Recently, stock price models based on Lévy processes with stochastic volatility were introduced. The resulting vanilla option prices can be calibrated almost perfectly to empirical prices. Under this model, we will price exotic options, like barrier, lookback and cliquet options, by Monte-Carlo simulation. The sampling of paths is based on a compound Poisson approximation of the Lévy process involved. The precise choice of the terms in the approximation is crucial and investigated in detail. In order to reduce the standard error of the Monte-Carlo simulation, we make use of the technique of control variates. It turns out that there are significant differences with the classical Black-Scholes prices.

In this article we show that the local structure of the projective representation space of a graded algebra can locally be described by quivers with an automorphism of their path algebra, a weighted twist. We describe the quotient spaces of these weighted twisted quiver settings and determine which of them are smooth.

In this note we prove that a smooth order satisfying the reverse geometric engineering conditions in stringtheory [5, §2] (in any compactifying dimension) is Auslander regular. Moreover, we classify theétale local structure of smooth orders over an isolated central singularity.

Given a Cayley-Hamilton smooth order A in a central simple algebra Σ, we determine the flat locus of the Brauer-Severi fibration of A. Moreover, we give a classification of all (reduced) central singularities where the flat locus differs from the Azumaya locus and show that the fibers over the flat, non-Azumaya points near these central singularities can be described as fibered products of graphs of projection maps. This generalizes an old result of Artin on the fibers of the Brauer-Severi fibration of a maximal order over a ramified point. Finally, we show these fibers are also toric quiver varieties and use this fact to compute their cohomology.