No.65 Fereign ExchaRge Market Maker's Optin}al Spread with Heterogeneous Expectations
Ryesuke Wada
June 2000
Department of Econoinics
Otaru University of Commerce
Foreign Exchange Market Maker's Optimal Spread with Heteregeneous Expectations
April 1,2000
Name:
Address
Phone:
Fax:
e‑mail:
Ryosuke Wada
: Otaru University of Commerce Dept of Economics
3‑5‑21 Midori, etaru Hokkaide, 047‑8501 Jap an
81‑134･‑23‑5109 81‑134‑27‑5213
[email protected]‑uc.ac.jp
Foreign Exchange Market Makets Optimal Spread with Heterogeneous Expectations
I. lntroduction
[[Erading volume and bid/ask spread have not shown any systematic relationships in foreign exchange market. The spread dees not change iR a way that existing models predict from the trading volume. Our papey aims at constructing a model with new analytical tools and using it to explain tihe varying correlation betweeit the volume and the spread. Heteyogeneity o￡ expectations is a key element. We characterize it by a distribution functien of a specific peint on expected time path on which actions hinge, We formulate a market maker's optimization problem and derive his optimal spread. WkD show this optimal bidlask spread increases, ceteris paribus, as the expectations become moye heterogeneous among the FX dealers. The market maker widens his spread, because by doing so he can exploit the higher degree of the expectation's heterogeneity. Attributing the change in the spread to the heterogeneity of expectations contrasts our model to the existing literature, According to its models, the larger spread is to protect open position from higher risks of volatke periods,
It can be shown that, as the expectations become more heterogeneous the volaeility increases as well,Wada(2000), Hence the spread and the volatility will show a positive correlation. On the other hand, the spread and the volurne will not show a clear correlation. This lack of correlation is obtained by identifYing sources of the trading volume. The trading volume is determined by the fundamentals' daily demand and supply as well as by frequencies of expectation renewals. These two elements can vary daily independently and their e￡Et}cts may be amplified or canceled out depending on the day. The correlation becomes obscure between the volume and the spread.
We apply our model on seemingly incoherent relationships between the spread
and the trading volume as are reported in Bollerslev and Domowitz(1993). The
existing literature, Admati and ?fieiderer (1988) and Subrahmanyam (1991), for
example, provide either positive or negative corre}ation. Therefore their models can not exp}ain the observed pattern of changing correlation, If we find effects of the heterogeneous expectations on both of the spread and the volume, the changing correlation is not puzz}in'
g.
In the following, we explain such seemingly incoherent changing patterns using our model in section II, A construction of the mayket maker's expected profit maximization problem and derivation of its solution are peresented in Appendix 1 and 2. The optimal solution takes a form of an optimal spyead. This optimal spread is larger as FX dealers' expectations become more heterogeneous. This is the key relationship to explain the empiyical observations.
II. lmplications of the Model
A. Majer Results en Empirical Observations
Bo}}erslev and Domowitz (1993) report lack of systematic relationship between spread and "market activity" in FX market, The market activity is meant to be a renewal of quotes. Its numbeT of times cou}d be a proxy fbr the trading volume.
Their findlngs indicate firstly that the spread is }arger in a period before }unchtime than in a period that follows even if these two periods may be similarly active.
Secondly if we compare lunchtime and late afternoon, lunchtime has fewer activities and much larger spread. Thus the spread and the volume do not show any systematic relation$hips erapirically,
'
The heterogeneity of expectations is the key element to reconcile the above observations, First}y our paper identifies banks' retail transactions as one of the sources ef uncertainty in inter‑bank transactions. Since the dealers adjust their positions eventually after they have xetail transactions, these retaii transactions give rise to the inter‑bank transactions. The retail transactiens mat￡er. So the dealers try to estimate an intraday pattem of the aggregate retail tTansactions.
Their estimates will be heterogeneous. A degree of the heterogeneity varies
depending on time and date. Their estirnates aye more heterogeneous in the
moming when banks' retail transactions aye taking place than the afternoon when
the xetail transactions are finishing. Hence, the expec￡ations are more
heterogeneous and the spread is larger in the moming, even though the moming and the afternoon may have the similar trading volumes,
In inactive periods such as lunehtime, weekends and days before holidays, competitions among market makers decxease. Many of them voluntarily drop out from the auction process. At the same time, the expected number of retail transactions and hence that of the inter‑bank transactions also decrease. If we compare the above inactive periods, then the lunchtirne is followed by the most uncertain period, The expectations are the most heterogeneous among those periods, Hence, the lunchtime has the largest spread.
B. Model's Characteristics
A process of price formation in the foreign exchange market poses a couple of theoretical difficulties which existing }iterature has not provided satisfaceory mode}s. One of the difficulties is continuous auction. In a continuous auction there
,
is not a spechic length of time to define demand and supply and it is not clear which priee is equalibyium among the series of realized trartsaction prices, It is hard to apply usual equalibrium analysis on the continuous auctioxx. As a tool to formulate the continuous auction as in Garman (1976) and Amihud and Mendelson (1980), our model defineS expected numbers of buyers and sellers per unit time in stead of demand and supply. In the continuous auction, it is not obvious how to compare heterogeneous expectations, There is not a specific point of time to do it.We identify the furst peak or bottom on the expected time path as a key value to compare heterogeneous expectations, The key value is the first local extremum on the expected time path. VLle characterize the heterogeneity of the expectations by a distribution function for this expected extxemum. Taking advantage of as many such extremums as possible increases the expected profu. Dealex's action to seek capital gain hinges on it. Therefore the distribution function describes distributions of actions,
Participants in a FX market are dealers. Some of them act as market makers.
For given values for bid and ask, sellers and buyersrandomly arrive at the market
maker and trade with him. These sellers and buyers are other dealers, Their
expected numbers of arrivals are determined by the distribution of the reservation
pxices and the competitions arnong the market makers. Each time an arrival occurs,
the market maker's position changes. Revenue or cost is incurred by the aryival.
The market maker's position follows a continuous tme Markev process. The levels of his position constitute "states". The arrival causes a transition between the states, The transition brings about ̀la reward'1 The market maker is not always passive in that he can ininate transactions with other market makers to adjust his position.
Therefore the process is controlled. This process is mode}ed as a continuous time coRtrolled Markov process with reward as in Yltshikevich(1977). The arrival process of se}lers and buyers is characterized by an "infuiitesimal matrix", which is a continuous time eountexpart to a Markov transition probabllity matrix. Elements in the infinitesimal matrix are the expected numbersof arrivals for given value$ ofbid and ask. Values of these elements aye determined by the distribution of the reservation prices and by the number of competitors. He ￡ries to maximize the expected daily profits by choosing the quotes and hence the spread. The model shows that this optimal spread increases, ceteyis paribus, as the expectations become more heterogeneous.
Garman (1976), Amihud and Mendelson (1980) and our model share the model specifications such as price sensitive randoin arrivals. We improve these preceding models by identifying sources of the price sensitive yandom arrivals,
Intra‑day fiow of the excess demand fbr FX by the economy's fundamentals wM ne￡ be balanced. The dealers as a whole adjust net total positiexxs and absorb ￡he excess demand. The transaction price makes the excess demandjust being absorbed, The dealers try to figure out patterns of the excess demand and to estimate forthcoming peaks and bottoms of the transaction prices. The heterogeneity of expectation comes firom diffexences in opinions with regard to the arMival patterns of retail transactions and the distribution of the expected peaks and bottoms.
C. Characteristics of Optimal Spread
1. IRteRsity Curves of Buyers and Sellers
In Appendix 1, ingredients of the maxket maker's expected profu maximization are
discussed, A nd in Appendix 2, flrom the arriva} intensity buyeTs and sel}ers as functiens
of the spread, the necessary condition of the optimal spread is derived, Wb discuss the
implications of the optimal spread.
seller's intensity seller's intensity
rnereheterogeneous .rt expeetatlen X/t/
/ / l‑
i/
pnce prlee
benchmadirk benchmark
velue value
Figurel:Seller'slntensityCurve Figure2:EffectofE{eterogeneousExpectations
seller's intensity
‑"‑‑‑‑‑‑‑‑‑‑‑‑higher aggregate arrivals
original level L' /
Jtt
r' ef x
Figure3:
pnce benehmark
velue
Effbet of Higher Aggregate Airriyals
The arriva} intensity is the expected number of arrivals for a given value of quoted price BeRchmark value is a weighted average of all the indications and the yepoxted last transaction prices. Dealers use the benchmark value as an index to signify the current transaction price, When one of the dealers contacts the market maker and ask fer prices for an immediate exeeution, probabllity that the dea}er accept quoted price depends on closeness to the behchmaxk value. Closer to the benchmark value, higher the probability that a transaction takesplace, hrrival intensity is higher (Figuerel), For a given c}oseness, if the dealers' expectations are more heterogeneous, t･hen the maxket maker sees higher probability of realizing transactions. So the higher deg]ree of expectation heterogeneity results in higher intensity. However, the highest illtensity is unchanged, if the total arrivals are unchanged (Figure 2), Then if the aggregate arrivals increase while the heterogeneity ofexpectations stay the same, then aryival intensity for the market makeur increase. Hewever the least competitive price to have arrivals stay the same (Figure 3). The change in a degree of competitions among the market raakers has the same effect as Figure 3. Even if the aggregate aryivals ever the entire market stay the same, the po￡ential arrivals to an market raaker who is still ready to trade.
The aggregate arrivals increase as the frequency of dealers' expectation renewals
increases. And also if the retail arrivals iRcrease, the aggregate arrivals increase. The
market maker faces such buyer's and seller's intensity curves. He tries to maximize
expected daily profits. For simplicity, we assume that the market maker chooses a pair
ofbuying and selling prices with the same arxiva} intensities(Figure 4)
seller's v intensity
inten$ity
buyer's lntensity
l
price
intensity
A
ny 7
benchmdirk
buying pnee
vdilue sellingprice
ffu)
. .B l/
l
o C haif gf spraed u
Figure 4 Intensity Curves and Choice of spread Figure 5 Determination of Optimal Spread
The higbest intensities of buyer's and seller's curve need not be the same, If we assume
￡he same retail arrivals, then the market maker's problem becomes choosing the pyofu maximizing Spxead.
2. Necessary Cendition for Optimal Spread
As benchmark value moves, the intensity curves shift for a given value of quoted price.
The market maker revises his buying and selling prices accordingly. We assume that the market maker chooses a price pair with the same aryival intensities, The solution of the expected profit maximization problem becomes choosing the optimal spread,.Let u be a half of the market maker's bid/ask spread. Then we expxess the arrival intensity as a function of u, Let f(u) be that function asigure 5). Using this function, the expected profit maximization is solved. The solution is derived in Appendix 2. The necessary condition for the optimal spxead is given by u f + f = O. This xesult is obtained for an asymptotic case such as time goes to infuiity, This is not urweasonable situation foy the FX market. It is not unusual that several ￡yansac￡ions take palace in one minute, If we consider second as unit of time, then it is reasonable to apply the asymptotic result on the intra‑day auction process. The neeessary condition for the asymptotic ease coincides that for the static case. Since the value of f(u) is an expected number of arrivals, rectangle ABCO in Figure 5 is the expected profit for an infinitesimal amount of time.
The necessary condition tells that choose the value of u such that the area ofABCO will
be maximum,
3.Comparative Statics of the Optmal Spread
effect of heterogeneity of expectation: A slope of f(u) becomes less steep. The optimal value of u becomes larger. Hence the optimal spread is larger.
e￡￡ect of aggregate arrivals: The slope of ￡(vt) becomes steepex while an intercept w!th
the horizontal axis is fixed, The optimal u increase and hence the spread increases, The aggregate arrivals are generated by two sources. The first is yenewals of dealers' expectations. The second is retail transactions with custorners. The second souxce is demand and supply from the macro economy. Arrivals from the first seurce are called
"heterogeneity arrivals" and the second "yetail arxivals",
4. Application on Intra‑Day Pattem of Spread (1) Morning and Aftemoon
The retail arrivals are supposed to show cleay intra‑day pattems. Their number is large in the morning toward noon lunchtime. Then there is a pause in the lunchtime. After the luuchtime, it surges again and tapers off toward the end of business hours.
The random arrivals create fluctuation of tsraitsactioxx prices, The larger expec￡ed number of arrivals is associated with larger varianee. (I]his is one of the mathematical characteTistics of Poisson process. We assume Poisson process for the arrival processes.) Hence, the txading volume and vo}atihry has a positive correlation. Since the larger fiuctuation is anticipated, a degree of expectation heterogeneity is higher. Hence, the optimal spread is larger for larger trading volume. Even if the trading volumes are same befoxe and after the }unchtime, a degree of the expectation heterogeneity may be different. In the aftemoon the xetail transactions come in a more predictable way because of communications between the customers and dealers. The heterogeneity is lower in the aftemoon. As a result, even if the trading volumes are the same, the optimal spread is sma}ler in the afternoon.
(2) Lunchtime
At lunchtime, many dealers voluntanly drop out from the auetion process. Retail transactions become lew. If an market maker is still ready to trade, buyer's and seller's intensity curves for him shift as fo11ows, They shift up in a manner of Figrtre3 due to the decreased level of competition. The optirnal spread is wider. Then the intensity curves shift down due to decrease in retail arrival$. The spread is narrowed. If the effect of decreased competitions is 1arger, then the optimal spyead is wider.
D. Conclusions
Auctions in FX market are continuous time. [[E:ansactions take place asynchronically,
Equalibrium analysis is difficult to apply Wk) introduce expected nu' mber of arrivals of
buyers and sel}eTs for an infinitesimal arnount o￡ time, These expected numbers are
called arrival intensities. The arrival intensities play a yole of demand and supply in the continuous auction. We construct "buyey's and se}ler's intensity curves", The maxket maker faces these curves, For gtven arrival curves, he tries to ll!aximize his expected pxofit. The expected profits are maximized with respect to bid/ask spread. We obtain the necessary condinon for the optima} spread.
The optimal spread incyeases as the dealers' expectations become more
heterogeneous. Even if the trading volumes are similar, the optimal spyead will be
larger if the degree of the heterogeneity is higher. If the degxee of competkion
among the market makers is lower, the optimal spyead is larger. Using these
implications of the model, we can resolve the seemingly incoherent changes of
correlation between the trading volume and the spyead,
Appendix 1. Con$truction of the Optiraization Problern
A. EnvironB3entsoftheMarket
1. [l]wo Groups of Dealers
Price" means spot foreign exchange rate. Auction participants are foreign exchange dea}ers. There are two groups of dealers. The first group consists of market makeys.
They quote their own prices alld stand ready to trade at them. The second group consists of those who do not quo￡e own prices. [[Eransactions take place between the first and the second groups or within ￡he fust group. All the dealers can assume epen position. Sizes of the open positions are subject to exogenouly imposed constraints. The market makers are allowed to assume }arger open positions. The constraints aye two transaction units fox the market makers and one unit for other dea}ers, We call those in the second group with stagle unit constraint "S‑dealer", when distinction is necessary fbr clarity. The FX market is wholesale maxket. All of the dealers have re￡ail customers and trade with them as well. All the dealexs are risk neutral. If S‑dealers erecognize possibility of capital gain, his position must be one unit open. Contrary the market makers' positions do not necessari}y refiect theix view. Their positions are exposed random fiuctuations in order to seek profits from bidlask spread.
2. Contmuous Auctions
Auction process goes on continuous time, A market makex quotes a pair of his buying and selling prices, wheneveer other dealers ask fbr quotations, These quoted prices are fox immediate executions. If one ofthe prices is good enough to the inquiring dealers, then transactions take place. [[}ransactions occur asynehronously. When a transaction takes place, we say "an arrival of buyer or seller" occurs and count one arrival, The FX market opens in the morning and ends late afternoon. The dealers' daily pxofits are evaluated at the end of business hours, When retail customers sleep, our market is
closed.
B. Ingredients for Market Maker̀s Optimization Problem
1.Choice va]riables
The rnarket maker acts to maximize his expected daily profits. His choice variables
are a pair of buying and selling prices, and his position. His expected profit
maximization problem can be viewed as an inventory control problem. The size of the
open position corresponds to an inventory. He can choose only either a price pair or inventory level. He can control either the prices or the inventory at one time,
He gives out the pair of buying and selling prices together when inquired, This is the practice called "two way quotation" and a rule of the market. The inquiring dealer decides which side to trade. The market maker could differentiate competitiveness of his prices so that a particular side is more likely to be chosen. Still he does not have complete control over his own position.Itis so unless he quotes absurd prices. Thus if he chooses prices then he cannot control his inventory If he wants to adjust inventory right away as he wishes, he can do so by trading with other market maker. In that case, the price applied is someone e}sès. Thus if he chooses his inventory, he cannot control the price.
2. A Little Lagged Inventory Control
The market maker's position is subject to a constraint. Meanwhile, when he has an arrival, a constraint may be violated. In order to reconcile such inconsistent situations, we distinguish "a desired " value and an actual value. And we allow the actual value to be diffbrent from the desired value momentarily. The constraint is deimed on the desired level of inventory. The mayket makey chooses actions so that the desired position satisfies the con$traint. If the constraint is violated due to the random ayrivals, he immediate}y trades with other market makers. Then the constraint becomes just binding.
3. Buyer's and Seller's intensity Curves
For a given value of quoted price, an expected number of aryivals ofbuyeys or sellers can be defined. The expected number of arrivals foy an infinitesimal amount of time is called
"arrival intensity". The arrival intensity is price sensitive, in a very much the same way as demand and supply curves. We call relationships between the price and the arrival intensity "an intensity curve", We have "buyer's intensity curve" and "seller's intensity curve". The market maker daces two intensity curves and chooses actions.
C. Generation of Arrivals of buyers and sellers
1.Inside and Outside Sources of Arrivals
It was defined that an a:rival of buyer or sel}er" implies that a transaction take place
with a matket maker. [Iknro $ources generate these arrivals, one within the market and
the other outside, 'l]he fust seurce is dealers̀ revises of heterogeneous expectatiens. The
second source is retail transactions. The second source is, in another words, demand
and supply from the macro economy.
2.Generat ieRs ofArrivals by Expectation Renewals
All the dea}ers have heterogeneous expectations aitd revise them from time to time, S‑
dealers assume open positions according to their expectations. When they revise expectations, they try to adjust their positions accordingly. Thus the expectation renewals generate arrivals.
The diffbrence in expectations can be expressed as the difference in the fust peak or bottom in an expected time path of transaction prices, Possible capital gain hinges on such extemums. Becorning a buyer er seller, depends on them. The di￡ference with this regard results in the diffbrence in actions, The hetexogeneity of expectations takes a form of distribution of such extremums. Wb call a distribution function for it
"heterogeneity distribution". We call aryivals generated by the expectation renewals
"heterogeneity arrivals". For a given quoted pair of prices, the heterogeneity distribution deteermines probability that the next heterogeneity arxival is a buyex or seller, The total numbers of the h.eterogeneity aryivals aye generated by the expectation renewals, These arrivals are sorted into buyers and selleers according to the heteyogeneity distribution,
4. Arrival Generatien by Retail [thra"sactions
The dealers engage in retail tsransactions with their customers as well as who}esale transaction in the market, Dealers are ready to ￡rade during the business hours whenever custoraer$ want. MeanwhiIe S‑dealers must have constructed the desired levels of positions according to their own expectations, The randomly arriving retail transactions disrupt the already constyucted positions, Had this occurred, the dealers would counterbalance retai} transactions to recover the desired positions. They trade in the market, Thus a sequence of the retail transactions changes !nto a sequence of the arrivals in the market, We call the aryivals generated by the retail transactions "retail arrivals",
5.Priee Insensinve Aggregate Retaik Axrivals
Arrival process is defined for the entire market and for the ipdividual market makers.
Wl) call retail aTrivals aggxegated over the en￡ire maTket "aggxegate xetail axrivals". It is assumed that ￡he dealers̀ customers do not respond to intra‑day price movements.
Hence the aggregate retail arrivals are price insensitive. Since the intra‑day price
movements do not infiuence retail customers, daily demand and supply from them would not be exactly equal, except for by chance. Therefore the daily aceumulative numbers of the buyers and the sellexs would not be equal. The dealers absorb the difference. The agg]regate of the entire dealers̀ position would be open ovemight.
D. Constructing lntensity Curves
1.Poisson Processes
We assume that the retail arrivals and the heterogeneity arrivals constitute Poisson processes. It means that the number of arrivals follows Poisson distribution, that its expected number fox a given time interval is propoxtionate to a leBgth of the interval and that its variance is equal to the expected value. The buyer's and sellex's retail arrivals constitute distinct two Poisson processes. The aggregate retail arrivals are prlce msensltlve,
The heterogeneity arrivals, buyers and sellers combined together, constitute a unique Poisson pxocess. We call these heterogeneity arxivals aggregated ovey the entue market "aggregate heterogeneity arriva}s". The aggregate heterogeneity aMriyals are
also price insensitive. This insensitlveness comes from that the frequency of the expectation renewals, not a price }eve}, determines the number of the aggregate heterogeneity arrivals. Thus, there are three Poisson pxocesses for the aggregate aryivals and none of therR are infiuenced by the price movements.
2. Price Sensinve Process for Individual Market Makers
A fraction of the aggregate arrival processes yeaches individual market makers. More precisely speaking, it is a fraction of a sum of Poisson processes, The flrraction of the corabined aggregate processes appears as two arrival processes of buyers and $ellers.
The fraction changes as the quoted price changes. Due to competitions, the arMival processes for a given market maker become price sensitive.
3. Price Search
The market makers always post "indications", These indicat!ons are intended to be
approximate values. An exact price applied in each transaction is known only when the
dealers inquitie the maxket makers about them. Besides the indications are net upda￡ed
continuously. Even if one dealer tries an extensive search, some of the quotations may
change befbre he cempletes the search. Thus, it is not clear what is the best price
available at a giv' en moment. The dealers shop around but complete. They accept prices randomly, The probability of the how good the price is compare with some benchmark value.
their searches are not acceptance depends on
4. Benchmark Vaiue
The dealers always rnonitour the market makexs̀ indications. And information vendors distribute real time transaction prices to the dealers. These reported ptices come from multiple sources and are anonymous. We assume that dealers use a weighted average of the indications and the last yeported prices as a benchmark value. It is weighted aveurage of pair pxices of the all the indications and the last repoyted transaction prices.
The weights are up to the individual dealers. The benchnark value signifies the current price to the dealers, The market makers use ￡he benchmayk value to set own prices. All the dealers use benchmark value to calculate retail prices, The retail prices are benchmark value plus or minus fixed margin.
5. Airrival Intensity Curves
The market makers have prices for immediate execution. These prices can change any moment. Hence they are not necessayily same as their indications. It is not clear what are the best prices in the market ￡or a given moment. So dealers use the benchmark value as a substitute for them. When dea}exs search fox the best price for the immediate execueion, they accept the market maker̀s price if they judge it is close enough to the benchmark value. For a given benchmayk value, it looks as if the inquiring dealers accept the price with some probability. This probability decxeases as the distance between the benchmatrk value and the market makex's price. This decreasing schedule itself becomes a relationship between the arrival intensity and the quoted price, "The buyer̀s intensity cuTve" is defined to the right of the benchmark value and is downward
sloping. "The seller's intensity curve" is defined to the left of the benchmark value and upward sloping,
D. Sirnplified lnventory Control Problem
1, Eq}}a} Intensity fbr Buyers aRd Selleys
We considex only rather simple case such that a chosen paix has the same arrival
intensity. It is assumed that the market maker chooses price pair with equal arrival
intensities. It means that when he has open position, he does not differentiate the
arrival intensities between buyers and sellers so that the position is more likely to move
in a particular direction. It is assumed that market makeys take other market makers̀ indieations as given. The indications are truly avai}able price pa!r when they are revised and quoted. The revisions of the indications occur with random intervals of time, Wb do not consider the cases such that the market makers manipulate own
indications so as to influence the dealeys' expectations.
2. Necessary ConditioB fbr OpeimEgity
The market maker shifts ayound his buying and selling prices whi}e keeping their
arriving intensities equal. Then the axriva} intensity curves are derived as fuBctions of
spread. Let u be the halfof the spread. And let q = f(u) be intensity curve as a function of
u. In the appendix the necessary condition for the expected pyofit maximization is
obtained,!tisgivenby f̀u+f=O.
Appendix 2. Derivation of Optimal Spread
1. Profits and Its Expected Vtilue
A market maker wants to maximize his expected profit for a time interval lo,T]. This maximization probleyfl can be formulated as follovLrs. Random variables chaBge values from time to time in contlnuous time. We use integrals to express traded quantities.
Zi'(t) and Z2'(t) are accumulative values that he bought and sold by time t. Hence dzi(t) and dZ2(t) are quaittSties traded ae each transaction at time t. S,' and S2" are buying and selling prices applied at time t. Using integration notation, J}i" Si'(t)dE[Zf(t) means total quantity bought for a time interval [e,T]. Similarly Jl3" S,'(t)dE[z,'(t)] means total quantlty sold during the iRterval [O,Tl. Z(t) is inventory level at time t. Posi￡ive value means long position. Wlr expresses the position's vafue at the end of the day. jE}, is information. Conditional expectatioR on profits for interval IO,T] is to be maximized. He can choose S,* aRd S2' and, when necessary, Z(t) is adjusted. He is risk neutral.
The stared variab}es are desired valvtes of the origlnal variables. It is alloxKTed that the actual values are different from the desired values temporarily. The market maker's posi￡ion is exposed raRdom shock of transactions v"rhile he has to satisfy the constraint on the position. Temporary discrepancy is allowed. Had it occurred, he trades with
other market maker and makes his position just binding.
2. 0mitting Retail 'IEraRsaction
The market maker has retail transactions too. Profits from the retail transactions
become imaplicit in the rnaxirnization problem. His retail prices altd wholesale prices are
always different by constant margiB. As the number of the retail traRsactions iRcreases,
his profits from the margin increase. This is exogenously given pyocess. Profits as market
maker is realized when he trades with other dealers.
e
,m,. e,s}E ‑El [.1[T ss (t) dzs (t) ‑ 1[T sr (t) dzr (t) ‑t‑ wz,( z(t) )l a,]
= gf,}. fl,sill{y[T ss (t) dE[z,* (t)j ‑ y[T sr (t) dE[zl* (,)]
+E[Wlr(Z(t))ilt,]}･ (1)
subjecttolZ*(t)is2, for OStST.
'
where VYIi･(Z(t)) is final reward a￡ time T, having Z(t) of position at time t.
3. Model Specifications
This is a model of a con￡rol}ed, continuous time Markov process vvrith finlte states.
This model is presented iR Yeshkevich (1977). Wk] look for an optimal policy amoRg
the time invariant policies. The position denoCed as Z(t) is the "state" variable. The arrival processes of buyers and sellers are Poisson processes. These random arrivals cause transitions ofthe state. As a traRsaction takes place, the statejumps from a given value to another. We introduce two simplifying assumptions.(A‑1) Each arrival's quantity is either one or two units. The transaction inc･urs cash fiow. Thejump ofthe state has an associated "reward". The position is subject to the constraint of 2 units. We consider slmple policy sgch that (A‑‑2) The market maker always chooses price pair with the same arrival inteRsities. It means that the next arrival is equally likely to be a buyer or a seller. We construct an "infinitesimal matrix ". This ls a continuous time counterpaTt of Markov transition rr}atrix. Let q be the infinitesimal matrix. The marke￡ maker chooses prices so ehat buyer's and seller's ayrival intensities are the same. Let q be that value of the inteRsity.
The quantity associated with each arrival is either oBe or tvtro units. We deBote
probabilities that the qttantity is one unit and two units by vi and v2 . Staee space is
{1,2,3,4,5}. Let Si deno￡e state i. [Irhese state are defined to correspond to the inventory
levels {2,1,O,‑1,‑2}. If buyer of one unit arrives, the inventory decreases by 1 and state
moves down by 1. Let Q be the infinitesimal matrix of the process. Using the notatioR defined above, Q is written as
‑1 vl v2 OO' i ‑2 vl v2 O
9:gAi!Eq v2 vi ‑2 vi v2 (2)
O v2 vl ‑2 1 O O v2 vl ‑1
The raw nllmbey is the state before the transitioll and the column numb' er is the state after that. The interpretatioB of the rr}atrix is that element ei,i+i means the expected number of seller's arrival who trades one unit vtThen the state is Si. rl]he element Q,･,j means expected nttmber of arrivals combining buyers aRd sellers together wheR the state is $. It is expected number of arrivals to force the market maker to leave the state $.
It has negative value. The probabiliey to stay at o' decreases by Qj,j for an infinitesimal amount of time. For 2' = 1,s, the elemeltt's vabue is ‑q. XVhen the state is at Si and Ss, the constraint is already has been just binding. If the position goes out the consLraint, he woul(l caBcel out excess quantity by cognterbalaRcing transactiens right away. So the states 1 has virtualiy only buyers and state s has oRly sellers. The value of the element is hal￡
4. 0mitting Expected Capital Gains
Since we consider the simple case such that the market maker chooses only the price pair with the same arrival intensities. If he "vLTants to maintain equal arrival intensities, he has to keep shifting around the price pair as the benehmark value, i.e. the cetrrent price level, fiuctuates. S,' and S,' have to shift arogBd. Then the market maker has capital gains and losses. We separately calculate pTofits from the bid/ask spread from the net capital gaiRs. Then it can be shown that we can eliminate the expected capital gains from the expected profu maximization pyoblem. If we separate the capital gains and bid/ask profits, the profits can be wyitten as follows.
￡￡{.,iPklu+fkCk (3)
where N is the number of transaction, Ck is the average of bid/ask when kth traksaction takes place and fk indexes whether it is purchase or saie. For simplicity, suppose each quantity is one itnit. Ik = 1 if it is sale and fk == ‑1 if 2t is purchase. gh is approximately
equal to the benchmark valtte. Since we consider the case the arrival iktensities of buyers and sellers are kept equal, lk =: 1 with probability i and hence EVlel = e. r]7he if we take expected valiie of equatioR (3), then it is given by uN + ￡",.,E[fhgk]. For each k, E[fkCkl= E[Ik]E[Ck] =O holds. The capital gains are expected to be zero, when the dealer maintaiRs equal arriva} inteRsities for buying and seliing prices and if he starts out with no inventory. For simplicity, we assume that his ini￡ial inventory is zero. Then we can elimiRate capita,I gains from the expected profit maximization pyoblem. As a result, finding a pair of S," and S2' is the same as finding the optimal bid‑ask spread
2u".
5. DiiEferential Equation of Reward
When the inventory increases by a seller's arrival, we call the transitioR of state
"upwardjump". Eachjump is associated with a reward. Ajump from Si to S2 means
one unit of sale. If the state is S2, the expected revvrard from the buyer's arrival is given by vi +2v2,. An upwardjurnp caused by selley's arrival incurs cost. We do not have to assign a negative reward to the upward jump, however. Except foT the last downward
jumps at the end of the day, the downward jump already signifies a profit, not just revenue. Let R be the s×1 vector and denote a vector of the expected rewards of
Jumps, l.e.,
vl + 2v2 vl ÷ 2v2
R=‑22Lq vi+2v2. (4)
1 o
Let the s×1 vecter W(t) deRote the expected profit for an time interval [t,T] where '
T is the end of the day. The ith element of W(t) is the expected profi￡ for the case that
at time t the state is Si. The expec￡ed profit satisfies the following differential equation.
We can calculate the expected pyofit as the solution ofthe following differential equation (Yushkevich 1977, Corollary 3.2 and Supplementary Remarks 5.),
Wt(t)=‑R‑eW(t) , (5)
with the boundary condition W(T)= Irplr. (4) is a,R example of Bellman's equation. The elements of R and Q are constant. Since the sum of the elernents in each rovtr of Q is zero, Q is singular. The solution of (s) is given by
'
Tiy(o)=(ygTesQds)R+wlr. (6)
wheye esQ i‑i: f+ sS/l + (S2Qi )2 + (S$/ )3 +...+ (Sff/)" +... and f is aB identity matrix. Since
Q is not invertible, the expression for Jl3"esQds cannot be simplified. J6i'esQds =T(f+
!ll9iL, + (T,C?., )2 + (T,(?, )3 +...+ (TQ,),h‑‑‑i +...) .
6. Intensity Curves as }i"uxxctions of Spread
The common arrival intensity of buyers aRd sellers is g. The value of the arrival intensity q depends on u. "le construct the intensity curve as fol}ows. WheR the dealers ask fbr market maker's quotations, ehey compare the quoted price with the beRchmark value. The benchmark value is a weighted average of all the indicatioits and reported last traRsaction prices. Therefore the arrival intensity monotonically decreases as the distance increases between the benchmark value and the quoted price. Then we can
derive a relationship betweeR this distance and the arrival inteRsity. We can do this
foy both buyers and sellers. We have two intensity curves. They are functions of the
distance defined as above. Next, fix the value ofthe intensity aBd, using intensity curves
find the values of distance. Take a sum of two values of distance. It is the bid/ask
spread. If yott divide it by 2, then the answer is u. Thek for a given va}ue of u, there is
unique value of arrival inteBsity. Thus we have aryival intensity curves as functions of
the spread or half of the spread u.
7. Necessary Condition for Optlmal Spread
As shown in (3), R is also a function of u. The expected proik is giveR by
equation (6) and this equation is maximized with respect to u. Next we find optimal spread. Let q = f(u) be an intensity curve as a fttnction of u, one half of the spread.
Substitute f(u) and (4) into (6) and differelttiate with yespect to u. Since ( JIY" esQds )R = T(qJ+g(T2q!A)+!g‑21¥i/91+...+q(TgkA.,)h‑'i+...)u2vL where viii ii.taR and 9=gA,the
derivative of the first teyra of equatioR (6) is given by till [ ygT esQdsR ]
,,, T{ fqt +(TqA )gt + (Tq2A.i )2g' + (Telll/ )3g' +...+ (T(qkA‑)hifd +... }.2v
+T{ +gi + q( T2q! A) + g( T3q!A )2 ‑i‑ ... + q( TgkA! )hrm' + ... }2v
== (TeTqg'u+q( y(]T esQds )}2vL (4‑13)
rl'he optimal value of u makes (7) equal to zero. TeTQ and Jgi" esQds become proportional to T, as T‑ oo. The p:oof is given later. Then for T= oo. equatioB (7) :oholds if
q'u +q =o. Thus the necessary condition for the optimal spread u' for the asymptotic case is giveR by
f'u+f == o. (8)
8. proof of asymptotic proportionality
ete is the probability distribgtion ofthe state iB which Z vgTould stay at time t, starting at time o from one of the states, The row gives the starting state and
the column gives the state at time t. etQ converges to a stationary distribution, as t ‑‑> oo. Let C be ￡he $tationary dis￡ribution which is associated with Q ,i.e., C ==
limt‑nv,..etQ.We can show that fJ3"esQds a}so coRverges to C. For a given E> e,, there exists T such that IletQ‑Cll <6 for t> T. H･II represents themaximum
of the absolute values of the elements in a ma￡rix. Using the facts that }JllesQds ==
fJUne esQds+lXesQds and fJges̀?ds‑C : lJll(es̀?‑C)ds, vTe }}ave limt‑m,..}Jll(eSQ‑C)ds ==
limt‑m>.. I Jg‑ (eSQ ‑ C)ds + limtm+.. e#(eSQ ‑ C)ds. Because limt‑.. Il} Jif (eSQ ‑ C)dsll =O and
limt‑,.. Il} Jl; (esQ ‑ C)dsl s limtrm,oo e Jl! II(esQ ‑ C)Eds g c, we have l} J3 esqds ‑ CII s E and
,ttmco ‑l y(ì eSQds =C :,ttm,., etQds (g)
Eequa}ity (g) shows TeTQ and Jl3" esQds become proportional to T, as T ‑ oo Divide (7)
by T, let T ‑‑> oo ,and in (g) replace t by T, then (7) becomes zero, if q'u +q =O.
Reference
Admati, A.R. and P. Pfleiderer, 1988, "A theory of intraday pattern: Vblume and price variability," Review of.l7Ynallcial Stuclies 1,3‑40
AJnihud, Ylikov and Mendelson, Haim, 1980 "Dealership Market, Market‑Making with Inventory," eloun2al of.l?rinancta1 Eleonomies 8, 31‑53
Bollersley Tim and Domowitz, Ian, 1993, "'Ilrading Pattems and ?rices in the Interbank Foreign Exchange Market," Zhe elburna7 ofKnanca Vbl,48,No4, 1421‑
1443
Garraan, Mark B, 1976 "Market Micyostructure," e7burnal af.EYnancial Eleonomaies 3, 257‑275
Subrahmanyan, A,, 1991, "Risk aversion, market liquidity and price efficiency"
Review of‑F]lrhoancial Studies 4 417‑441
'
A.A. Yushikevich, 1977, "Controlled Markov Models with Countable State Space and Continuous Time", 71he 71beay ofRrobabiZity andltsApplilrations, Vbl.22, No.2, 215‑235
Wkida, Ryosuke ,2000, "MicTostructure Model of 'IErading Vblume and Vblatility in
FX market", mimeo
τhis 韮）iscussion Paper Ser圭es is published by the Center 婁or Bus圭ness Creation（chan墓ed from the institute of Economic Research on Apri1 1999） a1｝d in七egrates two old ones published separately by the Depart紐ent of ￡cono田ics ar｝d the I）eparもmenも of Co田∬lerce．
Discusslon 菱）aper Series Institute of Eco員。醗ic Research Otaru University of Co田meree
｝｛o． Tiも1e
1．ホーキンズ＝サイモンの条件に関する諸説の統合について
2． ｝Iotivatio離 aτKi Causa｝ Infe罫ences in the 8udgetary Contro1
3．npo6．ηeMblynpaBJIe｝1｝｛月 pa6Qqe勇
c峯4渥。員 Ha r！pe双肩P縁∫…丁舅只x滋a冴7）｝｛epo
BOCTOKa（cOI4｝40」！orレIL｛eCK縁e acrleKTbl）
4． 1）yna．mic ？ax Inc三dence i籍 a Finite Horizon ｝lode1
5． Busi【｝ess Cycles with Asset Price ｝重onetary PO1重cy
6。 Contir｝uous Double−Sided Auet重ons
8ubbles and 七he 裂ole of
i轟 董〜Oreign ExchaBge 播arkets
7． ？he Existence of Ramsey 碧quilibrium w三も蓑 Coasu限萎）tion Externality
8． Noney， ｝leutrality of Consumption 了｝axes， and 6rowth in interte田pora1 0ptimiziR9 ｝≦◎dels
9． Product Returns i漁 もhe Japanese I）三sもribution Syste田；A Case Stu（玉y of a Japanese Wholesa18r s Return Reducもion Efforts 10． Dynamics， Consistent Con5ecもures a丑d He七eroge轟eous Agents in the Private Provisio丑 of Public （｝oods
11． 王nもra−industry Investment and I飢perfecも ｝歪apke七s A（｝eo田e七ric apProach三雛S三醗ple〔｝ene節a1莚qui1玉bri娃瓜 12． Sit−1）own 七〇 SplΣt：Flint 6｝i 碍orke三》s in 1937−1939
13． 7he Co田Ple田entarity be乞weerl Endogeno｛】s Σ｝Potecしion and Direct i／ore圭9n 玉nvesも醗ent
14． Consumpもion Taxaもion and Tax Pre茎）ayme1｝t aPproacむ in Dyna厩ic Gene「al equilibPiu獄 ｝fodels with Consu鐵ez Du1・ables
15． Eegulatory Syste田 and Superv三sio湾 of the F玉nancial Institu七io罫｝s in Japan
16． Financial Restructuring ar｝d the ｛｝。 S． Regulatory Fra凱ewor｝：
17． The Legacy of the Bubわ1e Economy in Japan：Peclining cross Sha把eholding and Capital Formation
18・ Stockoド難ersh重ρ i孕 the 蓼・ S・：Capita夏 Formaもio簸 a籍d Regulation 19． 1nterna．t，ional Joint Ventures and EndGger｝ous Proもectior｝
εL Politica1−Eco員。剛y Approach
20．G｝i社をめぐるアメリカ労働史研究：ファインとエッヅフかスの理場像の吟昧 21．卸売業の経営と戦略一一卸売流通研究会ヒアリング調査録（1）：日用 雑貨卸売企業
Auもho｛・／S
タ スク フ。タ，テぐイハ。ンカー
Yoshihiro 醤aL｝ζa
AHaTO．ηレ1蕗
捏1 ｝4 x a 莇 」： O B レ｛ q H
ΣHKyPKM
J軽n−ichi 1taya
擁iroshi Shibuya
Ryosuke 翼ada
Sadao Kanaya
忌 歪｝o矯oichi Sh亙notsuka Jun−ichi Itaya
Jeぞfery Alan Brunson
Jun−ich三 Itaya
＆ 1）ipankar I）asgupta もaixun Zhao
Satoshi Takata Laixun Zhao
Jun−ichi ltaya
Osamu Ito
Jane W． D A夢ista 藷iroO 註ojo
鍾a．rshall E． Blu臓e
La三xun Zhao
窩田聡
卸売流通研究会
（代表 高宮城朝刷）
Date Ju1．1992 Au菖．1992 Nov●1992
Jan．正993
Jun．1993
Aug．1993
Sep．1993
Hovの1993
琵ar．1994
Jun．1994
Oct．1994
Dec＿1994 Feb齢正995
遡ar．1995
甑y1995
トIay 1995
｝重ay 1995
｝lay 1995
鰹OV●1995
Feb．1996
Apr．1996
﹁
●
2 3
O O 4﹇b 22
︐
齢
曹
ρU7 り山2 ︐ ﹁ 8nぴ 2り乙 ● ． ・ nU− り4 つ03
3
O ． 9 Qり 4 Fδ3 り0 0ρ ﹁ O ● ρD7 00 903 3 ● ● ● Ωヲ 0 1 3 弓 4 O ︐ り6 3 4 4 ● O ●
0
45錠U 7 444 4
●
導 8
A 聾〇七e on the 王醗Paets of Price Shocks on Wage in Unionized Economies
Tr。nsf，r Pr1cing△nd th，賛at。re。f the subsidiary fi臓 The 王ncidence of a Tax on Pure in an A王truistic
Overlapping Generations Eco籟。蹄y
，S醗all Gove罫nmenも， in the 2圭sも Century
Characteristics and Reforms oぞ Public ｝至ea1七h I舞surance Syste日1 in 5apan
The Role of Local （｝overnmen七s in U罫ba罫｝ 1）eveloP羅e登t Po｝icy Opti蹴al Taxation and もhe Private P罫ovision of Pu毫｝11c 600ds
An 琶conometric Stu（iy of Trade Crea七ion and 璽｝rade Dユversion the EEC，LAFτA and C｝1霧A：A Simple Applieation oぞ the （｝ravity
ADynamic擁odel of Fisca19econstructlo瀦
The Japanese 響ay of Solvi舞g Fina穀cial Insもitution Failures The Federal Role． in Co醗mu！1ity Development in the U・S・
：Evoluもion vs． 1）evo1娃tion
註ent−Seeking Behavior in the 界ar of Aもtr三も三〇n
サハリン石油・ガス開発フ。ロジェクトと北海道経済の活性化第三愚
購買部門の戦略性と企業間連携について
窪｝he Formation of Cus七〇ms U舞ions 翫nd． もhe 奮ffect on 〔…overn∬とenも
POlicy Objectives
The Transiもioτ皇 of Postw乱r Asia−Pacific Trade Relations
地域型ベンチャー支援システムの研究 亙一道内製造業系ベンチャー企業のケーススタデ Compa【・ison Qf Agricultural Policy in the U． S． a農d the Japan US Healもh 王nsurance：霊｝ypes， PattePns of Cove罫age and
Constraints to Refor田
International Capit＆l F且◎ws and 巽＆tio罫｝a亙 ｝ia．croeconomic
Policies
F亙nancial Liberalization and Secu野it三zation in H◎using Finance and the Cねanging 801es of the Govern職ent
Social Effic重enc了 an（i もhe 選a寿ket Revo三ution， 三n ｛｝S Housing 罫inance
Government Expendi七ure and もhe 藝alance of 碧aン田ents：塾udget Deficit， Fi箔anc童a1 亙nte呂raもion， aτ｝d 奮co轟。賄ic I）iPloπ竃acy
A ｝｛三story of P郵GC an（呈 1ts 駐oles
Dyna田ic 菱｝rovision of Public Goods as Environ醗e捻taユ Exte！・na．liもies
ACo厭parative Static Analysis o歪l the aala籍ced Budget Inc童dence in the Prese漁ce oま『 Secも。罫幅S茎）ecific UnemPloy田ent
o ■
Laixur｝ Z｝1ao
Laixun Zhao Jun卿ichi Iもaya
Hiroshi Shibuya τ akashi 賛akahama
Yoshino！Li Akiyama
Jun−ichi Itaya ＆ David de ｝leza ＆ Gareth I）． ｝iyles
70shikazu Tateiwa
l）wayne A． 藝a．nks
Ja員e 胃。 D Arista
Syn−ya I獄ura
Gary Dymski ＆ Dore籍e 王senberg
｝｛iroshi Sh重buya
C． 韮）ξしvid 6usもafson
Toshihiro Ihori 邑 Jun−ichi Itaya
奎くoh Su醗三no
ln ｝fasahiro Endoh
｝iodel
Toshihiro Ihori ＆ Jun−ichi I七aya
Osa醗u Ito
Jane ｝（nodell
Jun−ichi Itaya 農 ｝｛ifoyuk玉 Sano
北東アジアー号ハリン研究会
俘藤 一 ｝Iasahiro Endoh
｝Iasa擁三罫0 奮ndoh
イー地域経済社会システム研究会
Hay 1996
Jun．1996 Sep．1996
Sep．1996 Sep．1996
Sep．1996 0ct．1996
2Uハ00ヲOJ O︾0θ 1
1
︐
，
tも C C OO
6 9 1 9
0
C t O
6 9 9 1 C t O
6 9 9 1
．
も C O
6 9 9 1 V
・O 馨 農U790ヴ 99 1
1
．
齢
VrOa 黄駈
7 9 1 9
●
a r
跡
7 9 1 9
，
P r A
7 9 9 1 2
●A P
77 90﹂90σ 1 1 ・
●
−t UC JO
7 9 正 9
O
C t O
8 9 9 y 1 a
麺
8 9 9 1 n
，J U 8 9 9 1
9
11 1 J
OQ890e 90σ − i ．
P
