3.1. Measuring the No-Arbitrage Band
The analysis follows the information share approach of Chu and Hsieh (2002). The theoretical futures price used to test for market efficiency is the Cost of Carry relationship, which is derived form an arbitrage strategy that consists of a long position in the index portfolio, with a price S and a short position in an equal amount of index futures, priced 0 at F . The hedged strategy will yield a flow of dividends over time, as well as a fixed 0 capital gain of F0 −P0. Since the position is riskless, it should earn the riskless rate of interest. To prevent profitable arbitrage, the theoretical equilibrium futures price at time t is thus:
) )(
) (
( )
(t S t e r d T t
F = − − (1) where F(t) stands for the theoretical futures price at time t for a contract expiring at time T;
S(t) is the spot price of the underlying asset at t; r is the risk-free interest rate; and d is the dividend yield on the stock index portfolio. The rate r is often refereed to as carrying charge, since it represents the opportunity cost of carrying the spot asset to maturity of the futures contract. The buyer of stock index securities incurs the opportunity cost of his funds but receives dividends. Therefore, the futures price should equal the cost of buying the spot index securities, including the opportunity cost, adjusted for dividends paid during the remaining life of the futures contract. As the futures contract approaches maturity, the futures price converges to the value of the spot index. Equivalently, the basis,
that is, the difference between the futures price and spot index value, converges to zero at expiration. The implicit assumptions underlying the cost-of-carry model include perfect markets, constant carrying charges, and constant dividend flow to the index stocks. Any price deviations form Equation 1 will be corrected as arbitrageurs sell the overpriced instrument and buy the underpriced one.
The impact of transaction costs is to permit the future price to fluctuate within a band around the formula value in Equation 1 without triggering profitable arbitrage opportunities. The width of the band derives from round-trip commissions in the stock and futures markets and from the market impact costs of putting on the trade initially. Most studies view commissions as fixed costs, although fees vary by groups of traders as well as by order size. Market- impact costs can be measured by bid-ask spreads that vary by trader. This study took an approach similar to Chu and Hsieh (2002) and measured arbitrage profit at different levels of transaction costs. The three levels of two-way transaction costs are specified as 0.20, 0.30, and 0.40% of theoretical futures price.
Equation 2 describes the no-arbitrage band for the futures price
C e
t S t F C e
t
S() r−d T−t ]− < ( )<[ ( ) r−d T−t ]+
[ ( )( ) ( )( ) (2)
where C stands for the total transaction costs of executing arbitrage including round-trip stock commission, round-trip futures commission, market impact in futures, and market impact in stocks2. If the futures price penetrates the upper bound, a long arbitrage trade will simultaneously short the futures and buy the spot. If the futures price drops below the lower bound, a short arbitrage will make the reverse transactions.
Using the cost-of-carry relationship, this article establishes ex-post and ex-ante no arbitrage conditions between the spot index and futures in Equations 3-5 (see the Appendix), as well as between ETFs and E-minis in Equations 6-11. The ex-post test
2 This assumes that the transaction costs are the same for long and short positions in fu tures and for purchases and sales in stocks. It is not crucial to the analysis.
focuses on the frequency and persistence of boundary violations. The ex-ante calculates arbitrage profit with explicit consideration of the transaction lag.
The ex-post no-arbitrage relationship between index futures and the spot index is
m where IDX(t) is the reported spot index, and Cc+m represents the transaction costs consisting of both commissions and market- impact costs of bid-ask spread.
Equation 3 implicitly assumes that arbitrageurs trade a basket of underlying stocks against index futures. Given the time lags in program trading, an ex-post boundary violation provides merely a mispricing signal but not realized arbitrage profit. To measure the ex-ante arbitrage profit, this study imposes a 5- min transaction lag for program trading.
The ex-ante profits for long and short arbitrage are calculated in Equations 4 and 5
m 5 min after an ex-post mispricing signal (Chu and Hsieh, 2002). Arbitrageurs using program trading can realize profits only if the violations lasts longer than 5 min.
Alternatively, traders can view ETFs as a cash proxy and arbitrage the mispricing between ETFs and E- minis. Suppose an arbitrage is entered at t and lifted at futures expiration date T. With the consideration of transaction costs, the no-arbitrage band between SPDRs and S&P 500 E- minis becomes Equation 6. For QQQs and NASDAQ 100 E- minis is in Equations 7.
There are a few notable differences between Equation 6 and 7 and the conventional futures pricing equation. First, prices of SPDRs are multiplied by 10 to make them comparable to futures prices; prices of QQQs are multiplied by 40 to the same reason.
Second, using the ETFs quote price for a better measure of transaction costs. ETF bid and ask prices [ETF )(t bid and ETF )(t ask] are used to calculate the no-arbitrage boundaries.
Here assumed that arbitrageurs buy at the ask and sell at the bid price when they trade ETFs. Because the market- impact costs have been explicitly considered, the transaction costs C in Equation 6 and 7 consist only of trade commissions. c
Equations 8 and 9 define the ex-ante profit of long and short arbitrage using SPDRs against S&P 500 E- minis, and Equations 10 and 11 for NASDAQ 100. This study assumes that arbitrageurs can trade at the next available ETF quote price and futures trade price immediately after a mispricing signal.
c
The sample period considered here is October 2, 2000 through May 29, 2001. Quote data of ETFs (SPDRs and QQQs) for this study were obtained from the NYSE’s Trade and Quote (TAQ) Database and trade data of Indexes (the S&P 500 Index and the NASDAQ 100 Index) and their E- minis contracts were from the Tick Data Database. In the TAQ data, only regular AMEX quotes were used. All prices were filtered.3 The
3 To minimize errors, we omit quotes if the TAQ database indicates that are out of time sequence or involve either an error. TAQ quotes were screened to remove zero and negative spreads, and spreads greater than
dividend data are from the CRSP daily database. As a proxy for the opportunity cost in the calculation of futures mispricing, monthly three- month Treasury Bill rates from web database of the Federal Reserved Board were used for the riskless rate of interest. In the intraday analysis, this article assumed that daily T-bill rates and dividend yields were continuous and constant intraday.
To form trading pairs, this investigation matched every reported index and ETF quote with the most recent E- mini trade prices. The number of matches is equal to the total number of index values reported or the number of ETF quoted for the corresponding period. To computing mispricing series, futures prices are synchronized with the spot values using a MINSPAN procedure suggested by Harris, McInish, Shoesmith, and Wood (1995). If there is no futures trade at the exact time of the reported spot value, the closest futures observations within the previous 7 seconds and the net 7 seconds are considered.
When only one futures trade meets this criterion, a pair is form. If both a leading and lagging futures trades are obtained, the closer trade is used to form the pair and the other one is discarded.
There are 104,788 spot- index and E- minis matches in the pre-decimalization period and 107,904 in the post-decimalization period. For pairs of SPDRs and S&P 500 E- minis, there are 206,622 observations in the pre-decimalization period and 224,602 in the post-decimalization period. For NASAQ 100, there are 109,706 spot- index and E- minis matches in the pre-decimalization period and 113,592 in the post-decimalization period.
For pairs of QQQs and NASDAQ 100 E- minis, there are 237,594 observations in the pre-decimalization period and 291,020 in the post-decimalization period.