Given the existence of the newswatchers and the underreaction that they
create, it is certainly natural to beginan examination of simple arbitrage
strategies with the sort of momentum-trading style that we have considered
thus far. However, once it is understood that the momentum traders must—
if they are the only arbitrageurs active in the market—ultimately cause
prices to overreact, we then ought to think about the effects of second-round
“contrarian” strategies that might be designed to exploit this overreaction.
To incorporate such contrarian strategies into our model, we assume that
there is a total risk tolerance of γavailable to engage in arbitrage activity. We
also continue to assume that all arbitrageurs have horizons of jperiods. But
there are now two arbitrage styles. A fraction wof the arbitrageurs are mo-
mentum traders, who use ∆Pt− 1 to forecast (Pt+j−Pt). The remaining (l−w)
are contrarians, who use ∆Pt− 1 −cto forecast (Pt+j−Pt). If we choose the lag
length cproperly, the contrarians will in equilibrium put negativeweight on
∆Pt− 1 −cin making these forecasts.
Suppose provisionally that one takes the fraction was fixed. Then the
equilibrium is a natural generalization of that seen above. In particular,
prices will be given by:
(9)PD z z zPPtt t t tzM
tiC
tci
ij=+− +− +⋅⋅⋅++++++−−−−
=∑
{( ) ( ) }/()(^12) 12 1
1
εεε
φφ∆∆
A UNIFIED THEORY OF UNDERREACTION 517
Figure 14.3. Cumulative impulse response and the information diffusion parameter.
The information diffusion parameter z takes on values of 3, 6, 9, and 12. The other
parameter values are set as follows: momentum traders’ horizon jis 12, the volatil-
ity of news shocks is 0.5, and momentum traders’ risk tolerance gamma is 1/3.