  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 Levy processes whose Levy densities decay as exp(-lambda|x|) or
faster, where lambda>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.
