Currency Hedging Strategies Using Histogram-Valued Data: Bivariate Markov Switching GARCH Models

Previous studies aimed at determining hedging strategies commonly used daily closing spot and futures prices for the analysis and strategy building. However, the daily closing price might not be the appropriate for price in some or all trading days. This is because the intraday data at various minute intervals, in our view, are likely to better reflect the information about the concrete behavior of the market returns and reactions of the market participants. Therefore, in this study, we propose using high-frequency data along with daily data in an attempt to determine hedging strategies, using five major international currencies against the American dollar. Specifically, in our study we used the 5-min, 30-min, 60-min, and daily closing prices of the USD/CAD (Canadian Dollar), USD/CNY (Chinese Yuan), USD/EUR (Euro), USD/GBP (British Pound), and USD/JPY (Japanese Yen) pairs over the 2018–2019 period. Using data at 5-min, 30-min, and 60-min intervals or high-frequency data, however, means the use of a relatively large number of observations for information extractions in general and econometric model estimations, making data processing and analysis a rather time-consuming and complicated task. To deal with such drawbacks, this study collected the high-frequency data in the form of a histogram and selected the representative daily price, which does not have to be the daily closing value. Then, these histogram-valued data are used for investigating the linear and nonlinear relationships and the volatility of the interested variables by various singleand two-regime bivariate GARCH models. Our results indicate that the Markov Switching Dynamic Copula-Generalized autoregressive conditional heteroskedasticity (GARCH) model performs the best with the lowest BIC and gives the highest overall value of hedging effectiveness (HE) compared with the other models considered in the present endeavor. Consequently, we can conclude that the foreign exchange market for both spot and futures trading has a nonlinear structure. Furthermore, based on the HE results, the best derivatives instrument is CAD using one-day frequency data, while GBP using 30-min frequency data is the best considering the highest hedge ratio. We note that the derivative with the highest hedging effectiveness might not be the one with the highest hedge ratio.


Introduction
Big data are understood as data in a gigantic size having a large volume of many varieties of information, compiled continuingly and in general at a relatively high frequency. However, the data collected can be either structured or non-structured, making it unable to be managed in the general database systems [1,2]. Presently, big data are utilized in many fields such as medicine, sciences, computer science, and business, in addition to as a financial investment. At present, global stock markets such as the New York Stock Exchange (NYSE) and the Nasdaq Stock Market (NASDAQ) commissioned IBM Netezza to collect ent between the upturn and the downturn periods. However, the MS-GARCH model is still not suitable for the estimation to obtain the HR and the optimal portfolio containing spot and futures contracts because it cannot provide the variance and covariance of the returns of the hedging instruments, which are key variables in the hedging equation for risk management.
The present study takes into consideration the probable existence of structural change, which demarcates the upturns and downturns episodes in the currency market. It, thus, employs the Markov switching CCC-GARCH model of Pelletier [15]; the Markov switching DCC-GARCH model of Billio and Caporin [16]; and the recent Markov switching dynamic copula GARCH models of Pastpipatkul, Yamaka, and Sriboonchitta [17] as a means to obtain the variance and covariance of spot and future returns of the international currency for calculating the hedge ratio and hedging effectiveness.
This study uses, for its analysis, the spot and future returns of the top five international currencies against the American dollar released at 5-min, 30-min, 60-min, and daily intervals for the one-year time period of 2018 to 2019. As the data used were of a high frequency, we reduced the series into the daily histogram-valued form following the concepts of González-Rivera and Arroyo [8] so as to simplify and quicken data processing. The histogram is particularly useful for dealing with large data sets when big data are more meaningful in some instances. In our case, conceptually, the optimal hedging strategies should be built on the basis of the best representative of the intraday trading prices rather than the daily closing price. In the conventional method, the solution to analyzing the daily high-frequency data is to reduce the collection of records (in the form of histogram-valued data) associated with each observation (taken every 5 min, 30 min or 60 min of the day) to one value, which may be represented by either the mean, mode, maximum, or minimum of all observations, called the histogram-valued data. However, with these representations, the variability across the records in the histogram is lost [18]. Thus, Dias and Brito [18] suggested finding the representative of the data by considering the quantile of the cumulative distribution of the histogram-valued data. González-Rivera and Arroyo [8] and Rakpho et al. [9] mentioned that by recording the high-frequency data as a histogram, all information of the data during the day is collected, and It is possible to choose reference points that better represent the histogram in each day. In this study, we aimed at finding currency hedging strategies using histogram-valued data and Markov switching CCC-GARCH, Markov switching DCC-GARCH, and Markov switching dynamic copula GARCH. More specifically, we aimed at calculating the HR from the conditional covariance matrices to achieve the hedging strategy and compared the performance of HR by considering HE. One of the main contributions of this paper is that it allows for a comparison for whether the results are different depending on the model, currency, and futures contract of the currency selected. In addition, to the best of our knowledge, this was the first attempt ever to investigate and determine currency hedging strategies using histogram-valued data and the Markov switching dynamic copula GARCH model.
The remainder of this paper is organized as follows. Section 2 introduces the methodologies considered in this study. Section 3 describes the data and discusses the related statistics in this study. Section 4 provides the model comparison results and compares the hedge performance across the currency markets. Finally, Section 5 is the conclusion.

Model
In this section, we first explain the histogram-valued data concept and the way to obtain the representative data from the histogram. Various econometric models used to forecast the dynamic volatility and dynamic correlation are briefly presented. Finally, we explain the procedure to compute the hedge ratio and hedging effectiveness for the currency markets.

Histogram-Valued Data
In this study, we consider high-frequency data, i.e., 5 min, 30 min, and 60 min and these data can be recorded in the form of a histogram [18]. Let Y = {y(1), . . . , y(T)} be the vector of the histogram-valued variable and y(t), t = 1, . . . , T be the histogram-valued data at time t which can be represented by the histogram H y(t) = I y(t)1, , I y(t)1 , p t1 ; I y(t)2, I y(t)2 , p t2 ; . . . I y(t)n t , , I y(t)n t , p tn t , where I y(t)i and I y(t)i are the lower and upper bounds of the sub-interval i, respectively, with i = {1 t , . . . , n t ) . n t being the number of sub-intervals for the t th observation, p ti being the probability or frequency associated with the sub-interval i at time t and Note that within sub-interval i, the values of y for each unit t are uniformly distributed.
To find the reference point that best represents y(t), the cumulative empirical distribution function is used to derive the y(t) and its inverse or quantile function [19,20] where a y(t)i = I y(t)i − I y(t)i .

Markov Switching(MS)-CCC-GARCH and Markov Switching(MS)-DCC-GARCH
Billio and Caporin [16] and Chodchuangnirun, Yamaka, and Khiewngamdee [21] introduced the extension of the CCC-GARCH and DCC-GARCH of Engle [12] to the Markov switching model of Hamilton [22]. These two models have similar structures, with the only difference being that MS-CCC-GARCH assumes the regime-dependent correlation matrix R s t to be constant in each regime, while the regime-dependent correlation matrix is considered to be varying over time in each regime, R t,s t , for MS-DCC-GARCH. Note that s t is a state variable, which follows a first-order Markov process and can assume only integer values of 0,1.., S; thus, the correlation matrix at time t, R t,s t , can be split in S regimes, and we have R t for s t = 0, s t = 1, . . . , s t = S.
In this section, only MS-DCC-GARCH is explained. Following Billio and Caporin [16], the model takes the following form is vector of the reference points that best represent the histogram at time t for spot and futures returns, respectively. U t is an independent and identically distributed process with zero mean and variance-covariance matrix I 2 . Q t is the time-varying variance-covariance matrix presented as where ε spot t and ε f utures t are the error terms following Student's-t distributions with degrees of freedom of v spot and v f uture for spot and futures, respectively. u spot and u f utures are the constant terms.
Thus, R t,s t becomes a 2 × 2 matrix of the regime-dependent time-varying correlation between spot and futures. We would like to note that if MS-CCC-GARCH is estimated, R t,s t is assumed to be constant for each regime. However, in the case of MS-DCC-GARCH, R t,s t is the time-varying conditional correlation matrix that can be predicted by the following process where Q t,s t represents the regime-dependent unconditional correlation and ξ t is the 2 × 1 vector of standardized residuals, ). It is worth noting that DCC just imposes a GARCH(1,1) on the conditional correlation and uses only θ 1,s t and θ 2,s t to add a dynamic behavior according to Billio and Caporin (2005). This model allows parameters θ 1,s t and θ 2,s t to be governed by s t , where s t = {1, . . . , S} is the latent and unobservable market state variable at time t, and S is a positive integer representing the total number of market states. In this study, we considered only upturn and downturn market regimes, thus s t = {1, 2} was assumed. The state variable s t evolves according to the following transition probabilities: where p ij = Pr(s t = j|s t−1 = i ), 2 ∑ j=1 p ij = 1. p ij is the probability of switching from state i to state j.

Markov Switching Dynamic Copula GARCH
In this subsection, the Markov switching dynamic copula (MSDC) GARCH model is explained. The Student's-t copula function is assumed for this model as it has been commonly and successfully used for fitting financial data [17]. This model is different from the MS-CCC-GARCH and MS-DCC-GARCH in two aspects. First, the joint distribution assumption of this model is a Student's-t copula, while multivariate Gaussian distribution or normal joint distribution is assumed for the MS-CCC-GARCH and MS-DCC-GARCH models. Second, the dynamic copula GARCH is extended to the Markov switching model; thus the dynamic conditional correlation is predicted by restricted ARMA (1,10) [13], which can be written as where ω 0,s t , ω 1,s t , and ω 2,s t are parameters. Λ(•) is the transformation function to ensure that the regime-dependent dynamic correlation R t,s t always remains in [−1,1]. This transformation function is represented by the following equation and Γ t is the forcing variable, which is defined as follows where F −1 is the quantile function of Student's-t distribution. U 1,t = F 1 (Ψ −1 y(t) spot (k)) and U 2,t = F 2 (Ψ −1 y(t) f utures (k)) are student-t cumulative distribution functions (CDFs) of Ψ −1 y(t) spot (k) and Ψ −1 y(t) f utures (k), respectively. Formally, the copula function is used to join different univariate distributions to form a valid multivariate distribution. According to the theorem of Sklar [23], the conditional bivariate joint distribution function of Ψ −1 y(t) spot (k) and Ψ −1 y(t) f utures (k) can be defined as where C t is the time-varying Student's-t copula function and Θ s t = {ω 0,s t , ω 1,s t , ω 2,s t } denotes the vector of the regime-dependent parameters in ARMA (1,10) in Equation (11). Assuming all CDFs are differentiable, the conditional bivariate joint density is then given by where θ s t is the vector of parameters to be estimated in the GARCH models in Equations (5) and (6), and f 1, y(t) f utures (k), respectively, constructed by the GARCH models. Thus, the full conditional likelihood function can be expressed as: where Φ s t = Θ s t , θ spot,s t , θ f utures,s t , and Pr(s t |Ω t−1 ) is the filtered probabilities [24], where Ω t−1 is all past information at time t − 1. Note that getting the filtered probabilities is a recursive process and the probabilities of staying in regimes 1 and 2 are predicted by In the estimation perspective, it is difficult to optimize and maximize the likelihood function in Equation (15); thus, we followed the two-step estimation or inference for the margins (IFM) of Joe and Xu [25] to partially resolve this problem, where the marginal densities and the copula density were estimated separately. This is to say, the GARCH process was firstly estimated to obtain the GARCH parameters. The obtained parameters are treated as a fixed parameter in the likelihood function in Equation (15), and then MSDC is estimated to obtain Θ s t and forecasts the regime-dependent dynamic correlation R t,s t .

Hedge Ratio and Hedging Effectiveness
To secure positions in a spot market, investors employ futures as hedging assets. For each spot contract, the hedge ratio tells us how many futures contracts should be purchased or sold [26]. Once h Spot,t , h Futures,t and R t,s t are obtained, the regime-dependent hedge ratio can be defined as After obtaining the optimal hedge ratio, we consider the reduction in the hedged portfolio's variance as the hedging effectiveness. The corresponding regime-dependent hedging effectiveness (HE) can be calculated by where var unhedged t,s t and var hedged t,s t are the regime-dependent variance of the unhedged portfolio (only spot) and the hedged portfolio at time t, respectively.
To compare the performance of HR obtained from different correlation and volatility models, Chang, González-Serrano, and Jimenez-Martin [5] suggested considering the HE values, because a higher HE indicates a larger risk reduction, meaning that the hedging method with a higher HE is regarded as a superior hedging strategy. Therefore, this study also uses HE to compare the performance of our bivariate conditional correlation models, namely Markov switching (MS) CCC-GARCH, Markov switching (MS) DCC-GARCH and the Markov switching dynamic copula (MSDC) GARCH. In addition, we also consider the classical models consisting of CCC-GARCH, DCC-GARCH, copula-GARCH, and dynamic copula (DC)-GARCH as other competing models.

Data
The data used in the present study are the futures and spot closing prices of the top five most traded currency pairs, namely USD/CAD, USD/CNY, USD/EU, USD/GBP, and USD/JPY at 5 min, 30 min, 60 min, and one day's intervals-thus considered as high-frequency data. The study period is from January 2018 to May 2019, which coincides with the time the UK is pursuing its withdrawal from the European Union (EU). Brexit is considered one of the major events bringing about shocks and high volatility in financial markets in UK, Europe, and the world, as well as the concern and interest of people worldwide. The futures prices here refer to the prices of futures contracts with an expiry date in June 2019. All historical time series were collected from the Bloomberg database for transformation into a logarithm of the change in price or return as The descriptive statistics of the variables for the investigation are provided in Tables 1 and 2.  Tables 1 and 2 present the descriptive statistics of the spot and future prices of the variables at different frequencies. Most currency spot and futures prices have negative means. The standard deviation (Std) of both the spot and future series is close to zero, indicating the low dispersal of data points or return levels. The largest Std for both the spot and futures belongs to the daily returns of JPY at 0.0073 and 0.0071, respectively, suggesting the high dispersal of return levels, with some distance away from the mean. Meanwhile, the smallest Std for both the spot and futures is found in the daily returns of CAD and CNY at 0.0002 and 0.0005, respectively. The skewness of some variables is less than 0, meaning that the returns are distributed skewed to the left, and if it is greater than 0, it is said to be negatively skewed. The largest skewness was presented in the 30-min spot returns of CAD. As a measure of the tailedness of the data distribution, the kurtosis of both spot and future returns at 5-min and 30-min intervals, including that of future returns at 60-min intervals, had values indicating that the distributions had heavier tails than a normal distribution. The kurtosis of 5-min JPY future returns, in particular, had a value as high as 15.3740, much higher than the highest point of the bell-shaped curve of a normal distribution. The remaining data series had kurtosis values lower than that of the normal distribution (=3), meaning that their distributions were lightly tailed. Figure 1 reveals the high-frequency data of the spot and future returns of five currencies to exhibit a high fluctuation over the whole sample period. The spot currencies seemed to be more fluctuating than the futures. Figure 2 presents the histogram-valued data in the first ten sample periods. We can see that the variation of the 5-min spot returns is not greater compared with the futures counterpart.

Results
In this section, we provide the results of the study. The estimation results of the GARCH models are firstly reported, then the obtained standardized residual of spot and futures are further used as the input of single-regime bivariate GARCH and two-regime bivariate GARCH models. Finally, all volatility models are compared using various criteria.

The GARCH Model Estimation Results
Tables 3-6 report the results from the estimation using the GARCH(1,1) model for spot and future returns at various frequencies, which will be used to determine the standardized residuals for further analysis of the relationship among the variables. The results tell us that only the parameter estimates of β 1 are statistically significant for all currencies in terms of spot and futures, as well as data frequencies, except for future returns of CNY. The coefficients of the intercept term are also statistically significant for the returns of EUR futures and CNY futures.     To validate the estimation model, we performed a test of autocorrelation and heteroscedasticity using the Ljung−Box Q-Statistic and the ARCH(1,1)-LM tests, respectively [27,28]. Evidently, these two econometric problems were not present in the spot and future returns of all currencies at all data frequencies. This means that the standardized residuals derived from the GARCH(1,1) model estimation were reliable and thus could be used for further estimation using the correlation models.
As this study took into consideration the probable structural change causing the occurrence of upturn and downturn episodes in the currency market, we proposed using the Markov switching CCC-GARCH, Markov switching DCC-GARCH, and Markov switching dynamic copula GARCH models as the mechanisms to find out the covariance of the spot and future returns of all currencies that were used for calculating the hedge ratio and the hedging effectiveness. However, we had to test the reliability of our proposed models to ensure achieving an accurate HR and HE by comparing them with the other five competing models commonly found in previous research works, namely the CCC-GARCH, DCC-GARCH, COPULA-GARCH, DC-GARCH, and MS-COPULA-GARCH. Therefore, we had eight models to estimate in order to get the optimal hedge ratio and hedging effectiveness.

Optimal Model Selection
In this sub-section, we compared the performance of the volatility models for all of the data frequencies. The results of the model performance comparison are shown in Tables 7-10, and the selection of the optimal model was based on the Bayesian information criterion (BIC) and Log-likelihood (LL). The model with the lowest BIC was preferred because the lower BIC corresponded to the lower variation in the error term. As evident in Tables 7-10, the MSDC-GARCH model had the lowest BIC for all cases. Thus, we will used its estimates to compute HR and HE.

Parameter Estimates of the Optimal Model
The parameter estimates from all models are presented in Table 11. This result implies that a structural change exists in currency spot and futures. Our finding is in line with Korley and Giouvris [29].

Testing the Efficiency of Hedge Ratios Obtained from Different Estimation Models
After getting the optimal model, we used its estimation results for calculating the hedging effectiveness: HE for the hedge ratio and HR for hedging with futures for international currencies. Furthermore, we assessed the extent of risk reduction by the hedging strategy using HE. To confirm that the selected MSDC-GARCH model could produce unquestionably reliable results, we calculated HE for comparison using the estimation results of the other seven competing models, and the overall results are presented in Table 12.  Table 12 shows the averages of the HE and HR values derived from the eight models considered in this study for all currencies and four data frequencies, which were at 5-min, 30-min, 60-min, and 1-day intervals. Our MSDC-GARCH model was indeed the best hedging model for all currencies and all frequencies as it provided higher HE values than other the models. From this model, the hedging effectiveness lay between a maximum of 8.99 % for CAD 30 min data and a minimum of 1.60% for JPY 60 min data. In contrast, we found that the average of HE was smaller for the CCC-GARCH in most cases and in particular, ranging from a maximum of 7.22% for EUR daily data to a minimum of 0.003% for CAD 5-min data. We also observed that with the use of CAD 30-min data, the HR of the MSDC-GARCH model was 0.301; indicating that in order to minimize risk, a long position of one dollar in CAD should be hedged by a short position of $0.301 in CAD futures contracts. Meanwhile, the HR of the CCC-GARCH model for EUR daily data and CAD 5-min data were 0.530 and 0.271, respectively. By comparing the HE across data frequencies, we found that the highest HE values for CNY, EUR, and GBP were from using 1-day future returns, while CAD had the highest HE using the 30-min returns, and JPY did so from using the 5-min returns.
To obtain a better picture of time-varying evolution, we illustrated the evolution of the time-varying hedge ratio between the spot and futures returns over the full sample period. These time-varying hedge ratios were computed from the MSDC-GARCH model with 5-min data (presented by the blue dotted line), 30-min data (presented by the black dashed line), 60-min data (presented by the red dashed line), and daily data (presented by the green dashed line) and are shown in Figure 3. We can observe that there was a substantial time-variation in the optimal hedge ratio for all currencies in most frequencies, arguably only for 5-min data, as there was some stability in the HR. Nevertheless, all time-varying hedge ratios exhibited a mean-reverting pattern and were stationary. . Time-varying hedge ratio measurements. Note: the hedge ratio measure plotted is the weighted average (HR t,s t =1 × Pr(s t = 1|Ω t )) + (HR t,s t =2 × Pr(s t = 2|Ω t )) with weights given by the smoothed probability of two regimes, i.e., Pr(s t = 1|Ω t ) and Pr(s t = 2|Ω t ): (a) CAD, (b) CNY, (c) EUR, (d) GBP, and (e) JPY.

Conclusions and Recommendations
This study introduces various bivariate MS-GARCH models for histogram-valued data to quantify HR and HE for five major international currencies. This study finds that the MSDC-GARCH model outperforms the others because it provides the highest HE values for all currencies and data frequencies. In addition, BIC also indicates that the MSDC-GARCH model outperforms other hedging methods. Therefore, with a more precise specification of the joint distribution of assets, we can effectively manage the risk exposure of portfolios. This result also confirms that the behavior of spot and future returns in the currency market can be explained appropriately by a nonlinear model.
MSDC-GARCH provides the best-performed hedge ratios and hedge effectiveness for risk reduction for all currencies, implying that MSDC-GARCH hedge strategies are suitable for those currency markets. Among the five currencies, the most efficient hedging currency is CAD with an HE of 0.889, while the least efficient hedging currency is JPY with an HE of 0.160.
Considering that the highest HE values estimated from the MSDC-GARCH model for JPY and CAD are from the 5-min and 30-min data, respectively, we can state that the data series for a particular variable (or currency in our case) and different frequencies will lead to a difference in the estimated values of HR and HE. Therefore, risk managers that commonly prefer using daily closing price or return for investment analysis and risk prevention have to be careful about using such data. With the use of big data in this study, we can conclude that daily data may not provide the best HE, and thus we should estimate HE using different frequencies of data.
The data used in this study are regarded as big data because of its enormous size and being collected at a high frequency-sometimes described as high-frequency data. The sizable data points result in the time-consuming and complicated nature of data management. This study uses the histogram-valued data as a solution for data management, but this approach has a drawback in that we cannot determine which data frequency will lead to the highest hedging effectiveness. In essence and to the best of our understanding, daily closing prices should be used carefully along with data at a wide range of frequencies [30].
For further study, we suggest applying the Smooth transition dynamic copula model of Yamaka and Maneejuk [31] to fit histogram-valued data and to quantify the HR and HE. Moreover, the copula approach is quite flexible for modeling the dependence between spot and futures; thus, we suggest considering different copula functions to capture the dependence structure of spot and futures and to improve the hedging strategy.
Author Contributions: Conceptualization, W.Y. and P.M.; data curation, N.P. and S.S.; methodology, W.Y. and N.P.; visualization, and writing-original draft, W.Y. and N.P.; writing-review and editing, W.Y. and P.M. All authors have read and agreed to the published version of the manuscript.
Funding: This work was funded by Center of Excellence in Econometrics, Chiang Mai University, Thailand.

Data Availability Statement:
The data used in the empirical analyses are available online at Thomson Reuter database. The data are available upon request.