C. The Nature of EquilibriumWith all of the assumptions in place, we are now ready to solve the model.
The only task is to calculate the equilibrium value of φ. Disregarding con-
stants, optimization on the part of the momentum traders implies:
φ∆Pt− 1 =γEM(Pt+j−Pt)/varM(Pt+j−Pt) (6)where γis the aggregate risk tolerance of the momentum traders, and EM
and varMdenote the mean and variance given their information, which is
just ∆Pt− 1. We can rewrite Eq. (6) as:
φ=γcov(Pt+j−Pt, ∆Pt− 1 )/{var(∆P)varM(Pt+j−Pt)} (7)
The definition of equilibrium is a fixed point such that φis given by Eq. (7),
while at the same time price dynamics satisfy equation (5). We restrict our-
selves to studying covariance-stationary equilibria. In the appendix, we
prove that a necessary condition for a conjectured equilibrium process to
be covarian cestationary is that φ<1. Such an equilibrium may not exist
for arbitrary parameter values, and we are also unable to generically rule
out the possibility of multiple equilibria. However, we prove in the appen-
dix that existence is guaranteed so long as the risk tolerance γof the
momentum traders is sufficiently small, since this in turn ensures that φis
sufficiently small. Moreover, detailed experimentation suggests that a unique
covariance-stationary equilibrium does in fact exist for a large range of the
parameter space.^12
In general, it is difficult to solve the model in closed form, and we have to
resort to a computational algorithm to find the fixed point. For an arbitrary
set of parameter values, we always begin our numerical search for the fixed
point at j=1. Given this restriction, we can show that the condition φ<1 is
both necessary and sufficient for covariance-stationarity. We also start with a
small value of risk tolerance and an initial guess for φof zero. The solutions
in this region of the parameter space are well-behaved. Using these solutions,
we then move to other regions of the parameter space. This procedure en-
sures that if there are multiple covariance-stationary equilibria, we would al-
ways pick the one with the smallest value of φ. We also have a number of
sensible checks for when we have moved outside the covariance-stationary
region of the parameter space. These are described in the appendix.
Even without doing any computations, we can make several observations
about the nature of equilibrium. First, we have:
510 HONG AND STEIN
(^12) Our experiments suggest that we only run into existence problems when boththe risk
tolerance γand the information-diffusion parameter z simultaneouslybecome very large—
even an infinite value of γposes no problem so long as zis not too big. The intuition will be-
come clearer when we do the comparative statics, but loosely speaking, the problem is this: as
zgets large, momentum trading becomes more profitable. Combined with high risk tolerance,
this can make momentum traders behave so aggressively that our φ<1 condition is violated.