7.5

2.5 -

-2.5 --

-5 --7.5 --10 --12.5 --15 --17.5 --

Jul-92 Jul-93 Jul-94 Jul-95 Jul-96 Jul-97 Jul-98 - Futuretospot -Basistospot-Spottofuture Basis to future

Figure 12.4 Granger causal flows between spot and future prices and the basis.

basis are used to investigate the Granger causal flows between spot and futures. During the long data period from 1 July 1988 to 26 February 1999, the bivariate ECM

is estimated using OLS with 4 years of data. The 4-year data window is rolled over daily, and each time the model is estimated simple /-tests on the significance of the coefficients show how the lead-lag relationship between spot and futures prices evolves over time. Figure 12.4 shows these /-statistics on a2 (future to spot), y, (basis to spot), p] (spot to future) and y2 (basis to future).

Note that after the structural break on 17 January 1995, when the dramatic fall in prices on 17 January 1991 drops out of the data, part of the error correction mechanism broke down. Since z = lnF-lnS the coefficient y, should be positive for error correction, but it becomes negative after the structural break. However, the /-statistics on y2 are very large indeed, and negative as they should be, so the error correction mechanism is currently working through changes in futures prices.

Figure 12.4 gives a very clear message that it is futures and not spot oil prices

that are being driven: there are very significant causalities from spot oil prices, There are ven-

and from the basis, into futures prices of crude oil on the next day. It is not significant causalities

surprising that futures are not good forecasts of spot prices in the crude oil from spot oil prices, and

market. In fact in any energy market, demand fluctuations produce an from the basis, into

immediate response in spot prices because of the inelastic supply curve. The futures prices of crude oil

subsequent effect on inventory levels changes the convenience yield, but it may °» !he next da>

rSJ = a0 + a,rSif , + a2ru i + yxzt x + £U,

rU = Po + pVs,r-l + P2rf,(-1 + llzt-\ + £2,r

take time for futures prices to respond.7 However, spot prices are difficult to predict, which is to be expected since demand fluctuations are governed by so many unpredictable quantities.

The generalization of an ECM to more than two variables is straightforward. The ECM has one equation for each variable in the system, where the dependent variable is the first difference, and each equation has the same explanatory variables: lagged first difference terms up to some order p, and up to r lagged disequilibrium terms corresponding to the r cointegration vectors. The full specification of an ECM in vector form is therefore

Ay, = a0 + B,Ay, ! + B2Ay,2 + . . . + BpAyt p + , , + ,. (12.6)

Each of the n equations in (12.6) has as regressors a constant, the lagged first differences of all variables in up to order p, and all lagged disequilibrium terms because of the term , ,. For large p this is a huge number of potential regressors, and it is unlikely that they would all be significant in every equation. OLS estimation of each equation separately will indicate which variables should be included in each equation, and when (12.6) has been specified effectively it may then be used to model the lead-lag behaviour between returns in the whole system. More details of short-run dynamics in cointegrated systems may be found in Proietti (1997) and in many of the texts already cited in this part of the book.

12.4 Cointegration in Financial Markets

It is only recently that market practitioners have found important applications of the vast body of academic research into cointegration in financial markets that goes back over a decade. In this respect financial analysts have been characteristically slow in adopting a new modelling approach. Only during the last few years have there been interesting practical developments. Asset management companies have been investing in quantitative research projects to base buy-and-hold and long-short strategies on cointegration. Commodity analysts model the lead-lag relationship between spot and futures returns using ECMs. The pricing and hedging of spread options can now be based on cointegration This section reviews some of the publications that are relevant for the implementation of cointegration modelling of financial markets.

12.4.1 Foreign Exchange

Any system of financial asset prices with a mean-reverting spread will have some degree of cointegration, even though the Granger causalities inherent in

7The closing time of the spot market is only half an hour later than the futures. It is unlikely that this time differential is the only reason for there to be such a strong causality from todays spot close to tomorrows futures close.

such a system contradict the efficiency of financial markets (Dwyer and Wallace, 1992). Two log exchange rates are unlikely to be cointegrated since their difference is the cross rate and if markets are efficient that rate will be non-stationary. There is, however, some empirical evidence of cointegration between three or more exchange rates: see Goodhart (1988), Hakkio and Rush (1989), Baillie and Bollerslev (1989a, 1994), Coleman (1990), Alexander and Johnson (1992, 1994), Chen (1993), MacDonald and Taylor (1994) and Nieuwland et al. (1994).

Two log exchange rates are unlikely to be cointegrated since their difference is the cross rate and if markets are efficient that rate will be non-stationary

12.4.2 Spot and Futures

Many financial journals (the Journal of Futures Markets in particular) contain papers on cointegration between spot and futures prices. Since spot and futures are tied together, the basis is the mean-reverting cointegration vector. The ECM has become the focus of research into the price discovery relationship, which has been found to change considerably over time. See MacDonald and Taylor (1988), Nugent (1990), Bessler and Covey (1991), Bopp and Sitzer (1991), Chowdhury (1991), Lai and Lai (1991), Khoury and Yourougou (1991), Schroeder and Goodwin (1991), Schwartz and Laatsch (1991), Beck (1994), Lee (1994), Schwartz and Szakmary (1994), Brenner and Kroner (1995), Harris et al. (1995) and Alexander (1999b). This research shows that there is considerable scope for futures traders to develop ECMs that will exploit this very strong cointegration relationship.

12.4.3 Commodities

Commodity products that are based on the same underlying, such as soya bean crush and soya bean oil, should be cointegrated if carry costs are mean-reverting. However, the evidence for this seems rather weak, and the academic argument that related commodities such as different types of metals should be cointegrated is even more difficult to justify empirically. Brenner and Kroner (1995) present a useful survey of the literature in this area and conclude that the idiosyncratic behaviour of carry costs makes it very difficult to use ECMs in commodity markets. High-frequency technical traders dominate these markets, and it is unlikely that cointegration between related commodity markets is robust enough for trading.

12.4.4 Spread Options

Modelling cointegrated assets with Brownian diffusion processes is one of the many interesting problems presented at the Finance Seminars in the Newton Institute at Cambridge University during the summer of 1996. Jin-Chuan Duan and Stan Pliska took up this challenge and have recently produced an excellent study of the theory of option valuation with cointegrated asset prices (Duan and Pliska, 1998). Their discrete time model for valuing spread options