# Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model

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2 The market maker sets the market price, p, and then takes the position (u + ) to clear the market. It is assumed that market making is a perfectly competitive profession, so that the market maker sets the price p such that, given the total order submitted, his profit at the end of the period is epected to be zero. II. Analysis: Since the noise traders order is eogenous, we need only to consider the optimal actions of the market maker and the insider. The market maker observes only the total order flow, u +. Given this information, he must then set the equilibrium market price p that gives him zero epected profits. Since his end of period profits are ( ν p)(u+), this implies that the price set by the market maker satisfies p = E ν u + ] (1) The information on the total order size is important to the market maker. The greater the total order size, the more likely it is that is large because the insider knows ν is greater than p 0. Thus, the market maker would tend to set p higher than otherwise. Similarly, if u + is low, the more likely it is that is low because the insider knows ν is below p 0 and is submitting a sell order. In this case, the market maker would tend to set p lower than otherwise. Thus, the pricing rule of the market maker is a function of + u, that is, P ( + u). Since the insider sets, it is an endogenous variable that depends on ν. The insider chooses to maimize his epected end of period profits, π, given knowledge of ν and the way that the market maker behaves in setting the equilibrium price: ma E π ν] =ma E (ν P ( +ũ)) ν] () An equilibrium in this model is a pricing rule chosen by the market maker and a trading strategy chosen by the insider such that: the insider maimizes epected profits, given the market maker s pricing rule; the market maker sets the price to earn zero epected profits, given the trading strategy of the insider; the insider and market maker have rational epectations, that is, the equilibrium is a fied-point where an agent s actual behavior is that epected by

3 the other. Suppose the market maker chooses a market price that is a linear function of the total order flow, P (y) =µ+λy. We will later argue that a linear pricing rule is optimal. If this is so, what is the insider s choice of? From () we have ma E (ν P ( +ũ)) ν] = ma = ma Thus, the solution to the insider s problem in (3) is E (ν µ λ ( +ũ)) ν] (3) (ν µ λ), since E ũ] =0 = α + βν (4) where α = µ λ and β = λ 1. Therefore, if the market maker uses a linear pricing-setting rule, the optimal trading strategy for the insider is a linear trading rule. Net, let us return to the market maker s problem of choosing the market price that, conditional on knowing the total order flow, results in a competitive (zero) epected profit. Given the assumption that market making is a perfectly competitive profession, a market maker needs to choose the best possible estimate of E ν u + ] in setting the price p = E ν u + ]. What estimate of the mean of ν is best? The maimum likelihood estimate of E ν u + ] is best in the sense that it attains maimum efficiency and is also the minimum variance unbiased estimate. Note that if the insider follows the optimal trading strategy, which according to equation (4) is = α+β ν, then from the point of view of the market maker, ν and y ũ+ =ũ+α+β ν are jointly normally distributed. Because ν and y are jointly normal, the maimum likelihood estimate of the mean of ν conditional on y is linear in y, that is, E ν y] is linear in y when they are jointly normally distributed. Hence, the previously assumed linear pricing rule is, in fact, optimal in equilibrium. Therefore, the market maker should use the maimum likelihood estimator, which in the case of ν and y being normally distributed is equivalent to the least squares estimator. This is the one that minimizes 3

4 E ( ν P (y)) ] = E = E ( ν µ λy) ] (5) ( ν µ λ (ũ + α + β ν)) ] Thus, the optimal pricing rule equals µ + λy where µ and λ minimize min µ,λ Recalling the assumptions E ν] =p 0,E 0, the above objective function can be written as E ( ν (1 λβ) λũ µ λα) ] (6) (ν p 0 ) ] =Σ 0,Eu]=0,E u ] =σ u,andeuν] = min µ,λ (1 λβ) ( Σ 0 + p 0) +(µ+λα) + λ σ u (µ+λα)(1 λβ) p0 (7) The first order conditions with respect to µ and λ are µ = λα + p0 (1 λβ) (8a) ( ) β (1 λβ) Σ 0 + p 0 +α(µ+λα) +λσ u p0 β (µ + λα) +α(1 λβ)] = 0 (8b) Substituting µ + λα = p0 (1 λβ) from (8a) into (8b), we see that (8b) simplifies to λ = βσ 0 β Σ 0 + σ u (8b) Substituting in for the definitions α = µ λ and β = 1 λ in (8a) and (8b), we have3 µ = p0 (9a) λ = 1 σ u (9b) 3 Note the typo in the solution for in Theorem 1of the Kyle paper. 4

5 In summary, the equilibrium price is p = p0 + 1 (ũ + ) (10) where the equilibrium order submitted by the insider is = σ u ( ν p0) (11) From (11), we see that the greater is the volatility (amount) of noise trading,, the larger is the magnitude of the order submitted by the insider for a given deviation of ν from its unconditional mean. Hence, the insider trades more actively on his private information the greater is the camouflage provided by noise trading. More noise trading makes it more difficult for the market maker to etract the signal of insider trading from the noise. Note that if equation (11) is substituted into (10), one obtains p = p0 + 1 ũ + 1 0) (1) ( ν p ( ) = 1 σ u ũ + p0 + ν Thus we see that only one-half of the insider s private information, 1 ν, is reflected in the equilibrium price, so that the price is not fully revealing. (A fully-revealing price would be p = ν.) To obtain an equilibrium of incomplete revelation of private information, it was necessary to have a second source of uncertainty, namely, the amount of noise trading. Using (11) and (1), we can calculate the insider s epected profits: E π] =E(ν p)] = E ( ν p0) 1 ( ν p0 Conditional on knowing ν, that is, after learning the realization of ν at the beginning of the period, the insider epects profits of ũ )] (13) E π ν] = 1 (ν p 0 ) (14) 5

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