The Value-at-Risk (VaR) framework [Jorion, 1996], [Dowd, 2005], [Christoffersen, 2011], [Danielsson, 2011] has growing importance for banks and regulators as it helps to quantify possible losses due to trading assets and portfolios. This framework helps us to measure the risk that future returns may exceed a certain VaR value with a certain probability over a period of time. Statistically the VaR is associated with the quantiles, that is the probability mass in the tails of the return distribution. The violations of the left tail indicate the risk for long trading positions, and the violations of the right tail indicates the risk for short positions.
Our research studies conditional VaR estimation via forecasting the mean and variance of the return distribution using GARCH models. This includes formulation of innovative GARCH models and implementation of corresponding calibration algorithms. Recently, we developed: normal mixture GARCH models and heavy-tail mixture GARCH models [Nikolaev, Boshnakov and Zimmer, 2013]. Normal mixture models help to achieve better fit to time-varying data compared to Gaussian models [Haas, Mittnik, and Paolella, 2004]. Heavy-tail mixture models capture more accurately data coming from more peaked distributions with longer tails (the excess kurtosis of the data distribution). These mixture models help us to describe more adequately aberrant changes in time series arising from occasional events. Especially for mixture GARCH we derive dynamic (temporal) training algorithms that are used for analytical optimization through generalised expectation maximisation.
The empirical investigations show that the developed mixture GARCH deliver improved forecasts compared to linear as well non-linear Gaussian and Student-t noise models with respect to the log-likelihood and several statistical characteristics. The analytical training leads to mixture GARCH models with quite similar statistical features to these achieved by Monte Carlo simulation methods [Ausin and Galeano, 2007].
The stochastic volatility models are also attractive for Value-at-Risk estimation, because they also allow efficient generation of the predictive distribution of returns and its quantiles. Currently we develop nonlinear filters for calibration of heavy-tailed Asymmetric Stochastic Volatility (ASV) models incorporating leverage effects. These filters approximate the moments of the volatility posterior via recursive numerical convolution, and thus they obtain the likelihood more efficiently than algorithms that use stochastic sampling.
VaR testing of state-space ASV models is carried out following a bootstrapping methodology [Rodriguez and Ruiz, 2009]. Bootstrap replicates of the return series are made by drawing samples from the standardised residuals. Next, the model is reestimated over the replicates, and samples from the predictive return distribution are obtained. The experiments are performed according to the methodology of Christoffersen . It involves precomputing the failure rates as the proportion of exceptions for the tails that exceed the corresponding value-at-risk, and computation of likelihood ratio statistics. The obtained results show that the ASV models generate reliable 1% and 5% VaR predictions for long as well as short trading positions.
The research into heavy-tail ASV and heavy-tail mixture GARCH models found that they produce accurate forecasts, which leads to adequate results in conditional VaR evaluation. Out calibration algorithms for such heavy-tail volatility models construct efficiently the predictive distribution, and make them useful tools for assesment of the risk exposure and for mitigating financial risks.
Nikolaev,N., Boshnakov,G. and Zimmer,R. (2013). Heavy-tail Mixture GARCH Volatility Modeling and Value-at-Risk Estimation, Expert Systems with Applications, vol.40, N:6, pp.2233-2243.
Nikolaev,N., de Menezes,L. and Smirnov,E. (2013). Nonlinear Filtering of Asymmetric Stochastic Volatility Models and Value-at-Risk Estimation, Technical Report 43-07-13, Goldsmiths College, University of London.
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Haas,M., Mittnik,S. and Paolella,M. (2004). Mixed Normal Conditional Heteroscedasticity, Journal of Financial Econometrics, vol.2, pp.211-250.
Dowd,K. (2005). Measuring Market Risk, 2nd ed., John Wiley and Sons, England.
Hartz,C., Mittnik,S. and Paolella,M. (2006). Accurate Value-at-Risk Forecasting Based on the Normal-GARCH Model, Computational Statistics and Data Analysis, vol.51, N:4, pp.2295-2312.
Pascual,L., Romo,J. and Ruiz,E. (2006). Bootstrap Prediction for Returns and Volatilities in GARCH Models, Computational Statistics and Data Analysis, vol.50, pp.2293-2312.
Ausin,M.C. and Galeano,P. (2007). Bayesian Estimation of the Gaussian Mixture GARCH Model, Computational Statistics and Data Analysis, vol.51, N:5, pp.2636-2652.
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