Dynamic Hedging with Stochastic Differential Utility

Abstract

In this paper we study the dynamic hedging problem using three different utility specifications: stochastic differential utility, terminal wealth utility, and a new utility transformation which includes features from the two previous approaches. In all three cases, we assume Markovian prices. While stochastic differential utility (SDU) has an ambiguous effect on the pure hedging demand, it does decrease the pure speculative demand, because risk aversion increases. We also show that in this case the consumption decision is, in some sense, independent of the hedging decision. In the case of terminal wealth utility (TWU), we derive a general and compact hedging formula which nests as special cases all of the models studied in Duffie and Jackson (1990). In the case of the new utility transformation, we find a compact formula for hedging which encompasses the terminal wealth utility framework as a special case; we then show that this specification does not affect the pure hedging demand. In addition, with CRRA- and CARA-type utilities the risk aversion increases, and consequently, the pure speculative demand decreases. If futures prices are martingales, then the transformation plays no role in determining the hedging allocation. Our results hold for a number of different price distributions. We also use semigroup techniques to derive the relevant Bellman equation for each case.