Consider first, the “fair price” of the futures contract implied by the spot-futures parity theorem. The theorem implies that, given the assumption of well functioning com-petitive markets, a constant, annual, risk-free rate of interest rf and a cost of carry c, no arbitrage should ensure that the following relationship between the futures and spot price of the underlying commodity holds at timet:
Ft,t+k=St(1 + k
365(rf +c)), (47)
wherec ∈ [0,1]. That is, given the exploitation of arbitrage opportunities, we should have that the cost of purchasing the underlying good at price St today and holding it untilt+k (given opportunity cost of capital and cost of carry) should be equal to the current futures priceFt,t+k. Of course, this relationship implies that as the maturity date approaches (i.e. ask →0) we have thatFt,t =St.
This relationship is an approximate one and will not hold exactly in reality: indeed, the risk-free rate and the cost of carry vary in time and are uncertain, and some goods are perishable and cannot be stored indefinitely. Nevertheless this relationship is useful for considering the rolling over of futures contracts, since if we keep a given futures contract in a portfolio, its residual maturity will decrease. The formula in (47) demonstrates this effect and the need to adjust the futures price series level if we want it to maintain the same residual maturity.
Upon the approach of the futures’ maturity, we also wish to extend the price series and obtain price data for each date. In order to do so we would have to close out our current position and then open a new position in the futures contract of the next maturity.
For example, suppose we are holding a futures contract that expires at timet+k andk is approaching 0. We could sell this futures contract and purchase a new contract on the same underlying good but that expires at timet+k+j. However in doing so we would clearly incur a loss since we have that:
1 + k
365(rf +c)<1 + k+j
365 (rf +c) (48)
by the spot-futures parity theorem. This is known asrollover riskand the difference in
the two prices is called thecalendar spread.
However, this loss for the trader should not be considered as part of the overall price series historical data we use for forecasting since it represents a predictable discontinuity in the series. Therefore typically futures price series are also adjusted for this calendar spread by the data provider. There are a few ways to go about doing this, each with their pros and cons:16
1. Just append together prices without any adjustment. This will distort the series, by including spurious autocorrelation.
2. Directly adjust the prices up or down according to either the new or old contract at the rollover time period. This can be done by simply subtracting the difference between the two price series, or multiplying one of the price series by ratio of the two (i.e. absolute difference or relative difference, respectively). This method works, but it causes either the newer or older contract prices to diverge further and further from their original values as we append additional contracts. Moreover, it leaves the choice of adjustment a rather arbitrary one.
3. Continuously adjust the price series over time. This method melds together the fu-tures contract prices of both the “front month” contract (i.e. the contract with the shortest time-to-maturity) with the contracts of longer times-to-maturity (called the “back month” contracts) in a continuous manner. This allows us the potential to create a continuous contract price which reflects an “unobserved” futures con-tract which maintains a fixed time-to-maturity as time progresses. Ultimately, we are free to choose a model whereby we can reconstitute the unobserved futures contract price by employing information in the prices of observed contracts of different maturities.
Example: Smooth transition model
Consider two futures contracts on the same underlying commodity, one with time-to-maturityk, the other with time-to-maturityk+j, where we assume that their prices, Ft,t+k and Ft,t+k+j, approximately satisfy the no arbitrage condition of the spot-futures parity theorem. Moreover, let ǫi,t for i = 1,2 be error terms
16See Fulks (2000), a widely disseminated PDF document available on the world wide web. Alterna-tively, Masteika et al. (2012) provides a more recent treatment of the relevant issues.
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satisfying the standard assumptions of a regression model. The price variables Ft,t+k,Ft,t+k+j, andStare observable, as is the current risk free raterf,t. The cost of carryct, is unobservable since it includes a convenience yield, and so we must estimate it. Either way, we can then write down the model:
Ft,t+k =St(1 + k
365(rf,t+ct)) +ǫ1,t (49a) Ft,t+k+j =St(1 + k+j
365 (rf,t+ct)) +ǫ2,t (49b)
Pt =αFt,t+k+ (1−α)Ft,t+k+j (49c)
where ǫi,t represents a residual deviation away from the spot-futures parity fair value, α = Kk, where K is an upper bound on k +j (that is it represents the time to maturity when the future is first issued) andj is sufficiently large so that the difference in futures prices aren’t negligible (typicallyj ≥ 30since futures contracts of different maturities are indexed by month).
Pt, therefore, represents our estimate of the unobserved contract which incorpo-rates the information in the front and back month contracts. Since the spot-futures parity doesn’t hold exactly,Ptreflects not just the spot priceSt,the risk free rate rf,t,and the cost of carryct; but also some residual error factorsǫi,t fori= 1,2.
The Bloomberg console allows the user to specify various criteria which modify how the continuous contract price series is constructed from the front and back month contracts. Any of the 3 methods above are available for use. In constructing the price series data employed in this paper I use a method similar to (3) above but simpler in its weighting. The continuous contract futures pricePtis equal to the front month contract price Ft,t+k until the contract has 30 days left to maturity, so that k = 30. At that point, the continuous contract reflects the weighted average between the front month and the next back month contract, with the weights reflecting the number of days left until maturity of the front month contract. That is,
Pt= k
d
Ft,t+k+
k−x d
Ft,t+k+j (50)
whered = 30represents the total number of days in the month andkis the number of days remaining in the month. Oncek = 0, the price is thenPt = Ft,t+j, until this new
front month contract again has 30 days left until maturity, orj = 30. If the difference in time-to-maturity for all contracts is fixed at30days (i.e. a different contract matures every month), then this scheme represents the reconstitution of an unobserved futures contract with a fixed time to maturity of30days, as time progresses forward indefinitely.