"Technical Analysis in Financial Markets"

(t,T] at time t with continuous compounding. The convenience yield can be described as the utility of having the asset in stock. The term (r_t^f+u_t-y_t) is called the cost of carry and (2.1) is called the cost of carry relationship. The daily return r_t^F of the futures contract, expressed as the log difference, is given by

r_t^F=r_t^S+(Δ r_t^f+Δ u_t-Δ y_t) (T-t)-(r_t-1^f+u_t-1-y_t-1). (2)

This formula shows that a change in one of the factors of the cost of carry has an impact on the futures price. Otherwise, the return of a futures contract is equal to the excess return of the underlying asset over the cost of carry.

Assume that we have two futures contracts, 1 and 2, with futures prices F_t⁽¹⁾ and F_t⁽²⁾ and expiry dates T₂ > T₁. It follows from (2.2) that two futures contracts traded in the same period have the same trends in prices. The futures price of contract 2 can be expressed in terms of the futures price of contract 1 as

F_t⁽²⁾=F_t⁽¹⁾ e^{(r_t^f+u_t-y_t) (T₂-T₁)}. (3)

Notice that r_t^f, u_t and y_t are numbers averaged over (t,T]. Thus in equation (2.3) it is assumed that r_t^f, u_t and y_t averaged over (t,T₁] is equal to r_t^f, u_t and y_t averaged over (t,T₂]. Formula (2.3) shows that if, as is usual, the cost of carry is positive, the futures price of contract 2 which expires later is higher than the futures price of contract 1 which expires earlier. But if the utility of having an asset in stock is high, e.g. when there is a shortage of the commodity in the short run, then the futures price of contract 2 can be lower than the futures price of contract 1. Thus the prices of different futures contracts can move at different price levels.

A long continuous time series of futures prices will be constructed, in order to be able to test technical trading strategies with long memory. The continuous time series must be constructed out of the many price series of the different futures contracts that have the same price trends, but move at different price levels. In particular roll over dates must be defined at which the price movements of the different contracts are pasted together. In practice most trading occurs in the second nearest contract, that is the futures contract that has the one but nearest expiration date. We investigated the liquidity of the cocoa futures contracts and decided to take as roll over dates the date one month before most of the practitioners switch to the next contract, so that the continuous time series always represents a high liquidity futures contract. Figure 2.1 exhibits graphically the roll over procedure used in this chapter.

Murphy (1986) suggests pasting the prices of two successive futures contracts to study price movements over a long period of time. But the pasting of prices will introduce price