Intraday arbitrage opportunities of basis trading in current futures markets: an application of the threshold autoregressive model Chien-Ho Wang Department of Economics, National Taipei University, 151, University Rd., San-Shia, Taipei, 23741 Taiwan. Tel.: 886-2-86747156 #67172, e-mail: wangchi3@mail.ntpu.edu.tw Wen-Chung Guo Department of Economics, National Taipei University, 151, University Rd., San-Shia, Taipei, 23741 Taiwan. Tel.: 886-2-86747156 #67156, e-mail: guowc@ntu.edu.tw 2014/04/07 0
Intraday arbitrage opportunities of basis trading in current futures markets: an application of the threshold autoregressive model Abstract This study applies the threshold autoregressive model to analyze arbitrage opportunities associated with basis trading in current futures markets. We derive an empirical model of basis trading arbitrage and then use a sample of five-minute intraday data for the Euro/US dollar and Japan Yen/US dollar for model testing. Although the currency spot rates and future prices are generally nonstationary and cointegrated, the basis variables emphasized herein are well-behaved stationary. We find that the threshold effect significantly affects future basis, suggesting nonlinear contrarian behavior of basis and arbitrage opportunity when the basis exceeds a threshold level. The results reveal that the basis is mean reverting, particularly when it falls outside a range near zero. Finally, we additionally present an analysis of out-of-sample performance and find that the threshold model outperforms the benchmark autoregressive model. Keywords: arbitrage; currency futures; threshold autoregressive model; basis. I. Introduction This study provides initial analysis of threshold models related to basis arbitrage trading, which is observed practically a common trading strategy. The basis is defined as the difference between the futures and spot prices, and has the advantage of being easily and immediately observable during trading hours. Basis trading is a fundamental arbitrage strategies, particularly for high-frequency trading (see, for instance, Aldridge (2013)). This strategy bets that the difference between the spot and futures prices reverts to its mean. This study is motivated by limited academic knowledge regarding the currency basis intraday behavior. The present investigation also elucidates a theory of contrarian trading in currency futures. 1
Suppose the reaction to positive information in the futures markets exceeds that in the spot markets, resulting in a higher basis value. This may lead to a contrarian basis behavior, because the futures price overreacts in comparison with the spot rate. In this case, arbitrageurs may sell the basis by selling the currency assets and purchasing futures contracts. Similarly, when futures prices overreact to negative signals, arbitrageurs observe a negative basis value and may buy the basis by buying the currency assets and selling futures contracts. This study aims to apply the threshold autoregressive model to test this contrarian theory, using a large sample of five-minute intraday data for two major currency futures, Euro/US dollar and Japanese yen/us dollar. Numerous related studies have addressed arbitrage behaviors. For instance, Dwyer Jr. et al (1996) applied a cost of carry model together with the TVECM method to analyze arbitrage behavior. Their study found an association between arbitrage and rapid convergence of basis. Meanwhile, Butterworth and Holmes (2002) examined spread trade for two index equities, FTSE 100 and FTSE Mid 250, in the UK market. Butterworth and Holmes discuss whether and how arbitrage opportunity may be found. Eun and Shim (1989) applied the vector autoregressive model to investigate the international transmission between the stock markets of nine developed countries. Several previous studies, such as Beck (1994), Antoniou and Holmes (1995), and Brockman and Tse (1995), also emphasized the role of basis information, proposing that futures may increase market efficiency because futures contracts trading data helps with information release. This study also closely resembles several existing empirical studies on threshold models. Most of these studies stress on equity markets. Among them, Chung et al (2005) used the threshold vector error correction model (TVECM) to investigate dynamic price relationships between ADRs and their underlying stocks, and supported the threshold effect. Park et al (2007) examined bivariate three-regime TVECM in natural gas markets, and suggested that the threshold effects depend on season and geographical location. Li (2010) studied dynamic hedging behavior with TVECM regression for the US S&P 500 and Hungarian BSI stock market and their index futures, and discovered that the threshold effect help hedging effectiveness only for emerging markets. This study is also related to intraday analysis on exchange rates conducted by Yu et al (2008), who analyzed panel probability distribution on foreign exchange rate. Yu et al (2008) identified trivial arbitrage associated with the British pound/us spot rate, and suggested that Monday and Friday effects do not exist. This investigation differs from early research in that basis variables are not non-stationary and thus the cointegration model such as VECM is not suitable. Instead, this study applies the threshold autoregressive model to study the threshold effect and its implication for arbitrage behavior. 2
The authors derive an empirical model of arbitrage based on basis trading, and then test that model using a sample of intraday data. Empirical results in present investigation extend the current literature in three ways. First, while currency spot rates and future prices are generally nonstationary and cointegrated, the basis variables emphasized herein are well-behaved and stationary. Henceforth, the study takes advantage of observed stationary variables without facing difficulties in interpretation when the cointegration model is used. Second, the threshold effect is found to strongly influence future basis. This result supports nonlinear contrarian behavior of basis and the existence of arbitrage opportunities when the basis exceeds the threshold. The return achieved from basis trading exhibits mean reverting behavior, particularly when it belongs to the outside regime. Finally, we additionally present an analysis on out-of-sample performance and find that the threshold model tends to outperform the benchmark autoregressive model. The remainder of this paper is organized as follows. Section II presents empirical models. Section III then describes data used in this study and analyzes the empirical results obtained from the threshold models. Better out-of-sample performance is also presented with the threshold effect. Finally, Section IV draws conclusions. II. Empirical models This section derives an empirical arbitrage model based on basis trading in currency future markets. St and Ft denote the spot rate and the future price of a currency (Euro/US dollar) during period t. The cost-of-carry model of future prices implies F S exp(( r r )( T t)), where r and 3 t t US r US represent the risk-free interest rates for the Euro and US dollar, respectively, while T represents the expiration time of future contracts. This formula leads to F S ( r r )( T t) S. Moreover, t t US t Bt=(Ft-St) / St denotes the basis as the percentage of a currency. Accordingly, we have B ( r r )( T t). This suggests that the basis is positive (negative) when the Euro interest rate t US exceeds the US interest rate, and converges to zero as the time-to-maturity approaches zero. Consider a basis arbitrage trading. If the futures price overreacts excessively to a positive signal relative to the spot rate, investors may consider a strategy involving negative positions of future
contract if the basis B t exceeds ( r r )( T t) by a considerable amount that covers the US transaction cost. The investors may engage in arbitrage and expect a positive return through the basis trading. Let the adjusted basis be Z B ( r r )( T t). Accordingly, t t US Z t shall be mean-reverting to zero when Zt 1. Similarly, if the future price overreacts to a negative signal, investors may consider a strategy incorporating positive future contract positions when Zt <- 1. Henceforth, we use a threshold model to describe two regimes that depend on whether Zt 1 exceeds or falls below threshold. This investigation extends the econometric model with threshold effect devised by Hansen (1996). The observation is divided into two regimes which depend on whether the adjusted basis Zt exceeds or falls below threshold econometric model:. The adjusted basis Zt is represented as the following Z t 1 1Zt 1 + t, if Z t 1,. (1) 2 2Zt 1 + t, if Zt 1 where Z t is the explanatory variable, t represents the error term, 1 and 1 is the estimated parameter for the first regime ( Z t 1 ), while 2 and 2 denote the estimated parameter for the second regime ( Z t 1 ). Our hypothesis is that basis trade arbitrage involves selling foreign currency futures whenever the futures price exceeds the spot price by a certain level, and uses a reverse trading strategy whenever the futures price falls short of the spot price by a specific amount. Specifically, when Z t 1, the proposed model expects basis to exert a stronger mean-reverting effect. Meanwhile, our hypothesis predicts that parameter i, i 1,2 lies in the interval 0, 1 and satisfy 1 1 2. The estimation adopts the method of Hansen (1996) to conduct a two-step estimation procedure for a threshold autoregressive model like (1). The procedure is described as below: In the first step, the sum of square errors (SSE), S( ), is defined as follows: 4
T 2 t 1 1 t 1 t 1 1 1 t 1 t 1. S( ) Z Z I( Z ) Z I( Z ) t 1 The authors find the arg min S( ) ˆ that minimizes the SSE. 1 Second, after obtaining the threshold value, the coefficients in two regimes using least square methods. ˆ is used to split data. The authors estimate III. Data and Empirical results Data Description This study uses currency futures data from the Chicago Mercantile Exchange (CME). Five minutes price data for March 2009 to March 2010 comprises 67569 observations on 249 trading days. The adjusted basis is calculated using one month interest rates as the risk-free rates. Table 1 lists descriptive statistics for all the study variables. The mean and median value of basis for Euro/US dollar is positive since the Euro interest rate exceeds that for the US dollar. Conversely, for the Japanese yen/us dollar, negative value of basis is observed for over 75% of the sample due to the low interest rate in Japan. [Insert Table 1 about Here] Before the authors conduct the estimation procedure, unit root tests were performed to clarify the existence of unit roots in the series. For the series of futures and spot prices, augmented Dickey-Fuller (ADF) unit root tests are selected. Regarding interest rates, Enders and Granger (1998) found that the interest rate series may include structural breaks, in which case the standard ADF tests are biased towards non-rejection of the null hypothesis. The Zivot and Andrews (1991) unit root test is chosen to detect unit roots robustness. 1 To avoid the few observations in regime 1, this study searches the threshold value in the closed interval 0.15 T, 0.85T, where T denotes the total sample number. Restated, we search the threshold value in the middle 70% of the total sample to avoid uninteresting cases when regime 1 contains a low percentage of the sample. 5
Table 2 lists the results of the unit root tests for the study variables. Consistent with most previous investigations, both spot exchange rates and futures prices are nonstationary, meaning these series accept the null hypothesis of unit roots. However, the basis and adjusted basis variables analyzed herein are stationary. Henceforth, all variables used in our econometric models are against the null hypothesis of unit root, and hence suitable for the subsequent regression analysis. Particularly, two dependent variables, adjusted Euro basis and adjusted Japanese yen basis all have a 99% level of significance to reject the null hypothesis of unit root by either the augmented Dickey-Fuller test or the Zivot-Andrews (1992) unit root test. [Insert Table 2 about Here] Empirical results Before estimating the threshold model (1), it is necessary to test the nonlinear relationship in the sequences of adjusted bases. The authors first implement the likelihood ratio test of Hansen (1996) to examine the existence of a nonlinear threshold relationship. Table 3 lists the test statistics for the threshold effect, together with p-values. This suggests that the two basis differences used in our sample all have (single) threshold effects with high significance. In fact, this study also estimated the model for two two thresholds and failed to find any significant second threshold. The results thus reveal that there is strong evidence of a single threshold effect in the autoregressive model. [Insert Table 3 about Here] Table 4 lists the estimation results of (1). The parameters 1 and 2 for two currencies have values between 0 and 1, indicating that all regimes generally exhibit mean-reverting behavior. Threshold has values of 0.0116 and 0.068 for the Euro/US and Japanese yen/us, respectively. This results reveals a positive threshold value and 1 1 2 0, consistent with our hypothesis regarding arbitrage using basis trading. Specifically, in the case of Japanese yen/us, 1 0.8036 6
and 2 0.3696 are both significant and clearly consistent with our hypothesis. Accordingly, the adjusted basis variable displays contrarian behavior of, especially when the adjusted basis exceeds a threshold level. This observation can be explained by the phenomenon whereby arbitrageurs may be profitable based on either sell the basis or buy the basis to induce the adjusted basis to trend toward to zero. Out-of-sample performance [Insert Table 4 about Here] The above analysis demonstrates that the threshold model tends to perform better in-sample than a simple autoregressive model. Regarding the robustness, Table 5 additionally presents an analysis on out-of-sample performance. The first two-thirds of the sample is employed for estimation using the linear autoregressive regression model and the threshold model. The square prediction errors for the last one-third of the sample then reveal that the threshold model has smaller mean and medium values of square prediction errors than does the simple model. The threshold models thus achieve superior out-of-sample performance to the autoregressive models. [Insert Table 5 about Here] IV. Conclusions This study provides econometric analysis of a nonlinear model dealing with foreign currency arbitrage, focusing on the basis variable, which is widely used in practice but previously has received little academic attention. An empirical model of arbitrage based on basis trading is derived to use the threshold model to estimate a sample of intraday data. Threshold effects are generally found, and suggest nonlinear reversion behavior of currency futures. This result reveals strong evidence of nonlinear contrarian behavior of basis and arbitrage opportunities when basis exceeds the threshold. Additionally, out-of-sample performance shows that the threshold model achieves better prediction 7
performance than the standard autoregressive regression model. A natural extension of the present analysis would be to employ more sophisticated pricing models that use trading volume and open interests data in currency futures markets. 8
Table 1. Descriptive statistics for variables used in this study Variables Mea SD. 10% First Median Third 90% n quartile quartile Euro basis (%) 0.0967 6.3216-3.0000-1.00 0.0000 1.0000 3.0000 Japanese yen basis (%) -3.7611 17.5225-8.9432-6.3651-3.3142-0.7921 0.9576 Adjusted basis for Euro/US dollar (%) Adjusted basis for Japanese yen/us dollar (%) Interest rate difference (Euro/US dollar) Interest rate difference (Japanese yen/us dollar) -0.0187 0.0012-0.0587-0.0408-0.0207 0.0001 0.0204 0.0418 0.0017-0.0063 0.0115 0.0378 0.0687 0.0959 0.2859 0.0547 0.1484 0.1563 0.1891 0.4888 0.5900-0.0756 0.0040-0.085-0.0788-0.0731-0.0681-0.0634 9
Table 2. Unit root tests variables ADF statistics Zivot-Andrews statistics Euro spot rate -1.71846-3.45157 Japanese yen spot rate -1.89836-4.98788* Euro futures price -1.71335-3.46404 Japanese yen futures price -1.91112-5.01398* Euro basis -16.4746** -15.7778** Japanese yen basis -20.0113** -20.4575** Adjusted basis (Euro) -3.81** -19.6130** Adjusted basis (Japanese yen) -12.7** -18.0422** Note: * and ** denote rejection of null hypothesis of unit roots under 5% and 1% respectively. 10
Table 3. Test for Threshold effects Dependent variables Hansen (1997) s P-value (5%, 1% critical F-statistics value) Adjusted Euro basis 880.6859*** 0.0000 (2.9957, 4.6050) ( 1 10 16 ) Adjusted Japanese yen basis 2939.4759*** 0.0000 ( 1 10 16 ) (2.9957, 4.6050) Note: *** denotes rejection of the null hypothesis of linear model 1% respectively. 11
Table 4. Estimation results of Threshold autoregressive models. Panel A. Euro Basis trading return as the dependent variable variables First regime: Basis trading return 0.0116 Second regime: Basis trading return > 0.0116 Constant -0.0019*** -0.0096*** (0.0002) (0.0001) Lagged Euro Basis trading return 0.7712*** 0.5791*** (0.0332) (0.0032) Panel B. Japanese yen Basis trading return as the dependent variable variables First regime: Basis trading return 0.0686 Second regime: Basis trading return > 0.0686 Constant 0.0099*** 0.0446*** (0.0002) (0.0007) Lagged Japanese yen Basis trading 0.8036*** 0.3696*** return (0.0052) (0.0068) Note: ***, ** and * denote significance at 1, 5 and 10% levels, respectively. The numbers in the parenthesis are the standard deviations. 12
Table 5.Comparison of Out-of-sample performance between threshold models and linear models Out-of-sample evaluation Dependent variables Threshold model mean square errors Linear autoregressive model mean square errors Sum of mean square errors Sum of mean square errors Euro (adjusted basis) 26.2049 45.3671 Japanese yen (adjusted 30.8356 36.8193 basis) 13
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