Stock return volatility using Bayesian symmetric and asymmetric GARCH

In economics, measurement is the assignment of numbers to one or more properties of objects, events, and situations based on a rule in order to generate reliable information about those objects, events, or situations. Measurement and understanding of economic reality are two sides of the same coin. Indeed, measurement alone assigns a meaningless random number, and understanding without measurement remains merely a philosophical act. Uncertainty indicates limited knowledge and the impossibility of the accurate description of current or future conditions. The valid measurement of uncertainty and forecasting its future values ​​are very important for credit institutions. This is because, in addition to average returns, decision makers are also sensitive to return uncertainties and the consequent risk. To measure and express uncertainty, we can use the probability distribution of the results or the possible occurrences of the desired situation. But this description is usually difficult or impossible due to the complexity of estimating the probability distribution. For this reason, simpler criteria and approximations are used instead of distributing the probability of the situation. Thus, the relatively simple concept of volatility is used to measure uncertainty that plays a central role in the financial theory, risk management and pricing. Any model proposed to measure volatility must meet the implicit adequacy criteria and be useful for policy-making in this market.

An important issue in providing a proper statistical analysis of uncertainty is the estimation of the parameters of volatility models (time series type). Research for the presentation of econometric models that can predict volatility has paid off with the introduction of the ARCH model by Engel (1982), which uses the classical maximum likelihood technique. Despite this initial success, the estimation of these models, which is widely performed by the maximum likelihood method, has major weaknesses. In this regard, it is possible to know the asymptotic properties of unit root tests in the presence of ARCH effects, abnormal asymptotic distribution of estimators due to the wide tail feature of financial data distribution and how to choose volatility model based on information criteria regardless of the degree of uncertainty of models; only the interrupts that are set are noted. The consequence of these cases is the creation of unfavorable results in the field of prediction and the invalidity of testing hypotheses. Due to the importance of modeling and predicting volatility in financial markets, the present study uses the Bayesian inference method. This method, in addition to solving these problems, enables researchers to assess the probability of the model being accurate. In order to make the modeling more consistent with the reality of financial data, in this study, the t-distribution is used as the marginal distribution of returns in Bayesian GARCH models (linear Bayesian GARCH model, Bayesian GJR-GARCH models and nonlinear Bayesian EGARCH).

The results of this study obtained by the use of the Bayesian factor show that the most suitable model for the equation of average stock prices is a model with random movements around a fixed value. This means that stock prices follow a random geometric step process on a daily basis. According to Bayesian GARCH model in Tehran Stock Exchange, with a probability of 68%, the volatility half-life is about 27 days, and with a probability of more than 50%, there is a leverage effect in this market. However, the results showed that, for all models, the probability of volatility damping is higher than the probability of the immortality of volatility. In nonlinear models, due to the effects of leverage, the probability of damping was higher than the linear model, and this is an indication for the predictability of these models compared to the linear model. In addition, the results indicate that the daily volatility of stock prices has leverage effects. Both Bayesian GJR-GARCH and Bayesian EGARCH confirmed these effects. Also, using the Bayesian information deviation criterion, GJR-GARCH model is selected as the best model to predict the stock market volatility.

In order for the volatility model to be sufficient, it should combine basic items such as theoretical concepts, policy perspectives, mathematical concepts and techniques, empirical facts and data. In addition, rules must meet certain requirements to perform reliable stock market volatility measurements. These requirements depend on the nature of the stock market and the circumstances in which the measurements are made. The fact that measuring the stock price index uncertainty requires a model means that uncertainty cannot be measured by simply calculating the probability distribution of stock price index (or return) data. However, predicting the stock return uncertainty is further complicated by the general fact that uncertainty cannot be measured directly and must be inferred from market price behavior. This means that uncertainty cannot be measured in the same way that temperature is measured with a thermometer: because it is a hidden variable. The only thing is that, if prices fluctuate sharply during the day, there is probably a high uncertainty. As a result, measuring uncertainty requires statistical modeling, which requires some assumptions. In this paper, Bayesian method was used to have valid estimates for volatility. This method is philosophically distinct from other methods of statistical inference. In this method, all unknowns, even parameters, are assumed to be random variables whose probabilistic distributions are determined by the researcher's beliefs about their possible values. Because Bayesian inference approaches start from previous beliefs about parameters, it seems very subjective, and this is a challenging issue. However, most Bayesian and non-Bayesian inference results are very similar, especially when using obscure backgrounds, but this similarity does not mean the same thing because the main difference between Bayesian and non-Bayesian approaches is in interpreting the results. Bayesian method is very important in the analysis of financial markets because, in this field, the volume of the background information of researchers is relatively high and failure to use such a volume of information seems illogical.