Abstract: Financial asset volatility models can be divided into two broad categories: generalised autoregressive conditional heteroskedasticity (GARCH) models and stochastic volatility (SV) models. Currently, GARCH-type models are unique in terms of both the depth of theoretical research and the breadth of empirical application, and have been widely used in financial market analysis. However, the non-exponential form of GARCH models used in the existing research is constrained by the positive conditional variance and does not consider the presence of zero return. The log-form log-ARCH class of models ensures the positivity of the fitted conditional variance, but the logarithmic operations of this class of models cannot occur at zero values and are meaningless if the return is equal to zero. Therefore the model is not able to fully utilise the sample data, which in turn results in a lack of accuracy in explaining the problem. There are two general cases where zero return occurs. In the first case, the probability that the actual return is equal to zero is zero, but zero may still occur in the observed return calculation due to issues such as missing trades, discrete approximation errors (rounding errors), missing values and other data. In the second case, the probability that the actual rate of return is zero is not equal to zero, and market conditions affect the probability that the rate of return is zero.
   In order to estimate exchange rate volatility more accurately, this paper models the foreign exchange data containing zero return. The main work of this paper lies in, firstly, applying a log-GARCH model, which is not restricted by a positive conditional variance, to fit the exchange rate market yield data, and also using the ARMA model form to represent the log-GARCH model. Second, widely using treatment of replacing the zero return with the smallest non-zero absolute value yields biased estimates, this paper proposes a treatment framework for handling data containing zero returns, i.e., treating zero values as missing observations. Then, the log-GARCH model containing missing observations is estimated unbiased by combining the QMLE method of SUCARRAT et al.(2016) and the expectation maximisation (EM) algorithm. Finally, an empirical analysis is conducted to compare the differences in volatility estimation results under two different treatments of zero returns-the non-zero-value instead of zero-value approach and the treating-zero-value-as-missing-value approach.
   The sample selected for this paper includes data on the GBP-RMB exchange rate price, the JPY-RMB exchange rate price, the AUD-RMB exchange rate price, the USD-HKD exchange rate price, the USD-JPY exchange rate price, the AUD-USD exchange rate price, the GBP-USD exchange rate price, the GBP-JPY exchange rate price, and the GBP-AUD exchange rate price. The number of zeros in the sample data ranges from 2 observed zeros for the GBP-RMB exchange rate (0.1% of the sample size) and 1 observed zero for the AUD-RMB exchange rate (0.1% of its sample size) to 732 observed zeros for the USD-HKD exchange rate (20.2% of its sample size), with the number of zeros occurring in each set of exchange rate return data varying, and the reasons for the occurrence of each of these zeros varying.
   For yield series with more zeros, the difference in the estimates obtained under the different methods is larger; for yield series with fewer zeros, the difference in the estimates is smaller. The presence of zeros increases the sensitivity of the yield series to market changes. The effects of different treatments on the volatility estimation results are significant, and the estimation results obtained from the method of using non-zero values as missing values are closer to the real situation of the market.

Key words: exchange rate volatility, log-GARCH model, ARMA expression, missing value

摘要： 在实际外汇市场中,由于诸如交易缺失、舍入误差等原因使得收益率序列中出现零值,常见GARCH族模型无法对含零收益率数据进行有效拟合,导致波动率估计结果产生较大偏差。为了更准确地估计汇率波动率,本文对含有零收益率的外汇数据进行建模。首先运用不受条件方差为正限制的log-GARCH模型对汇率市场收益率数据进行拟合,同时提出一个处理含有零收益率的数据处理框架,即将零值视为缺失的观测值。然后通过结合QMLE方法和期望最大化(EM)算法对含缺失观测值的log-GARCH模型进行无偏估计。最后通过实证分析比较零收益率两种不同处理方法——非零值代替零值方法和将零值视为缺失值方法下波动率估计结果的差异。研究结果显示零收益率的存在会增加波动率的估计偏差,将非零值作为缺失值方法得到的估计结果更接近市场真实情况。

关键词: 汇率波动, log-GARCH模型, ARMA表达, 缺失值