
INTRODUCTION
Commodity futures markets fulfill the key economic functions of allowing for hedging and price discovery. In these markets, two important questions arise.
First, are futures prices interconnected across the maturity curve? In theory, they should be linked through the costofcarry relationship. In practice, such market integration requires crossmaturity arbitrage. Buyiiksahin et al. (2009), however, document that even the three largest U.S. commodity futures markets did not witness substantial trading activity in longerdated derivatives until 20032004 (crude oil) or later (corn and natural gas). This empirical fact suggests the possibility of changes in crossmaturity linkages in the past fifteen years.
Second, assuming that differentmaturity futures prices are interconnected, where in the term structure do price shocks originateand which other parts of the term structure do they reach? Is the direction of the shocks' propagation stable over time? A conventional view takes the physical market as the place where the absolute (or "flat") price emerges as a function of the supply and demand for the underlying asset. In turn, the derivative market allows for relative pricing: futures prices derive from the spot price. Price shocks should thus spread from the underlying asset to the derivative instrument. Yet, amid a massive increase in fardated commodity futures trading after 2003, might one not expect to also observe price shocks propagating from the far end to the short (physical) end of the futures curve?
There is, to our knowledge, no theoretical model studying these questions in a setting where multiple maturities of futures contracts trade simultaneously. In this paper, we investigate market integration and shock propagation empirically through the prism of the theory of information.
We introduce the concepts of "forward" and "backward" information flows across the term structure, and we rely on net transfer entropies to construct a novel type of directed graph linking all the parts of the term structure. Our laboratory is the New York Mercantile Exchange's (NYMEX) market for West Texas Intermediate (WTI) light sweet crude oil futures. This market provides an ideal setting for our analysis: among all commodity markets in 20002017, WTI futures boast the highest level of trading activity and the greatest number of farout delivery dates.
The theory of information, first outlined in a seminal paper by Shannon (1948), studies the measure, storage, and quantification of information. A key concept, in this theory, is "entropy"the amount of uncertainty associated to a random variable. In our paper, the information entropy H([R.sub.t]) of the price return R on a crude oil futures contract of maturity r is a quantity that captures the degree of uncertainty associated to [R.sub.t]. In other words, H([R.sub.t]) measures how much we don't know about that oil futures' price returns.
In this setting, we use the concept of "mutual information" to investigate futures market integration across maturities. When two variables are interdependent (as is the case, for example, for two times series of futures price returns with two different maturities), the mutual information measure gives the amount of entropy that is reduced (i.e., the amount of uncertainty that is resolved) compared to the case where the two variables are independent (Shannon and Weaver (1949); Schreiber (2000)). Computing the mutual information between contracts is thus analogous to assessing their integration or return comovements. In contrast to other methods such as Pearson correlations, this probabilistic approach does not require making any assumption about the functional form of the relationship between the variables under consideration.
We find substantial variations over time in the amount of mutual information shared by crude oil futures with different delivery dates. In general, intermediatematurity contracts (six months to two years) share relatively more mutual information than other contracts do. For all contracts, crossmaturity mutual information increases dramatically after 2003 (amid tight oil supply conditions, a dramatic growth in backdated crude oil futures trading, and the onset of commodity markets' financialization) and reaches a peak at the top of the oil price boom in Summer 2008. It falls back sharply in 2012 (to pre2005 levels) and drops further in 2013 and 2014 (to pre2002 levels). It has since recovered dramatically. Taken together, these termstructure findings point to a puzzling resegmentation by maturity of the WTI market in 20122014.
We also investigate the propagation of price shocks across the futures term structure, relying for this purpose on the concept of "transfer entropy" (Schreiber (2000)) that allows for dynamic analyses and for the determination of directionality. It enables us to answer the following question: does a shock to the price return of a futures contract with maturity r at time t beget a shock at time t+\ to a futures contract with another maturity? Determining directions is important, as it allows us to ascertain whether price shocks evolve from shortterm to longterm maturities or vice versa. Focusing on directionality relates our work to extant studies of Granger (1969) causalitybut in a nonparametric world. The technique can also be compared to the exploration of volatility spillovers (as in Adams and Gluck (2015) or Jaeck and Lautier (2016)) while allowing for nonlinearitiesa crucial advantage given ample evidence that the dynamics of crossmarket contagion (i.e., shock propagation) are nonlinear. (1)
On average across our 20002017 sample period, we find that contracts with maturities up to 21 months emit more information entropy than do more backdated contractsa pattern consistent with the traditional view of how futures market function. A dynamic analysis, however, reveals that the amount of entropy emitted by other parts of the curve is nontrivial and can be high at times. Moreover, the directions of the entropy transfers (from neardated to fardated contracts or vice versa) is not stable over time. In particular, an analysis of information entropy flows that run "forward" (i.e., from neardated to furtherout maturities) vs. "backward" (i.e., from backdated to nearerdated contracts) shows that the backward flows are actually higher than their forward counterparts in 2008, i.e., during a 12month period encompassing the peak of the 20072008 oil price boom and the subsequent price collapse after the Lehman crisis.
Finally, we utilize those nonparametric measures to define a metric that allows us to build an original type of directed graphs. The latter complement our other tests and provide a powerful visualization toolas well as a means to detect anomaliesfor our highdimensional data. Indeed, insofar as all the futures prices that we study create a system, the latter is complex: it comprises many components that may interact in various ways through time. To wit, on any day in our sample, after discarding illiquid maturities, there remain 33 different WT1 futures delivery dates: hence, we have 528 pairs of maturities to examine after accounting for directionality. Moreover, such linkages may change through time as a result of evolving market conditions or trading practices. Finally, chances are few that the relationships between different maturities are always linear.
A graph gives a representation of pairwise relationships within a collection of discrete entities. Each point of the graph constitutes a node (or vertex). In our analysis, a node corresponds to the time series of price returns on a futures contract for a given maturity over a specified period of time. The links ("edges") of the graph can then be used in order to describe the relationships between nodes. More precisely, the graph can be weighted in order to take into account the intensities and/or the directions of the connections. We do both on the basis of information theory.
There are several ways to enrich the links of a graph. In the finance literature on commodity markets, for example, the connections between the nodes have been tied to the correlations of returns (e.g., Lautier and Raynaud (2012)), variance decompositions of return volatilities (Diebold et al. (2018)), or the activities of futures market participants (e.g., Adamic et al. (2017)). Here, we rely on the theory of information in order to determine the intensities of the links and to obtain their directions. To our knowledge, such an application is original in both the finance and commodity literatures.
The use of graph theory in this context allows us to examine precisely where the information entropy is transferred in the futures price system, and how far throughout the term structure it flows in practice. To that effect, we construct a reference graph that depicts the average behavior of the system in 20002017. In part, we find that this benchmark graph supports a conventional view of how a futures market operatesspecifically, the notion that price shocks are thought to form in the physical market (here represented by the short maturities) and transmit to the paper market (here made up of furtherout maturities). At the same time, however, we find that intermediate maturities send out substantial amounts of information entropy not only to furtherdated contracts but also to neardated ones. Furthermore, a dynamic analysis shows that there are sometimes major changes in the organization of the crossmaturities connections. The biggest such rearrangement is in Fall 2008, with the direction of information flows "flipping" entirely (i.e., originating at the far end of the curve and reaching even the shortest maturities). To the best of our knowledge, while this kind of reverse informationflow pattern is theoretically conceivable, our analysis provides the first empirical evidence of its...
Shock Propagation Across the Futures Term Structure: Evidence from Crude Oil Prices.
Author:  Lautier, Delphine H. 
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COPYRIGHT GALE, Cengage Learning. All rights reserved.
COPYRIGHT GALE, Cengage Learning. All rights reserved.