Dynamic Hedging with Stochastic Differential Utility

Proof. Differentiate inside the expectation again:
"
∂ 2 J
= E U ¡ 00 W θt
∂w 2 T −t
"
∂ 2 J
∂x 2 = E
U 00 ¡ W θt
T −t
·
∂ 2 J
∂w∂x = E U ¡ 00 WT θt
−t
¢ µ ∂W θt
T −t
∂w
¢ µ ∂W θt
¢ ∂W θt
T −t
∂x
T −t
∂w
2
+ U ¡ #
¢ 0 WT θt ∂ 2 WT θt
−t
−t
∂w 2
2
+ U ¡ #
¢ 0 WT θt ∂ 2 WT θt
−t
−t ;
∂x 2
∂WT θt
−t
+ U ¡ ¢ 0 WT θt ∂ 2 WT θt
−t
−t
∂x
∂w∂x
Since all the terms that multiply U ¡ 0 WT −t¢ θt are zero, the result follows by
applying Proposition 3.
;
¸
.
Observe that all second derivatives are negative, since U ¡ WT −t¢ θt is assumed
concave.
Now if we replace the relationships that we obtained in the optimal hedging
equation, we get the following more compact hedge ratio:
Proposition 4 Given our assumption on the budget constraint and utility,
the optimal hedging ratio is given by
θ TV
t = − exp [−r (T − t)] (υ t υ 0 t) −1 ·
υ t σ 0 tπ + J w
J ww
m t¸
. (11)
Proof. Replace terms.
The main discovery is that the first and big term outside the brackets
reduces to a deterministic expression, which does not depend upon the form
of the value function. Also, adding a discount factor in the model would
change the HJB equation accordingly, but would not have effects on the
18