Abstract
This paper uses extreme value theory and exponential generalised autoregressive score models to estimate the tail extremes of financial return series. The peak-over-threshold method based on the generalised pareto distribution is combined with the EGAS models and the nonparametric quantile method is used to determine the thresholds in the POT method, which is used to calculate the value-at-risk of financial markets and to perform backtesting. The empirical analysis was conducted on the soybean oil, rapeseed oil, and palm oil futures indices in the Chinese futures market. The study demonstrated that the EGAS-POT models based on nonparametric quantile thresholds can effectively characterise tail risk and provide a feasible measure of risk for investors.
1. Introduction
In recent years, China’s economy has made remarkable achievements. At the same time, as the Chinese futures market gradually integrated with the international futures market. In 2020, 6.153 billion lots and 437.53 trillion yuan were traded in China’s futures market, accounting for 13.2% of the total volume in the global futures market. Palm oil, soybean oil, and rapeseed oil on China’s futures market will rank second, fourth, and ninth, respectively, in the global trading volume of agricultural futures and options. Chinese vegetable oil futures is one of the most widely used varieties in the Chinese market. Moreover, Chinese vegetable oil futures are typical in agricultural futures market. There is a substitute or complementary relationship among all varieties, and it is closely related to spot price. In addition, the price fluctuation of vegetable oils, oilseed, and oilseed meal has a wide range of influence, which has direct or indirect impact on the living consumption of residents, processing of food enterprises, and production of livestock enterprises. Since soybean futures were listed in 1993, oil and oil futures have gradually realized the integration of upstream and downstream of the industrial chain and the diversification of derivatives, such as futures and options. By introducing foreign traders to participate in the trading of palm oil futures and options and other kinds of instruments, they can hedge and avoid risks to a certain extent. China oil futures price is also a leading indicator to monitor the price fluctuation of oil agricultural products and reflect the change of consumer price level. The price fluctuation not only has the transmission mechanism from futures to spot market but also has the cross-market and cross-period relationship. In the context of accelerating the financialisation of agricultural products and preventing systemic financial risks, the role of agricultural futures market in managing price risks is more prominent. Therefore, it is of great theoretical and practical significance to study the risk of vegetable oil futures.
In the context of globalisation, risk measurement has always been an important part of financial risk management and investors and regulators have studied quantitative tools for financial risk, such as value-at-risk [1] (VaR), for this purpose. However, as extreme events have had a huge impact on financial markets, the volatility in international financial markets has increased, the risk of a global economic downturn is growing, and uncertainty about the outlook has increased dramatically. The tails of traditional metrics describing the distribution of returns are underperforming.
The behaviour of the tails of financial risk has been assessed by many scholars. McNeil [2], Jondeau and Rockinger [3], and Da Silva and Mendez [4] showed that the tails of returns on asset returns have extreme values and do not follow a normal distribution and that their empirical distribution is characterised by spikes and thick tails. Therefore, classical parametric methods based on the assumption of a normal distribution are not suitable for estimating risk in financial markets. One of the ideal alternative parametric methods is extreme value theory (EVT), and methods based on EVT can be used on the basis of VaR. Embrechts [5] discusses the application of EVT to risk modelling. Manfred Gilli and Evis Kellezi [6] apply EVT to measure tail risk in six stock market indices, The results show that extreme value theory is effective for estimating extreme events in financial markets and that the POT method allows better use of information in the data sample to understand the details of financial market data. McNeil [7] showed that direct application of EVT can overestimate or underestimate VaR as the financial asset return series do not satisfy the assumption of independent homogeneous distribution, and there is heteroskedasticity. Therefore, the original income sequence needs to be processed.
Starting from Engle’s autoregressive conditional heteroskedasticity [8] (ARCH) model. The problem of constant variance of time series variables in traditional econometrics is solved. More models have been developed to model volatility, such as the generalised autoregressive conditional heteroskedasticity [9] (GARCH) model, the exponential generalised autoregressive conditional heteroskedasticity [10] (EGARCH) model, and the asymmetric power generalised autoregressive conditional heteroskedasticity [11] (APARCH) model. These models perfectly interpret the volatility characteristics of time series in financial markets, such as asymmetry and leverage. As financial markets become more complex. Creal, Koopman, and Lucas [12] creatively proposed a unified framework for modelling time-varying parameters, namely the generalised autoregressive score (GAS) models, which provides a new option for modelling financial asset return volatility. Nortey et al. [13] modelled the extreme values of stock index volatility in Ghana by the autocorrelation of the return series was corrected and the conditional heteroskedasticity in the presence of collections. Applying EVT to fit the tails of daily stock return data for Ghana, the study showed that among the methods used to estimate the parameters, the maximum likelihood estimation (MLE) method provides more accurate estimates. Taking the top 10 sector indices of the SSE as an example, Ping [14] developed a GAS volatility model to predict and compare the out-of-sample VaR forecasting effects of the model. The study showed that the GAS volatility model for time-varying volatility modelling can effectively use the valid information of the distribution, and VaR performs better. To analyse the correlation between different types in the financial market, Rongda and Jianjun [15] developed a multivariate GAS model to analyse the interaction between the dependence balances and volatility between the prices of crude oil and gold, and the results show that the predictive power of volatility and correlation in the multivariate GAS model outperforms the DCC-GARCH model. Lazar and Xiaohan [16] introduced intraday information into the GAS model in quantile regression setting to estimate risk. The results show that the GAS model, augmented by the implemented volatility metric, consistently outperforms other models across all indices and various probability levels.
Traditional time-series models, such as AR model, MA model and GARCH family model, were mostly used in previous studies. In addition, they tend to predict the return series or measure the risk of the return series of stock index. There are few researches on the risk measure of the return series of agricultural futures and futures index in the financial market. In this paper, the time-varying parameters based on the score function are introduced into the EGARCH model for the futures index of Chinese vegetable oils and fats, and the most suitable residual distribution is selected. A nonparametric method is proposed to select the threshold value for the extreme value of the filtered standardized residual sequence, and the EGAS-POT model is established by combining the extreme value theory. Compared with the traditional threshold selection method, the utility of the model in risk measurement is investigated and tested back. Based on the empirical analysis results, some reasonable suggestions are put forward for the risk management of vegetable oil futures in China.
2. Materials and Methods
2.1. EGAS Volatility Model
The GAS model also goes by the name of dynamic conditional score (DCS) model, Score driven (SD) model, or dynamic score (DySco) model, is a time-varying volatile parametric models driven by observations that allows the model parameters to vary as the score function of the log-likelihood function changes. The dynamic behaviour of the time-series process is portrayed through the dynamics of the parametric variables leading to the variables and exogenous variables. The EGAS model, on the contrary, builds on the GAS model using the logarithm of the conditional variance instead of the conditional variance, allowing for asymmetries in positive and negative asset returns on volatility, thus allowing the dynamics of the impact of positive and negative returns on volatility to be captured effectively.
Assuming that is the financial time-series observation, is the time-varying conditional standard deviation, which represents the volatility of the time-series data, and is the information set at moment t − 1, then the observation probability density function is as follows:
Then the expression for the EGAS model based on time-varying volatile is as follows:where when t = 1, is the unconditional standard deviation, is the standardized residual series, and are time-varying coefficient matrices, reflecting the time-varying nature of the fluctuations and the aggregation and mean recovery of the fluctuations, respectively, usually and can be taken as 1. is the constant vector; is the information matrix; is the deflation matrix; in general, takes 0, at this time; is the deflation matrix is the unit deflation matrix; is the score function corresponding to , is the core driving term of the EGAS fluctuation model.
When the standardized residuals obey a different distribution, the expression for the score function changes as well. For financial time series, the distributions often assumed are: the Gaussian (Normal) distribution (N), the standard student T distribution (ST), the generalised error distribution (GED), and the skewed student T distribution (SKST), where the probability density of the skewed student T distribution is as follows:where , , , is the beta function, is the symbolic function, and is the variable with the mean value of 0 and the variance of 1. Parameter represents the kurtosis of the skew t distribution. When is smaller, the kurtosis is larger, and the thick tail of the spike is more obvious. is an asymmetric coefficient, indicating the skewness of the skew t distribution. If > 0, the distribution is right, and if < 0, the distribution is left. The skew t distribution includes normal distribution, skew normal distribution, and t distribution. When the parameter and = 0, the distribution is normal distribution. When parameter . At this time, the distribution is skew normal distribution. When the parameter = 0, the distribution is t.
When the standardized residuals obey SKST, the score function is as follows:
As can be seen from the score function, the skewness parameter and the kurtosis parameter determine the value of the score function, and as the value of increases, the score function becomes more sensitive to extreme values; the skewness parameter reflects the sensitivity of the score function to shocks on the left side, when the distribution is left-skewed, the score is relatively more sensitive to shocks on the right side; when the distribution is right-skewed, the score is relatively more sensitive to shocks on the left side; when the distribution is symmetrical, the response of the score to shocks on both sides is symmetric.
2.2. Extreme Value Theory
Extreme value theory was introduced by Gnedenk [17]. Also known as the law of small numbers is primarily concerned with the prediction of rare events. The theory aims to investigate the distribution of the extremes of a sequence, using the generalised pareto distribution or generalised extreme value distribution (GEV) to approximate the tail distribution of losses. First applied in hydrology, seismology, and climatology, it is commonly used to analyse probabilistic rare cases. With the increasing refinement of the theory, EVT research has been applied to science and technology, engineering and other fields, with Longin [18] pioneering the use of EVT in risk management with good results. In financial engineering, for the tail characteristics of risk loss distribution, it is usually used to analyse events with rare probability. It can rely on a small amount of sample data to obtain the change of extreme value in the overall distribution when the overall distribution is unknown and has the ability to estimate beyond the sample data.
There are two methods for the application of extreme value theory: Block maxima (BM) method and peak-over-threshold (POT) method, as the use of the BM method leads to the absence of extreme data in the block and the loss of extremely valuable extreme information. This paper uses the POT method, which is more widely used in practice. Clément Dombry and Ana Ferreira [19] show that the POT method is preferable when considering MLE, and the estimation results are more convincing for extreme data values.
Assume that the data of the random variable sequence is , ,…, is independently and identically distributed, and the distribution function of the random variable is , let be the maximum value of the random variable sequence data, by setting the threshold value , and all the observed data above this threshold value form a data group , with this data group as the object of modelling, such that , then we have
The derivation of the conditional probability formula leads to the following:
When the threshold is taken to be relatively high, the suprathreshold distribution will converge to the GPD. The GPD expression is the distribution function approximating the generalised Pareto distribution.where and are the shape and scale parameters, respectively. When ≥ 0, ≥ 0, indicating a thick tail of the distribution function, and the presence of extreme values; when < 0, The probability density function of the GPD is known, and hence the log likelihood function of a sequence of random variables known to obey an independent distribution:
When is determined, the estimates of and are obtained by MLE according to equation (8), and the shape parameter reflects the tail of the distribution. At the same time, the number of observations of the random variable series data that exceed the threshold data can be obtained, denoted as , and the new expression can be obtained by replacing the value of with the frequency according to equation (6):
2.3. VaR Estimation Based on the POT Method
VaR is the maximum possible loss to an investor owning a single asset or portfolio of assets at a certain confidence level (99%, 95%) and holding period. Its essence is to calculate the tail quantile of the yield distribution, where the long VaR corresponds to the lower tail quantile of the yield distribution and the short VaR corresponds to the upper tail quantile of the yield distribution. The expression of is
Taking equation (9) into equation (10) gives the following:
3. Empirical Analysis
3.1. Data Analysis
The empirical part uses the Chinese futures market Soybean oil (Y8888), Rapeseed oil (OI8888), and Palm oil (P8888) futures indices as raw data (data source: Flush iFinD), with the sample space selected from 4 January 2010 to 31 May 2021, and the first-order difference of the logarithm of the daily closing price is used as the daily log return for ease of processing, that is, , where denotes the closing price on day t and denotes the closing price on day t − 1. Figure 1 shows a graph of the daily return series.
Descriptive statistics for the daily return series are shown in Table 1.
As shown in Table 1, from the description of daily logarithmic returns, the average returns of soybean oil, rapeseed oil, and palm oil futures are all near 0, with a range of 0.117006, 0.201763, and 0.134392, respectively. The excess kurtosis coefficient is greater than 0, that is, the logarithmic rate of return series has a peak. The skewness is less than 0, indicating that there are different degrees of left skewness, indicating that there are more huge falls in the market than huge rises, that is, there is a negative skewness vegetable oil futures all exhibit a left-skewed and spiky distribution that does not obey a normal distribution. This is the same result as Balaban’s [20] study on the distribution characteristics of daily stock returns and their asymmetry. The test used to test the daily returns for smoothness is the Jarque-Bera [21] test, and the results are shown in Table 2. The original hypothesis is rejected because the significance is much less than the critical value of its significance level of 1%, and the series does not have a unit root, and is a smooth series.
In order to further test the distribution characteristics of the sample series, Figure 2: normal Q-Q plots of daily returns was described. The scattered points on the log-return normal Q-Q plots were curved at both ends and distributed outside the 95% confidence level interval of the normal distribution, indicating that the distribution of log-return is thick-tailed. To sum up, the original sample sequence follows the distribution with sharp peak and thick tail deviating to the left, so the predicted results of VaR calculation method based on normal distribution are too conservative.
Second, part of the efficiency of financial markets also reflects the general autocorrelation between raw returns, so using metrics directly is not feasible. Figure 3: ACF plot shows that there is no autocorrelation in the daily return series, and the autoregressive conditional heteroskedasticity test on the return residual series shows that there is a strong ARCH effect in the residual series through the ARCH-LM test. The POT method requires the sequence of random variables to meet the requirement of independent identical distribution, so for the original data with volatility aggregation and leverage effect, so the volatility model needs to be constructed to filter the return series, and the residual series of each set of returns obeying independent identical distribution is found.
3.2. Model Parameter Estimation
In this study, the EGAS (1,1) model was chosen to filter the data for each set of return series and to compare the fit of the model under various hypothetical distributions according to the AIC criterion and the SC criterion. The AIC and SC estimation results of the model are shown in Table 3:
As can be seen from Table 3, the hypothetical distribution with skewed, spiky and thick-tailed characteristics is significantly better than the symmetric hypothetical distribution, and the hypothetical distribution of raw returns is chosen as SKST.
According to the parameter estimation results in Table 4, the parameter was significantly greater than 0, indicating that the return rate series had obvious time-varying fluctuation characteristics. Parameter is close to 1, which means that the return rate series has a strong agglomeration, , which means that the unexpected news impacts the fluctuation of return, and 0, it proves that this kind of shock is positive, that is, the occurrence of fluctuation is usually followed by a larger fluctuation in the later period. In terms of distribution parameters, the palm oil futures parameter is larger, reflecting more extreme risk exposure in historical data.
3.3. The POT Method
The residual series obtained by constructing the EGAS (1,1)-SKST model satisfies the POT method requirements such that the residual series and the distribution function is fitted with GPD model for data above the threshold extremes. In performing model estimation, the upper and lower tail thresholds of the residual series are first determined. Caeiro [22] used different methods for the selection of the thresholds a comparative analysis was carried out.
In this paper, we adopt a novel nonparametric method for selecting the threshold . Applying Grevenko’s theorem [23], the empirical distribution function is related to the overall distribution function, thereby improving the accuracy of the quantile of the original data series.
Step 1. If the number of data in the residual series is , let the parameter = 100, extract 100 data item by item in the residual series in time order, select the 2nd to 101 data for the 2nd time, ..., select the to data for the time, and so on, to obtain (N − M + 1) data.
Step 2. Based on the confidence level , the quantile values of each group of data were calculated separately using the historical simulation method combined with the principle of Mouchel [24] and through Holger Drees [25] in a comparative study of random and deterministic thresholds, so that = 10% to obtain (N − M + 1) quantile values.
Step 3. The average of the selected quantiles is noted as . The value closest to s is found in the new interest sequence, which is noted as the threshold value , and is used as the threshold value in the POT method.
The nonparametric method not only effectively avoids the subjective judgement of thresholds based on image methods that lead to over or improper fitting, but for the value of avoids the situation in the POT method where extreme values are piled up leading to too large a choice of thresholds and a small sample of extreme value data. The results based on the nonparametric threshold selection are shown in Table 5.
The results of the nonparametric method of fitting are shown in Table 6.
In order to test the fitting effect of the POT method, we further give Figures 4 to 9 GPD fitting diagnostic plots for the fitting of the residual series. Observe that the data points in the graph are concentrated in each distribution curve except for individual data, proving that the POT method fits the extreme data of the new coupon series well and can accurately reflect the tail characteristics of the true loss distribution of the extreme risk of the returns series.
In order to compare the validity of the nonparametric method of threshold selection, the Hill estimator [26] and the mean excess function [27] (MEF) were used to estimate the VaR values by Figure 10 selecting the threshold and performing a comparative analysis. The Hill plots Figure 10 and MEF plots are shown in Figures 10 to 12.
The thresholds were selected by observing Hill plots and MEF plots, and the results of the threshold selection are shown in Table 7.
Define the nonparametric method to select the threshold value of VaR is VaRn, Hill estimation method and MEF method to select the threshold value of VaR is VaRm, respectively, at the confidence level of 99% and 95% of the VaR value, The estimation results are shown in Tables 8 and 9.
3.4. VaR Backtesting
The test for VaR takes the Kupiec test [28], which is a very widely used method of posterior analysis, by constructing a likelihood ratio statistic to test the estimated loss value and the actual loss value, which passes the test within a certain acceptance range, and under the original hypothesis, the statistic obeys a distribution with degree of freedom of 1. The smaller the statistic , the larger the P value, indicating that in the model, the more accurate and the higher the credibility. When , its validity passed the post hoc test.where is the number of days to failure and is the total number of days observed. is the frequency of failure and is the confidence level.
The backtesting for the raw returns of the three futures indices are presented in Tables 10 and 11 for the long and short positions under the two threshold selection methods, respectively.
From Tables 10 and 11, it can be seen that the palm oil futures index is exposed to greater risk than the other two futures indices at the 99% and 95% confidence levels. The reason is that the gap between China’s production and demand ranks among the top in the world, and China is excessively dependent on imports of palm oil. The EGAS-SKST-POT model under the threshold selected using the nonparametric method, the number of days to failure is closer to the theoretical number of days and the VaR values of long and short positions under the model pass the model backtesting, proving that the model is feasible. The use of this approach enables a significant increase in the accuracy of the model, and out-of-sample forecasting ability. In contrast, the thresholds based on the Hill estimator and the empirical mean-excess function deviate significantly and the VaR values for long and short positions under the model do not all pass the model backtesting.
4. Conclusion
The main objective of this paper is to apply EVT and the EGAS model to the Chinese vegetable oil futures index by targeting the characteristics of aggregation, persistence and asymmetry of daily returns in the Chinese stock market. First, the standard residual series based on the SKST distribution is inscribed through the EGAS(1,1) model, and a nonparametric quantile-based approach is adopted to select the threshold and apply the POT method in extreme value theory to calculate VaR values and perform backtesting. The study shows that the nonparametric method proposed in the article is able to select suitable thresholds and that suitable estimates can be obtained by fitting the GPD distribution for data where the new series exceeds the threshold, as well as the feasibility of the EGAS model on the China Futures Index.
Based on the empirical analysis of the value at risk of soybean oil, rapeseed oil, and palm oil futures indexes, the following suggestions are put forward:
First of all, in order to prevent the price fluctuation and risk of vegetable oil futures market, we should give full play to the price discovery and hedging functions of futures market. In the current situation of increasing price volatility in the futures market, it is necessary to strengthen effective monitoring of vegetable oil futures prices, improve the financial market supervision system, and prevent abnormal price volatility from negatively affecting other futures varieties. By monitoring and real-time analysis of abnormal events such as agricultural futures market information and other market information and policy changes, effective supervision of risk events can be achieved, and then timely warning can be given before the risk may break out, and timely response can be made after the risk occurs.
Second, strengthen the construction of futures market, vigorously develop futures market, increase trading varieties, control prices, give play to the unity and advance advantage of futures and options market information, and develop in the direction favorable to the development of financial market. Establish cross-sector agricultural product market risk early warning organization coordination mechanism. To solve the problems of incomplete and asymmetric information of agricultural products market, we should construct a multilevel matching and linkage information system of agricultural products market. Build agricultural market information sharing platform, form multidepartment organization and coordination mechanism.
Finally, as a result of the palm oil futures price volatility compared with other oil futures price volatility is larger, the sino-us trade friction and imposing import tariffs under the background of U.S. soybeans, necessary policy interventions can be taken to reduce oil futures price volatility, focus on palm oil futures varieties, broaden the import channel, implementing multiple imports of palm oil, add palm oil strategic reserve. Participants and investors in relevant industries should enhance policy attention and market sensitivity, comprehensively consider the factors of market fluctuations, and further strengthen rational production and investment awareness. In addition, the government should improve laws and regulations related to palm oil production and trade and reasonable market structure to ensure the safety and virtuous cycle of vegetable oil market and related markets.
Data Availability
The data used to support the results of this study are available from the Flush iFinD.
Conflicts of Interest
The authors declare that they have no conflicts of interest related to this work.