NOISE TRADER MODELS Noise trader: speculators who extrapolate trends and downplay fundamentals Evidence for noise trading: 1. Some stocks trade at a persistent discount to their book value 2. 3. Large changes after rumour Stock market crashes Assumptions of Noise trader models: 1. Two assets: risk free asset (return = r) and a risky stock (price at time t = P) 2. Discrete time model: t = 0. 1, 2 .... 3. There is only one share of stock (#-RUM) 4. Two types of traders: fundamentalists and noise traders 5. H (D Fundamentalist Optimization Problem: at any t fundamentalists choose 1, to maximize: U())= M(r+ EPI - me(1+ r)) - NYE,(put1 - Empt+1)2 U: a mean variance optimization 1: the amount of the stock the fundamental investor wants to hold Pro Prey: Stock price at t and t+1 : risk free rate E. P: best predictor of Py at time t (forecast of Pry conditional on all information available at time t E. Pity - P, (1 + r): expected gains in excess of buying risk-free asset with P, E (Pr+1 - E, Pr+1)2: conditional variance 7 > 0: a parameter>Noise Trader Optimization Problem: at any t noise traders choose ], to maximize: V(n) = m(r+Ep+1 - p(1+ r)+ p.) - myE.(Pit - Expel) U: a mean variance optimization 7: the amount of the stock the fundamental investor wants to hold 1-X8 (ET: noise traders expect an additional return at t via pr * Optimal Portfolio Weights: r+ Ep+ - (1+r)p + p * Expression for pr: Sav 1+r At = additional return at time t p* = best predisctor of p = % of noise trader4. Consider the noise trader model of DeLong et alia. (10) a. Derive the effect on the stock price p. if there is a fall in u, the percentage of noise traders in the market; assume p > p* > 0. Also in a sentence provide an economic explanation for the effect. (10) b. Derive the effect on the stock price p: if there is a fall in conditional variance of. Also in a sentence provide an economic explanation for the effect