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 D t+1 in t+1, D t+1 =F t +G t, P t 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 Z t (i) unit of non-traded asset that pays N t+1 per unit in subsequent period.

4 3. Investors observe stock dividend D t, price P t, payoff of non-traded asset N t, own endowment of non-traded asset Z t, and stock dividend F t. Informed investors observe G t. I 1 t = { D, P, N, F, G, Z ’ }, I t 2 = { D, P, N, F, Z 2 } All stocks are normal with mean zero and constant variance, and mutually independent except E[D t N t ] =σ DN. 4. Investors max. Expected U. E [-e -λw t+1 ∣ I t i ] 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, b 1, b 2, σ R (1)2, σ R (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 θ 1 =0 and （ ii ） For (small) and, constants. and,

10 B. Return-Order Imbalance Theory

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

12 3. A discretionary liquidity trader with a 2Z 1, either split his demands equally among two periods on trade in period 1 or 2. A nondiscretionary liquidity trade of Z 2 at date 2. Z 1,Z 2 ~N(0,v ε ) 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 v ε is 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(P 2 —P 1,Q 1 )/Var(Q 1 )>0. And this coefficient is increasing in the risk aversion coefficient R. 3. The expectation of the price changes, P 2 —P 1 conditional on the contemporaneous and lagged imbalances, Q 2 and Q 1 respectively, is linear in these variables. The coefficient of Q 2 is positive while that of Q 1 is negative.