ارزیابی ارزش در معرض خطر به همراه تئوری ارزش شدید مبتنی بر موجک : شواهد حاصل از بازارهای نوظهور

This paper introduces wavelet-based extreme value theory (EVT) for univariate value-at-risk estimation. Wavelets and EVT are combined for volatility forecasting to estimate a hybrid model. In the first stage, wavelets are used as a threshold in generalized Pareto distribution, and in the second stage, EVT is applied with a wavelet-based threshold. This new model is applied to two major emerging stock markets: the Istanbul Stock Exchange (ISE) and the Budapest Stock Exchange (BUX). The relative performance of wavelet-based EVT is benchmarked against the Riskmetrics-EWMA, ARMA–GARCH, generalized Pareto distribution, and conditional generalized Pareto distribution models. The empirical results show that the wavelet-based extreme value theory increases predictive performance of financial forecasting according to number of violations and tail-loss tests. The superior forecasting performance of the wavelet-based EVT model is also consistent with Basel II requirements, and this new model can be used by financial institutions as well.

Value-at-risk (VaR) became a common tool for financial forecasting with the Riskmetrics-exponentially weighted moving average (EWMA), made famous by Morgan in the early 1990s. This model is actually a special case of Bollerslev’s [1] generalized autoregressive conditional heteroskedasticity (GARCH) model. More than 100 volatility models were developed during the last 28 years [2]. One of the most important features that made the conditional volatility models popular is their ability to capture many of the typical stylized facts of a financial time series, such as time-varying volatility, persistence and volatility clustering. However, conditional volatility models cannot capture extreme movements, as these models are based on past volatility rather than the extreme observations. Extreme value theory models can capture extreme movements, and the forecasting performance of these models is better than that of GARCH type models [3]. There are some important reasons to model the returns with extreme value theory. First, the distribution of returns is heavy-tailed or leptokurtic for most of the financial returns. Second, the right and left tails of returns are not symmetrical, and extreme value theory models can be applied to each tail with different parameters, as opposed to GARCH and other volatility models. The methodology of forecast combination was introduced by Bates and Granger [4]. They urged that we should combine forecasts as a weighted average of the individual forecasts. Forecast combination can be estimated by a combination of individual forecasts or by a combination of models. Individual forecasts’ combination can be estimated with artificial intelligence techniques. Liu [5] and Ozun and Cifter [6] proposed neural networks for a combination of individual forecasts. Liu [5] combined neural networks with historical simulation and GARCH(1, 1) models and found that the combination of historical estimation with GARCH and neural networks significantly improved forecasting performance. Ozun and Cifter [6] combined Hill [7] type EVT with historical simulation and GARCH models with neural networks. They found that the combination of EVT and other models improves forecasting performance. Hybrid models as a combination of EVT with conditional volatility models were proposed by McNeil and Frey [8]. This model uses a two-stage approach. In the first stage, a GARCH type model is applied to residuals. In the second stage, EVT is applied to standardized residuals. McNeil and Frey [8] found that the conditional EVT procedure gives a better one-day-ahead forecast than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. Wavelets can also be used to estimate hybrid financial forecasting models. The combination of wavelet transform and GARCH models was introduced by Chi and Kai-jian [9], Lai et al. [10] and [11], He et al. [12] and [13], and Tan et al. [14]. Chi and Kai-jian, Lai et al., and He et al. [12] proposed wavelet-decomposed value-at-risk, and He et al. [13] proposed wavelet denoising ARMA–GARCH models. This paper typically used Kupiec [15]’s test for backtesting, and although their model is superior to conventional ARMA–GARCH models, the number of violations is greater than that of ARMA–GARCH models. Tan et al. [14] combined wavelet transform with ARIMA and GARCH models and applied this model to one-day-ahead electricity price forecasting. They found that their model is far more accurate than other forecasting models. Yamada and Honda [16] used wavelet analysis to predict business turning points of the Nikkei 225 index and found that wavelet analysis can capture business peaks and troughs (minimum points) as an alternative structural break analysis. Bowden and Zhu [17] combined wavelet analysis with structural breaks and applied this combined model to the agribusiness cycle. By using wavelets, they added the business cycle feature to structural break analysis. In this paper, wavelet-based extreme value theory (EVT) is introduced for univariate value-at-risk estimation. A wavelet-based EVT model is proposed as a combination of wavelets and the EVT model following the approach of McNeil and Frey [8]. In the first stage, wavelets are used as a threshold in generalized Pareto distribution, and in the second stage, EVT is applied with wavelet-based thresholds. The relative performance of this new hybrid model is compared with conventional volatility models for one-day-ahead forecasts, and wavelet-based EVT is benchmarked against the Riskmetrics-EWMA, ARMA–GARCH, generalized Pareto distribution, and conditional generalized Pareto distribution models. This new model is applied to two major emerging stock markets: the Istanbul Stock Exchange (ISE) and the Budapest Stock Exchange (BUX). It is found that the wavelet-based EVT model increases predictive performance of financial forecasting according to the number of violations and tail-loss tests for emerging markets. The remainder of the paper is organized as follows. Section 2 provides value-at-risk methodologies. Section 3 describes the data on daily index returns. Section 4 presents empirical results for the forecasting performance of the models. Section 5 concludes the study.

Value-at-risk (VaR) and conditional volatility models have become common tools for financial forecasting. However, conditional volatility models cannot capture extreme movements, as these models are based on past volatility rather than the extreme observations. On the other hand, extreme value theory models can capture extreme movements and forecasting performance of these models better than conventional volatility models. In this paper, wavelets and EVT are combined for volatility forecasting and wavelet-based threshold estimation adds the business cycle feature to extreme value theory. The filter length is determined as 64 days (26) in wavelet-based GPD. Since threshold estimation is one of the main problems in generalized Pareto distribution, this approach can also be considered an improvement in non-parametric statistics. This combined feature increases the predictive out-of-sample forecasting performance. The main contribution of this paper is to propose a combined EVT model and compare the predictive performance of this model with conventional conditional volatility models. This hybrid model is tested on two major emerging stock markets: the Istanbul Stock Exchange (ISE) and Budapest Stock Exchange (BUX). These two markets are selected since global stock investors frequently take positions in both markets. The relative performance of wavelet-based EVT is benchmarked against the conditional volatility and extreme value theory models. The comparison of the performance of wavelet-based EVT to that of the traditional volatility models shows that the wavelet-based EVT model is the most appropriate one among other value-at-risk models according to the number of violations and Christoffersen [38] tail-loss tests. The number of violations in the wavelet-based GPD model dramatically decreases, and the model is statistically significant for all Christoffersen [38] tail-loss tests for both stock markets. The superior forecasting performance of the wavelet-based EVT model is also consistent with Basel II requirements. Therefore, financial institutions can also estimate market value-at-risk by using wavelet-based EVT models for their single assets. For future research, this model can also be extended to multivariate cases for portfolio value-at-risk estimation.