Extreme Value GARCH modelling with Bayesian inference

Extreme value theory is widely used financial applications such as risk analysis, forecasting and pricing models. One of the major difficulties in the applications to finance and economics is that the assumption of independence of time series observations is generally not satisfied, so that the dependent extremes may not necessarily be in the domain of attraction of the classical generalised extreme value distribution. Even when the dependence satisfies conditions for the sequence to be within the domain of attraction of the generalised extreme value distribution, the traditional modelling approach does not necessarily give full insight into the form of the dependence. The generalized extreme value distribution can be combined with either other time series models or covariates to capture such dependence. This study examines a conditional extreme value distribution with the added specification that the extreme values (maxima or minima) follows a conditional autoregressive heteroscedasticity process. The dependence has been modelled by allowing the location and scale parameters of the extreme distribution to vary with time. The resulting combined model, GEV-GARCH, is developed by implementing the GARCH volatility mechanism in these extreme value model parameters. Bayesian inference is used for the estimation of parameters and posterior inference is available through the Markov Chain Monte Carlo (MCMC) method. The model is firstly applied to relevant simulated data to verify model stability and reliability of the parameter estimation method. Then real stock returns are used to consider empirical evidence for the appropriate application of the model. As with most extreme value modelling applications, the shape parameter is the most difficult parameter to estimate. This study also investigates the sensitivity and stability problems in an extension of the GEV-GARCH model to allow a time varying shape parameter. It is demonstrated that a non-constant extreme shape parameter with a GARCH type time varying structure typically leads to over-parameterisation and consequent estimation difficulties. A comparison is made between the GEV-GARCH and traditional GARCH models. Both the GEV-GARCH and GARCH show similarity in the resulting conditional volatility estimates, however the GEV-GARCH model differs from GARCH in that it can capture and explain extreme quantiles better than the GARCH model because of more reliable extrapolation of the tail behaviour.