Chapter 3

1. Chapter 3 Secondary Market Making ---Order Imbalance Theory and Strategies

2. A. Information Asymmetry in return-volume

3. (A)Assumptions 1.Two trade securities, risk-free bond and stock. Bond unlimited supply at a constant nonnegative interest rate, r. Stock pays dividend Dt+1 in t+1, Dt+1=Ft+Gt,Pt ex-dividend price. 2.Two classes of investors, 1 and 2. with population weights of w and 1-w. Each investor endowed shares of stock and income from non-traded assets. In period t investor i has Zt(i) unit of non-traded asset that pays Nt+1 per unit in subsequent period.

4. 3.Investors observe stock dividend Dt , price Pt , payoff of non-traded asset Nt, own endowment of non-traded asset Zt , and stock dividend Ft . Informed investors observe Gt . I1t = { D, P, N, F, G, Z’ }, It2 = { D, P, N, F, Z2 } All stocks are normal with mean zero and constant variance, and mutually independent except E[DtNt] =σDN . 4.Investors max. Expected U. E [-e-λwt+1∣Iti] risk aversion λ=1 5.Trades are of hedging trades and speculative trades.

5. (B)Model 1.Equilibrium price and volume • Proposition I: The Economy defined above has an equilibrium in which investor i’s stock holding is

6. and the ex dividend stock price is where and a, b1, b2, σR(1)2, σR(2)2 and r are constants. Investors trade for expectation of future stock return or his exposure to non-trade risk. Trade volume V,

7. 2.Dynamic relation between return and volume （i）Return generated by public information Investors demand unchanged → white noise （ii）Return generated by hedging Investors demand changed → Sell, current period return↓, next period return↑ Buy, current period return↑, next period return↓ （iii）Return generated by speculation Investors demand changed → Sell, next period return↓, current period return↓ Buy, next period return↑, current period return↑

8. Proposition II: （i） where, , volume normalized by unconditional mean. constants. （ii） + higher-order terms in where,

9. Proposition III: （i）For θ1=0 and （ii）For (small) and , constants. and ,

10. B. Return-Order Imbalance Theory

11. Assumptions 1. A security trades at date 1 and 2, liquidation payoff 2. Two types of utility maximizing traders：informed traders learned realization ofθ prior to trade at date 2 and uninformed “market makers” without knowledge of θ . No agent learnedε.

12. 3. A discretionary liquidity trader with a 2Z1, either split his demands equally among two periods on trade in period 1 or 2. A nondiscretionary liquidity trade of Z2 at date 2. Z1 ,Z2~N(0,vε) are mutually independent and independent of θandε. 4. The mass of informed traders is M and market makers 1-M. Both have negative exponential utility, risk aversion R.

13. (B) Model 1.Equilibrium prices： 2. Holdings： 3. Market Equilibrium：

14. Lemma 1： Given that the discretionary liquidity trader split his order across periods, the unique linear equilibrium is Order Imbalances：

15. Proposition I： As long as the long term risk from holding the asset vεis sufficiently high, the following results hold： 1. In equilibrium, the discretionary liquidity trader split his order across the two period, so that equilibrium order imbalances are positively autocorrelated. 2. Lagged imbalances are positively related to price changes, Cov(P2—P1,Q1)/Var(Q1)>0 . And this coefficient is increasing in the risk aversion coefficient R. 3. The expectation of the price changes, P2—P1 conditional on the contemporaneous and lagged imbalances, Q2and Q1 respectively, is linear in these variables. The coefficient of Q2 is positive while that of Q1 is negative.