Dynamic Hedging with Stochastic Differential Utility

of the wealth or because the margin account, R increases as in the standard
case. Furthermore, notice that the SDU utility adds a penalty through the
presence of the additional term in the numerator: If the curvature of the
value function increases, both numerator and denominator increase, making
the net effect unknown (in the standard case, risk aversion would decrease.)
The sign of R does not change under SDU. In short, the global net effect
is that the extended risk aversion increases because of the presence of the
mentioned term in the denominator. This is, of course, an implication from
Proposition 6 (Comparative Risk Aversion) in Duffie and Epstein (1992).
With this in mind, Duffie (1989) arguably calls the first term between
brackets in equation 7 as the pure hedge demand, andthesecondtermas
the pure speculative demand 15 . The term pure hedge demand comes from a
uniperiod model, where we want only to minimize risks, that is, we minimize
the variance of our position, without preoccupations with the return. In this
case, θ t =(υ t υ 0 t) −1 υ t σ 0 tπ t (see additional discussion in Section 4.1).
Ifthespotcommitmentiszeroattimet, the hedger may still be willing to
buy futures through the pure speculative demand term. This also would be so
if covariance between spot and futures prices were null - υ t is orthogonal to σ t
-, meaning that futures contracts do not provide any protection against spot
price fluctuations 16 . Also, if the covariance between futures and spot prices
increases in absolute value, one increases the position under hedge, since the
role of protecting against undesirable fluctuation in prices increases.
15 Adler and Detemple (1988) call them, respectively, as the Merton/Breeden informationally
based hedging component, and as the mean-variance component.
16 Here we are not taking into account equilibrium concerns.
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