Journal of Fmanclal Econonucs 9 (1981) 47–73. North-Holland Publishmg Company
OPTIMAL
DEALER PRICING UNDER TRANSACTIONS AND RETURN UNCERTAINTY*
Thomas HO
New York University, New York, NY 10006, USA
Hans R. STOLL
Vanderbdt University, Nashville, TN 37240, USA
Received October 1979, final version received September 1980
The paper exammes the optimal behavior of a smgle dealer who 1s faced with a stochastic demand to trade (modeled by a continuous time Pmsson Jump process) and facmg return risk on his stock and on the rest of his portfolio (modeled by diffusion processes). Usmg stochastic dynamic programmmg, we dertve the optimal bid and ask prices that maxnmze the dealer¡¯s expected utility of termmal wealth as a function of the state m which he finds himself. The relationshIp of the bid and ask prices to inventory of the dealer, mstantaneous variance of return, stochastic arrival of transactions and other variables is examined.
1. Introduction
Dealers in securities markets stand ready to trade immediately fixed amounts of securities at stated bid or ask prices. In this paper the optimal bid and ask prices are determined such that the dealer¡¯s expected utility of terminal wealth is maximized. Our concern is primarily with the risk assumed by the dealer, which in this case arises not only from uncertainty about the return on his inventory but also from the uncertainty about when future transactions will occur (which affects how long he must bear return uncertainty). We consider a single dealer trading a single stock and facing a downward sloping stochastic demand for the service of immediacy provided by the dealer.
This paper is related to an earlier paper by Stoll (1978a); indeed the economic setting of the problem is much the same. What is different is the nature of uncertainty (primarily the introduction of transactions uncertainty), explicit treatment of a multiperiod strategy for the dealer, and the in- troduction of the demand side. The paper should also be placed in the context of the growing literature on the micro-structure of securities markets.
*We wish to thank Thomas Copeland, E. Phlhp Jones, Robert Schwartz and the referee for their helpful comments
48 ¡®1:Ho and H.R. Stall, Optimal dealer pricing
This literature is concerned with the difference between transactions prices and ¡®true¡¯ underlying prices as determined by the available information set. One approach to this issue is to propose price or return generating processes that are consistent with our understanding of market structure but do not necessarily derive from any principles of individual economic behavior. The end product of such research is the relationship between the observed distribution of returns and the ¡®true¡¯ underlying distribution. The work of Clark (1973) Blattberg and Gonedes (1974) Westerfield (1977), Oldfield, Rogalski and Jarrow (1977) Schwartz and Whitcomb (1977) and Goldman and Beja (1979) is in this spirit. The latter paper is most closely related to our paper since it is concerned with the impact of the specialist on the process by which transactions prices adjust to the evolving ¡®true¡¯ price of the stock. However, they do not build an explicit micro-economic model of the
dealer.
An alternative approach is to specify the demand for and supply of the
services of intermediaries – i.e., dealers – whose compensation is reflected in a transaction price different from the ¡®true¡¯ price. For an extensive review of the literature, see Cohen, Maier, Schwartz and Whitcomb [CMSW] (1979). On the supply side, work began with the paper by Demsetz (1968), and additional theoretical contributions were made by Tinic (1972) and Stoll (1978a). There are also a host of empirical papers on bid-ask spreads that have arisen out of this or closely related approaches. In addition to Demsetz and Tinic, see Tinic and West (1972), Benston and Hagerman (1974) Stoll (1978b), and Smidt (1979), for example. Papers by Copeland (1976), Epps (1976), CMSW (1978), and Goldman and Sosin (1979) can be viewed as the basis for the demand for dealer services because they specify why investors trade and how trading affects prices. However, the role of intermediaries is not explicitly considered.
In an excellent paper, Garman (1976) does consider the role of the intermediation process, but he is more concerned with the equilibrium price of securities rather than the equilibrium price of the services of inter- mediaries. We\take the dealer¡¯s opinion of the ¡®true¡¯ price of the stock to be exogenously determined by his information set and ask how the dealer prices relative to his ¡®true¡¯ price whereas Garman is concerned with the necessary behavior of transactions prices to meet an exogenous inventory objective of the dealer. We adopt Garman¡¯s concept of stochastic supply of and demand for securities but view both the demand to sell securities and the demand to buy securities as demands for dealer services. Unlike Garman and recent papers by Amihud and Mendelson (1979) and Mildenstein and Schleef (1979), based on Garman, we do not require our dealer to maximize expected profits. Indeed our key concern is with the risk the dealer faces, and how this affects his willingness to supply dealer services. In addition we introduce stochastic processes for the ¡®true¡¯ return on the dealer¡¯s stock and the other
7: Ho and H.R. Stall, Optimal dealer pricing 49
components of his portfolio. In the face of stochastic transactions and stochastic returns, the dealer maximizes expected utility of terminal wealth by adjusting bid and ask prices through time. We present a dynamic programming solution to this more general problem. Mildenstein and Schleef also employ dynamic programming but they assume a risk premium for the dealer rather than deriving the premium in their analysis.
This paper is limited to the behavior of a single dealer making a market in a single stock. The generalization to many stocks is not difficult but adds needless complexity and is not considered in this paper. The determination of market bid and ask prices when there are several¡¯ dealers (or, for that matter, individuals placing limit orders) is also considered elsewhere.¡¯
The next section of this paper discusses the assumptions underlying our model. The dynamic programming problem is then stated and a solution presented. The model is used to analyze the dealer¡¯s optimal bid-ask spread and price adjustment strategy under symmetric as well as asymmetric demand conditions. A numerical illustration is also presented. The model is then used to analyze the decision to become a dealer in a particular stock. Finally the paper is summarized.
2. Assumptions
Transactions are assumed to evolve as a stationary continuous time stochastic jump process as in Garman (1976). A different process is allowed for purchases by the dealer and sales by the dealer. The process may be represented as follows:
Dealer sales (public purchases):
dpr——-z (la)
/l,dt
1 – E,,dt
Dealer purchases (public sales):
&dt
1- R,dt
(lb)
where Q is the jump size (number of shares in a transaction), and A,,& are the intensities of processes, representing the average number of public purchase or public sale transactions respectively per unit time. &,dt and i,,dt can be interpreted as the probabilities of a dealer sale or a dealer purchase over the next instant.
¡®A simplified version of our approach and some results are m Ho and Stoll (1980).
50 7: Ho and H.R. Stall, Optimal dealer pwmg
The use of a Poisson jump process to model transactions is quite natural. In actual markets, dealers are obligated to trade a minimum quantity (i.e., 100 shares), and transactions occur at discrete time intervals.
The dealer determines a price of immediacy, b, should a public sale (dealer purchase) order arrive and a price, a, should a public purchase (dealer sale) order arrive. The dealer does not directly quote b, a. Rather he quotes bid
and ask prices which are defined as
where p is the dealer¡¯s opinion of the true price of the stock at the time he sets the bid-ask quotation. The public¡¯s opinion of the true price is denoted by p*, and the two need not be the same.
The dealer is assumed to trade only with the public in a passive way. That is, he sets prices and must wait until a transaction occurs. He cannot immediately dispose of inventory by trading with a dealer of last resort. The arrival rate of dealer purchases, &, and dealer sales, A,, depends on. the dealer¡¯s buying fee and selling fee respectively. Linear demand curves are shown in fig. 1. By raising b, the dealer can reduce the probability of further public sales and by lowering a, he raises the probability of public purchases. Of course public purchases and sales are also influenced by the difference between p and p*. This difference is assumed to be parameterized in the functions i, and A,.
$ PuMic sellers)
Fig. 1 Average pubhc demand to buy from dealer, &, and average pubhc demand to sell to dealer, i,. when pubWE opmton of ¡®true¡¯ puce (p*) and dealer¡¯s opmlon of ¡®true¡¯ prxe (p) ddfer Eealer sets bid price (p,,) and ask price (p,) and regulates demand at ic and A:
7: Ho and H.R. Stall, Optimal dealer pricing 51
In addition to uncertainty about the timing of subsequent transactions, the dealer faces uncertainty about the return on his existing portfolio. In the absence of a transaction, portfolio growth, dX, is represented by a stochastic differential equation of the following form:
dX =r,X dt+XdZ,, (3)
where rx is the mean return per unit time; dZ, is a (non-standardized) Wiener process with mean zero and instantaneous variance, &.
The dealer¡¯s wealth is divided into three components: (1) Cash
Cash is accumulated when the dealer sells securities (short selling is permitted) and paid out when the dealer buys securities. Any balance in the cash account earns (or pays) the risk free rate of interest, r. The change in the value of the cash account, F, is
dF=rFdt-(p-b)dq,+(p+n)dq,. (4)
Uncertainty in the cash account is due to uncertainty about transactions not to any uncertainty about the interest to be earned.
(2) Inuentory
The dealer¡¯s inventory consists of shares of the one stock in which he makes a market. The change in the value of the inventory account, I, is
dl=r,ldt+pdq,-pdq,+IdZ,, (5)
where notation is as in (3) and (4). Shares enter inventory at p whereas the associated cash flow in the cash account is (p-b) or (p +a) per share. The difference reflects the dealer¡¯s profit. Uncertainty arises both from transac- tions uncertainty reflected in dq, and dq, and from uncertainty about the return on the stock which is reflected in the instantaneous variance of, of the Wiener process, dZ,. Over time, the dealer experiences discrete jumps in inventory due to transactions of Q shares at price p. There is an offsetting
jump of slightly smaller size (smaller by the transaction cost, a or b) in the cash account. The fluctuations in value between jumps are due to return uncertainty represented by the last term in (5), and this return uncertainty is the underlying source of risk facing the dealer.
In the real world, fluctuations in return are reflected in the price of the stock (absent dividend payments). In our model this would require a complex accounting through time of the relation between return fluctuations and
52 T Ho and H.R. Stall, Optimal dealer pricing
price fluctuations. However, allowing the price to fluctuate introduces no new element of uncertainty that is not already reflected in return uncertainty. Thus we adopt the convention that the true price, p, is kept constant. This implies that return uncertainty shows up as a continuous stock dividend that keeps the true price constant. It should be emphasized that this assumption of constant p is an accounting convention and that no sources of uncertainty
are ignored. The change in shares outstanding could ultimately affect the demand to trade in the dealer¡¯s stock; but the effect would be diffused across all stocks, and given the short time horizon of the model, would be unlikely to be consequential to the dealer¡¯s decision on how to set his spread today.
(3) Base wealth
The dealer has base wealth Y On the day he starts as a dealer F,=O and I, =O, and base wealth is his wealth. The dealer is assumed to have one consumption point at time T, and base wealth is an optimally diversified portfolio with respect to that time horizon. Base wealth serves as collateral to back up the borrowing of shares or money that arises in the course of acting as a dealer. The change in base wealth is
dY=r,Ydt+ YdZ,,
(6)
where notation is as in (3). Unlike Garman (1976) we are not concerned with bankruptcy of the dealer. We assume the time horizon is sufficiently short and the collateral is sufficiently large to make bankruptcy an un- important consideration in light of the Wiener process in which sudden jumps in return are not possible and in light of the dealer¡¯s ability to adjust prices. The effect of bankruptcy m a model with extended time horizon and price jumps deserves further investigation.
The objective of the dealer is to maximize the expected utility of his total wealth, ELI(W.), at time 7; his horizon, where
W,.=F,+I,.+ Y,.
(7)
The world ends at Tand the dealer is assumed to liquidate his inventory and base wealth at their market values without transactions costs.2 While consumption occurs once in our model, the multiperiod aspects are reflected in the random shocks to the opportunity set arising from numerous transactions and price changes prior to the horizon date. We seek the optimal strategy for choosing the values of a and b that maximize the dealer¡¯s preference function. This involves weighing the costs and benefits of
¡®We could impose an arbitary liquidation fee whtch would affect our mathematical results but would not add greatly to our understandlng of the risks and appropriate dealer strategy in the periods prior to the consumption point
T. Ho und H.R. Stall, Optimal dealer pricmg 53
being a dealer. The costs arise from the non-optimal portfolio the dealer stands ready to accept. Not only is his degree of diversification and level of risk not optimal but the presence of transactions uncertainty makes un-
certain when and how much his portfolio will be unbalanced. The benefits arise from the fee he is able to charge for his service of immediate trading. Because the model assumes a single dealer in the stock, that dealer has the ability to earn monopoly profits.
The optimal strategy is complicated by the multiperiod framework which permits the dealer to adjust a and b as he moves through time – usually in response to inventory changes. Although stochastic, these inventory changes are in turn influenced by the bid and ask prices. This is known as a closed loop control problem because the optimal values of a and b depend on the observed state variables (F,I, Y) as well as on time, t. In other words the optimal dealer strategy we seek is a fincrion that specified the choice of a and b for any position (described by t, F, I, Y) in which the dealer finds himself. The appropriate procedure for such a problem is dynamic programming.
Two points about the relationship of our analysis to the literature on portfolio theory and capital asset pricing should be emphasized. First, ours is a partial equilibrium analysis that is concerned only with the individual dealer¡¯s portfolio problem. No assumptions are made about equilibrium asset prices and no solutions are offered to equilibrium asset pricing in the presence of a dealer. Second, our analysis does not assume multiperiod consumption. In our model, as in standard one period portfolio models, the dealer costlessly liquidates all assets at the horizon date. The model differs in the multiperiod evolution of the opportunity set. Despite the single con- sumption horizon, time dependencies in the opportunity set, induced by the dealer¡¯s bid and ask pricing strategy, make it impossible to reduce our problem to the standard one period problem. We leave to future research the
development of a more complete multiperiod portfolio analysis of the dealer and the integration of dealer behavior into the theory of equilibrium asset pricing.
3. Statement of dynamic programming problem
For any state of the world which is described by time and the values of the state variables – dealer¡¯s cash, inventory and base wealth – the dealer chooses a and¡® b to maximize the expected utility of his terminal wealth. Define the optimal performance function J( ) as
J(t,F,I,Y)=max[EG¡¯(W,)It,F,Z,Y]. o.b
(8)
54 7: Ho and H.R. Stall, Opttmal dealer pncing
The function J( ) is the solution to the maximization problem (8) for an optimal bid and ask strategy from t, to 7: It can be viewed as a derived utility function which describes the result of optimal dealer behavior conditional on any observed t, F, I, Y Since there is no intermediate consumption prior to IT; the fundamental recurrence relation implied by the principle of optimality of dynamic programming is simply that
and
maxdJ(t,F,Z, Y)=O,
a.b
J(xF,Z, Y)=U(W,).
(9a)
(9b)
In other words J must meet the condition that the maximized increments to J are always zero; for if they were not, one could increase derived utility by an alternative bid-ask strategy. Also, at time IT;J must give the same level of utility as the elementary utility function, U.
Writing out the partial differential equation implied by (9a) – Bellman¡¯s
equation
– gives
max (dJ/dt) = J, + LJ
o,b
+~~~{UJ(F+PQ+~Q,Z-PQ, a,b
+&t-J(F-PQ+bQ,Z+pQ,
Y)-J(F,Z, Y)]
Y)-J(F,Z, y)])=O, (10)
where J, is the partial derivative of J with respect to time and L is defined as
the operator that takes the differential with respect to time of the mean returns and Wiener risk components of the function J using Ito¡¯s Lemma. Thus3
LJ = J,rF + J,r,Z + J,r,Y+iJ,,a:Z2 +$J,,a$ Y2 + J,,a,,Z X
(11)
where rsf and C: are the instantaneous variance of dZ, and of dZ, respectively, and ~,r is the covariance between dZ, and dZ,, and where subscripts on a function indicate the variable with respect to which a partial derivative is taken. As is evident in (4) (5) and (6) the mean returns and Wiener risks do not depend directly on a and b, and therefore LJ does not involve maximization over a and b. Thus J, + LJ simply represents the total
3A source on the dynamtc programming solutton to problems in stochasttc control IS Davts (1977). Davis also uses the operator Lin the sense we do To illustrate the operator L, consider
a stmplitied problem. Suppose y=f(t,x) where dx is gtven by (3). Usmg Ito¡¯s Lemma and takmg expectattons gives E dy =f,dt +f,Edx +fJx,E(dX)2 =f, dr +fxE[r,X dt + X dZ,] + 5 /xxX2a: dt. Noting that E dy=dEy [see Davts (1977)] and that E(X dZ .)=O, It follows that dEy/dr =f, + Lf where Lf=f,r,X +ffxx us X2, whtch is analogous m form to (11).
7:Ho andH.R.StallO,ptimal dealer pricing 55
time derivative of derived utility when there are no transactions. (Of course a and ¡®b indirectly affect Y, I and thereby influence the dealer¡¯s ultimate return and risk.)
More important to the dealer and to our problem are the last two lines of (10) which represent the maximized increments to J( ) resulting from transactions the value of which are PQ and which occur with probability A,dt in the case of dealer sales or I,dt in the case of the dealer purchases. The dealer also adds his fee, aQ or bQ, to his cash account. These increments do depend on a and b. The independence of the processes, dq, and dq,, allows us to write the effect as the sum of the two increments. The time argument in J( ) has been suppressed.
It is convenient to restate the problem in terms of time remaining to the horizon date 7: Let 7 denote this amount of time, and replace t in (8) by 7. In other words we now think of J as a derived utility function that depends on the time remaining to the horizon date. Thus when s=O,
J(O,F,I,Y)=U(W). (12) The derived utility at the horizon data is just the utility of wealth at that
point. The transformation of the time variable also implies that
J,= -J,.
(13)
Next consider the last two lines of (lo), and since p is arbitrary let p= 1. Because aQ and bQ are small relative to Q, we make the following first-order approximations:
J(F+Q+aQ,I-Q, Y)=J(F+Q,I-Q, Y)
+ J&F + Q,1-Q, Y)aQ, (14a) J(F-Q+bQ,I+Q, Y)=J(F-Q,I+Q, Y)
transaction on J; the approximation on a transaction.
We assume the same linear purchases:
&=l(a)=a-pa, i,,=i(b) = c(- /lb.
is only with respect to the fee collected
+J,(F-Q,I+Q,
Y)bQ. (lab)
Note that there is no approximation with respect to the effect of a
demand
relation
for dealer
sales and
Wa)
(15b)
56 T Ho and H.R. Stall, Optimal dealer pricing
This assumption of symmetric demand is made for expositional clarity and
will be relaxed later. Now define the sell operator, S, as
SJ=S[J(F,Z, Y)]=J(F+Q,Z-Q, Y), U6a)
and define the buy operator, B, as
BJ=B[J(F,Z, Y)]=J(F-Q,Z+Q, Y).
(16b)
The sell operator acting on J describes the dealer¡¯s derived utility after a sale by the dealer but before including the selling fee. Utility will decrease if the sale drives the dealer farther away from his desired portfolio by increasing a short inventory position. Utility will increase if the sale reduces a long inventory position. A corresponding result holds on the buy side.
Substituting (13), (14) and (15) into (10) and using (16) allows us to restate (10) as follows:
J,=LJ+max
{[l(a)aQSJF-L(a)(J-SJ)]
0.b
+ [A(b)bQBJ, – ;L(b)(J – BJ)]}.
(17)
This equation says that the change with respect to time remaining of derived utility depends on the return and risk of the dealer¡¯s current wealth (LJ), over which he has no direct control, and the maximized value of two terms that reflect the net contribution to the dealer¡¯s derived utility from dealer sales and purchases respectively, over which he has control. The essence of the dealer¡¯s problem is contained in these two terms.
Consider the last term of (17), which represents the net benefit from a purchase by the dealer. The expected gross benefit (in utile dimensions) from a dealer purchase is
WPQBJ,,
where L(b)bQ is the expected revenue from purchase transactions; and BJ, is the marginal utility of cash after a purchase transaction, i.e., BJF converts expected revenues into ¡®utiles¡¯. Since Q and BJF are given and since A(b) has been assumed to be linear, the expected gross benefit is a quadratic function of b and is drawn in fig. 2. The curve passes through the origin, is at a maximum when b =u/2/? [from (15)], and is zero again at b=u//?. Because of the assumed symmetry in demand conditions and on the temporary assum- ption that BJF= SJ, the curve is identical for dealer sale transactions. The assumption BJ, = SJ,, which implies that the marginal derived utility of cash after a sale equals the marginal derived utility of cash after a purchase, is made for expositional purposes only and is not used in the formal develop- ment of the model,
7: Ho and H.R. Stolf, Opfimal dealer pricing
Fig. 2. Dealer¡¯s expected (gross) benetit functton for a purchase or sale transactton
= I(x)xQBJ,, with x = a, b, where a, b are the selling and buying fee respectively, j.( .) is the mean arrival rate of transaction, Q is the transaction size, and BJF is the marginal utrhty of the fee after the transaction. Dealer¡¯s expected (gross) cost func!ion for a purchase, condrtional on
positive inmal mventory = (J – BJ)i(b), where (J – BJ) > 0 IS the decrease on the dealer¡¯s derived utility resulting from a purchase. Dealer¡¯s expected (gross) cost function for a sale conditional on positive initial inventory=(J -S.l)l(a), where (J- SJ)
b* = c(/2p + (J – BJ)/2BJ,Q. (18a)
In other words, set b at the level which maximizes expected profits associated with the stochastic arrival of public sellers plus a risk premium term. The risk premium term ultimately depends on the dealer¡¯s fundamental attitude toward risk and the level of return and transactions uncertainty of the stock,
58 7: Ho and H.R. Stall, Optimul dealer prwing
all of which enter into J and L?J.4 On the other hand, when Inventory is positive, utility is increased by a sale transaction, i.e., i.(u)(J – SJ) < 0 because (J - SJ) < 0. The new position after a sale, SJ, would be superior to the old position, J. Thus the ¡®cost¡¯ function for a sale transaction is negative and upward sloping as shown in fig. 2. The larger is u, the lower is the probability of a dealer sale and therefore the less negative is the expected cost (the smaller is the expected increase in utility). The optimum sellmg fee, u*, is a set to maximize the difference between the ¡®benefit¡¯ curve and the ¡®cost¡¯ curve for sales. Analogously to the case of a purchase this solution is
u* = c(/2,8+ (J - SJ)/ZSJ,Q.
(18b)
Because the dealer has positive inventory, b* >a* in fig. 2, as one would expect. When dealer inventory is zero, the two ¡®cost¡¯ functions coincide and are downward slopmg, in which case a* = b*. As inventory changes, the ¡®cost¡¯ functions shift in opposite directions and a and b change in opposite directions. For given p, this implies that both the bid price (pb) and ask price (p,) fall in response to inventory increases and both rise in response to inventory decreases, as shown in fig. 3. We shall see later that the spread, the difference between p, and pb, is to an important degree invariant with respect to inventory changes.
pa
¡®b
b
1
¡®b
Fig 3 Adjustment of btd prxe (p,,) and ask pr~e (p,) relative to ¡®true¡¯ pr~e, p, wth constant spread, s = (I+ h, and changmg prtce adjustment, d = a – b
¡°Carman¡¯s dealer would set h*=a/2P, and the analysis m that case IS, for example, concerned wth the time until bankruptcy. Amlhud and Mend&on allow the dealer to adjust his price ~ in our model, a and h – to avold fadure. But they mamtam the assumption of risk neutrahty.
7: Ho and1f.RStalOlp,tImaI dealer prrcmg 59
Until now our methodology has been quite standard. We have formulated the dynamic programming problem and stated the solution – (18) – in terms of marginal conditions on the derived utility function. However, we wish to go beyond the qualitative conclusions contained in our discussion of fig. 2 and eq. (18) and analyse the specific determinants of the dealer¡¯s bid- ask strategy in a multiperiod environment. This requires a solution for J.
We now proceed to a final statement of the dynamic programming problem by performing the maximization required in (17). This has in fact been done in (Isa) and (18b). Substitution of (18) into (17) and using (15) yields
J, = LJ + [cN,Q – fl(J – SJ)]¡®/4PSJ,Q + [ctBJ,Q -B (J – BJ)]2/4fiBJ,Q.
(19)
The problem is to solve (19) for J subject to (18) and the terminal condition, (12). The solution for J can then be substituted into (18) to yield the overall optimal values of a and b as functions of the state variables, F,I,Y
4. Solution
Unfortunately to our knowledge there is no closed form solution for equations of the form (19). However by using Taylor¡¯s series expansion we can, in principle, come arbitrarily close to the solution. Indeed this pro- cedure has certain advantages since it allows one to see the nature of the solution as a function of time to the horizon date. Thus write .J( ) as a Taylor¡¯s expansion around 5= 0,
J(T,F.I,Y)=~~(F,Z,Y)+~~(F,I,Y)~+~~(F,I,Y)~~+…,
(201
where higher-order terms are assumed to be small enough to be neglected. It will be evident from the methodology that follows that higher-order terms could be incorporated. However, for the purpose of this paper the increased complexity that results is not offset by a corresponding increase in under- standing of the dealer. The interpretation of (20) is that J( ) depends on how much time remains in the dealer¡¯s planning horizon. When r=O, only the first term of (20) is relevant. When r>O but sufficiently small, only the second term is relevant, and so on.¡¯
The solution to (20) is found by finding the different functions 4¡¯( ), c$¡®( ) and +¡®( ). Th e superscripts on the $ are notations associating each function with the degree of the derivative of J with respect to T. Eq. (20) holds for all values of 7.For t=O, J is given by (12). Therefore
#¡®(F,I, Y)= u(W), (21)
¡®As a practical matter the omltted hlghel order terms are not hkely to be large for the short plannmg horizons over which most dealers operate
60 7: Ho and H.R. Stall, Optind dder pricing
the utility of terminal wealth. In other words if the dealer is at maturity the optimal performance function is just his utility function.
For r ~0, (19) must be satisfied. Substituting for J,, J and J, in (19) from (20) and evaluating the resulting expression at T=O gives the solution for 47 k6
(22)
where
The operator, L, has been defined in (11). Its application in this case leads to
where
TW =W¡¯IW) +rrUIW+r,(Y/W ),
and U¡¯ >O and U¡± ~0 are the lirst and second derivatives of U(W). In other words LU( W)= Lq5¡¯ represents the positive contribution of mean return to utility when there is only a short instant of time remaining plus the negative contribution of return variance. The term, fZ, is the maximum expected profits from either sales or purchases (which are the same under symmetric demand assumed here).¡¯ The term 2U¡¯fl thus represents the contribution to dealer welfare of the profits he expects to make from purchase plus sale transactions. Transactions uncertainty does not enter into 4¡¯.
6From (20),
(J,Ir=O)=@, (JJr=OJ=@=U, (J,)r=O)=r¡¯$P/W=U¡¯. Thus (19) evaluated at r=O ts
Because U= SU =BU and Sb¡±=BU¡¯= LJ¡¯ (since at the hortzon date there IS no rusk and the utrlity of a dollar IS the same whether held in inventory or cash),
4¡¯ = LU +2U¡¯(~x¡¯Q;4@), whtch is (22).
¡®Usmg (15a) expected profit per sale transaction of size Q 1s
A,Q = IQ – /IQa¡±
The value of u whtch maxtmizes thts expression IS u=ai2P Substitutmg thrs tn the expected protit expresston ytelds maxtmum expected profit equal to a¡¯Q,¡®4P a value denoted by II. Fmally substttutmg the value. ~128, m the demand relatton ytelds the opttmal arrival rate at the prtce whtch maxtmtzes expected proftt, I.e.. a/2. Corresponding terms arise m the case of dealer purchases.
7: Ho und H.R. Stall, Optimal dealer priung 61
The function $¡¯ is found in a similar way. Substitute for J,, J and J, in (19) from (20), take the derivative with respect to time, and evaluate at t =O. The solution uses the solutions for 4¡± and 4¡¯, and involves derivatives and moments of third and fourth order. To simplify the problem, we now assume the dealer¡¯s elementary utility function is quadratic so that we may restrict ourselves to a mean-variance world.8 The solution for 4¡¯ is then¡¯
~2=~l~U(W)+U¡¯rn+2U¡±n2+2U¡±~r,W+)U~~~~Q2.
(23)
This expression is a negative adjustment to the dealer¡¯s derived utility for aspects of return uncertainty and the effect of transactions on portfolio return uncertainty. The first term of (23) reflects the compound effect of return and return risk resulting from twice applying the operator Lto U(W). The term is written out in the footnote and inspection will indicate that it is normally negative.¡± The second term is the increase in utility resulting from the interest earnings on the transaction profit. Because U¡±
a= (s+d)/2, b=(s-d)/2. 6. The hid-ask spread
From (18a) and (18b) the dealer¡¯s spread is
s = IX//I+ (J – SJ)/2SJ,Q + (J – BJ)/2BJ,Q.
(24)
The first term in (24) is the spread which maximizes expected revenues per share from sale and purchase transactions from the symmetric linear demand functions. The more inelastic the demand for dealer services the larger the spread.¡¯ 2 The remaining terms are ¡®risk premiums¡¯ for a sale and purchase transaction, which assume optimal dealer behavior in the future. The sum of the risk premiums is required because the dealer sets the spread without knowing whether the next transaction will be a purchase or sale.
¡°Smidt (1979) has called the mtd-pomt of the dealer¡¯s spread (p+d), the market prrce, and he has called our p the equihbrrum prtce. He wews the dealer as adjusting his inventory by altering the difference between ¡®market price¡¯ and the ¡®equihbrium price¡¯, I.e., by altering our d.
12The papers of Garman (1976) and Amthud and Mendelson (1979) are concerned wrth thus term or the relatea expresston for profits.
7: Ho and H.R. Stall, Optimal dealer pricing 63 Substitution of the solution for J into (24) yields the following optimal
dealer spread :
s=r-fZ(Q/W) afz+jZ(Q/W)af[(r,-ri-GG,)+Z(r,+2fI/W)]t2, P
where Z = U¡± W/U¡¯, the coefficient of relative risk aversion, and G, = rr +$oi, which is the instantaneous growth in the variance of I. The spread equation actually omits certain small second order terms. The full equation and the procedure for deriving it are in a mathematical appendix available from the authors.
We can now see what factors determine the spread and how the spread is changed as the horizon of the dealer is lengthened. When r=O, the dealer does not look beyond the current moment and is concerned only with the fee he can collect from a purchase or sale. Thus only the risk neutral, spread, a/p, already discussed above, is relevant.
At the time horizon increases, the second term of (25) which is the lirst- order risk adjustment, is relevant; and the spread is higher since all variables are positive. This term is one half the risk premium term derived in Stoll (1978a) where a one period framework was relied upon. The difference arises from the fact that we have maximized expected utility of a monopolistic dealer wheras in Stoll (1978a) demand conditions are not considered, and bid and ask prices are determined that maintain the dealer¡¯s original expected utility of terminal wealth. As was the case in Stoll (1978a), the first-order risk adjustment term does not depend on the dealer¡¯s inventory. Rather it depends on factors which are relatively stationary through time, i.e.:
(1) The dealer¡¯s attitude toward risk as reflected in the coefficient of relative risk aversion, Z.
(2) The relative value of the transaction for which the bid-ask quotation is made, Q. (Recall p= 1.) Larger transactions require larger spreads.
(3) The risk of the stock as measured by the instantaneous variance, uf. Note that as far as the spread is concerned ¡®total risk¡¯ not just ¡®systematic risk¡¯ matters. With respect to base wealth, an efficient portfolio, the risk of inventory is measured by the covariance between the return on inventory and the return on base wealth, crIY. However that risk is offset by the expected excess return, r,-r, and therefore does not appear in the final expression, (25). Because the dealer moves to a sub-optimal portfolio, the variance matters with respect to that fraction of his wealth, Q/w which exceeds the optimal investment in the stock.
In fact, inventory position does not appear anywhere in (25) (except as it affects rw). The reason is that the dealer adjusts the bid and ask prices
(25)
64 7: Ho and H.R. Sroll, Optimul de&r pr~cmg
equally whenever there is an inventory change. Thus when he acquires shares, he lowers his bid price (raises b) to discourage additional sales to him; and he lowers his ask price (lowers a) to encourage additional purchases from him. In other words he places himself on the margin where he is indifferent to a public sale or purchase because the higher compensation derived from a sale to him is equal in utility to the lower risk (cum smaller compensation) from a purchase from him.
Lengthening the dealer¡¯s time horizon makes the last term of (25) relevant and causes the spread to be still larger since the term is positive. For a given time horizon, one might expect the spread to be less under our multiperiod model than under the one period model of Stoll (1978a) since the dealer has an opportunity to adjust prices as his inventory changes. However, this is not the case. In a one-period model, one transaction occurs; and the dealer holds the risky position to the horizon. In our multiperiod model, additional transactions as well as dealer price adjustments may occur. The uncertainty
of these additional transactions (which increases the uncertainty of the dealer¡¯s portfolio return and profits) more than offsets the risk reduction the dealer chooses to achieve by changing his prices. The single period approxi- mation simply ignores these additional risks.
As in the case of the first-order risk adjustment, the second-order risk adjustment depends on the dealer¡¯s attitude toward risk, the transaction value, and the stock¡¯s return variance, which are multiplied by the bracketed term. The first term inside the brackets is a risk premium plus the growth in the variance of I. The second term is the coefficient of relative risk aversion multiplied by the return from two sources – the expected return on his total portfolio (rw) and expected dealer profits expressed as a fraction of his wealth (2lZ/W). Since the expected returns and growth in variance of I are likely to be small relative to the dealer¡¯s horizon, the principal effect comes from 2lI/W: the larger the monopoly profits today the more the dealer stands to lose tomorrow, and therefore the larger the second-order risk adjustment.
Probably more surprising than the terms which enter (25) are the terms that do not enter. We had expected the transactions processes to play a more explicit role. They do not enter at all in the first-order risk term. They enter the second-order risk term through II, but this is simply an adjustment for the expected growth in dealer wealth from his monopoly profits. There is no term for the variance of the transactions process. The absence of such a term in the spread equations is due to two factors. First, in this model transac- tions uncertainty is not undesirable per se because there is no direct adverse effect of a below average number of buyers or sellers. We have not assumed a minimum cost of doing business, for example. Transactions uncertainty is undesirable only because it increases the uncertainty of the return on the dealer¡¯s portfolio by making uncertain how large an unbalanced position
T Ho and H.R Stall, Optimal dealer prrcrng 65
must be held. The effect is seen in (23) in the term (c42)a~Q¡¯. Second, the reason that this term does not appear in (25) is analogous to the reason that inventory does not appear. For any I, the dealer sets his bid and ask price so that he is indifferent on the margin as to whether the next transaction is a purchase or sale. This is true whatever is the probability of the next purchase or sale even if those probabilities were to differ because of asymmetric demand. As a result the term, (42)c7fQ2, does not enter the spread. Just as current inventory does not matter for the spread, the expected change in inventory does not affect the spread.13
7. Price adjustment
On the assumption of symmetric demand, the price adlustment variable
derived from (18a) uld (18b) is
d=t((J-SJ)/SJ,Q)-+((J-BJ)/BJ,Q). (26)
The principal function of d is to promote adjustment of the dealer¡¯s inventory when it is temporarily thrown out of balance by the random arrival of transactions. Substituting the solution for J into (26) yields
d= -Z(I/W)a:r-Z(I/W)a:[r,-r+G,+Z(r,+2n/w)]?¡¯, (27)
where certain small terms have again been neglected. The full equation is in a mathematical appendix available from the authors.
Eq. (27) is identical to (25) except for the fact that 21 replaces Q. Whereas the spread depends on the size of the transaction yet to come (Q), the price adjustment depends on the inventory which has been acquired (I). Eq. (27) is an inventory response equation that specifies the price adjustment variable be negative (positive) when inventory is positive (negative). When d
130ne mtght ask whether transacttons uncertamty would be more crtttcal of one mtroduced sertal dependence m the process for purchases and sales. Postttive sertal dependence m purchases and in sales would be like Increasing the transactton stze Q, and would therefore mcrease the spread Negattve sertal dependence would reduce the spread Whether any addtttonal effects would enter IS not at all obvious, and we have not worked out thts case.
66 7: Ho and H.R. Stall, Optrmal dealer pricing 8. Asymmetric demand
Asymmetric demand arises if the dealer¡¯s perception of the true price, p, differs from the public¡¯s perception, p *. For example if p* >p, the expected number of dealer sales at p exceeds the expected number of dealer purchases at p as in fig. 1. Suppose the demand curves for dealer sales and purchases differ both in their slope and intercept,
2,= a,- Aa, (15c)
A,=ab -&b.
(154
Asymmetric demand alters the spread equation by changing the first term in (24) or (25) from a/P to (cz,/2~,+a,/2/$). However, if this term – the risk neutral spread – and expected dealer profits remain the same under symmetric as under asymmetric demand, there is no additional effect on spread through the risk premium terms in (25).14 For example suppose that ao=ab= 100 initially, that a, rises to 120 and a,, falls to 50, and that the appropriate changes occur in /?, and fib to hold constant expected profits and the risk neutral spread. Then there is no additional effect through the risk terms despite the fact that the expected number of public sellers at p now
exceeds the expected number of public buyers at p (a,= 120 vs a,=50). This is because the dealer knows today that he can follow a price adjustment strategy in the future that makes him indifferent to this asymmetry.
What is crucially affected by asymmetric demand is the price adjustment variable, d. Just as the dealer adjusts prices if current inventory is non-zero so he adjusts prices for expected imbalances in demand which could drive future inventory to undesirable levels and thereby increase the dealer¡¯s risk to undesirable levels. Let d¡¯ denote the price adjustment under asymmetric demand. Following the procedure used in deriving (27), except that we now assume (15~) and (15d), gives
d¡¯ = Ca,IW,-ath%J -tZ(QIW)b:((a,-aClb)/2)t2 +4 (28)
where d is given by (27).
The first term is simply the price adjustment necessary to maximize
expected profits. This may require inventory accumulation because the stock is a good buy (p>p*) or inventory reductions because the stock is a poor buy (p
stock and the dealer¡¯s attitude toward risk.
These price adjustments do not eliminate inventory accumulation or
reduction because the source of this inventory change is the favorable or unfavorable price of the stock in the opinion of the dealer. The price adjustment compensates the dealer for the loss of portfolio diversification and change in portfolio risk resulting from the additions to or reductions from inventory. If the dealer always sets p =p*, where p* is defined as the price at which the expected demand for dealer sales is equal to the expected demand for dealer purchases, LX=,,a,; and the case of asymmetric demand would not arise.¡°j
9. Numerical illustration
Consider a dealer with a planning horizon of t =0.04 of a year, appro- ximately two weeks. For simplicity assume he faces symmetric demand. What will be his spread and price adjustment for the following assumed values of the exogenous variables?
a= 2000 per year. Expected number of purchase transactions plus expected number of sale transactions at optimal dealer fee assuming symmetric demand.¡±
/I = 100,000 per year. Determined to give risk neutral spread of x/P = 0.02.
2 = – U¡± W/U¡¯ = 2. Coefficient of relative risk aversion, assumed constant.
r6The problem then is how to determme expected demand One possibthty IS to base thts expectation on recent transacttons thereby causmg p=p* to be based on the evolutton of transacttons.
¡°Under risk neutral pricing and symmetrtc demand the expected arrival rate of purchases or sales ts a/2. The number chosen ts based on Smtdt (1979) who found stx of twelve NYSE stocks had fewer than 6000 transactions per year and on the assumption that the speciahst partrctpated as a dealer m one third of those transactions.
68 7: Ho und H.R. SM. Optrmcrl dewier pricing
0; =0.50 per year.¡®*
r, = 0.20 per year, rr = 0.22 per year, r = 0.10 per year. 0: = 0.06 per year, (T,,,= 0.05 per year,
We also assume the dealer begins anew from alternative initial positions on which he has not earned profits so that F = -I and W= Y
Q II¡¯ 000
IO
0 000
0.415
0 I12 0 447
0.240 0 379
-005
0 000
0 208
0.1 I2 0.224
0.240 0 240
0.543 0 272
0.0
0.000
0.000
0 05
0.000
-0208
0 IO
0.000
-0416
0 20
0.000
-0833
040
0.000
– 1.668
On the basis of these assumed values (25) and (27) now become
s1 =f(Q/W)~+~(Q/W)[0.99+0.21/W+40Q/W]t2, d = – (I/W)t – (l/¡®W)[0.99+0.21/W+40Q/WJz2,
(250 (27¡¯)
where s1 =s-LX/P. The risk neutral spread, x/p=O.O2 in this illustration, is invariant to the variables of interest to us, and we look only at the risk premium terms. It is obvious from the equation that, under the assumption
of constant relative risk aversion, only the ratios, Q/W and I/l/t: are important. Values of s1 and d for alternative values of these ratios are given in table 1. In the equation, the spread depends on I because the level of I affects portfolio return, rw, in (25). However, as the table shows, this effect is less than one basis point and our earlier assertion that for practical purposes the spread is independent of inventory position is correct.
Table 1
Price adjustment ((1) and risk premium component of spread (,$I). stated m percent, by transaction value as a fraction of wealth (Q/W) and Inventory value as a fraction of wealth
I ¡®W
– 0.40 -020
;¡¯ 0.000 0.000
I.668 0.830
-0
0.543 0.543
(II W) Dealer¡¯s horlron IS assumed to be r =0.04 years
005 :; 0.112 0 112 1.786 0 894
0112 0II2 0112 0.112 0112 0.000 -0224 -0448 -0 897 ~ 1796
0240 0.240 0240 0240 0240 0.000 ~ 0.240 -0480 -0.961 ~ 1.924
0544 0.544 0544 0544 0545 0000 -0.272 -0 544 – 1089 -2 180
0.10
~I 0239 0.239 1914 0.958
0 20 fi¡¯ 0 542 0 543 2 170 1.086
;¡¯ 1341 1.342
2 682
___
¡°The annual
dealer from mformational trading, which IS not exphcltly modeled here. By standing ready to trade at bid or ask prxe, the dealer mcurs losses when mvestors have mformatton Since the dealer¡¯s positIon tends to mcrease Just before a price declme and decreases Just before a price Incrra\e. hl\ risk ISgreater than ImplIed by the underlylng variance of the stock¡¯s true return
0.40
1.343 1.342 0671
I 343 1.343 1.344 I 344 1.345 1346
0.336
0.000 -0 336 -0672 -1 345 -2692
return barlance IS on the
high side m part to capture the special risk of the
7: Ho and HA. Std. Optimal dealer priang 69
It is evident from the table that for a wealthy individual the risks of making a market in a single stock can be small. For example for Q/W=O.O5 the risk premium in the spread is only 0.122 percent ¡®for values of I/W between -0.40 and +0.40) or $6.10 on a $5000 transaction. This implies a proportional buying or selling fee net of monopoly profits of only 0.061 percent or $3.05. We expect that for certain classes of dealers, such as certain NYSE specialists, block positioning firms, or underwriters, the ratio, Q/W may be considerably higher. And in such cases the risk premium term in the spread can increase significantly as the table shows. An obvious recommendation for improving the quality of securities markets is to increase the amount of wealth devoted to market making either by raising wealth of the dealer or by having more dealers. However one cannot compel an
increase in a dealer¡¯s wealth, and institutional restrictions (as on the NYSE) as well as clerical and informational economics of scale may limit the number of dealers in a stock.
The values in table 1 depend on the time horizon of r =0.04. Doubling
r to 0.08 of a year would raise the spread (for Q/W=O.O5) to 0.248 percent or $12.40 on a $5000 transaction. The spread more than doubles because the second-order term becomes relatively more important. Nevertheless its contribution to sr is only 0.048 percent, which implies that the degree of approximation achieved by eq. (10) is still adequate. While the variance of inventory return, a:r, increases as t and the variance of inventory due to transactions, (2, +A&, increases as t, the combined effect on portfolio risk, which appears in the second-order term increases faster than T.
The table is helpful in understanding the behavior of bid and ask prices implied by our model. Let a prime on a variable indicate the risk premium component of the variable. Now consider a particular transaction size, say Q/W=O.O5, where sr=0.00112. At I=0,d=0,and
and
p;=p+a¡¯=p+(s¡¯+d)/2=1.00056,
p;=p-b¡¯=p-(s¡¯-d)/2=0.99944.
Suppose the dealer acquired $5000 of inventory so that I/W=O.O5. He would lower his prices by setting d = -0.00224 so that
and
JFE D
pa= p + (sl + d)/2 = 0.99944,
pb = p – (as1- d)/2 =0.99832.
70 T Ho and H.R. Stall, Optimal de&r prwng
The dealer has lowered bid and ask prices by the value of the spread, which, in the absence of monopoly profits, causes not only his bid price but also his ask price to fall below his opinion of the true price. Because the dealer has monopoly power it is unlikely that the ask price will in fact fall below p. Noting that s=si +a/jI and a//3=0.02, the actual ask and bid price cor- responding to the previous example when I/W=O.O5 are
and
p,=p+(s+d)/2=1.00944,
&,=p – (s- d)/2 =0.98832.
If, by chance, no additional transaction occurs before the horizon date, the model assumes the dealer liquidates his inventory at p, thereby yielding an expected fee of b. If additional transactions occur that cause ending inventory to be zero, the dealer earns no risk premium because in effect he has borne no risk. Of course, prior to the horizon date, there is uncertainty about his ending inventory which increases the dealer¡¯s portfolio risk, and this causes him to demand a risk premium today.
10. Demand conditions and the decision to become a dealer
Our model is perhaps most applicable to the NYSE where each stock has only one dealer – the specialist. Consider the question of whether a newly listed stock would always find a specialist willing to make a market in the stock. That decision depends on the demand conditions and on the risk of being a dealer. The demand conditions in turn depend on the fundamental desires of investors to trade the stock and on the alternative sources of immediacy. There may be competing markets which reduce c( and increase p to the specialist. Furthermore the demand for dealer services depends on the ease with which investors could ¡®make their own immediacy¡¯ by using limit orders and borrowing or lending. For example instead of selling immediately
to the specialist at the specialist¡¯s pb an investor could enter a higher limit price and borrow funds while he waits for his transaction to be executed.
The cost to investors of these competing sources of immediacy influence the elasticity of demand for dealer services.
In- terms of our model an individual will not become a dealer if the spread he must set to cover his risk is so large as to cut off the demand for his service. Under symmetric demand the spread required by the dealer is given
7: Ho und H.R.StallO,ptimal dealer pricing 71
by (25). Starting from a zero inventory position the spread at which the expected arrival of transactions is zero is 2u//? (see footnote 7). Thus the condition for being a dealer is that the required spread be less than the spread at which trading is zero,
+iZ(Q/W)af[(r,-r+GI)+Z(rW+21TI/W)]r2.
(29)
This condition can always be satisfied for transaction size, Q, small enough. But suppose there is some minimum Q the dealer must be ready to trade. Then the condition can be violated if c(, the transactions arrival rate when the dealer¡¯s fee is zero, is sufficiently small. For example when cr=200, Q/Y
= 0.1, z= 2, C$= 0.5 per year, r= 0.10 per year, ry= 0.20 per year, /I= 100,000, and r=O.O4, the dealer would not make a market, for in that case expected profits would not offset the risk. It is also clear that condition (29) is less likely to be satisfied the larger an individual¡¯s Z or the smaller his wealth. Regulations to increase the transactions size for which a bid-ask quotation must be honored would reduce the willingness of individuals to become a dealer. Since the NYSE specialist also earns brokerage on limit orders he executes, one may see an individual declare his willingness to be a specialist without at the same time being willing to act as a dealer. In such circumstances the provision of immediacy by the specialist is likely to be particularly poor.
11. Summary
We have considered the optimal behavior of a single dealer in a single stock who is faced with a stochastic demand for his services (modeled by a continuous time Poisson jump process) and faces return risk on his stock and on the rest of his portfolio (modeled by diffusion processes). As time unfolds and transactions occur, the dealer is able to set his bid price and ask price relative to his opinion of the ¡®true¡¯ price of the stock so as to maximize the expected utility of terminal wealth. This stochastic dynamic programming problem is solved for the dealer¡¯s derived utility function; and using this solution, we are able to specify how the dealer sets the bid and ask price at any moment of time as a function of the state in which he finds himself. Indeed the solution could be programmed on the computer.
We show that the spread, the difference between the ask price and bid price, is given by a risk neutral spread that maximizes expected profits for the given stochastic demand functions plus a risk premium that depends on transaction size, the return variance of the stock and the dealer¡¯s attitude toward risk. Consistent with the one period model of Stoll (1978a), the spread does not depend on the dealer¡¯s inventory position. However dealer
72 7I Ho and H.R. Stall, Optimal dealer pricing
price adjustment does depend on inventory. When inventory increases both bid price and ask price decline, and the converse is true when inventory decreases.
Despite the dealer¡¯s ability to adjust prices in response to inventory changes as time unfolds, the dealer¡¯s risk is greater than implied by the one period model of Stoll (1978a). This results from the fact that the uncertainty of the demand to trade with the dealer is not totally eliminated or offset by the dealer¡¯s pricing strategy.
Other issues investigated are the case of asymmetric demand for trading with the dealer and the conditions under which an individual chooses or refuses to make a market in a stock. In an inactive stock, when the dealer is required to trade a minimum amount, the expected profit from trading may not be enough to offset the risk.
Although couched in terms of a dealer, our model can apply to any individual who is determining his reservation price for a stock as a function of his opinion of the stock¡¯s value and the degree to which the stock will cause his portfolio to be unbalanced. Thus the placement of limit orders by individuals could, for example, be modeled by our approach.
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