Pseudo-diffusions and Quadratic Term Structure Models

The non-gaussianity of processes observed in financial markets and relatively good performance of gaussian models can be reconciled by replacing the Brownian motion with Lévy processes whose Lévy densities decay as exp(−λ|x|) or faster, where λ > 0 is large. This leads to asymptotic pricing models. The leading term, P0, is the price in the Gaussian model with the same instantaneous drift and variance. The first correction term depends on the instantaneous moments of order up to three, that is, the skewness is taken into account, the next term depends on moments of order four (kurtosis) as well, etc. In empirical studies, the asymptotic formula can be applied without explicit specification of the underlying process: it suffices to assume that the instantaneous moments of order greater than two are small w.r.t. moments of order one and two, and use empirical data on moments of order up to three or four. As an application, the bond pricing problem in the non-Gaussian quadratic term structure model is solved. For pricing of options near expiry, a different set of asymptotic formulas is developed; they require more detailed specification of the process, especially of its jump part. The leading terms of these formulas depends on the jump part of the process only, so that they can be used in empirical studies to identify the jump characteristics of the process.