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9. The Informed Trader’s Problem: Profit Maximization • Kyle's Bayesian Learning model assumes that informed investor demand x can be expressed as a simple linear function of v: x = α + βvwhere α and β are simple coefficients. • Similarly, the security’s price p, set by the market maker or dealer, is also assumed to be a simple linear function of demand: p = μ + λ(x + u), where μ and λ are also simple coefficients. • Thus, informed investor demand x is a linear function of true security value v and the security price p is a linear function of the sum of informed and uninformed investor demand X = (x + u). • The informed trader’s problem is to determine the optimal purchase (or sale) quantity x: (1) • To maximize the informed trader’s profits,: (2) • Note that λ must be positive for the second order condition (the second derivative must be negative to hold for maximization. We rearrange terms to obtain: (3) (4) • which is linear in v as Kyle proposed it would be. Now, we see that our coefficients α and β are simply: • Thus, the informed trader linear demand function x = α + βv is set based on the dealer pricing function.

10. Dealer Price Setting • We now explore our coefficients α and β. • The dealer observes total order flow X = x + u and sets a single market clearing price p= E[v│x + u] where x = α + βv. • Since v and X are normally distributed, we apply the Projection Theorem to p: (5) • This dealer pricing function has a straightforward interpretation. • The sensitivity of the dealer price to total share demand is a function of the covariance between the stock’s value and total demand for the shares. • Thus, if the dealer believes that total demand for the stock increases dramatically with its intrinsic value v (unknown to him, but known to the informed trader), the price that the dealer sets for shares will be very sensitive to total demand. • If the informed trader dominates trading, the dealer will set the price of the security mostly or entirely as a function of total demand for the security. • Sensitivity to total demand will diminish as uninformed demand volatility increases. • As total demand deviates more from expected demand, share prices will increase.

11. Informed Trader Demand and Dealer Price Adjustment • Since x = α + βv, and Σ0 is the variance of asset payoffs v, the variance of informed trader demand VAR[x] will equal 20. This means that VAR[x+u] = Σ0 + u2, which means that the dealer pricing equation is: (6) • β is the slope of a line plotting a random dependent variable X or (x + u) with respect random variable v: (7) • Kyle suggested a linear relationship between the security price and its demand: p = μ + λ(x + u). This implies a slope λ equal to: (8) • which implies μ = p + λ(-x - u) and: (9) • Next, we will use α and β coefficients from above to demonstrate that μ = E[v]: (10) • Now, we will rewrite λ, substituting in for β: (11)

12. Solving for Demand and Pricing Coefficients • We for λ, starting by multiplying λ and the right hand side of equation 11 by the denominator of the right hand side of the equation: (12) • Simplify the left hand side, then multiply both sides by λ and simplify further by subtracting 1/4Σ0 from both sides: (13) (14) (15) (16) • λ is positive and that our second order condition for profit maximization has been fulfilled. •  is the dealer price adjustment for total stock demand or the illiquidity adjustment. • Σ0/u2 is the ratio of informed trader private information resolution to the level of noise trading. • The dealer price adjustment is proportional to the square root of this ratio, increasing as private information Σ0 is increasing and decreasing as noise trading increases. This means that if the dealer determines that the informed trader resolves a substantial level of risk relative to the amount of noise trading, the level of dealer price adjustment will be large.

13. Informed Trader Demand • Informed investor demand coefficients α and β are:  The informed trader demand function is: (17) • Informed trader transaction sizes will increase as the variance of uninformed noise demand for shares σu2 increases. • This increased noise volume will better enable informed traders to camouflage their information advantage over the dealer. • As the informed trader’s information improves relative to the dealer, the informed trader will seek to camouflage his advantage by reducing his trade volume. The informed trader will earn his profits by maintaining more profit on a per share basis rather than on transaction volume.

14. Dealer Price Setting • Recall from equation (7) that the market maker sets the price at p, which we will rewrite using the result of equation 16: (18) (19) p = μ + λ(x + u) = E[v] + (x+u) • Notice that the dealer price p is the security’s expected value E[v] conditioned on total demand [x+u] for the security. • Higher noise or uninformed trader uncertainty reduces the security price unless total demand (x+u) is negative. • The opposite is true for value or cash flow uncertainty, which increases the informed trader's informational advantage. • The market maker sets the price at p, such that the informed trader buys (sells) whenever v > E[v] (v < E[v]), and buys (sells) more aggressively as this difference increases.

16. Illustration: Kyle's Adverse Selection Model • Suppose that the unconditional value of a stock in a Kyle framework is normally distributed with an expected value equal to E[v] = \$50 and a variance equal to Σ0 = 30. • An informed trader has private information that the value of the stock is actually v = \$45 per share. • Uninformed investor trading is random and normally distributed with an expected net share demand of zero σu2 equal to 5,000. • The dealer can observe the total level of order volume X = x + u where u reflects noise trader transactions and x reflects informed demand, but the dealer cannot distinguish between x and u. • The ability of the informed trader to camouflage his activity is directly related to σu2 and inversely related to x. • What would be the level of informed trader demand for the stock? We solve for x as follows, using equation 17:

17. Informed Demand and Dealer Pricing Coefficients • The informed trader would wish to sell an infinite number of shares to earn a \$5 profit on each, but cannot because the dealer would correctly infer that his share sales convey meaningful information, and the dealer's price revisions would lead to slippage. • Thus, at what level does the dealer set his price, given the total demand X = [x + u] = -64.5497 + 0 that he observes? First, we solve for parameters in the dealer pricing equation: .

18. The Dealer Price • The Dealer sets her price as follows: