key: cord-0644351-rcoan7so
authors: Bo, Shi
title: Dynamic Risk Measurement by EVT based on Stochastic Volatility models via MCMC
date: 2022-01-24
journal: nan
DOI: nan
sha: 6308780018962a14f8f6662d378bc0455b011cbc
doc_id: 644351
cord_uid: rcoan7so

This paper aims to characterize the typical factual characteristics of financial market returns and volatility and address the problem that the tail characteristics of asset returns have been not sufficiently considered, as an attempt to more effectively avoid risks and productively manage stock market risks. Thus, in this paper, the fat-tailed distribution and the leverage effect are introduced into the SV model. Next, the model parameters are estimated through MCMC. Subsequently, the fat-tailed distribution of financial market returns is comprehensively characterized and then incorporated with extreme value theory to fit the tail distribution of standard residuals. Afterward, a new financial risk measurement model is built, which is termed the SV-EVT-VaR-based dynamic model. With the use of daily S&P 500 index and simulated returns, the empirical results are achieved, which reveal that the SV-EVT-based models can outperform other models for out-of-sample data in backtesting and depicting the fat-tailed property of financial returns and leverage effect.

variables in the SV model. Accordingly, plentiful econometric methods have been proposed to estimate stochastic volatility patterns. The critical advantage of SV models over ARCH-type models is a white noise process added to evaluate changes in underlying volatility dynamics (Shephard et al. (1996) ; Kim et al. (1998) ). However, it is argued that financial time series data (e.g., stock returns and foreign exchange returns) exhibit several properties that deviate from the normality assumption and may have leverage effects. Thus, some researchers have suggested using non-normal conditional residual distributions for stochastic volatility modeling, including the student's t-distribution Harvey et al. (1994) , Generalized hyperbolic skew Student's t-distribution Nakajima and Omori (2012) , skewed distributions Harvey et al. (1994) . They first considered the fat-tailed property of asset returns on the underlying SV model, instead of the leverage effect. The distributions are correlated with the variance process, also called the leverage effect by Jacquier et al. (2004) , which can solve the problem.

Since the Basel Accord II, Value at risk (VaR) measuring the maximum possible loss of a portfolio of assets at a given confidence level over a given future horizon has become the benchmark of market risk measurement in both academia and industry. In general, the prediction of VaR is based on the distribution of the error term of the financial returns and the volatility forecast. Arising from the time-varying property of the financial market, the conventional assumption of unconditional normal distribution is no longer applicable, according toConsigli (2002) . Over the past few years, a number of studies have begun to primarily estimate and forecast VaR using a wide variety of volatility models. However, they also did not consider the fat tail of the model, the leverage effect and the extreme cases that can lead to considerable exceptional exceedance under extreme situations similar to the COVID-19 pandemic. For instance, Liu et al. (2018) employed a HAR-related models combined with extreme value theory to forecast VaR, which did not consider the leverage effect though the extreme cases were examined. Afzal et al. (2021) adopted DCC-GARCH for VaR estimation, which considered time-varying dynamics in VaR estimation, instead of the extreme cases and the correlation between returns and volatilities. Assaf (2017) selected several stock indices in international markets for one-day-head forecasting of VaR using SV and regime switch models and backtested them, whereas the fat-tail nature of stock returns was poorly reviewed. As a result, the original hypothesis was unexpectedly rejected at the backtesting stage. Furthermore, Yang et al. (2017) employed the MCMC method for parameter estimation of the SV with leverage model and performed VaR estimation, whereas they still ignored the extreme cases. In brief, the fat tails and leverage effects of the financial reporting series should be considered for the estimation of VaR, and the extreme cases should also be on the table.

In this paper, a novel method is purposed, and daily S&P 500 and simulated data sets are used and divided into training and test sets, respectively, according to different time periods, combined with GARCH and SV models using MCMC for parameter estimation, the portrayal of potential volatility dynamics and the addition of fat tails and leverage effects. On that basis, we also attempt to use each model combined with extreme value theory that can deal with the fat-tailed phenomenon more effectively by fitting the tails of the distribution directly to the data Bekiros and Georgoutsos (2005) . Besides, the goodness-of-fit test presented by Choulakian and Stephens (2001) is performed to test the reasonable degree of parameter fitting. Lastly, VaR forecasting and backtesting is performed to compare each method. As indicated by the empirical results, SV-EVT models could be practicable for VaR estimation according to their backtesting results; to a certain extent, it addresses too much exceedance. The VaR under SVtl-EVT model performed better than all other models, and all SV-based models performed better than GARCH-based and empirical method, but all models are somewhat overstretched under exceedance clustering. In-depth challenging research may be required to further optimize the dynamic model for more extreme scenarios including COVID-19 ever (four meltdowns in ten days in March 2020) that has triggered considerable uncertainty in the market. This paper is organized as follows. In Section 2, three stochastic volatility models are introduced, including SV with student-t, SV with correlated error and SV with student-t and correlated error and its MCMC sampling procedure. In Section 3, extreme value theory is outlined, as well as its application in risk measurement. In Section 4, the estimation results of all models are presented. In Section 5, backtesting methods and the Value at Risk backtesting results are presented. Section 6 draws the conclusion of this paper.

This paper starts with introducing the vanilla SV model with linear regressors. Subsequently, the analysis is conducted with three generalized models, including the SV model with Student's t errors (SVt), the SV model with correlated error (SVl), and their combination, the SV model with Student's t errors and leverage (SVtl). Lastly, we close the section after discussing Markov chain Monte Carlo (MCMC) sampling.

Let y t = (y 1 , . . . , y n ) T be a vector of observations. The SV model is outlined below:

where N (0, 1) is the normal distribution with mean 0 and variance 1, and ε t and η t are independent. X = x 1 , . . . , x n is a n × K matrix including in its t-th row the vector of K regressor at time t. The h = (h 1 , . . . , h n ) represents the log-variance with h 0 ∼ N µ, σ 2 / 1 − ϕ 2 . The β = (β 1 , . . . , β K ) are regression coefficients. We denote θ = (µ, φ, σ) as the SV parameters: µ is the long-term level, φ is the persistence and σ is the standard deviation of log-variance.

The basic model is restrictive for numerous financial series for its fat-tailed tendency. One of the extend of the basic model is to allow fat-tails in the mean equation innovation. Formally,

t ν (0, 1) is the student's t distribution with ν degree of freedom, mean 0 and variance 1. ε t and η t are independent. The only difference between Equation 1 and Equation 2 is that the observations are t distributed. In addition, when the degree of freedom ν goes to infinity, Student's t converges in law to the standard normal distribution. The Figure 2 .1.1 clearly shows that the probability density with tail plot of simulated y t generated from SVt has fatter tail than y t from SV model, which is more reasonable in the finance world. The basic model with zero correlation between ε t and η t also can be extended to allow for a so-called "leverage effect" via correlation between the mean and volatility error terms. The new parameter is introduced as ρ, indicating correlation between asset's returns and asset's volatility. If ρ is negative, then a negative innovation in the levels, ε t , will be associated with higher contemporaneous and subsequent volatilities Jacquier et al. (2004) .

where the correlation matrix of (ε t , η t ) is espressed as:

Thus, Equation 1 now is a special case of Equation 3 with ρ = 0. 

where the correlation matrix of (ε t , η t ) is identical to Σ ρ in Equation 4. Naturally, SVtl acts as a better choice, since it considers a long-term empirical fact that the increase in volatility following a previous drop in stock returns, which may cause the returns more fluctuated, and modelled by the negative correlation between error terms of returns and volatility; it is termed leverage effect as well. Thus, given the leverage effect and fat-tailed property of financial series, we can more effectively fit the real financial return data through the SVtl model. According to Figure 2 .1.1, the density of SVtl model has thicker tail and lower peak, which implies that it has more frequent extreme values and volatile compared to SVt and basic SV model.

For stochastic volatility models, various estimation procedures have been proposed to overcome the problem caused by unsolvable likelihood function. For instance, The parameters of the model were first estimated using the generalized moment estimation (GMM) method, which was first proposed by Hansen (1982) and then applied byMelino and Turnbull (1990) . However, the GMM method has poor finite sample size. Subsequently, Harvey et al. (1994) estimated the parameters with the use of the pseudo-maximum likelihood estimation (QML) method for the SV model using the Kalman filter, whereas the QML method has poor finite sampleability since that none of the mentioned methods requires a specific distribution. For this reason, the estimation performance is dependent on the sample size. However, in Bayesian methods, besides using sample information, prior information before sampling is incorporated to improve the estimation accuracy of parameters. The Markov chain Monte Carlo (MCMC) method, an effective means to achieve Bayesian inference, has been extensively employed in the study relating to SV models. Jacquier et al. (2002) first applied MCMC to SV models and showed that the estimation accuracy of MCMC estimators was better than that of GMM estimators and QML estimators. This shows that the MCMC method can effectively solve the parameter estimation problem of SV models. In this paper, the parameters of the SV model are estimated using MCMC.

MCMC combines the features of Markov chain and Monte Carlo estimation. To be specific, the former MC is Markov chain, and the latter MC refers to Monte Carlo estimation, where Markov chain can be used to draw a random sample (θ (1) , . . . , θ (m) ) from the posterior distribution p(θ|x), while Monte Carlo estimation is a method to estimate the posterior mean E(g(θ)|x) from the sample mean g(θ). Notably, given different initial values, the Markov chain converges after a period of iterations; however, before convergence, the distribution of each state in the first k iterations is not smooth. In practice, the first k iterations that have not reached the smooth distribution should be removed, and only take the next m k iterations for estimation, this process is called burn-in .We denote θ = (φ, σ, ρ, µ, β, ν) as the SVtl parameters, stock returns y = (y 1 , . . . , y n ) T , and the unobservable log volatility as h = (h 1 , . . . , h n ) .

Accordingly, the conditional likelihood function of the model can be expressed as:

Joint prior probability density 1 of parameters to be estimated θ and unobservable parameter n t=1 p (y t | θ, h t ) are writtern as

The joint posterior probability density of θ and h is proportional to the product of their prior probability and the conditional likelihood function according to Bayes' theorem:

Thus, the prior distribution of each parameter should be given in the stochastic volatility model. A common strategy is to select a vague prior here, e.g., µ ∼ N (0, 100), since the likelihood usually comes with sufficient information about this parameter. In accordance with the basic principle of MCMC parameter estimation, the stationary distribution is independent on the initial distribution, and the Markov chain is considered to be convergent if the marginal distribution of the states at the respective moment is stationary after a sufficient number of iterations. Thus, the posterior distribution of the parameters does not change significantly with the prior distribution of the parameters. For the prior and posterior distribution for the respective SV model, we strictly follow theJacquier et al. (2004) . Next, the following are selected as the priors µ ∼ N ormal(0, 100)

For the MCMC sampling algorithm, let θ = (φ, σ, ρ, µ, β, ν), y = {y t } n t=1 , h = {h t } n t=1 . For the prior distributions of µ and β, this paper assume µ ∼ N (µ 0 , ν 2 0 ), and β ∼ N (β 0 , σ 2 0 )

We draw the random samples from the posterior distribution of (θ, h) given y for the SVtl model using MCMC method Jacquier et al. (2004) , as follows:

1. Initialize θ and h.

6. Generate ν | φ, σ, ρ, µ, β, h, y.

7. Generate h | θ, y.

After the sampling is completed and all parameters converge, a complete SV model can be built, and the estimated volatility and returns can be acquired, which are subsequently transformed to the standard residuals. Thus, after obtaining the pair of time-varying volatilities and accounting for fat tails and leverage effects, the fitting of the extreme values should be conducted continuously.

Value at risk (VaR) represents the maximum potential loss of a portfolio of financial assets for a given level of confidence (α). Let P t be the price of the financial assets at time t and its log return at time t is

This paper assume that the dynamics of Y are given by

where the innovations Z re a strict white noise process with mean 0 and 1 variance. Let the density function of this return series be f (x), then the VaR at confidence level α can be expressed as:

Next, the formula for calculating the dynamic VaR of the return on assets Y t can be expressed as V aR t α :

where µ is the return forecast at day t. σ t is the volatility forecast at day t. V aR α (Z) denotes the value-at-risk of the residual term Z t at quantile α.

Different methods have been proposed to determine the VaR. One of them is a parametric model that can be employed to predict the return distribution of a portfolio. If this distribution is known in closed form, the value at risk is just the quantile of this distribution. Under non-linearity, Monte Carlo simulation or historical simulation methods can be employed. The advantage of the parametric approach is that the factors can be updated using a general model of changing volatility. Once the asset or portfolio distribution has been chosen, the predicted volatility can be adopted to express the future distribution of returns. Thus, the conditional predicted volatility measure σ t+∆ can be employed to determine the value-at-risk for the next period, where ∆ is period length. In our case of this paper, historical simulation approach is adopted to determine VaR.

Using empirical distribution has been recognized as the simplest method to determine VaR. First, the empirical distribution F e n . For data points {l i } i=1,...,n , the empirical distribution is the mass 1 n at each l i including repetition. We say V aR α (F e n ) = l ( nα ) , where nα = min[k ∈ N | k ≥ nα], then we sort data points in ordered values l (1) ≤ l (2) ≤ . . . ≤ l (n) . To prove that l ( nα ) , if there is no repetition, for example, we have

Since F e n is a step function:

Thus, l ( nα ) is the value-at-risk.

An ARCH process introduced by Engle (1982) (autoregressive conditional heteroskedasticity) refers to a time series variance model. An ARCH model is adopted to express a constantly changing and potentially unstable variance. Although it is possible for an ARCH model to be employed to describe a gradual increase in variance over time, it is most commonly adopted in situations where there may be short-term increases in variance. GARCH refers to an extension of the ARCH model that combines a moving average component with an autoregressive component. GARCH is the "ARMA equivalent" of ARCH, where ARCH has only one autoregressive component. The GARCH model allows for a broader behavior of more persistent volatility. The GARCH model can be written as:

where y t is considered log return series, α 0 , α 1 ≥ 0 to avoid negative variance, and for inference we would also assume that the t are normally distributed. However, given the fat-tailed property of the financial return series being consistent with the SV model, we use GARCH with student t innovation, t ∼ t ν (0, 1). Table 1 lists the results of parameter estimation by MLE (maximum likelihood estimation). According to the t-value and P-value, all parameters are significant. 

When a dynamic model is used, the volatility of asset returns is heteroscedastic and it is not appropriate to determine VaR directly from the distribution instead of converting asset returns Y t into standard residuals Z t . The estimates of the conditional mean and standard deviation series (μ t−n+1 , . . . ,μ t ) and (σ t−n+1 , . . . ,σ t ) can be determined from Equation 2, 3, 5 and 15. Subsequently, the standard residuals for the respective model are determined as:

The Figure 3 .3 clearly represents that after standardization, the points is more closed to the 45 degree line, thus indicating that it is relatively normally distributed, and the interval for residuals are smaller than original data 2 . In general, to find VaR, it is commonly assumed that the series of asset returns shows a conditional normal distribution with time-varying variances. This finding can be recognized as an improvement over the VaR calculated under the assumption that returns follow a normal distribution in general, whereas the VaR calculated under the assumption of normality of the error term is still not highly accurate since tail characteristics are not sufficiently considered. Accordingly, the VaR method should be supplemented by other methods (e.g., extreme value theory). Stress tests are commonly performed in accordance with the extreme value theory (EVT) Gencay and Selcuk (2004) . Extreme value theory does not use the overall distribution of the series, but fits the tails of the distribution directly using sample data, which can accurately express the quantile of the tails of the distribution and effectively solve fat tails. Instead, standard residuals are adopted to fit the tails.

The extreme value theory consists of Block Maxima model and POT (peaks over threshold). The POT model supposes the distribution function of the standard residual series {Z t } is F (z), and denote by u the threshold of a certain sufficient threshold, define F u (y) as the conditional distribution function of the random variable Z over the threshold u, which can be expressed as:

for 0 ≤ y < x 0 − u, where x 0 is the right endpoint of F . Balkema and De Haan (1974) and Pickands III (1975) represented for a large class of distributions F that it is possible to find a positive function β(u) such that

When u is large sufficiently, for a conditional distribution function F u (y), there exists a G ξ,β (y), so:

In Equation 19 ξ is the shape parameter, when ξ ≥ 0, y ∈ [0, ∞]; when ξ < 0, y ∈ [0, −β/ξ]. β represents the scale parameter. The distribution function G ξ,β (y) is termed the generalized Pareto distribution. According to Equation 19, the probability density function G ξ,β (y) of the generalized Pareto distribution can be obtained, so that for a given sample {z 1 , . . . , z n } that fits the generalized Pareto distribution, its log-likelihood function is:

After determining u, the estimates of β and ξ are obtained by maximum likelihood estimation based on Equation 20 using the observations of {Z t }. Let N u = {#n|Z n > u} N n=1 be the number of samples in the sample greater than the threshold value u, then it yields:

Subsequently,by substituting Equations 19 and 21 into equation 17, it yields:

For a given confidence level α, VaR of the POT can be obtained by Equation 11. Lastly, the dynamic VaR model can be concluded based on EVT-POT-SV-GARCH 3 (σ t,G represents the estimated volatility from GARCH model, and the subscript f ore implies forecasted.):

To estimate the parameters of the POT model, the first step is to select a reasonable threshold parameter u and then estimate the parameters ξ and β using the maximum likelihood estimation method. A high threshold will result in too little excess data and may increase the variance of the estimated parameters. In contrast, once a low threshold is chosen, the estimation accuracy will be increased but biased estimates will be produced. However, there has been no uniform method for selecting the "threshold" thus far. The problem of how to reasonably determine the threshold value to achieve the optimal partitioning of the sample to balance the correlation between bias and variance remains unsolved in the existing research of extreme value theory. In this paper, the threshold is estimated primarily mainly using the mean excess function method McNeil and Frey (2000) . The mean excess function can be written as:

where X (1) < X (2) < . . . < X (n) is the curve formed by the excess mean graph for point (u, e(u)) , and by selecting an appropriate threshold u, so that e(x) is approximately linear when x ≥ u 0 . If the excess mean figure is upwardsloping when x ≥ u 0 , the data shows a GP distribution with positive shape parameters ξ, and then the distribution is converted into a fat-tailed distribution. If the graph represents downward-sloping when x ≥ u 0 , the data originates from distributions with thin-tails and shape parameters ξ < 0. If the line is horizontal, the data originates from exponential distribution, ξ = 0. According to the excess mean method and Figure 3 .3.1, the threshold and estimated parameters via MLE for each model can be determined as Table 2 . To further test the validity of the POT fitting method, we following Choulakian and Stephens (2001) also need to check the goodness-of-fit. We adopt method from Choulakian and Stephens (2001) . The basic principle is to calculate the Cramer-von statistic W 2 and Anderson-Darling statistic A 2 in accordance with the parameters estimated by POT, and then to find the corresponding P-values. The model selection is appropriate when the P-values corresponding to both W 2 and A 2 are greater than 0.1. Based on the parameters estimated above, the goodness-of-fit is determined by testing the corresponding statistics as listed in Table 3 . Obviously, all corresponding P-values are greater than 0.10, thus, according to the selection criteria of the tail data proposed by Choulakian and Stephens (2001) , the POT model should be fitted to the tail data. Likewise, the fit to the tail data of S&P500 is reasonable. Thus, in general, the selection of the threshold is appropriate, and it is feasible to use the POT model to fit the tail data, and the VaR analysis of the SV and GARCH model based on the POT method is reasonable. 

The behavior of the S&P 500 is examined, and a sample from 01/4/2011 to 12/30/2016 acts as the training data resulting in 1509 data points, and that from 01/3/2017 to 12/31/2020, 1000 data points in total, acts as the test data to evaluate model. The price data originates from WRDS dataset. The price index is transformed by the first difference of the log price data to generate a series, which is close to the percentage return of continuous compounding. According to French et al. (1987) and Poon and Taylor (1992) studies, stock index prices were not adjusted for dividends, and they reported that the inclusion of dividends had an insignificant effect on the estimation results. Returns are calculated on a continuously compounded basis and expressed as a percentage, so they are determined as y t = 100 × ln( pt pt−1 ), where p t represents the index in day t.

The summary statistics are presented in Table 4 . According to the above table, the return series has negative skewness, thus indicating that generates frequent small gains and few extreme or significant losses in the given time interval. Also, it is leptokurtic with kurtosis 4.504, which implies that returns distribution is relatively peaked and possess fat tails. It is easy to assume that the existence of this phenomenon is common in U.S. equity market. Figure 4 also confirms the above assumption in terms of the log returns and estimated volatility. From 2011 to 2012, the maximum daily loss even has reached approximately 6.734.

(25) Figure 4 : Daily log-returns of S&P 500 and Estimated Volatiliy Furthermore, the ADF statistics is -12.017. On that basis, the series is considered to have no unit root and the returns are stationary. The J.B. statistics (1346.57) indicates that the return series deviates from normal distribution, corresponding to the skewness and kurtosis, and the Ljung-Box statistics shows weak serial correlation.

In general, the joint posterior distribution of model parameters and potential quantities marks the goal of Bayesian analysis. To inspect it, we can investigate summary statistics and various visualizations of marginal posterior distributions. Table 5 lists the posterior mean and standard error of three stochastic volatility models: According to several empirical studies, the leverage effect measured with the correlation coefficient ρ is expected to be negative Yu (2005) ; Omori et al. (2007) . We can observe that the leverage effect, ρ, in SVl and SVtl model are negative, which conveys a signal that the leverage effect asymmetry exists. In addition, the strong persistence can be indentified in φ, in all three models, φ reaches nearly 0.94. The µ tends increase from SVt to SVtl model, thus revealing that after considering leverage effect and fat tails, the long-run log-variance level up-regulated. Besides, the convergence issue arouse our attention. From Figure 5 , the top row shows the posterior of the daily volatility (in percent) 100 × exp(h/2) through its median (black) and 5% and 95% quantiles (gray). The remaining panels summarize the Markov chains of the parameters µ, φ, σ, ν and ρ. To be specific, the middle row presents trace plots and the bottom row shows prior (gray, dashed) and posterior (black, solid) densities. For the sampling procedure, 20000 MCMC draws are sorted after a burn-in of 2000. It is noteworthy that convergence is achieved for all parameters. 

The value-at-risk VaR measure promises that the actual return will only be worse than the VaR prediction of α × 100 in time. Given the time series of past ex ante VaR forecasts and past ex post returns, the "hit sequence" of VaR violations can be defined as:

The hit sequence returns 1 on day t + ∆ if the loss on that day is higher than the VaR predicted in advance for that day. If no VaR is violated, then the hit sequence returns 0. When backtesting the proposed model of this paper, we construct a sequence {I t0+j∆ } J j=1 spanning J days, indicating the time of past violations , and then sum this sequence to determine the total number of days exceeded, N J t0 = J j=1 I t0+j∆ . The most straightforward aims at comparing the number of observed exceptions with the number of expected exceptions. Given the properties of the binomial distribution, a confidence interval can be set for the expected number of exceptions. First, we need the test statistic, for large J, it yields:Ẑ

thenẐ ≈ N (0, 1) by central limit theorem. The 1 − β confidence interval (CI) can be expressed as:

Subsequently, the number is counted if exceedanceN J t0 over [T 0 , T J ]. If this number lies outside the 1 − β CI, the null hypothesis (the proposed model of this paper is accurate) at the 100(1 − β)% confidence. For instance, if we have α = 0.95, J = 250, β = 0.05, we will wantN J t0 ∈ (5.75, 19.25). Here, ifN J t0 ≥ 20, we say model is too optimistic or VaR is not high enough. IfN J t0 ≤ 5, the model is considered to be too pessimistic or VaR is considered to be too high. The following three test statistics derived from Christoffersen (1998) are also implemented including the unconditional, independence, and conditional coverage 4 . The idea of Christoffersen (1998) is to separate out the particular predictions being tested and then test each prediction separately. The first of these predictions was that the model produced the "correct" frequency of exceedances, described here as the prediction of correct unconditional coverage. The other prediction is that the exceedances are independent on each other. This latter prediction is important because it suggests that the exceedance cases should not cluster together over time. To explain the approach of Christoffersen (1998) , we briefly explain the three tests.

Kupiec (1995) introduced a variant of the binomial test, termed the proportion of failures (POF) test. The POF test works in combination with the binomial distribution method. Given the above test, we gain some interest in testing whether the proportion of violations obtained from the proposed model of this paper, called π, is significantly different from the committed proportion p, which is called the unconditional coverage assumption. To test this, the probability of an i.i.d. Bernoulli (π) hit sequence is written as:

where T 0 and T 1 denote the number of 0s and 1s in the sample. π is estimated by pi = T 1 /T -i.e., the proportion of violations in the observed sequence. By plugging this estimate back into the likelihood function, the optimized likelihood is yielded:

(30) Under the unconditional coverage null hypothesis of π = p, where p denotes the known VaR coverage, the likelihood is expressed as:

The unconditional coverage hypothesis through the likelihood ratio test is checked as:

Asymptotically, as T goes to infinity, this test will be distributed as a χ 2 of 1 degrees of freedom. Substituting in the likelihood function, we have:

Reject or accept the VaR model either by adopting a specific critical value or by determining the p-value associated with test statistic.

Christoffersen (1998) designed a test to verify whether the probability of observing an anomaly on a given day is dependent on whether an exception has occurred. Different from the unconditional probability of observing an exception, Christoffersen's test only measures dependence between consecutive days. In Christoffersen's interval prediction (IF) method, the test statistic for independence is expressed as:

where • n 00 = Number of periods with no failures followed by a period with no failures.

• n 10 = Number of periods with failures followed by a period with no failures.

• n 01 = Number of periods with no failures followed by a period with failures.

• n 11 = Number of periods with failures followed by a period with failures. and • π 0 : Probability of having a failure on period t, given that no failure occurred on period t − 1 = n 01 /(n 00 + n 01 ) • π 1 : Probability of having a failure on period t, given that a failure occurred on period t − 1 = n 11 /(n 10 + n 11 ) • π: Probability of having a failure on period t = (n 01 + n 11 )/(n 00 + n 01 + n 10 + n 11 )

Lastly, we are interested in testing both the VaR violations for independence and the average number of violations for correctness. Conditional coverage tests can be performed to jointly test for independence and correct coverage:

again following χ 2 1 distribution. Christoffersen's method allows us to test the coverage and independence hypotheses. Furthermore, if the model fails both hypotheses, his method can test the respective hypothesis separately and thus determine why the model fails. Table 6 lists the results of backtesting. * indicates that VaR exceedance of the model is inside the confidence interval and ** implies that the test rejects the null hypothesis. The significance level 5% is used to calculate VaR. If LR uc is statistically significant, the number of expected and actual observations below the VaR estimate is statistically the same. In addition, as revealed by the rejection of the null hypothesis, the calculated VaR estimates are not sufficiently accurate. According to the LR uc test statistics, and at 5% significance levels, , VaR model based on empirical method is rejected, whereas all other models had statistical significance, thus verifying the dynamic VaR model of this paper to be reasonable and valid. However, according to LR ind and LR cc , , the VaR model based on all models are rejected, whereas SVtl-EVT model has the minimum test statistics of 6.7 (the critical value is 5.991) with respect to LR ind and LR cc , thus indicating that SVtl-EVT model has the least effect of exceedance clustering. It is not hard to image that the rejection of all models in this time interval, since the entire time period selected contains the late 2020s at which COVID-19 pandemic causes stock market collapsed. Moreover, expect for empirical method, the exceedance of all models is inside the 95% confidence interval of the whole period, and the SVtl model has the suitable exceedance of 38. As demonstrated by the above results, SV models combined with EVT is practicable, SVtl outperforms other methods in the period, and all SV-based models outperform other models. Further improvements are required to cope with the case of exceedance clustering. According to the simulated data listed in Table 7 , the empirical and SVl-EVT model are rejected by LR uc and LR cc with too little and too much exceedance. The similar conclusion can be observed. The SVtl-EVT model still has the best performance with lowest LR cc statistics, and not rejected by all the tests. The number of exceedance is inside the confidence interval. 

In this paper, a method combining SV and GARCH models with extreme value theory is proposed to determine and backtest Value-at-Risk. To process time-varying volatility, fat-tails and leverage effect of financial series, the extended SV models, SVt and SVtl, are presented. The parameters of SV models are estimated using MCMC algorithm, which has higher estimation accuracy than GMM and QML methods, with 20000 simulation points and 2000 buin-in. After the sampling procedure, all parameters are successfully converged. To fit the extreme tail characteristics of the financial asset returns, the POT method is incorporated from extreme value theory to capture the tail distribution of the residuals, and the appropriate threshold parameter u is selected using excess mean. Moreover, the parameters of the POT model are obtained using the method of maximum likelihood estimation, and then the empirical study is conducted. As indicated by the results, the combined model of VaR estimation is effective and reasonable for the backtesting through binomial, independent, conditional and unconditional coverage, and the degree of fit of the financial data tails by goodness-of-fit test.

The models are applied for S&P 500 and simulated returns, and then they are adopted to predict future returns and volatility. Subsequently, based on forecasted daily return and volatility, the one day Value-at-Risk is calculated. SV-EVT models are indicated to be practicable for VaR estimation according to their backtesting results. The VaR under SVtl-EVT model has higher performance than all other models, and all SV-based models outperformed GARCH-based and empirical method in the data period. Furthermore, all models are overstretched under exceedance clustering to a certain extent, whereas SVtl still outperforms others and generates the minimum number of exceedance. Further research may be required to further optimize the dynamic model (e.g., using SVJt and SVLJt (SV model with leverage, jump and fat-tails) models) to cope with extreme situations include COVID-19 that has led to dramatic uncertainty to the market. 

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Modelling financial time series

Statistical aspects of arch and stochastic volatility

Stochastic volatility: likelihood inference and comparison with arch models. The review of economic studies

Multivariate stochastic variance models. The Review of Economic Studies

Stochastic volatility model with leverage and asymmetrically heavy-tailed error using gh skew student's t-distribution

Bayesian analysis of stochastic volatility models with fat-tails and correlated errors

Tail estimation and mean-var portfolio selection in markets subject to financial instability

Forecasting the value-at-risk of chinese stock market using the harq model and extreme value theory

Value-at-risk analysis for measuring stochastic volatility of stock returns: Using garch-based dynamic conditional correlation model

The stochastic volatility model, regime switching and value-at-risk (var) in international equity markets

Value at risk estimation under stochastic volatility models using adaptive pmcmc methods

Estimation of value-at-risk by extreme value and conventional methods: a comparative evaluation of their predictive performance

Goodness-of-fit tests for the generalized pareto distribution

Stochastic volatility with leverage: Fast and efficient likelihood inference

Leverage, heavy-tails and correlated jumps in stochastic volatility models

Large sample properties of generalized method of moments estimators

Pricing foreign currency options with stochastic volatility

Bayesian analysis of stochastic volatility models

Extreme value theory and value-at-risk: Relative performance in emerging markets

Residual life time at great age. The Annals of probability

Statistical inference using extreme order statistics. the Annals of Statistics

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

Expected stock returns and volatility

Stock returns and volatility: an empirical study of the uk stock market

On leverage in a stochastic volatility model

Evaluating interval forecasts

How relevant is volatility forecasting for financial risk management?

Backtesting portfolio risk measures: A duration-based approach

Vevaluating density forecasts with applications to financial risk management

Multivariate density forecast evaluation and calibration in financial risk management: high-frequency returns on foreign exchange

Techniques for verifying the accuracy of risk measurement models