Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 5183914, 9 pages

https://doi.org/10.1155/2017/5183914

## Dynamic VaR Measurement of Gold Market with SV-T-MN Model

^{1}Library, Chongqing University of Technology, Chongqing, China^{2}School of Science, Chongqing University of Technology, Chongqing, China^{3}School of Accounting, Chongqing University of Technology, Chongqing, China

Correspondence should be addressed to Bao Yang

Received 16 May 2017; Revised 18 September 2017; Accepted 26 September 2017; Published 23 October 2017

Academic Editor: Francisco R. Villatoro

Copyright © 2017 Fenglan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

VaR (Value at Risk) in the gold market was measured and predicted by combining stochastic volatility (SV) model with extreme value theory. Firstly, for the fat tail and volatility persistence characteristics in gold market return series, the gold price return volatility was modeled by SV-T-MN (SV-T with Mixture-of-Normal distribution) model based on state space. Secondly, future sample volatility prediction was realized by using approximate filtering algorithm. Finally, extreme value theory based on generalized Pareto distribution was applied to measure dynamic risk value (VaR) of gold market return. Through the proposed model on the price of gold, empirical analysis was investigated; the results show that presented combined model can measure and predict Value at Risk of the gold market reasonably and effectively and enable investors to further understand the extreme risk of gold market and take coping strategies actively.

#### 1. Introduction

Gold is one of the most brilliant and beautiful of metals. The price of gold has been the concern of investors for the special value-added property. With the gradual liberalization of the price of gold, gold itself supply and demand, the dollar exchange rate, interest rates, and other complex factors will cause greater volatility in prices. O’Connor et al. [1] have investigated physical gold demand and supply, gold mine economics, and analyses of gold as an investment. Also they have researched gold market efficiency, the issue of gold market bubbles, gold’s relation to inflation, and interest rates. The volatility of gold price increases the loss of earnings. In this case, it is more important to measure and control the gold spot market effectively.

VaR (Value at Risk) is the main method of risk management at present. It indicates the maximum loss level of financial assets at a given confidence level for a certain period of time in the future. Generally, the income series is regarded as a product of fixed variance normal distribution, but it does not satisfy the time-varying financial markets. The GARCH [2–4] model captures the time-varying and sequence correlation of price volatility for the characteristics of “peak-tail” and “volatility clustering” of financial income series. But GARCH models define the conditional variance as the deterministic function of the square of past observations and the variance of the previous condition. The estimation of the conditional variance is directly related to the past observations. Therefore, the estimated volatility series is not very stable when there are anomalous observations, and GARCH model for long-term volatility prediction ability is relatively poor. Another model to characterize volatility is stochastic volatility (SV) models proposed by Taylor (2003) [5], which introduce an autoregressive equation with implied volatility into the model, making the models more useful, and characterize the essential characteristics of financial markets. Zhen-long and Yi-zhou [6], Jing-dong and Chi [7], and other authors compared the GARCH models and SV models to describe the volatility of financial time series, and the SV model can fit the financial sequence better. Su-hong and Shi-ying [8] proposed an improvement of the error term of the standard SV model, which can further enhance ability of the SV model to describe the financial sequence. Therefore, this paper chooses the SV-T model to describe volatility of the gold return series. With developments of Gibbs sampling technique and computer technology, Jacquier et al. [9] firstly used the Markov chain Monte Carlo (MCMC) method to estimate parameters of SV model in 1994. Subsequently, Hui-ming et al. [10, 11] further studied the MCMC method based on Gibbs sampling to obtain the best SV model parameters estimation results. Since the SV-T model is difficult to estimate and the out-of-sample volatility prediction, the SV-T model is transformed into a linear state space without loss of any information [12–17]. Standard Kalman filter can obtain the better state estimation only in the linear Gaussian state space. But the tail distribution of the yield at the same time of linearization is no longer Gaussian distribution, and it obeys the logarithm of left partial long tail square distribution. The improper model error may be generated when a single standard normal distribution approximation is used, so Gaussian mixture distribution approximation is proposed. Anderson and Moore (1979) [18] proved that an arbitrary random variable can be approximated by a finite Gaussian mixture distribution, and the approximation error can be arbitrarily small when the normal number of the factor is large enough. In this paper, we estimate parameters of SV-T model using the MCMC algorithm firstly and then estimate the parameters of Gaussian mixture distribution using EM algorithm. Finally, we propose a new method for estimating the parameters of Gaussian mixture model by means of Lemke (2006) [16]. Approximate Filter (AMF) algorithm is applied to realize out-of-sample prediction of volatility.

VaR is a main measurement for the future specific period of time under normal fluctuations in the case of the maximum possible loss. The inadequate consideration of financial market causes the tail risk underestimated. Extreme Value Theory (EVT), which does not consider the whole of the distribution of return series, directly uses the tail of data to fit its distribution, and it can deal with the thick tail phenomenon more effectively and measure the risk loss under extreme conditions [19–25]. At present, the extreme value of the domestic financial risk measurement of financial assets rarely focuses on SV models. Therefore, in this paper, we use the SV-T-MN model to describe time series volatility of financial series and combine with the extreme value theory to fit tail distribution of standard residuals and then establish a new financial risk measurement model, dynamic VaR forecast model based on EVT-SV-T-MN. Finally, we make an empirical analysis on daily closing price of AU99.99 in Shanghai Gold Exchange to guide investors to fully understand the extreme risks of gold market and take active countermeasures.

The article includes five parts. The second part introduces the SV model of gold price volatility, and the third part introduces VaR model combining SV model and extreme value theory. The fourth part analyzes daily closing price data of AU99.99 of Shanghai Gold Exchange. The fifth part is the summary of full text.

#### 2. SV-T-MN Model Based on State Space

The core of measuring VaR of gold market returns is to accurately predict its volatility. In this paper, the SV model is used to capture “peak-tail” characteristics of volatility of gold market return series. Geweke (1994) proposed a thick tail SV-T model:

is the yield of gold price in period , is the logarithm fluctuation, and is the persistence parameter, which is a direct impact on the current volatility of future fluctuations in the intensity. Where obeys a normal distribution with mean 0 and variance , follows a standard distribution with degrees of freedom is the drift level of the wave equation. Due to the fact that there is a degree of freedom in the distribution of the error term in the thick tail SV-T model, it is difficult to estimate the parameters and to predict the volatility out of the sample.

In this paper, by taking the logarithmic square transformation of metric equation without loss of any information, it is linearly equivalent to state space model. The SV-T-MN model is established in two steps. In the first step, the SV-T model is linearized and transformed into the form of state space. In the second step, the logarithm of linearization T-squared distribution is approximated by Gaussian mixture distribution, and the Gaussian mixture model (SV-T-MN) is obtained as follows:

In the above model, represents , represents , represents , is the weight of the mixed normal distribution, , is the factor mean, is the factor variance, and is the number of the Gaussian mixture distribution factor. The estimation of all parameters of SV-T-MN model is carried out in two steps. In the first step, the estimation of parameters , , , and of SV-T model is obtained by MCMC method. The second step obtains its estimate by EM algorithm [26] for , , and in SV-T-MN model.

When the stochastic perturbation term satisfies a single normal distribution, the Gaussian mixture model of (2) degenerates to the general linear Gaussian state space model, and the outlier prediction usually adopts standard Kalman filtering algorithm [14]. For prediction of Gaussian mixture model, standard Kalman filtering algorithm can be extended to exact filtering algorithm, but the cost is exponentially increasing with time, which is difficult to be used in the real long-term observation sequence [26]. Therefore, this paper uses the Approximate Filter of (AMF ()) algorithm proposed by Lemke [16] to realize out-of-sample prediction of volatility. The iterative process of this algorithm is controlled by a parameter ; not only is it used effectively in the actual medium and long-term sequences, but also its prediction accuracy is very close to the exact filtering. -degree myopia filter algorithm is divided into three steps.

*Step 1. *When , for the first values of the observed sequence, we calculate its one-step prediction and update of each iteration process component by using the exact filtering algorithm and then get the mixed one-step prediction and update.

*Step 2. *When , the exact filtering algorithm is initialized, then the mixed one-step prediction and updating are initialized at to , and the exact filtering algorithm is run again. Similarly, we can get mixed one-step prediction and update at to .

*Step 3. *When , continue to repeat Step 2. We can get the volatility of gold price return at each time point and further predict the VaR at different time points; we can realize the simulation of dynamic risk value.

#### 3. Dynamic Extreme VaR Modeling Based on SV-T-MN Model

A new method of extreme value theory (EVT), which is POT (Peaks Over Threshold) model, does not focus on the discussion of extreme values (maximum or minimum) but focuses on the excess. POT model can be used to capture the information of heavy tails in a random sequence, by investigating the observational behavior of a sample over a certain threshold. And the model can be used to obtain the variance of the global extreme by extrapolating the sample data in the case of unknown population distribution. It overcomes the limitations of the traditional method which cannot exceed the sample data for analysis. In the measurement of financial returns data tail risk has better application effect [27, 28]. The POT model and the stochastic volatility model are combined to predict the risk value of the gold return sequence.

Since the POT model requires the income series to have independent and identically distributed properties, McNeil (2000) [29] found that the standardized residuals of financial data can effectively satisfy the conditions of independent and identically distributed. Therefore, the data are normalized to obtain the independent and identically distributed standard residuals and tail-fitted with . The standard residual of gold return sequence is , where is the conditional mean of the return sequence and is the conditional variance. Suppose that the distribution function of the standard residual sequence is and is a sufficiently large threshold. When , there are and is the overthreshold conditional distribution function of the random variable [30].We can get . The conditional distribution of can be obtained from the generalized Pareto distribution (GPD) which is well approximated by the Pickands [31] limit theorem. The GPD distribution can be expressed as

There are two unknown parameters in the GPD distribution, which are the shape parameter and the position parameter , where and when , ; when , . VaR is the extreme quantile of loss distribution function. When is known, is also available. The density function can be deduced from generalized Pareto distribution and log-likelihood function [30] is expressed as

The maximum likelihood method is used to estimate the shape parameter and the position parameter . However, it is possible to estimate the shape parameter and the position parameter more accurately, on the assumption that an appropriate threshold is selected. Because when the threshold is too high, the sample size is too small to meet the needs of parameter estimation and it is too low to meet the distribution of “thick tail” feature, the average excess function method [32–34] is often used to select the threshold. When a high threshold is assumed, the excess return is subject to the generalized Pareto distribution, where , the excess mean value exceeding the threshold is , and for any , we define the overaveraging function . For any , . The excess mean function is a linear function of for fixed , and the excess mean function [34] is as follows:where is the number of samples exceeding the threshold and is the corresponding sample value. When , the excess mean function graph is linear with respect to , and if the overfunction graph is tilted upwards, the data is subject to the GPD distribution with the shape parameter being positive, and the distribution is thick tail distribution. When distribution of is known, can be used to obtain the risk value of the income sequence at the given confidence level . The proposed model based on EVT and SV-T-MN model is as follows:The detailed modeling procedure is shown in Figure 1. There are three main steps in the proposed model. The first step is to estimate the parameters with MCMC and EM. The second step is to calculate the volatility based on EKF or AMF. The final step is to calculate the excess mean value function by using maximum likelihood estimation.