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Unformatted text preview: number of possible candidates for (n1, n2, n3) and have obtained the best
fit with (4,1,3). 35 model. We simply conclude that the effects the model predicts can be of
the same magnitude as those seen in the data.
In closing, we note that many intraday phenomena remain unexplained.
For example, a number of studies have shown that mean returns also vary
through the day. [See, for example, Jain and Joh (1986); Harris (1986);
Marsh and Rock (1986); and Wood, McInish, and Ord (1985).] In our
model, prices are a martingale, so patterns in means do not arise. This is
due in part to the assumption of risk neutrality. [Williams (1987) analyzes
a model that is related to ours in which risk aversion plays an important
role and mean effects do arise.] Developing additional models that produce
testable predictions for transaction data is an important task for future
research.
Appendix
Proof of Lemma 1
Consider the informed traders’ decisions. The ith informed trader chooses
xti, the amount to trade in period t, to maximize the expected profits, which
are given by Given the form of the price function in Equation (3), this can be
written as
Suppose that informed trader i conjectures that the market order of the
other n  1 informed traders is equal to
Then the total order
and the ith informed
flow is
trader chooses xti, to maximize which is equal to It is easily seen that the expected profits of the ith informed trader in
period tare maximized if xti is set equal to The Nash equilibrium is found by setting the above equal to
and solving for β t18 We obtain
36 We now determine the value of A, for a given set of strategies by all
traders. Recall that the total amount of discretionary liquidity demands in
if the jth discretionary liquidity trader
period t is
where
0 otherwise. The zeroprofit condition
trades in period t and where
=
for the market maker implies that Substituting Equation (A6) for β t, we obtain a cubic equation for λ t. The
unique positive root gives the equilibrium value for the assumed level of
liquidity trading. This is the value given in Equation (5). n
Proof of Lemma 2
As shown in the proof of Lemma 1, when nt traders are informed, each
places in period t a market order of
shares, where [We assume here that
= 1 and φ t = φ.] Now consider a deviant
trader who acquires information in addition to the other n traders. He will
demand xt, shares, where xt, maximizes where ii, is the total liquidity demand in period t. This is maximized at or substituting for β t, The expected profits earned by the deviant trader, π d( nt, Ψ t), will be 18 It is straightforward to prove that the equilibrium among informed traders is always unique. 37 Table 8
An example of the nonexistence of an equilibrium For each number of informed traders, n, this table gives π (n, 1), the profits of each informed trader if no
discretionary liquidity trader trades in that period; π (n, 2), the profits of each informed trader if both
discretionary liquidity traders trade in that period; π (n, 1.4), the profits of each informed trader if only
discretionary liquidity trader B trades in that period; and π (n, 1.6). the profits of each informed trader if
only discretionary liquidity trader A trades in that period. Similarly, for each n and each of these four
values of Ψ, the table gives the equilibrhtm value of λ t, which measures the cost of liquidity trading under
the assumed traders’ composition. which is Substituting for β t and λ (nt, Ψ t), we can simplify this to This completes the proof. n
An example of the nonexistence of an equilibrium
Assume that there are two discretionary liquidity traders, A and B, with
and var( YB) = 0.4. Assume that the cost of trader i observing
in period t the signal
is 0.11 for all i and that
= φ = 10,
again for all f. Finally, assume that the variance of the nondiscretionary
liquidity trading is 1. Is it an equilibrium for both discretionary liquidity
traders to trade in the same period? If they do, the total variance of liquidity
trading in that period will be 2, and in all other periods it will be 1. Table
8 shows the profits earned by informed traders as a function of the number
informed and of the variance of the liquidity trading. It also shows the
value of λ in each case.
From the first two columns of the table it follows that in equilibrium
there will be one informed trader in the market if Ψ = 1 and c = 0.11, and
three informed traders when Ψ = 2. This creates a problem, since λ(3,2) =
0.169 > λ(1,1) = 0.151. Taking as given, neither of the two discretionary
38 liquidity traders will be content to trade in the period they are assumed
to trade in. If each assumes that he can move to another period without
affecting other traders’ strategies, then each will want to move to one of
the periods with λ = 0.151.
Having shown that it is not an equilibrium for both A and B to trade in
the same period, we now show that it is also not an equilibrium for each
discretionary liquidity trader to trade in a different period. From the third
and fourth columns of the table it follows that if each discretionary liquidity
trader trades in a different period, then there will be two informed traders
in each period. However, in the period in which A trades, λ = λ(2, 1.6) =
0.161, while in the period in which B trades, λ = λ(2, 1.4) = 0.172. If B
takes the strategies of all other traders and λ t as given, he will want to trade
in the period in which A is trading.
A statistical test of hypothesis 2
Note first that
Thus, Hypothesis 2 is equivalent to the hypothesis that if t ∈ H and t' ∈ L, then
(A15)
Since Equation (A15) is equivalent to
(A16)
Η Denote the estimates of λ and λ , obtained by the regression described
after Hypothesis 1, by
. Suppose that these are estimated out of
sample. Also assume for simplicity that both t + 1 and t' + 1 are periods
with low trading volume. 19 Then Hypothesis 2 can be tested by comparing
.To see this, note that
L (A17)
The covariance term is zero (since
is estimated out of sample), and the
second term on the righthand side is the same whether the trading volume
is high or low at time t, as long as the trading volume is low in period t
+ 1. Thus, Equation (A16) can be tested by comparing Clark, P. K., 1973, A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices,
Econometrica, 41,135155.
Foster, F. D.. and S. Viswanathan, 1987, Interday Variations in Volumes, Spreads and Variances: I. Theory,
Working Paper 87101, Duke University, The Fuqua School of Business, October. 19 The case in which both are periods with high trading volume is completely analogous. Otherwise, the
discussion can be modified in a straightforward manner. 39 French, K. R., and R. Roll, 1986, Stock Return Variances; the Arrival of Information and the Reaction of
Traders, Journal of Financial Economics, 17,526.
Glosten, L R., and P. R. Milgrom, 1985, Bid, Ask and Transaction Prices in a Spet Market with
Heterogeneously Informed Traders, Journal of Financial Economics, 14,71100.
Harris, L., 1986, A Transaction Data Survey of Weekly and Intraday Patterns in Stock Returns, Journal of
Financial Economics, 16,99117.
Jain, P. J., and G. Joh. 1986, The Dependence Between Hourly Prices and Trading Volume, working paper,
University of Pennsylvania, Wharton School.
Kyle, A. S., 1984. “Market Structure, Information, Futures Markets, and Price Formation,” in International
Agricultural Trade: Advanced Readings in Price Formation, Market Structure, and Price Instability, ed.
by Gary G. Storey. Andrew Schmitz, and Alexander H. Sarris. Boulder and London: Westview Press, 4564.
Kyle, A. S., 1985, Continuous Auctions and Insider Trading, Econometrica 53,13151335.
Marsh, T. A., and K. Rock, 1986, The Transaction Process and Rational Stock Price Dynamics, working
paper, Berkeley, University of California.
Williams, J., 1987, Financial Anomalies Under Rational Expectations: A Theory of the Annual Size and
Related Effects, working paper, New York University, Graduate School of Business Administration.
Wood. R. A., T. H. McInish, and J. K. Ord, 1985, An Investigation of Transaction Data for NYSE Stocks,
JournaI of Finance, 40,723741. 40...
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This note was uploaded on 02/22/2013 for the course ECON 101 taught by Professor Svendsson during the Spring '12 term at Stockholm School of Economics.
 Spring '12
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