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Tracking value-at-risk through derivative prices

The focus of this work is on the problem of tracking parameters describing both the stochastic discount factor and the objective / real-world measure dynamically, with the aim of monitoring value at risk or other related diagnostics of interest. The methodology presented incorporates information from derivative prices as well as from the underlying instrument’s price over time in order to perform on-line parameter inference. We construct a parametric model of the stochastic discount factor which is introduced based on empirical results in the literature (Aı̈t-Sahalia and Lo, 2000; Jackwerth, 2000; Rosenberg and Engle, 2002, for example). This is used in a sequential Monte Carlo algorithm for tracking the parameters of this and of an objective density over time. Further, two new techniques for pricing European options in the framework are discussed. In applying this approach to price data, Variance Gamma and Normal Inverse Gaussian models of the underlying price process have been discussed. These are for illustrative purposes and other models could easily be also considered. Both models appear to track realistically; detailed results are presented for the Variance Gamma model. These cover the value at risk estimates, expected price change estimates and parameter estimates.