«Efficient Price Discovery in Stock Index Cash and Futures Markets Pascal ALPHONSE * ABSTRACT. – This study is concerned with the aggregation of ...»
ANNALES D’ÉCONOMIE ET DE STATISTIQUE. – N° 60 – 2000
Efficient Price Discovery
in Stock Index Cash
and Futures Markets
Pascal ALPHONSE *
ABSTRACT. – This study is concerned with the aggregation of infor-
mation in the French stock index cash and futures markets. The results
indicate that deviations from the equilibrium relationship linking cash
and futures prices originate mainly from information arrivals in the
futures market and that at least 95 % of the price discovery is achieved in this market. Finally, the results appear relatively independent of the particular choice of price measurement, transaction prices versus quotes.
Découverte du prix efficient dans les marchés au comp- tant et à terme d’indice boursier RÉSUMÉ. – Ce travail analyse l’agrégation d’informations nouvelles dans les prix au comptant et à terme de l’indice boursier CAC 40. Les résultats indiquent que l’information nouvelle arrive essentiellement sur le marché de futures et est, ensuite, transmise sur le marché des actions. Il apparaît qu’environ 95 % de la découverte du prix est réalisé sur le marché de futures. Ces résultats apparaissent relativement indé- pendants du choix de la mesure indicielle retenue, derniers prix négo- ciés versus dernières cotations disponibles.
* P. ALPHONSE: GERME-ESA, Université de Lille 2, 1 place Déliot, BP 381, 59020 Lille Cedex, France.
I thank B. BIAIS, C. GOURIÉROUX (the Editors), A. FRANÇOIS-HEUDE and an anonymous referee for valuable comments. All remaining errors are mine.
1 Introduction Besides the traditional role of risk sharing assigned to futures markets, these markets play an important role in the aggregation of information (see, for example, GROSSMAN , BRAY  and BRANNEN and ULVELING ). The case of stock index futures is analyzed in SUBRAHMANYAM  and in KUMAR and SEPPI . Futures and cash markets contribute to the discovery of a unique and common unobservable price that is the efficient price. The contribution of each market to the price discovery depends, at least in part, on the microstructure of these markets, including the level of transpa- rency, the liquidity supply mechanism, the rules governing the priority of orders, the constraints on short sales and the settlement mechanism.
Special features of the French stock index cash and futures markets may play a non-trivial role in the price discovery process. The major French stock mar- ket, the “Règlement Mensuel” (hereafter, the RM) is based on a fixed date sett- lement that operates once a month. Thus, the RM works like a futures market with delivery at the end of the month. This feature reduces the traditional dif- ference between stock and futures markets. As a consequence, and contrary to most of the US studies, the empirical results presented in this paper are much more related to differences in the assets traded, a single asset versus a basket of securities (in the spirit of the papers of SUBRAHMANYAM , CHAN  and KUMAR and SEPPI ) than to differences induced by the settlement schedules.
Furthermore, stocks and futures are traded in different market structures, a pure limit order market for the stocks and a floor market for the futures.1 As suggested by BIAIS, FOUCAULT and SALANIÉ , this difference affects the liquidity suppliers’ behavior and has an impact on the level of liquidity of the market. In particular, it appears that the cost of liquidity is larger in the floor market than in the limit order market due to the absence of incentive to undercut on large ask prices (overbid on low bid prices) in the floor market. On the contrary, limit order markets appear to entail efficient risk sharing and competitive pricing. This may affect the order submission behavior of informed traders in a way that is favorable to the stock market. As a consequence, traditional results indicating that information is first aggregated in the futures market (e.g. STOLL and WHALEY ) may in fact reflect, at least partially, differences in market microstructure. In this paper, we provide evidence of price discovery by stocks and futures traded in market architectures that differ significantly from the ones analyzed in other large market places.
On the empirical ground, most of the studies emphasize some kind of causality between futures and cash markets returns (e.g. STOLL and WHALEY ) and/or between futures and cash markets return volatility (e.g. CHAN, CHAN and KAROLYI ).While these analyses provide some evidence about
1. Since 1998, the financial futures exchange is also organized as a limit order market.
178 the relationship between the cash and futures markets, they do explicitly take into account neither the equilibrium relationship between cash and futures prices nor the price discovery process. A more satisfactory specification may be built on the observation that these two characteristics are linked to two particular forms of the cointegration property of futures and cash prices, the error correction form (hereafter, ECM) and the common trend form. WAHAB and LASHGARI  provide an in-depth analysis of the relationship between stock index cash and futures markets using an ECM framework. Nevertheless, the evidence presented in their paper should be interpreted with caution due to the use of non-synchronous daily closing prices. The analysis of SHYY, VIJAYRAGHAVAN and SCOTT-QUINN  is more closely related to our work because it deals with the French market and uses intraday data. Furthermore, the empirical analysis presented in their paper suggests that traditional results concerning the lead-lag relationship between cash and futures markets may be caused by a stale price effect due to non-synchronous trading among stocks.2 While this result is appealing, it should also be considered with caution because SHYY, VIJAYRAGHAVAN and SCOTT-QUINN  use data on the second nearest contract which is characterized by a very low level of activity.3 As far as the choice of the data is concerned, part of our work may be viewed as an extension and analysis of the robustness of the results of SHYY, VIJAYRAGHAVAN and SCOTT-QUINN  to the case of the first nearest contract which is characterized by a high level of activity. Finally, HASBROUCK  uses the common trends representation of a set of cointegrated variables to measure the contribution to the efficient price innovation from a particular market.
A cointegration analysis based on intraday data from the French stock index cash and futures markets is developed in this paper. The main results indicate that the futures leads the spot and contributes largely to the price discovery, suggesting that news are first aggregated in the prices of the futures and then transferred to the stock market. An analysis of the price adjustment to past desequilibrium supports this view.
The remainder of the paper is organized as follows. Section 2 presents a simple model of price discovery. The data are presented in section 3 and the results are presented in section 4. Section 5 concludes the study.
2. This stale price effect results in index autocorrelation. In order to analyse this effect, SHYY, VIJAYRAGHAVAN and SCOTT-QUINN  use quotes from the futures and cash markets in addition to transaction price data. Nevertheless, several studies argue that the role of infrequent trading should be limited. For example, LO and MACKINLAY  show that the level of infrequent trading that would be consistent with the level of index autocorrelation is much larger than the observed level of infrequent trading that characterizes most stock indexes. STOLL and WHALEY  and DE JONG and NIJMAN  show that the infrequent trading effect could be identified and eliminated without altering the lead/lag results. Finally, BOSSAERT  and CHAN  suggest that informational effects as well as infrequent trading may cause the index autocorrelation.
3. Activity data provided by MATIF SA, the futures exchange, show that the trading volume of the second nearest contract considered by SHYY, VIJAYRAGHAVAN and SCOTT-QUINN  is about twenty times lower than the trading volume of the first nearest contract.
179 EFFICIENT PRICE DISCOVERY IN STOCK INDEX CASH2 A Simple Model of Price Discovery
It is well known that prices of related securities like prices in spot and futures markets cannot diverge without bound because they are linked by an arbitrage relationship. The link between this arbitrage relationship (and the associated cost-of-carry pricing model) and the existence of a cointegration relationship between the spot and futures prices has been extensively presented in the literature (e.g. WAHAB and LASHGARI ). This literature suggests that inference concerning the spot and futures price dynamics should be based on Error Correction Models (ECM) where the error-correction components indicate i) the proportion of desequilibrium from one period that is corrected in a later period and ii) the relative magnitude of adjustments in each market towards equilibrium.4 Another important insight from this literature states that the spot and futures prices should share a common stochastic trend because spot and futures markets are essentially two different places to trade the same underlying commodity. Based on this feature, HASBROUCK  proposes a measure of price discovery that is used in this study. In the following, the ECM and the common trend model is presented in the context of a very simple model of price dynamics inspired from HASBROUCK .
Suppose, first, that a claim on some future cash flows is traded in two different markets, a cash market and a futures market. Obviously, it exists only one true value for this claim and we call this value the efficient price. As standard in the finance literature, we assume that the efficient price follows a random
µt = µt−1 + ηt (1) where the increments ηt are assumed to reflect new information about the future cash-flows.
The actual cash and futures price dynamics is driven by this common random walk and it may be specified by an unobservable component model:
Pt = K + µt 1 + t (2) where Pt = (St Ft ) is a (2 × 1) price vector, St and Ft are the spot and the futures prices observed at time t, K is a (2 × 1) vector of means, 1 is a unit vector and t is a vector of disturbances which is assumed to be a zero-mean covariance stationary stochastic process. The process t allows spot and futures prices to deviate from the efficient price due to market specific phenomenon such as a liquidity pressure. Price discovery refers to the impounding of (new) information into spot and futures prices and is related to the contributions to the efficient price increments from the spot and futures markets.
Suppose, for example, that the informational content of orders and trades (other than arbitrage-based orders and trades) in one market is larger than the
4. See BANERJEE, DOLADO, GALBRAITH and HENDRY  or LÜTKEPOHL , chap. 13 for a general analysis of cointegration and see BRENNER and KRONER  for an analysis of the link between arbitrage and cointegration.
180 informational content of orders and trades (other than arbitrage-based orders and trades) in the other market. These asymmetric contributions to the common efficient price discovery create a price discrepancy between the two markets that causes some investors to submit arbitrage orders. Therefore, arbitrage appears as an error-correction mechanism that prevents the spot and futures prices from diverging without bound. Furthermore, in our informational framework, the deviation from the cash and carry relationship disappears only when all the relevant information is shared by the two markets and impeded in the spot and the futures prices. As a result, cash and futures prices ultimately reflect the same information, as suggested by the common (unobservable) efficient price model.
Inference concerning the error correction mechanism may be obtained from an ECM for price changes and inference concerning the common (unobservable) efficient price may be obtained from the vector moving average (VMA) representation for the actual data. Additional motivation for these models is that ECM and VMA are alternative forms of a cointegrated system. In our bivariate framework, and due to the arbitrage constraint, St and Ft are expected to be cointegrated of order one, yielding a unique cointegrating vector.
Following JOHANSEN , the ECM for price changes is given by:
p−1 Pt = πi Pt−i + α(β Pt− p − β K ) + et (3) i=1 where the πi are (2 × 2) coefficient matrices, α is a (2 × 1) vector of coefficients for the cointegrating vector, β is the (2 × 1) cointegrating vector and et is a (2 × 1) vector of serially-uncorrelated disturbances with Cov(et ) =.
Note that in our spot-futures framework β should be equal to (1 − 1).
Furthermore, the coefficients of the vector α should indicate the proportion of the mispricing observed in a given period that is corrected in a later period as
well as the relative magnitude of the adjustments in the spot and futures markets. The VMA specification is directly obtained from the ECM and is given by:
In the present case, the matrix sum (ψ0 + ψ1 + ψ2 + · · ·) forms a (2 × 2) matrix whose rows must be identical because the long term response of each price to an innovation must be identical if the prices are not to diverge.
Denoting ψ one of these (identical) rows, the variance of the efficient price
may be obtained as:
where i refers to the relevant market ( i = s, f for the spot and the futures markets, respectively). Due to the possibility of cross-correlation in price innovations, the matrix may not be diagonal and may be replaced by its Cholesky factor. In this case, the price contribution of the market i is obtained by the
(ψF)2 Si = (8) (ψF)I (ψF) where F is the Cholesky factor ( F is the lower triangular matrix such that = F F ) and where ση is now given by (ψF)I (ψF).