Admati And Pfleiderer-A Theory Of Intraday Patterns - Volume And Price Variability

in our model prices are a martingale so patterns in

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 zero-profit 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 right-hand 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, 41,135-155. Foster, F. D.. and S. Viswanathan, 1987, Interday Variations in Volumes, Spreads and Variances: I. Theory, Working Paper 87-101, 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, 17,5-26. 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, 14,71-100. Harris, L., 1986, A Transaction Data Survey of Weekly and Intraday Patterns in Stock Returns, Journal of Financial Economics, 16,99-117. 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 53,1315-1335. 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, 40,723-741. 40...
View Full Document

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.

Ask a homework question - tutors are online