We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Lévy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Lévy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Abstract. We study generalized complex manifolds from the point of view of symplectic and Poisson geometry. We start by showing that every generalized complex manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson manifolds, to prove a local structure theorem for generalized complex manifolds which extends the result Gualtieri has obtained in the "regular" case. Finally, we begin a study of the local structure of a generalized complex manifold in a neighborhood of a point where the associated Poisson tensor vanishes. In particular, we show that in such a neighborhood, a "first-order approximation" to the generalized complex structure is encoded in the data of a constant B-field and a complex Lie algebra.

OverviewThese are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The research program outlined in this talk was realized in a series of articles [1]- [4]. The orbit method for unipotent groups in positive characteristic is discussed in [1]. The results on character sheaves discussed in parts I and II of this talk are proved in [3]. The results described in part III are proved in [2] and [4]; the former studies L-packets of irreducible characters and the latter is devoted to the relationship between characters and character sheaves on unipotent groups over finite fields.We will outline a theory that combines some of the essential features of Lusztig's theory and of the orbit method.

Let G be a connected unipotent group over a finite field F q . In this article, we propose a definition of L-packets of complex irreducible representations of the finite group G(F q ) and give an explicit description of L-packets in terms of the so-called admissible pairs for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F q ) is a power of q, confirming a conjecture of Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F q ) (where G is a unipotent algebraic group over F q ), the geometry of G, and the theory of character sheaves.

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in [Gua], [H3] (we call these "generalized Lagrangian submanifolds" in our paper), we introduce and study three other classes of submanifolds and their relationships. For generalized complex manifolds that arise from complex (resp., symplectic) manifolds, all three classes specialize to complex (resp., symplectic) submanifolds. In general, however, all three classes are distinct. We discuss some interesting features of our theory of submanifolds, and illustrate them with a few nontrivial examples. Along the way, we obtain a complete and explicit classification of all linear generalized complex structures.We then support our "symplectic/Lagrangian viewpoint" on the submanifolds introduced in [Gua], [H3] by defining the "generalized complex category", modelled on the constructions of Guillemin-Sternberg [GS] and Weinstein [W2]. We argue that our approach may be useful for the quantization of generalized complex manifolds.