Dynamic Hedging with Stochastic Differential Utility

Of course, replacing equation 17 into equation 7 gives:
θ TWSDU
jt
= −e −r(T i−t) (υ t υ 0 t) −1 υ t σ 0 t
6 CONCLUSIONS
"
π jt − R −1
jt
P I
i=1 ω itπ it
P I
i=1 ω itRit
−1
In this paper we study the dynamic hedging problem using three different
utility specifications: stochastic differential utility, terminal wealth utility,
and a proposed utility which links both approaches. In all cases, we assume
Markovian prices, as in Adler and Determple (1988). As a consequence of
this assumption, we escape from a myopic hedging problem at each time.
Furthermore, depending on the specification of the utility function, we must
use different Hamilton-Jacobi-Bellman, HJB, equations.
Stochastic differential utility, SDU, where we maximize consumption over
time as in Ho (1984), impacts the pure hedging demand ambiguously, because
SDU parameters add both in the denominator and the numerator of the
optimal ratio.
#
.
We see that SDU decreasesthepurespeculativedemand,
because risk aversion increases.
We also show that consumption decision
is independent of the hedging decision in the sense that we can split the
program into two independent programs, one for the consumption and the
other for the optimal hedging. In this case, if the drift of futures prices is
zero, there is no obvious impact on the optimal hedge.
Inthesecond-typeutilitycase,wederive a general and compact hedging
formula, which nests all cases studied in Duffie and Jackson (1990). This formula
may include the following particular assumptions found in DJ: Gaussian
28