The Time Pattern of Hedging and the Volatility of Futures Prices

Abstract

The paper proposes a multi-period model of hedging which allows for a futures position to be revised within the cash holding period. Within this framework, we assess the robustness of the two-period theory of hedging when generalized to many periods. We characterize the normal time path of a hedge and the way it is affected by the requirement that futures accounts mark to market daily. Finally we show how the resolution of production uncertainty over time affects hedging behavior and determines the volatility of futures prices. In the large literature on hedging in futures markets the principal analytical accounts of hedging are framed in a two trade date setting. An agent commits himself now to a future purchase or sale of a good whose cash price later is uncertain. This risk is hedged by purchase or sale of an appropriate number of futures contracts and holding this futures position until the cash transaction is realized. In the present paper we formulate a multi-period model of hedging which allows for the futures position to be revised within the cash holding period. The resulting analysis captures an important institutional reality that is absent in a two trade date model, namely the fact that the day to day fluctuations of the futures price will generate a series of random cash flows over the period during which a futures position is held. This is due to the requirement that futures accounts mark to market daily. Within this framework we address a number of issues. First we assess the robustness of the two period theory of hedging when generalized to many periods. Second we characterize the normal time path of a hedge and contrast futures and forward contracts from the perspective of an individual hedger. Finally, we show how the resolution of production uncertainty over time affects hedging behaviour and determines the volatility of futures prices. While the literature on asset demand in a multiperiod setting is large, futures markets are specifically treated in a limited number of studies. Grauer and Litzenberger (1976) are concerned principally with characterizing the relation of spot and futures prices in a for a storable, periodically produced commodity. They offer a rationale for positive stock holding in markets where spot price exceeds the futures price. Richard and Sundaresan (1980) are also concerned primarily with characterizing price behaviour. Since they posit identical consumers, in equilibrium individual holdings of futures are zero. There is no hedging activity in the sense that we understand it. Breeden (1980b) analyzes the optimal futures position sizes in a multiperiod setting. His framework is more general than ours in that his is a multi-good, multi-asset model where preferences