Dynamic Hedging with Stochastic Differential Utility

The practical advantage is to produce neat formulas for our problem. In
theoretical grounds we are assuming that we can add the certainty equivalent
in the HJB equation when we are maximizing the utility of the final wealth.
Our procedure lies in the remarkable consequence of the forward-looking
nature of the Bellman equation under SDU, dispensing with state variables
reflecting past decisions (see Duffie and Epstein, p. 373, 1992).
Then, the optimal hedging ratio is given by:
Proposition 8 The optimal futures position strategy is θ TWSDU , where
(J ww + J wx )+k(J)(Jw 2 + J w J x )
t = −
(J ww +2J wx + J xx )+k(J)(Jw 2 +2J w J x + Jx) × 2
θ TWSDU
(υ t υ 0 t) −1 ·υ t σ 0 tπ t +
J w + J x
(J ww + J wx )+k(J)(J 2 w + J w J x ) m t
¸
.
Proof. In this case the relevant HJB 22 is
A d J + 1 2 k(J) ¡ J 0 z t
ΣJ zt
¢
=0.
Then the proof follows exactly the same lines to derive equation 7.
This expression is similar to the formula found in Section 3. In addition,
because this is a concave transformation,thelemmasprovedinSection4
must hold. We just point out the following:
22 See appendix C for a proof.
24