Abstract: Accurately characterizing gold price volatility and predicting gold market risk are crucial for investors, financial institutions and regulators. Many scholars have applied GARCH family models to characterize gold market volatility and measure gold market risk, but existing research on the gold market has ignored the impact of zero return on risk measurement results. Zero returns can have an impact on the modelling of volatility in the gold market, which in turn can have some impact on the results of risk measures. The current literature on the study of zero returns in financial markets outlines two possible reasons for the emergence of zero returns. The first is that the probability of the actual return being equal to zero is zero, yet it may still be zero when calculating the observed return because of missing trades, rounding errors, and other data issues. The second is that the probability of the actual return being zero may not be equal to zero, the probability of the return being zero may be affected by market conditions, and the probability of zero may change with market conditions.
The current literature does not provide an in-depth study of the type of probability that the return is zero, although it does consider the case where a zero return exists. In addition, there is no research literature on zero return in the gold market. In this paper, we fully consider the type of probability that the return is zero in the sample, construct a zero-corrected GARCH model to deal with the zero probability, and obtain the formula for calculating VaR and ES under the zero-return-containing rate.
This paper takes the six groups of gold price data of gold 9995 (Au9995), gold 9999 (Au9999), gold 100g (Au100g), gold deferral (Au(T+D)), the price of gold in physical gold in Chow Tai Fook (AuZDF), and Shanghai and New Zealand gold 12 (NYAuTN12) in the exchange as the object of study. The first five groups of products in this paper are selected from the closing price data from 4 January 2012 to 28 January 2022.NYAuTN12 started to be listed on 14 October 2019, so the closing price data from 14 October 2019 to 25 March 2022 is selected. The proportion of the number of zero returns in the sample data, in descending order, is 7 zero observations (0.27%) for Au(T+D), 10 zero observations (0.41%) for Au9999, 16 zero observations (0.65%) for Au9995, 44 zero observations (1.80%) for Au100g, and 32 zero NYAuTN12 observations (5.36 per cent), and 847 zero observations (39.6 per cent) for AuZDF.
In this paper, six sets of gold return data are modelled, taking full account of the different characteristics of zero returns in the six return series. Three logit models, Constant, ACL(1,1) and Trend, are constructed to estimate the six sets of zero return series. In this paper, the SIC information criterion is used to determine the optimal model corresponding to each group of zero-containing return series, and the Constant model fits the best for four return series, Au9995, Au9999, Au100g, and Au(T+D); the ACL(1,1) model fits the best for AuZDF; and the Trend model fits the best for NYAuTN12.
The results of the study find that: (1)The conditional zero probability of each set of returns for Au9995, Au9999, Au100g, and Au(T+D) is constant, the conditional zero probability of AuZDF returns is time-varying and smooth, and the conditional zero probability of NYAuTN12 returns is time-varying and non-smooth. (2)The effect of zero return on VaR is highly nonlinear and dependent on the density function of wt. In the case where the zero probability is constant, the zero probability does not have a large impact on the VaR estimate. However, when the zero probability is time-varying, it causes a significant bias in the VaR estimation and may shift the VaR upward or downward. (3)When the zero probability is constant, it does not have a large impact on ES estimation. When the zero probability is time-varying, the effect on ES is generally monotonic. Specifically, for time-varying and smooth zero probabilities, the ES tends to be shifted upward, while for time-varying and non-smooth zero probabilities, the ES tends to be shifted downward. Without corrections for time-varying zero probabilities, risk estimates will be significantly biased.

Key words: gold market, zero return, risk measurement, value at risk, expected shortfall

摘要： 本文以上海黄金交易所中具有代表性的六组黄金产品Au9995,Au9999,Au100g,Au(T+D),AuZDF和NYAuTN12的收益率序列为研究样本,充分考虑样本中收益率为零的概率(零概率)类型,构建零修正的GARCH模型对零概率进行处理。研究发现,零收益率对VaR的影响是高度非线性的,无论是时变且平稳的零概率还是时变且非平稳的零概率都会对VaR估计造成偏差,可能会使VaR向上或向下偏移。时变零概率对ES的影响总是单调的,对于时变且平稳的零概率,ES往往会向上偏移;而对于时变且非平稳的零概率,ES往往会向下偏移。因此,构建零修正的GARCH模型来对时变零概率进行处理,对优化我国黄金市场风险度量有着重要意义。

关键词: 黄金市场, 零收益率, 风险度量, 在险价值, 预期损失