jumps in the continuous time series, because the prices of two different contracts move at different levels. These price jumps can have an impact on the results and may trigger spurious trading signals if technical trading rules are tested. Therefore a continuous time series must be constructed in another way.
Figure 2.1: Roll over scheme. The time axis shows the roll over dates from Dec. 1, 1993 until March 1, 1995. The arrows above the time axis show in which period which futures contract is used in constructing the continuous futures price series.
The holder of the long position in a futures contract pays a time premium to the holder of the short position. According to (2.1) the time premium paid at time t is
TPt=Ft-St=(e(rtf+ut-yt) (T-t)-1) St.
(4)
According to (2.4) the time premium that must be paid will be less when the duration of the contract is shorter other things being equal. However, (2.4) also implies that if a continuous time series of futures prices is constructed by pasting the prices of different contracts, at each pasting date3 a new time premium to the time series is added, because at each pasting date the time until expiration will be longer than before the pasting date. This time premium will create price jumps and therefore an upward force in the global price development. In fact, if the return of the underlying asset is not greater than the cost of carry a spurious upward trend can be observed in the continuous price series, as illustrated in figure 2.2, which may affect the performance of long memory trading strategies. Therefore we constructed a continuous time series of futures prices by pasting the returns of each futures contract at the roll over dates and choosing an appropriate starting value; see figure 2.2. For this continuous series, discontinuous price jumps and spurious trends will disappear and the trends will show the real profitability of trading positions in futures contracts.
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