoming trade, which was a sale to the dealer of $28.5 million. At the time of the trade, the dealer was long $1 million, as reflected in her inventory. Canonical models of price formation assume that incoming orders are the only instrument by which a dealer can adjust inventory levels and update prior information. If one assumes that this were the case, and as the dealer buys $28.5 million, her inventory at entry four should be $29.5 million long (the next incoming trade). Instead, the dealer is short $1.5 million at entry four, which implies that her inventory declined by $30.5 million between the third and the fourth trade. This decline is reflected in the last column, qtout + γt. It captures the inventory evolution that incoming order flow did not generate. This column is expressed as the sum of two components because qtout reflects the optimal amount that the dealer should trade given the information available at the time of the incoming trade. It is a first order condition. Any deviation from qtout must be a result of new information, and is reflected in γt. Therefore, the part of inventory changes not generated by incoming trades is the sum of planned and unplanned outgoing trades, qtout + γt.

Table 1.

Inventory Control: First Five Entries of Lyons’ (1995) Data set

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Note: This table shows the first five entries of the price (second column), incoming order flow (third column), and inventory (fourth column) variables from the data set. The last column captures the part of inventory evolution that is not due to incoming order flow, which reflects the optimal outgoing trade (qout), and deviations driven by new information (γ). Lyons’ (1995) data: New York-based dollar/deutsche mark dealer, August 3–7, 1992.

The dealer’s choice variables, pt and qout, are determined to solve the following familiar dynamic optimization problem:

J(It,xt,μt,Kt) = maxoutpt,qtE{(1ρ)[v˜tIt + Kt + ytct] + ρJ×(I˜t+1,x˜t+1,μ˜t+1,K˜t+1)},(5)

subject to the following evolution constraints:

Inventory:E[I˜t+1|Φti] = Itδ(μtpt)xt + qtout,(6)
NoiseTrading:E[x˜t+1|Φti] = 0,(7)
Information:E[μ˜t+1|Φji] = μt,(8)
Capital:E[K˜t+1|Φji]=Kt + ptδ(μtpt) + ptxt(μt + αqtout)qtoutct.(9)

This structure includes four variables that have yet to be fully defined:

  • (i) ρ, (ii) μt, (iii) Kt, and (iv) c(It).

  • (i) The term ρ>0 represents the probability that the market disappears forever without any active trading during period t, an assumption in which the model follows Foucault (1999); and Madhavan and Smidt (1993).

  • (ii) In forming her expectation of the asset’s true value, μt, the dealer uses her knowledge of the asset’s distribution plus two information signals. The true value of the traded asset is assumed to follow a random walk: vt = vt-1 + θt where θN(0,σv2). Each period the dealer updates her previous forecast of vt, μt–1, using two new information signals. The first signal, st, is extracted optimally from the amount of incoming order flow. The second signal, k(γt–1), is the information that triggered the additional interdealer trades in the previous period. The weight on these signals depends on the time between incoming trades. If the time is very long, then the information associated with previous interdealer trading is relatively stale and that information gets a lower weight. This structure can be represented by assuming that var(st) = σw2 and var(k(γt1)) = σw2Δτ, with Δτ being the clock time elapsed between events t–1 and t. As the appendix shows, this gives an updating as a function of:

μtμt1 = (Δτ/Δτ)st + (1 + /1 + Δτ)k(γt1).(10)

In equation (10), at the moment the dealer is setting pt, st has just arrived because it is based on the incoming order itself (qjt). The quantity shock signal k(γt–1) also signals the value of vt, but it arrives between t–1 and t, and hence it is assumed to have precision that decreases (that is, variance increases) as the clock-time elapsed from event t–1 to t increases.11

The appendix also shows that the estimate of the full-information asset value, μt, generates an unbiased estimate of the liquidity trade, Xt. We denote this statistic as E[Xtt] = xt.

The term Kt represents the dealer’s accumulated trading profits. In the equation, α captures the price impact of a marginal increase in the dealer’s outgoing quantity, which is assumed as modeling outside prices explicitly requires a general equilibrium framework that normally mutes dealer-level pricing effects.12

The term ct(It) represents the cost of holding inventory. This cost is assumed proportional to the variance of the dealer’s overall asset position, c(It) = ωσw2(It). This can be motivated by risk aversion or by the risk associated with breaching the position limits to which all dealers are subject. The appendix follows Madhavan and Smidt (1993) directly in developing the variance of wealth as a function of deviations from an optimal inventory level: σw2(It) = ϕ0 + ϕ1(ItId)2 where Id is the optimal hedge ratio of the risky assets held by the dealer. This allows quadratic inventory carrying cost as a function of inventory itself:

ct = ω[σW2] = ω[ϕ0 + ϕ1(ItId)2].(11)

Equations (5)–(9), and (11) comprise the optimization problem. The appendix shows the model solution to be:

p = μ + β(α/(1 + δα))(IId) + (1 + δα(1β)2δ(1 + δα))x(12)
qout = (A1/A1α)[(IId) + δ(μp) + x](13)
I = I + β(IId)(1+β)2x,(14)
Δpt = ψηtqjt + β(α/1 + δα))(qt1out + γt1 + qt1) + ψ(1ηt)γt1+(1 + δα(1β)2δ(1 + δα))Δxt.(15)

Equation (12) shows the price of the dealer as a function of the estimated asset value, (μt), the deviation from optimal inventory, (ItId), and the liquidity shocks (xt). In equation (13) the outgoing quantity shows that as the price impact of outgoing trades goes to zero, that is, α→0, outgoing trades fully adjusts inventories to the optimal level (in the appendix, A1<0 is shown). In this case, the price will depend only on the estimate of v and the liquidity demand. In equation (15), st is the information from incoming order flow (qjt) and the elapsed time is measured by η = Δτ/1 + Δτ. This equation shows that the increment in dealer price contains information-driven components from both the current incoming order (ηst), and the previous inventory shock ((1–ηt)γt–1), both weighted by the Bayesian updating term, ψ. The (qt1out + γt1 + qt1) term captures component of the price change attributable to inventory pressure—it is the change in the inventory. Finally, the dealer changes her price due to the noise-trading component (Δxt).

Intuitively, the dealer would like to maintain inventory at the optimal level, but as a market maker she must accept incoming orders that constantly disturb her inventory position. As incoming orders arrive, she tries to restore balance to her inventory with qt1out and price changes. Adjusting back to the optimal level Id via qt1out implies absorbing the costs from the outgoing order’s price impact (a). Adjusting inventories via price-induced orders implies absorbing the certain loss to the informed dealers, via δ(μtpt). The coefficients in equation (15) reflect the balance between these competing losses. Furthermore, the price is centered on the best guess of vt, which is derived from two information sources, st and k(γt–1). The respective coefficients reflect the information extraction, which involves weighing these signals by the time elapsed between events.

A Comparison with Existing Models

This section shows how the model presented nests the previous dealer-level frameworks. Restricting the model to no outgoing trades, and consequently no inventory shocks, the solution would be equation (16). This is the Madhavan and Smidt (1993) pricing behavior for an equity market specialist;

Δpt = st + ζ1(ItId) + ζ2xtγtqtout0tT.(16)

This model suggests, however, that these restrictions could shut down other avenues of inventory management that may be available to specialists, as suggested by Madhavan and Sofianos (1998). Romeu (2005); Bjonnes and Rime (2000); Yao (1998); Lyons (1995); and Madhavan and Smidt (1991); postulate that prices are set according to:

pt = μtα(ItId) + γDt.(17)

Equation (17) yields the price change as:

Δpt = β0 + β1qjt + β2(It1qj,t1 + qt1out + γt1) + β3It1 + β4Dt+β5Dt1.(18)

With the data used here, Romeu (2005) shows that structural breaks present in equation (18) coincide with systematic differences in the length of inter-transaction time (Δτ). Previous studies using canonical dealer pricing models have indeed noted that inter-transaction times imply changes in the precision of incoming order flow, however, there are, in fact, changes in both informative variables (qjt, γt). The model presented here shows why inter-transaction times could cause breaks. Rewriting equation (18) consistent with this paper’s data generation process, note the omitted term in brackets weighed by (1-ηt) below:

Δpt = φ0 + φ1qjt + φ2(qjt1 + qt1out + γt1) + (φ3φ2)It1extraneousterm+φ5Δxt + (1ηt)[φ4k(γt1)φ1qjtomittedterm]

The data generating process under the hypothesis of multiple instruments places zero weight on lagged inventory (the extraneous term), which would tend to bias (φ3–φ2) toward zero. However, the estimated coefficient φ2 captures not only the inventory effect, but it partially reflects information from γt–1, which is contained in the inventory term. Thus, the omitted term would normally transmit information from γt–1 to prices, but its absence drives the inventory term to partially reflect this information. Hence, the variation in the informativeness of γt–1 will affect the inventory term. When inter-transaction times are long (Δτ→∞ and (Δτ/1 + Δτ) ≡ η→1), the omitted term should be irrelevant. At such times, one should expect the incoming order flow coefficient (φ1) to be significant, and var(k(γt–1))®¥, hence γt–1 will be mostly noise, and uncorrelated to price changes. This would in turn make φ2 less correlated with the information effect in Δp, as the inventory term picks up the information in γt-1 in lieu of the omitted term. Hence, one would expect to see the inventory effect dampened at these times. When inter-transaction times are short (Δτ→0 and η→0), one would see the order flow coefficient (φ1) become less significant, whereas the coefficients on the inventory terms would be more significant, and pick up the inventory effect more clearly.

II. Data Considerations

This section discusses the data sources employed in testing the model, and then presents the data graphically to motivate both the new inventory and the asymmetric information effects predicted here, as well as those predicted by canonical models.

The data set consists of one week of a New York-based foreign exchange dealer’s prices, incoming order flow, inventory levels, and transaction clock times for one week of trading in 1992. Hence, pt, qjt, It, and Δτ(and η) come directly from the recordings of a Reuters Dealing trading system. Out of the 843 transactions, four overnight price changes are discarded as the model at hand deals exclusively with intraday pricing. A few measurement errors are present in transaction clock times, and these are treated with a dummy variable in the estimation.13 Table 2 presents descriptive statistics. One observes that the dealer keeps the average inventory at $2.1 million, however, it deviates as much as ±$50 million. Given a median incoming order of roughly $3 million, reversing a one standard deviation swing in inventory necessitates about five sequential incoming trades, which suggests very active inventory management.14

Table 2.

Descriptive Statistics

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Note: This table shows descriptive statistics for the dealer’s inventory and incoming order flow.

Table 3 shows the observed incoming trades received by the dealer, as well as bilateral trades that our dealer initiates with other dealers in the foreign exchange market. The table shows on average 20 outgoing trades per day initiated by our dealer. These, however, are conceptually different from qout, which represents an outgoing quantity planned at the time of price setting that captures alternatives to shading the incoming transaction price for inventory control. Thus qtout is unobservable in that it represents the dealer’s commitment to an outgoing trade at the moment of price setting only. At this moment, she commits irreversibly to trading at a price whose optimality depends on being able to trade qtout; this model suggests that the price set by the dealer would be different if qtout were not available for inventory control. Observed outgoing quantities differ from the planned qtout because the dealer reoptimizes in response to unanticipated information, frictions, or differences in the trading venues utilized to execute the outgoing trade. Although they are unobservable, the model solution provides equations that allow estimation of qtout and γt. Table 3 shows that the spread on both incoming and outgoing trades is tightly maintained at the market’s convention of 3 pips. Diverging from this spread is frowned upon by others in the market, as it is interpreted as failing to provide predictable over the counter liquidity. Hence, point estimates of the model that imply widening or narrowing the spread should be interpreted as theoretical constructs that in practice manifest themselves in other ways, for example, as shifts in the midpoint of the spread.

Table 3.

Observed Incoming Order Flow and Outgoing Trades

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Note: This table shows observed trades made by the dealer (not including a small amount of brokered trades). Note that outgoing refers to trades that the dealer is observed initiating. This is conceptually different from qout, which represents an outgoing quantity planned at the time of price setting that captures alternatives to shading the incoming transaction price for inventory control. Lyons’ (1995) data: New York-based dollar/deutsche mark dealer, August 3–7, 1992.

The fundamental question of interest is how dealers set prices, that is, equation (15). Its estimation requires decomposing the inventory change so as to get at the outgoing orders, qtout and inventory shocks. Because γt is driven by new information, the model solution reflects this information in our dealer’s estimate of the liquidation value of the asset. That is, price changes depend on updating priors using two sources of information: the incoming order flow, and the unexpected outgoing order flow (γt-1). Canonical models typically employ incoming order flow as a source of information; however, the use of γt-1 as a source of information is new. To get a feel for this variable, Figure 2 superimposes cumulative daily unexpected order flow on the price, and Figure 3 does the same for cumulative daily inventory shocks (that is, cumulative daily γt-1).

Figure 2.
Figure 2.

Canonical Models’ Information Effect: Incoming Order Flow and Price

Citation: IMF Staff Papers 2010, 002; 10.5089/9781589069121.024.A007

Note: This figure superimposes price on cumulative incoming order flow, August 3–7, 1992.
Figure 3.
Figure 3.

New Information Effect: Cumulative Inventory Shocks and Price

Citation: IMF Staff Papers 2010, 002; 10.5089/9781589069121.024.A007

Note: This figure superimposes price on cumulative unexpected inventory shocks, August 3–7, 1992.

The vertical lines represent the end of each day of the five-day sample (Monday through Friday). The correlation of two signals with price seems to vary. For example, on Monday and Wednesday, incoming order flow appears to be a more precise signal of price than inventory shocks, whereas on Friday the opposite seems to be true. In the model, elapsed clock-time affects the relative precision between these signals. Table 4 reports the daily correlations and average inter-transaction clock-time. Although these are cumulative signals, Friday gives an example of short inter-transaction clock-time, and higher correlation in the (cumulative) inventory shocks than (cumulative) order flow shocks. Hence, these signals seem to compliment each other and are weighted by inter-transaction time in the model.

Table 4.

Information Effect: Daily Correlation of Order Flow Variables with Price

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Note: This table shows the daily correlation between price and the order flow variable used to update priors. The first column shows incoming unexpected order flow and the second inventory shocks correlations for each day, August 3–7, 1992. The last column shows daily mean elapsed inter-transaction time.

Reporting errors imply mean absolute value transaction time.

III. Estimation

The framework presented provides sufficient identifying relationships so as to permit an almost direct system estimation of the model solution. Hence, only leveling constants, an autoregressive error on the inventory equation, and bid-ask bounce dummies on the pricing equation are added. Table 5 lays out the system of equations given in the model solution (the first column), with the empirical implementation of the solution (the second column), and the parameters recovered from each equation (third column). The first equation in the system, the inventory evolution, yields the optimal inventory level. The second equation identifies the optimal outgoing order qtout and γt. This is simplified as:

Table 5.

System of Estimable Equations

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Note: This table compares the algebraic solution to the model (in the first column) with the estimable equations these imply (in the second column). The final column shows testable restrictions on the model parameters. Row (1) shows inventory evolution, row (2) shows outgoing quantity, and row (3) shows price changes. The bottom shows the structural parameter measuring the expected cost of liquidity at the time of price setting, â. The system is estimated simultaneously using seemingly unrelated nonlinear least squares.
q^t1out = c3(It1 + I^d + qjt1),withI^d=c1(1c2)and(qt1out + γt1)(ΔIt + qjt1).(19)

Solving for γt-1 by adding and subtracting c3It, yields:

(ΔIt + qjt1)c3(It1 + I^d + qjt1)=(1c3)(ΔIt+qjt1) + c3(ItI^d).(20)

Hence, the transformation of equation (20) allows the estimation of the proportion of incoming trade that is expected to be traded out, c3, as a moving average of the net outgoing order flow (ΔIt + qjt–1), and the deviation from target inventory (ItÎd). Moreover, in the pricing equation (the third row of Table 5), removing expected outgoing trade, as well as the incoming trade, from the inventory change identifies the outgoing trade shock γ^t1. However, as equation (20) is a function of terms such as ΔIt. that are already present in the pricing equation, it is necessary to transform it so as to eliminate multicollinearity. Thus, equation (20) is simplified for the pricing equation to:

(1c3)(ΔIt + qjt1) + c3(ItI^d)=(1c3)ΔIt + qjt1 + c3(ItI^dqjt1)(q^t1out).(21)

One can express equation (21) in a more conceptual way using q^t1out:

(1c3)ΔIt + qjt1 + c3(ItI^dqjt1)=(1c3)ΔIt + (qjt1q^t1out).(22)

Equation (22) identifies γ^t1 as a weighted function of the inventory change that the dealer did not trade, less the part of the last incoming order that the dealer did not trade out. Grouping the terms on ΔIt in equation (22) with the inventory effect permits estimation of the system without multicollinearity in the pricing equation.

In estimating the incoming order flow’s information content, prior models use either order flow or its unexpected component. This study uses order flow directly in the price equation, so as to maintain comparability to foreign exchange market studies, such as Lyons (1995); however, estimation is robust to either measure.15 In addition, the model predicts that the only difference in the informativeness of incoming and outgoing order flow is due to the clock time between trades, η. Thus, the solution allows the identification of the information effect from the different components of equation (22) as the inter-transaction times are observed. Hence, as the model solution predicts identical coefficients on these terms, the components of γt–1 outlined above are accordingly constrained to have the same coefficient as incoming order flow after accounting for η.16 Two direction-of-trade dummy variables are included to capture the fixed costs such as order processing costs, and pick up the base spread for quantities close to zero. These variables equal unity if the incoming order is a purchase (that is, the caller buys), and negative one if the incoming order is a sale (that is, the caller sells). The elapsed time in between transactions is measured to the minute, and estimates are robust to monotonic transformations of η.17 Finally, scaling constants are included in all three equations, and the first equation is estimated with an AR(1) error to control for autocorrelation. The system is estimated simultaneously using Seemingly Unrelated nonlinear least squares. Table 6 shows the estimations of the model. Table 7 presents canonical model estimates of the same data as a basis for comparison.

Table 6.

Price Formation with Multiple Instruments

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Note: This table estimates the system of equations imposing all identifying restrictions. Estimation is robust over subsamples of this data set, including around the approximate break points found in previous canonical model estimations, α measures the expected price impact of augmenting the planned outgoing trade by $1 million in pips, and Id measures the implicit optimal inventory level used by the dealer. All estimates multiplied by 105, p-values in italics, Lyons’ (1995) data set.
Table 7.

Canonical Model Estimates

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Note: This table reproduces canonical microstructure estimates using the Lyons’ (1995) data set. All estimates multiplied by 105. Estimating over the two halves of the sample reveals that the simultaneous presence of inventory and information effects predicted by canonical models are significantly not different from zero (See Romeu, 2005). Hence, while inventory and information appear to be present in the data, canonical model predictions are overturned as predicted in the text.

The estimations in Table 6 indicate that the model fits the data fairly well. The main results are the significant and properly signed coefficients on the information and inventory effects, c11 and c12, as well as the predicted inventory evolution and outgoing trade estimates, c1, c2, and c3. Single-market maker model estimates are presented in Table 7 as a basis for comparison. Note that these estimates are not robust to subsample estimation. Specifically, model predictions of inventory effects are rejected in the first half of the sample, and similarly, predicted information effects are rejected in the second half of the sample.18 The model presented here is robust to subsample estimation, notwithstanding the lower p-values of estimated coefficients in the first subsample. Moreover, all three equations in the system are jointly significant as predicted, and the estimates fail to reject the testable restrictions. The model predicts that the dealer plans to trade out roughly one-third of each incoming trade (ĉ3 = 0.34) each time she quotes a price. Additionally, the model estimates the dealer’s target inventory at about 2 million (Îd = 2.09). From Table 2 the average inventory is 2.16, which is statistically indistinguishable from our dealer’s observed average.19

Asymmetric Information

The asymmetric information component (c11) is significant and larger than single-market maker model estimates given by β1 in Table 7 (105 multiply the pricing equation coefficients). One way to interpret the estimates is that the dealer widens her spread by 3.5 pips per $10 million of incoming order flow or inventory shocks (twice c11, as orders are quoted based on absolute size). These estimates indicate a more intense asymmetric information effect than previously estimated; not just because of the higher estimated effects, but because there are two sources of private information—both incoming and outgoing order flow—both pushing price changes. In terms of economic significance, the estimates suggest that the marginal $1 million order pushes the dealer’s price by about 2 basis points, given the average exchange rate in the sample of roughly 1.5 deutsche marks per U.S. dollar, or 2 percentage points per excess $1 billion traded. This is higher than market-wide estimates of the price impact of $1 billion of excess order flow, which fluctuate around half a percent (see Evans and Lyons, 2002). However, these latter estimates are not comparable because of the inherent difficulties of linearly interpolating one dealer’s behavior to the market-wide equilibrium. These difficulties are particularly acute as the dealer generating these data predominantly provides interdealer liquidity, not end-user liquidity. The hot potato hypothesis of Lyons (1997) would suggest that this dealer pushes prices in response to excess order flow more than others, who have access to end-users that absorb order imbalances.20 Put crudely, a foreign exchange position is like a hot potato. Liquidity providers such as our dealer pass it around, pushing prices until an end user is found who is willing to hold the offsetting position. In single dealer market maker models, even if pure inventory pressures were perfectly explained, there is a component of inventory change driven by new information. Inventory theory cannot explain this information-driven inventory component. This component is one of multiple signals that, according to the model, vary in precision depending on elapsed clock-time. This suggests that incoming order flow can be relatively less informative at different times, and should be weighed accordingly. Hence, estimations that assign all information-driven price changes to the (at times, noisy) incoming order flow may mute its true informative impact.

Inventory Effects

Turning to inventory effects, comparing coefficient estimates of the canonical single-market maker model and the model shown here presents difficulties because the dealer’s pricing decision is affected differently by inventory. Instead, it is useful to compare structural parameter estimates reflecting the dealer’s bid-shading in response to inventory pressure. Canonical models’ inventory specification depend crucially on the linear price relationship pt = μt − α(ItId) + γDt, as shown in equation (17).21 That pricing assumption yields two inventory terms:

β2It + β3It1β2(It1qjt1 + qt1out + γt1)+β3It1β2<0(qjt1 + qt1out + γt1) + (β2 + β3)It1<0;|β2|>β3.(23)

The estimate in Table 7 of β^3 = 0.72 from equation (23) is the canonical model’s (absolute) structural price adjustment per 1-million dollar deviation from the desired inventory level (that is, β^3 is the empirical estimate of the canonical model parameter α in equation (17)). In the model presented here, the analogous relationship is given in the first-order conditions specified by equation (12), where β(α/(1 + δα)) is our structural inventory effect on prices. A direct estimate of our model’s parameter α (the inventory evolution parameter in equation (14)) is β^ = (c^21), as shown in Table 5. This yields β^ = 0.34. Moreover, (α/(1 + δα)<1 for the range of α>0 and δ>0 consistent with our model. Hence, the total inventory effect in our model is β multiplied by a factor that approaches unity from below. That is, to arrive at the equivalent measure of the canonical inventory effect in equation (12) one must multiply β^ = (c^21) by a factor of at most, one. Hence, in comparing the price impact per million dollar deviation from the desired inventory level in equation (17) against equation (12), prior model estimates of inventory costs are at least two to three times larger than the estimates presented here. Hence, as changing price is but one of multiple instruments used to control inventory costs, inventory accumulation is not as important in explaining price changes.

Expected Cost of Outgoing Trades and the Base Spread

The use of multiple increasing-marginal-cost instruments to manage inventory requires having an expected cost of the outgoing trade at the time of price setting. This expected cost is estimated at α^ = 0.35 pips. This measure reflects the dealer’s expected marginal cost of trading out an extra million dollars, that is the dealer’s opportunity cost of changing the spread in response to a $1 million incoming trade. In principle, the dealer’s alternative is to change the price to offset the inventory carrying cost, estimated to be at most 0.34 pips per million, as discussed above. Hence, the estimates suggest that trading out excess inventory has a higher marginal cost for the dealer than accepting incoming trades, and the estimated proportion of excess inventory that is traded out, C3, is 0.33, meaning that for each incoming dollar, the dealer expects to trade out one-third. Finally, C4 measures the effective spread for qjt close to zero. It suggests that after having controlled for information and inventory effects, the baseline spread is roughly 2.5–2.8 pips (twice c4 times 10–5). Note that these estimates are approximately equal to the median interdealer spread observed in the foreign exchange market of 3 pips.

Fed Intervention

The last 5 percent of recorded trades occurred while the Fed intervened to support the dollar. In Figure 2, the sharp appreciation on the last day reflects the market reaction to the intervention involving dollar purchases totaling $300 million after the close of European markets. The Fed does not reveal the exact start time and there are too few observations to meaningfully estimate the intervention in isolation.22 Wald tests fail to reject equality between estimates of the model with and without the intervention period (that is, 95 percent of the sample, vs. 100 percent).

Table 8 shows the impact of the intervention on the estimated parameters. The intervention increases the asymmetric information effect of incoming order flow (c11) by over 8 percent, while the change in the estimated inventory effect (c12), as well as in other model parameters, is negligible. The dealer price appreciation recorded during the Fed intervention period, which presumably would be induced by Fed purchases of dollars, serves as a rough check on market wide studies of market liquidity. While the exact start time is not revealed, the $300 million intervention moved the market price between 20 and 32 pips before falling back. At the lower end of the range, this concords with estimates of between 5 and 8 pips per 100 million from Evans and Lyons (2002) (5 pips per $100 million), and Dominguez and Frankel (1993b) (8 pips per $100 million). At the higher end, 12 pips per $100 million implies a market-wide elasticity closer to the estimates of dealer costs in this study.

Table 8.

The Impact of a Federal Reserve Intervention

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Note: This table shows cost comparisons for the $300 million Fed intervention on August 7, 1992. The exact start time and sequence of the intervention is unknown. According to the Wall Street Journal, August 10, 1992: “The Federal Reserve Bank of New York moved to support the U.S. currency … as the dollar traded at 1.4720.” This is the most precise documentation available of the intervention start, and that price corresponds to 12:32 pm. Other times selected because of reports of a mid-day start, and because between 12:26 and 12:32 pm, the price jumped 36 pips, suggesting a possible intervention start there.

IV. Conclusions

This study attempts to empirically model market making when multiple instruments for inventory control and asymmetric information gathering are available, in an empirically tractable way. The model presented shows the ability to call others in the market and unload her unwanted inventory on them impacts information gathering and inventory costs. In the model, outgoing orders are not a panacea for inventory control—these are modeled with price impact (that is, increasing marginal costs). Hence, the market maker will equate the marginal loss of trading unwanted inventory to incoming calls with the marginal price impact (that is, the loss of trading unwanted inventory in outgoing calls) and with the marginal loss of the inventory imbalance (that is, the marginal inventory carrying cost). In addition, these outgoing calls do not occur in a vacuum. As long as events transpire during the outgoing call period, the dealer will learn through trading at those times and update her beliefs. These updates bring about price changes that neither inventory costs nor incoming order flow can explain. And foreign exchange dealers are just one example of two-tier market makers that smooth costs over multiple instruments. This paper argues that one should consider where dealers or specialists might be substituting away from conventional inventory costs when modeling price setting. Moreover, price-induced order flow is one of a multiplicity of informative instruments available to market makers.

The estimations support the proposed model and provide several novel empirical results. Generally, these indicate that previous studies overemphasize the role of price changes in inventory management, as no other instruments are considered. This omission biases, downward the role of information in price changes, can make inventory effects appear insignificant, and tightens the bid-ask spread. The data generating process modeled here suggests that information effects are also biased downward in canonical estimations, as the dealer infers asset values from multiple signals that vary in their precision. Canonical estimates fail to correct for the varying precision of the informative flows in two-tier markets, and hence, the information effect is biased downward as it overweighs the signals at uninformative times. The estimates also suggest that at the time of price setting, planned outgoing trades are one-third of the difference between dealer’s current and optimal inventory positions, and a Fed intervention increases the informativeness of order flow, and lowers the cost of liquidity for the dealer. It also lowers inventory costs and tightens the spread.

Finally, the model suggests a comment on the broader relation between portfolio flows and asset prices. The presence of inventory effects suggests that part of observed price changes is transitory. However, with multiple instruments, dealers exhaust the gains from sharing a large inventory position with less price impact. As a result, the transitory component of price changes is less important than the information components from the multiple instruments. Hence, while both transitory and permanent effects are present in the data, the model favors a permanent impact of portfolio flows on prices.

Appendix. Model Solution

Inventory Carrying Cost

Following Madhavan and Smidt (1993), the dealer holds a portfolio of three assets. She only makes markets in the first, a risky asset with a full information value denoted by vt, which evolves as a random walk. Write this value as:

vt = vt1 + θt,θN(0,σv2).(24)

The second is an exogenously endowed risky asset that is correlated with the first, and generates income yt. The third is capital, the risk-free zero-return numeraire, denoted by Kt. The distribution of the two risky assets is:23


The dealer’s total wealth is:

Wt = vtIt + Kt + yt,(26)

with It being the dealer’s inventory or risky asset position. Our dealer manages inventory because she pays a cost every period that is proportional to the variance of her portfolio wealth, which includes the cash value of the inventory. One can motivate this cost, for example, by risk aversion or marginally increasing borrowing costs. Assume that the dealer incurs a capital charge due to the γ shocks. That is, any gains (losses) entering into the dealer’s wealth due to γ are subtracted (added) from (to) the dealer’s capital, Kt at a cost vt.24 Incorporating this charge, at trade t the dealer’s wealth position is given by:

Wt = vt(E[It|Φt1] + γt1) + (E[Kt|Φt1]vtγt1) + yt.(27)

This assumption implies that the dealer only pays the inventory carrying cost on the expected wealth, and the inventory carrying cost due to quantity shocks is canceled by the capital charge. The appendix shows that the inventory cost is a function of the deviations from the optimal hedge ratio of the risky assets, given by Id. This hedge ratio optimally smoothes the dealer’s wealth, and enters the inventory cost as:


From equations (25) and (26) the variance of the dealer’s portfolio is

σWt2 = σv2It2 + σy2 + 2Itσvy.(29)

Add and subtract (σvy/σv2)2 into equation (29) to get:

ct = ω[σy2(σvyσv2)2 + σv2It2 + 2Itσvy + (σvyσv2)2]=ω[(σy2(σvyσv2)2) + σv2(Itσvyσv2)2],(30)

which is the right-hand-side of equation (28) with coefficients:

Id = (σvyσv2)ϕ1 = σv2ϕ0 = σv2(σvyσv2)2.(31)

This assumption implies that the dealer only pays the inventory carrying cost on the expected wealth, and the inventory carrying cost due to quantity shocks is canceled by the capital charge.

Dealer’s Beliefs

Given market demand qjt, the dealer creates a statistic based on the intercept of the demand curve, which is independent of her price. Denote this statistic as Dt.

Dt = qjt + δpt = δ(vtpt) + Xt + δpt = δvt + Xt.(32)

From the signal of market demand Dt the dealer forms two statistics. The first is an innovation in the full information value of the risky asset, which shall be denoted as st. The second is a signal of the liquidity demand, which is denoted as (lower case) xt, and will depend on the estimate of full information value, μt.

wt = δ1Dt = vt + Xtδ;E[wt] = vt(33)
xt = Dtδμt,E[xt]=Xt(34)

Consistent with rational expectations, assume that the dealer’s previous estimate, μt–1 is the steady state distribution over the true asset value vt, and that the variance of μt is proportional to the variance of wt. Hence, one can write σμ2 = Ωσw2.. Given the variance of wt, form a signal to noise ratio given by:

ϒ = σv2σw2,withσw2 = δ2σx2.(35)

The dealer uses the recursive updating of a Kalman filter to form the expectations over vt. This implies that she updates the prior belief μt–1 using the current order flow wt. The resulting posterior, μt, converges to a steady-state distribution whose time varying mean is an unbiased estimate of the true value of vt. The recursive equations to generate this estimate are given by:

Ω = ϒ + ϒ2 + 4ϒ2,(36)

Hence, if the dealer had only information based on the incoming order, she would use the following estimate, which is denoted as μtz, as the estimate of vt:

μtZ = Ωwt + (1Ω)μt1.(37)

Note, however, that the dealer also receives information for updating μt–1 through a linear function of the inventory shock that is denoted by k–1). Given k–1), an unbiased estimate of vt is given by:

μtγ = Ω[μt1+κ(γt1)]+(1Ω)μt1=μt1+Ωκ(γt1),(38)

where the same Kalman filter algorithm as defined above is used. Hence there are two signals of vt at the time of setting the price. Given the assumption, the variance of μtγ is a linear function of the variance of μtz. That is,

var(μtZ)=σZ2,var(μtγ) = σZ2*Δτ,(39)

where Δτ is the elapsed clock time between incoming order (t–1) and t. The optimal signal for the dealer is then:

μt = ημtZ + (1η)μtγ = η[Ωwt + (1Ω)μt1] + (1η)[μt1 + Ωκ(γt1)].(40)

with η = (Δτ/1 + Δτ). Now grouping and rearranging:

μtμt1 = ηΩ(wtμt1) + (1η)Ωk(γt1)=ηΩ(δ1Dtμt1) + (1η)Ωk(γt1).(41)

Since wt = δ1Dt = vt + Xtδ,

μtμt1=ηΩ(δ1(qjt + δpt)μt1)+(1η)Ωk(γt1).(42)

Add and subtract δμ to get:

μtμt1 = ηΩδ1[qjtδ(μtpt) + δ(μtμt1)] + (1η).Ωk(γt1)(43)

Solving for (μ1–μt–1) yields,

(μtμt1)[1Ωη] = ηΩδ1[qjtδ(μtpt)] + (1η)Ωk(γt1),(44)

which gives the final relationship for the updating:

Δμt = ξ1st + ξ2k(γt1),(45)

where st = qjt–δ(μtpt) is the unexpected order flow, and

ξ1 = ηΩδ(1Ωη)&ξ1η>0;ξ2 = (1η)Ωδ(1Ωη)&ξ2η<0(46)

Hence, ξ1 and ξ2 are inversely related with respect to η, and as inter-transaction time is longer, more weight is placed on the unexpected incoming order flow signal st. Here, κ(γt–1) is assumed to be some simple linear function: κ(γt–1) ω0 + ω1γt–1, where ω0 may be assumed zero if desired.

The Dealer’s Problem

The dealer’s problem is reproduced here:

J(It,xt,μt,Kt)=maxpt,qtoutE{(1ρ)[v˜tIt + Ktct] + ρJ(I˜t+1,x˜t+1,μ˜t+1,K˜t+1)},(47)

subject to the following evolution constraints:

E[I˜t+1|Φti] = Itδ(μtpt)xt + qtout,(48)
E[x˜t+1|Φti] = 0(49)
E[μ˜t+1|Φti] = μt,(50)
E[K˜t+1|Φti] = Kt + ptδ(μtpt) + ptxt(μt + αqtout)qtoutct.(51)

For expositional simplicity, in what follows the expectation operators on the evolution equations and the time subscripts are dropped, and a forward lag is denoted by a “superscript.” The first order conditions are given by:

p:δE[J1(I,x,μ,K)]+(δμ2δp + x)E[JK(I,x,μ,K)] = 0,(52)
qout:E[J1(I,x,μ,K)](μ + 2αqout)E[JK(I,x,μ,K)] = 0.(53)

Substituting equation (53) into equation (52), and assuming for now that E[JK(I’, x’, μ’, K’ (I confirm this later), price is:

p = μ + x2δ + αqout.(54)

Denote from here on the value function without its arguments for notational simplicity, maintaining the convention that J(’) is the forward lag of J(). Furthermore, in what follows a subscript denotes the derivative of the function with respect to that argument. The envelope conditions for this problem are:

JI() = (1ρ)μ + ρE[JI()]2ωϕ1(IId)[(1ρ) + ρE[JK()]],(55)
Jx() = ρ(E[JI()]pE[JK()]),(56)
Jμ() = (1ρ)IδρE[JI()] + ρE[Jμ()] + ρ(δpqout)E[JK()],(57)
JK() = (1ρ) + ρE[JK()].(58)

Based on the envelope conditions, it is conjectured that the value function takes on the functional form:

J˜(I,x,μ,K) = A0 + μI + K + A1(IId)2 + A2x(IId) + A3x2.(59)

Using the conjecture, and the evolution equations, taking the derivatives with respect to I and K updating:

E[J˜I()] = E[μ+2A1(IId)+A2x] = μ+2A1(IId),(60)
E[J˜I()] = E[1] = 1.(61)

Plugging equations (60) and (61) into equation (53) yields the optimal outgoing quantity:

Atα(IId) = qout.(62)

Substituting equation (62) into equation (54) for qout yields the pricing equation:

p = μ + A1(IId) + x2δ.(63)

Taking the evolution equation for inventory, equation (48), one can substitute equation (63) in for p and solve for I’ to get:

I = I + β(IId)(1+β)2x,(64)


β = (A1(1 + δα)αA1(1 + δα))A1 = (βα(1 + β)(1+δα)).(65)

Given the inventory evolution of equation (64), one can solve for the optimal pricing policy function, recognizing that relationship in equation (65) simplifies the implicit function of A1 multiplied by (1 + β) to A1(1 + β) = β(α/1 + δα)), and substituting:

p = μ + β(α/(1 + δα))(IId) + (1+δα(1β)2δ(1 + δα))x.(66)

Taking first differences of equation (66), and substituting in:

Δp = Δμβ(α/(1 + δα))qjt1 + β(α/(1 + δα))(qt1out + γt1)+(1+δα(1β)2δ(1+δα))Δx.(67)

Substituting the relationship for the updating of the μt given by equation (45) yields:


Next the conjectured functional form of equation (59) is confirmed. Begin by taking the envelope condition for x, equation (56), and solve for coefficients A2 and A3 of the conjectured functional form’s derivative, which is:


Substituting the optimal policy functions into equation (56), as well as the updated derivatives of the conjectured functional form which are given by equation (60) and (61) yields: A2 = –ρA1(1 + β), and A3 = ρ(1 + δA1(1 + β))/4δ. Continuing, the envelope condition on I in equation (59) can be solved with the conjectured functional form’s derivative, which is given in equation (60). This yields A1 = [(–ωϕ1/1–ρ(1 + β))]. An economically sensible solution requires A1<0, hence, using the definition for A1, it is required that:


This implies β∊(–1,0). As β → –1, the right-hand-side of equation (70) goes to negative one. As β→0, the right-hand-side of equation (70) is positive. Hence, as equation (70) is a continuous function, by the mean value theorem ∃β∊(–1,0). Therefore, equation (70) holds.

The Informed Trader’s Problem

This section shows that the conjectured behavior of the informed trader is optimal given the dealer’s optimal solution for price setting. This proof adapts the Madhavan and Smidt (1993) proof that conditions exist such that any deviation from the conjectured result would be suboptimal. The informed maximizes her terminal wealth after observing the liquidation value of the asset, and facing the same stochastic probability of a trading event occurring as the dealer of the previous section. Hence, prior to trading at time t, the informed faces a probability (1–ρ) of no trade occurring, in which she keeps her expected wealth, vtBt + Ct, with B and C representing the endowments of risky asset and capital, respectively. In the alternative, the informed trades, and updates her stocks to Bt + Qt and Ct–ptQt, respectively. We show that for Δ different from zero, Qt = δ(vt–pt) + Δt is suboptimal. The informed observes the dealer’s price, which is a function of her order through its effect on the dealer’s inventory and information. Taking the information effect first, using wt = δ–1Dt, the dealer’s signal, with Dt = δ(vt–pt) + Xt + Dt + δpt, the introduction of a nonzero deviation yields a distorted signal, wt’ = wt’ + δ–1Δt. This, in turn, yields price as an increasing function of the deviation:


where pt would be the price prevailing if Δt = 0 held. As in the case of the dealer, denote by V(vt, pt, Bt, Ct) the maximum expected wealth given the state, represented by the price, asset liquidation value, and the capital and inventory stocks. The informed trader chooses the optimal quantity for the order, which, by construction, allows the problem to be expressed as:


With transitional equations, E[vt + 1] = vt, Bt + 1 = Bt + Qt, and Ct + 1 = Ct + PtQt, with Pt = pt + λΔt, and Qt = δ(vt–pt) + Δt. Turning to the transitional equation for the notational base price, note that the price next trade depends on the trader’s current quantity through information and inventory effects, which in turn in a function of Δt. Hence, we can restate the dealer’s solution consistent with equation (66) as:


where, Et + 1t)] = μtt) by iterated expectations, and from equation (40), μtt) = μt + λΔt), which implies that the dealer’s expectations of the liquidation value are adjusted by the nonzero Δt “excess” trade if the informed deviates. From equation (66), we can rewrite the expectation of μt as μt = pt–ζ1(ItId)–ζ2xt. Using the expression derived for μtt) and equation (73), we have that price evolves by pt + 1 = pt + λΔt + ζ1(It + 1It + ζ2(xt + 1xt). Taking expectations, and using equation (13) for qout, we have:


Here, we can assume without loss of generality that the informed trader does not have a priori knowledge about our dealer’s inventory levels.25 However, this equation shows the full impact of a deviation affects the future price both through changes in the dealer’s expectation, and through her inventory pressure. Note that in the event that the marginal cost of trading out to other dealers is zero (that is α = 0), only the information channel is relevant, as the inventory adjustment is complete, illustrating the dichotomy between multiple instruments in this approach and canonical models. Omitting time subscripts, using superscripts to denote one-period ahead, the first-order condition for equation (72) is:


Taking the envelope conditions:


These suggest a conjectured functional form for the value function of:


with derivatives, v = B + 2A(v–p), p = –2A(v–p), B = v, and C = 1.

The transitional equations yield E(v’–p’) = (v–p)(1–(α/A1–α)ζ1δ)–Δ((α/A1–α)ζ1 + λ, which we can substitute into the first-order condition, and set the deviation to zero, which in turn gives a condition for A:


Taking the envelope condition for v, and substituting in the expected values with the use of the evolution equations, a second condition is imposed on A:


Note that equations (81) and (82) are analogous to the restricted case presented in Madhavan and Smidt (1993), where differences will appear in both the wedge associated with the inventory adjustment due to qout, in this case, (α/A1–α), and the scaling of the updating coefficient, Ω by the elapsed time fraction. As indicated in the Introduction section, the model presented would yield the informed trader of Madhavan and Smidt (1993) if the aforementioned effects are restricted away.

Since δλ = Ωη, we can express the conditions imposed by equations (81) and (82) as finding a δϵ(0, ∞) such that the function below satisfies:


Equation (83) represents a continuous function in δ, directly and indirectly through both Ω and β. We can express δζ1 = δA1(1 + β) = β(δα)/(1 + δα), and it is straightforward to show that as δ → 0, Ω → 0, and δζ1→0, and we can express (α/A1–α)ζ1δ = –δαβ/(1 + δα(1 + β)). Hence, as δ → 0, equation (83) is positive, and converges to ½ρ>0. Moreover, as δ → ∞, Ω→1, β → –1, and δζ1 → –1, and (α/A1–α)ζ1δ → ∞. Applying L’Hopital’s rule, it can be shown that equation (83) becomes negative, and hence, by the mean value theorem, ∃δ∊(0, ∞) equation (83) holds.


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Rafael Romeu is an economist with the IMF Western Hemisphere Department. The author is grateful to Roger Betancourt, Michael Binder, Robert Flood, Richard Lyons, Carol Ostler, Carmen Reinhart, and John Shea.


Lyons (2001) gives a broad treatment of these differences. In discussing inventory control, O’Hara (1995) also singles out foreign exchange dealers’ ability to lay off orders on one another.


More generally, see O’Hara (1995) on the empirical difficulties of predicted inventory effects.


A pip is the smallest price increment in a currency. The value depends on the currency pair. The data used here are dollar/deutsche mark, so a pip is DM 0.0001.


This amount observed concords with studies of intervention; for example. Evans and Lyons (2002) estimate 5 pips and Dominguez and Frankel (1993a) estimate 8 pips per $100 million.


Asymmetric information effects in equity markets are found by Froot, O’Connell, and Seasholes (2001); and Froot and Ramadorai (2001). Examples in foreign exchange markets include Evans and Lyons (2002); Froot and Ramadorai (2002); and Rime (2000). Examples in bond markets include Massa and Simonov (2003).


Madhavan and Sofianos (1998) find that New York Stock Exchange specialists engage in selectively trading to balance inventory.


An extensive description of the foreign exchange market’s institutional make-up can be found in Lyons (2001). Foreign exchange is traded bilaterally, over-the-counter, and privately, via computer emailing systems called Reuters Dealing. There are also electronic brokers similar to bulletin boards, provided by Reuters or EBS. Most large trades are done via the Reuters Dealing system, and the spread is fixed by convention.


One might argue that as Δτ→0, the dealer has less time to carry out planned transactions, but she can always elect to not answer the incoming calls until the part of planned transactions she wants done are satisfied. Furthermore, the increasing frequency of incoming calls and shortening of inter-transaction time would itself be a source of new information for the dealer, as suggested by Easley and O’Hara (1992). Indeed, Lyons (1995) finds evidence supporting that longer inter-transaction clock times increases the informativeness of incoming order flow, as interpreted in this study.


For example, the Evans and Lyons (2002) assumes that dealers submit bids simultaneously and transparently, which in equilibrium implies that prices be based on common information only. This paper avoids such restrictions because the focus is on interdealer price dynamics, but this comes at the expense of the market-wide price determination of such models.


The data are for the dollar/deutsche mark market from August 3–7, 1992. See Lyons (1995) for an extensive exposition of this data set. The transaction clock time measurement errors show up when the sequential order of the trades is not consistent with the clock-times, for example trade 2 cannot have occurred earlier than trade 1.


Lyons (1995) finds evidence that observed outgoing bilateral interdealer trades and brokered dealer trading are used to control inventory in the context of a canonical dealer-pricing model. These do not include a small amount of brokered trading (which occurs at 5 percent of the sample), which the dealer also engages in.


For example, Hasbrouck (1991a); and Madhavan and Smidt (1993) use the unexpected component of incoming order flow, and estimate this measure as a residual of a vector autoregression. In the case of the foreign exchange data used here, these autoregressions tend to have little explanatory power, making the residual almost identical to the incoming order flow.


Estimating the model with independent information coefficients on incoming order flow and gamma is possible, and support the restriction imposed here. However, under such estimations some inventory terms cannot be grouped as presented here, and collinearity prevents satisfactory estimations of the inventory effect, hence these estimable forms are not used.


Some measurement error in the time stamps leads to the inclusion of a dummy interacted with the absolute value of the clock time (which turns out to be insignificant).


Note that Romeu (2005) documents evidence of model misspecification and structural breaks present in these estimates of the canonical dealer-pricing model used here for comparison.


A Wald test fails to reject equality of the mean to the target with a p-value of 0.94.


Lyons (1996) describes this dealer as a “liquidity machine” in reference to the interdealer market.


For example, this pricing relationship forms the basis of Madhavan and Smidt (1991) or Lyons (1995).


Quoting the Wall Street Journal, August 10, 1992: “The Federal Reserve Bank of New York moved to support the U.S. currency as the dollar traded at 1.4720.” This is the most precise documentation available of the intervention start, and that price corresponds to 12:32 pm. Other times are selected because of reports of a mid-day start (hence, 12:02 pm), and at 12:26 pm the price jumps 36 pips, suggesting a possible intervention start at that point.


Note that this is a one-period-ahead conditional distribution, as the unconditional distribution would have a time-varying variance.


This assumption simply eases the exposition of the problem at hand, and keeps it in a discrete time framework. As discussed, γ has a time-varying variance. This complicates calculating the variance of the portfolio—this would involve moving the entire model to a continuous time framework. Because of the discrete-time arrival process of incoming calls, this would make for a cumbersome solution with little added payoff in relation to the problem of how dealers set prices on incoming orders. It would not, however, change the model’s conclusions regarding price setting with multiple instruments.


This is consistent with other inventory models and evidence from financial markets. In the alternative, it is straightforward to show that including a nonzero expectation on either of the last two terms in equation (74) leaves the pricing equation unaltered.

IMF Staff Papers, Volume 57, No. 2
Author: International Monetary Fund. Research Dept.