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 usedtechnique for measuring future expected risk for both financial and non-financialinstitutions. VaR, the measure of the worst expected loss over a given horizon at a givenconfidence level, depends crucially on the distributional aspects of trading revenues.Existing VaR models do not capture adequately some empirical aspects of financial datasuch as the tail thickness, which is vital in VaR calculations. Tail thickness in financialvariables results basically from stochastic volatility and event risk (jumps). Those twosources are not totally separated; under event risk, volatility updates faster than undernormal market conditions. Generally, tail thickness is associated with hyper volatilityupdating.Existing VaR literature accounts partially for tail thickness either by includingstochastic volatility or by including jump diffusion, but not both. Additionally, thisliterature does not account for fast updating of volatility associated with tail thickness.This dissertation fills the gap by developing analytical VaR models account forthe total (maximum) tail thickness and the associated fast volatility updating. Thoseaspects are achieved by assuming that trading revenues are evolving according to amixed non-affine stochastic volatility-jump diffusion process. The mixture of stochasticvolatility and jumps diffusion accounts for the maximum tail thickness, whereas the nonaffinestructure of stochastic volatility captures the fast volatility updating. The non-affinestructure assumes that volatility dynamics are non-linearly related to the square root of current volatility rather than the traditional linear (affine) relationship. VaR estimates areobtained by deriving the conditional characteristic function, and then inverting itnumerically via the Fourier Inversion technique to infer the cumulative distributionfunction.The application of the developed VaR models on a sample that contains six U.Sbanks during the period 1995-2002 shows that VaR models based on the non-affinestochastic volatility and jump diffusion process produce more reliable VaR estimatescompared with the banks’ own VaR models. The developed VaR models couldsignificantly predict the losses that those banks incurred during the Russian crisis and thenear collapse of the LTCM in 1998 when the banks’ VaR models fail.