A convergent series representation for the density of the supremum of a stable process

We study the density of the supremum of a strictly stable L\'evy process. We prove that for almost all values of the index $\alpha$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A. Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.Comment: 12 page

Probability measures, L\'{e}vy measures and analyticity in time

We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\'{e}vy measure and the third method uses the analytic continuation of the L\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6114 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Variance-optimal hedging for processes with stationary independent increments

We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace- or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.Comment: Published at http://dx.doi.org/10.1214/105051606000000178 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Geometric Asian Option Pricing in General Affine Stochastic Volatility Models with Jumps

In this paper we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain close-form solutions for some relevant affine model classes.Comment: 20 page

Asymptotic analysis and explicit estimation of a class of stochastic volatility models with jumps using the martingale estimating function approach

We provide and analyze explicit estimators for a class of discretely observed continuous-time stochastic volatility models with jumps. In particular we consider the class of non-Gaussian Ornstein-Uhlenbeck based models, as introduced by Barndorff-Nielsen and Shephard. We develop in detail the martingale estimating function approach for this kind of processes, which are bivariate Markov processes, that are not diffusions, but admit jumps. We assume that the bivariate process is observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We prove rigorously consistency and asymptotic normality based on the single assumption that all moments of the stationary distribution of the variance process are finite, and give explicit expressions for the asymptotic covariance matrix. As an illustration we provide a simulation study for daily increments, but the method applies unchanged for any time-scale, including high-frequency observations, without introducing any discretization error