Analytical Estimation of Value at Risk Under Thick Tails and Fast Volatility Updating

Despite its recent advent, value at risk (VaR) became the most widely used technique for measuring future expected risk for both financial and non-financial institutions. VaR, the measure of the worst expected loss over a given horizon at a given confidence level, depends crucially on the distributional aspects of trading revenues. Existing VaR models do not capture adequately some empirical aspects of financial data such as the tail thickness, which is vital in VaR calculations. Tail thickness in financial variables results basically from stochastic volatility and event risk (jumps). Those two sources are not totally separated; under event risk, volatility updates faster than under normal market conditions. Generally, tail thickness is associated with hyper volatility updating. Existing VaR literature accounts partially for tail thickness either by including stochastic volatility or by including jump diffusion, but not both. Additionally, this literature does not account for fast updating of volatility associated with tail thickness. This dissertation fills the gap by developing analytical VaR models account for the total (maximum) tail thickness and the associated fast volatility updating. Those aspects are achieved by assuming that trading revenues are evolving according to a mixed non-affine stochastic volatility-jump diffusion process. The mixture of stochastic volatility and jumps diffusion accounts for the maximum tail thickness, whereas the nonaffine structure of stochastic volatility captures the fast volatility updating. The non-affine structure assumes that volatility dynamics are non-linearly related to the square root of current volatility rather than the traditional linear (affine) relationship. VaR estimates are obtained by deriving the conditional characteristic function, and then inverting it numerically via the Fourier Inversion technique to infer the cumulative distribution function. The application of the developed VaR models on a sample that contains six U.S banks during the period 1995-2002 shows that VaR models based on the non-affine stochastic volatility and jump diffusion process produce more reliable VaR estimates compared with the banks' own VaR models. The developed VaR models could significantly predict the losses that those banks incurred during the Russian crisis and the near collapse of the LTCM in 1998 when the banks' VaR models fail.