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

# Suppose that all discretionary liquidity traders

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: cntually dominates, lowering λ t. 11 Proof. Define that is, the total variance of discretionary liquidity demands. Suppose that all discretionary liquidity traders trade in period t and that the market maker adjusts λ t, and informed traders set β t accordingly. Then the total trading cost incurred by the discretionary traders is λ t(h)h, where λ t(h) is given in Lemma 1 with Consider the period t* ∈ [T', T"] for which X,(b) is the smallest. (If there are several periods in which the smallest value is achieved, choose the first.) It is then an equilibrium for all discretionary traders to trade in t*. This follows since X,(b) is decreasing in h, so that we must have by the definition of t*, λ t(0) ≥λ t.(h) for all t ∈ [T', T"]. Thus, discretionary liquidity traders prefer to trade in period t* . The above argument shows that there exist equilibria in which all discretionary liquidity trading is concentrated in one period. If there is an equilibrium in which trading is not concentrated, then the smallest value of A, must be attained in at least two periods. It is easy to see that any small change in var for some j would make the λ t different in different periods, upsetting the equilibrium. n Proposition 1 states that concentrated-trading patterns are always viable and that they are generically the only possible equilibria (given that the market maker uses a linear strategy). Note that in our model all traders take the values of λ t as given. That is, when a trader considers deviating from the equilibrium strategy, he assumes that the trading strategies of other traders and the pricing strategy of the market maker (i.e., λ t) do not change.9 One may assume instead that liquidity traders first announce the timing of their trading and then trading takes place (anonymously), so that informed traders and the market maker can adjust their strategies according to the announced timing of liquidity trades. In this case the only possible equilibria are those where trading is concentrated. This follows because if trading is not concentrated, then some liquidity traders can benefit by deviating and trading in another period, which would lower the value of λ t in that period. We now illustrate Proposition 1 by an example. This example will be used and developed further in the remainder of this article. Example. Assume that T =5 and that discretionary liquidity traders learn of their demands in period 2 and must trade in or before period 4 (i.e., T' = 2 and T" = 4). In each of the first four periods, three informed traders trade, and we assume that each has perfect information. Thus, each observes in period t the realization of . We assume that public information arrives at a constant rate, with var( δ ) = 1 for all t. Finally, the variance of the nondiscretionary liquidity trading occurring each period is set equal to 1. 9 Interestingly, when nt = 1 the equilibrium is the same whether the informed trader ties λ t as given or whether he takes into account the effect his trading policy has on the market maker’s determination of A,. In other words, in this model the Nash equilibrium in the game between the informed trader and the market maker is identical to the Stackelberg equilibrium in which the trader takes the market maker’s response into account. 12 We are interested in the behavior of the discretionary liquidity Faders. Assume that there are two of these traders, A and B, and let var(YA) = 4 and var( YB) = 1. First assume that A trades in period 2 and B trades in period 3. Then λ1 = λ4, = 0.4330, λ2, = 0.1936 and λ3, = 0.3061. This cannot be an equilibrium, since λ2, < λ3, so B will want to trade in period 2 rather than in period 3. The discretionary liquidity traders take the λ ’s as fixed and B perceives that his trading costs can be reduced if he trades earlier. Now assume that both discretionary liquidity traders trade in period 3. In this case λ1, = λ2, = λ3, = 0.4330 and λ3 = 0.1767. This is clearly a stable trading pattern. Both traders want to trade in period 3 since λ3, is the minimal λ t. 1.3 Implications for volume and price behavior In this section we show that the concentration of trading that results when some liquidity traders choose the timing of their trades has a pronounced effect on the volume of trading. Specifically, the volume is higher in the period in which trading is concentrated both because of the increased liquidity-trading volume and because of the induced informed-trading volume. The concentration of discretionary liquidity traders does not affect the amount. of information revealed by prices or the variance of price changes, however, as long as the number of informed traders is held fixed and is specified exogenously. As we show in the next section, the results on price informativeness and on the variance of price changes are altered if the number of informed traders in the market is determined endogenously. It is clear that the behavior of prices and of trading volume is determined in part by the rate of public-informat...
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