Dynamic Hedging with Stochastic Differential Utility

4.3 EQUILIBRIUM
Assume exponential utility, such that u i (w) =−e −γ i w stands for the agent’s
Von Newmann-Morgenstern utility for terminal wealth 20 . Then, we obtain:
θ ∗ it = −e −r(T i−t) (υ t υ 0 t) −1 ·
υ t σ 0 tπ it − 1 γ i
m t¸
,
where T i represents the terminal date of agent i.
Market clearing, P I
i=1 θ it =0,impliesthat:
m t = υ t σ 0 t
P I
i=1 e−r(T i−t) π it
P I
i=1
e −r(T i −t)
γ i
P I
= υ t σ 0 i=1 ω itπ it
t P I
, (15)
ω it
i=1 γ i
where ω it ≡
e−r(T i −t)
P Ii=1
e −r(T i −t) .
First, m t is proportional to each individual spot position, whose weights
are given by ω i . If the covariance is high, that means that the futures contracts
provide a good hedge, increasing the demand for hedging. Finally m t
is proportional to the risk aversion of investors. Higher levels of risk aversion
correspond to higher m t , that is, an increasing in the pure speculative
demand.
Sufficient conditions for having m t =0are: (a) P I
i=1 ω itπ it =0,that
is, there is no excess demand for hedging; (b) υ t σ 0 t =0,inwhichcasethe
futures provide no hedge; or (c) γ i =0forsomeagenti. In case (a) agents
can costlessly insure themselves since there exists always someone who desires
to take an opposite position, and then speculators are unnecessary in this
market.
20 This section is closely similar to section 4 in DJ.
22