The intraday data used for exploring the change of TAIEX futures-spot dynamic relationship before and after the reduction of tick size are extracted from the Taiwan Economic Journal Data Bank (TEJ) and are computed in five-minute intervals.
In order to have better liquidity to trade and quote, this study takes the nearby contract into account at any given time until the first trading day prior to the maturity date of the nearby contract. We adopt the next maturing contract from the first day prior to the maturity date of the nearby contract, because the volume of the next maturing contract usually surpasses that of the nearby contract on that day. To form trading pairs, this investigation matches every reported index with the most recent futures trade prices prior to or at exact every five minute. Since there is a delay before the first trade of each stock on a new trading day and the bid-ask spread widens and quotes are older at the end of each trading day, futures and spot will have large and continuing deviations2. Therefore, we follow Henker and Martens (2005) to delete the first 30 and the last 10 minute of each day, leaving 47 observations from 9:30 to 13:20 per day (Henker, Thomas and Martin Martens (2005)). The sample period extends over two-year trading days from May 1, 2004 to December 31, 2005. The sample period is divided into two sub-periods according to the reduction of tick size on March 1, 2005. We eliminate the data on January 13, 2005, because of a large number of missing data after 10:00 am. Therefore, the first sample before the reduction of tick size from May 1, 2004 through February 28, 2005 comprises 203 trading days with 9,541 observations. The second sample after the reduction of tick
2 See, for example, Aggarwal and Park (1994) for the effects of the staleness of the index at the start of the day.
size from March 1, 2005 through December 30, 2005 comprises 213 trading days with 10,011 observations.
4.2 Construction of the TAIEX Index Futures MPE
Futures and spot prices are connected by the following cost-of-carry model:
) ( ,
,i ( ti t) r T t
t
S Div e
tF
= − − (1)∑
=−
+ −
×
= T
t
t t
d r Div
τ
τ ) (τ )
1 365
( (2) where Ft,i stands for the theoretical futures price on day t in 5-minute interval i for a contract expiring at time T, Divt is the present value of the cash dividends that will be paid during the remaining life of the futures contract from the 50 daily largest companies about 69% of overall market value on the stock market, and r is annualized one-month post office deposit rate as the risk-free rate of interest. rt is the effective interest rate of r. The rate rt 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 meaning the difference between futures and spot prices converges to zero at expiration. The implicit assumptions underlying the cost-of-carry model include perfect markets and constant carrying charges. Any price deviations from Equation (1) will be corrected as arbitrageurs sell the overpriced instrument and buy the underpriced one. Furthermore, we take the logarithm on the Equation (1) and define the percentage mispricing error (MPE) as
)
4.3 Identifying Arbitrage Opportunities and Transaction Costs
Arbitrage opportunities can be identified as follows. Sell program: sell stocks and purchase futuresi t i
t L
MPE, <− , (5) An arbitrage buy program is triggered while the MPE penetrates the upper bound,
. An arbitrage sell program is triggered while MPE falls below the lower
bound, . We introduce the Threshold Vector Error Correction Model (TVECM) to identify the upper and lower bound for arbitrage. The existence of transaction costs and other market imperfection factors might cause the error correction effects on the price adjustment be significant only when the deviation of prices between futures and spot is larger than a certain threshold.
i
Ut,
i
Lt,
−
4.4 Linear Vector Error Correction Model (VECM)
Let xt be a p-dimensional I(1) time series, with n observations, with d as the maximum lag length. A linear VECM of order d+1 can be written briefly as
t
difference sequence with finite covariance matrix Σ= ) . Note that assumption that the errors ut are i.i.d. Gaussian. Let these estimated parameters be denoted ~) and be the residual vectors.
4.5 Threshold VECM for Futures and Underlying Spot
Consider now an extension of Equation (1), provided by:where γ is the threshold parameter. Note that this paper uses the absolute value of error correction term as a threshold variable. In addition to the merit of parsimony in the modeling of threshold effect, the assumption is reasonable since transaction costs tend to be symmetric for either long or short position in the futures for its arbitrage.
Alternatively, this may be written as
t
and 1(.) denotes the indicator function. The existence of the threshold effect is confirmed if 0<
P
(wt-1(β
) ≤γ
) <1, otherwise the model simplifies to linear cointegration.The threshold VECM of futures and spot can be estimated using the maximum likelihood method proposed by Hansen and Seo (2002). Under the assumption that the errors ut are i.i.d. Gaussian, the likelihood function is
∑
=order to maximize the log-likelihood, to hold (β, γ) fixed and to compute the constrained MLE for (A1
,A
2,Σ). This is just OLS regression:
( ) ( ) ( ) ( )
search. Hansen and Seo (2002) suggest calibrating this region based on the consistent estimate β~ obtained from the linear model. Set ~), letw , and construct an evenly spaced grid
ton
[ γ
L,γ
U]
. Let[ β
L,β
U]
denote a (large) confidence interval for β constructed from the linear estimate β~ (based, for example, on the asymptotic normal approximation) and construct an evenly spaced grid on[ β
L,β
U]
. The grid search4.6 Tests for Threshold Effects
Let H0 represent the class of linear VECM in Equation (6), and H1 represent the class of two regime threshold VECM in Equation (9). These models are nested, with
the constraint H0 being the models in H1 which gratify . Our test will
0 (linear cointegration) with H1 (threshold cointegration).
In order to assess the evidence, both linearity and the threshold VECM are tested by using the Lagrange Multiplier (SupLM) test developed by Hansen and Seo (2002). The LM statistic employed is: Seo’s (2002) is employed to calculate the asymptotic critical values and p-values.
4.7 Conditioning Mispricing Errors on Volatility
We hypothesize that the MPE is reduced after the reduction of tick size, but that any reduction might be caused by the lower market volatility or be offset by the higher market volatility in the second sample period. To test the robustness of these probable results, we implement the methodology of Jones and Lipson (2001). The goal is to test whether the reduction of tick size diminishes the average mispricing error that triggers arbitrage while controlling for changes in the volatility and other control variables. We estimate the regression in Equation (20) for all trades prior to the reduction of tick size and use the estimated coefficients to calculate MPE prediction errors under the second sample period in Equation (21).
reduction where MPEi,t is the absolute value of percentage mispricing error on day t and intraday period i, while the superscript indicates the minimum price increment at the
period, VOLAi,t is the futures volatility in the 30 min prior to the every trade at time i, BUYi,t is an indicator variable with value one if the arbitrage trade is a buy program and zero otherwise, and SHORTi,t is an indicator variable with value one if the arbitrage trade involved short selling and zero otherwise. We do not control for the prevailing spread since the reduced spreads are the reasons why we expect the mispricing error to be smaller when arbitrage programs are initiated.
The coefficient estimates from Equation (20) are then used to predict the MPEs after the reduction tick size and, to compute the prediction errors,
t i t
i t
i reduction
post t i reduction
post t
i
MPE VOLA BUY SHORT
e
ˆ, − = , − −α
ˆ−β
ˆ1 , −β
ˆ2 , −β
ˆ3 , (21) where the hats for the coefficient estimates indicate the previously estimated coefficients are used. We can now, while controlling for volatility, test whether hypothesis of the test isH
0 :e
ˆi,pret −reduction =e
ˆipost,t −reduction , while the alternativehypothesis is