Incorporating discontinuities in value-at-risk via the poisson jump diffusion model and variance gamma model

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order toobtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion processwith discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuousmodel that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model).These models have been shown to have some success in capturing certain characteristics of return distributions, a fewbeing leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (AllOrdinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaRestimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovationfrom option pricing techniques, which concentrates on the more tractable characteristic functions of the models.Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performsand also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P.Morgan?s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the otherperformance criteria examined. Overall, no model was clearly superior according to all the performance criteriaanalysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and VarianceGamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than thatcurrently employed by Riskmetrics.