The leading term of asymptotics of prices and sensitivities of barrier options and first touch digitals near the barrier for wide classes of L\’evy processes with exponential jump densities, including Variance Gamma model, KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In particular, it is proved that option’s delta is unbounded for processes of infinite variation, and for processes of finite variation and infinite intensity, with zero drift and drift pointing from the barrier. Two-term asymptotic formulas are also derived. The convergence of prices, sensitivities and the first two terms of asymptotics in Carr’s randomization algorithm is proved. Finally, it is proved that, in each case, and for any $m\in Z_+$, the error of Carr’s randomization approximation can be represented in the form $\sum_{j=1}^m c_j(T,x)N^{-j} +O(N^{-j})$, where $N$ is the number of time steps. This justifies not only Richardson extrapolation but extrapolations of higher order as well.

We develop a class of no-arbitrage dynamic term structure models that are extremely parsimonious. The model employs a cascade structure to provide a natural ranking of the factors in terms of their frequencies, with merely five parameters to describe the interest rate time series and term structure behavior regardless of the dimension of the state vector. The dimension-invariance feature allows us to estimate low and high-dimensional models with equal ease and accuracy. With 15 LIBOR and swap rate series, we estimate 15 models with the dimension going from one to 15. The extensive estimation exercise shows that the 15-factor model significantly outperforms the other lower-dimensional specifications. The model generates mean absolute pricing errors less than one basis point, and overcomes several known limitation of traditional low-dimensional specifications.