Friedrich Hubalek

Old and New Examples of Scale Functions for Spectrally Negative Lévy Processes

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.

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.

Esscher transforms and the minimal entropy martingale measure for exponential Lévy models

In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale transform for exponential processes, and the minimal entropy martingale measure for exponential Lévy models, and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail.

Optimal expected exponential utility of dividend payments in a Brownian risk model

We consider the following optimisation problem for an insurance company Here U(x) = (1−exp(−γx))/γ denotes an exponential utility function with risk aversion parameter γ, C denotes the accumulated dividend process, and β a discount factor. We show that – assuming that a certain integral equation has a solution – the optimal strategy is a barrier strategy. The barrier function is a solution of the integral equation and turns out to be time-dependent. In addition, we study the problem from a different point of view, namely by using a certain ansatz for the value function and the barrier.

A multivariate view of random bucket digital search trees

We take a multivariate view of digital search trees by studying the number of nodes of different types that may coexist in a bucket digital search tree as it grows under an arbitrary memory management system. We obtain the mean of each type of node, as well as the entire covariance matrix between types, whereupon weak laws of large numbers follow from the orders of magnitude (the norming constants include oscillating functions). The result can be easily interpreted for practical systems like paging, heaps and UNIXs buddy system. The covariance results call for developing a Mellin convolution method, where convoluted numerical sequences are handled by convolutions of their Mellin transforms. Furthermore, we use a method of moments to show that the distribution is asymptotically normal. The method of proof is of some generality and is applicable to other parameters like path length and size in random tries and Patricia tries.

Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non-Gaussian Ornstein–Uhlenbeck process describes some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis of IBM™ stock from the New York Stock Exchange and Microsoft Corporation™ stock from Nasdaq during a history of five years.

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

In the present paper we provide a semiexplicit valuation formula for Geometric Asian options, with fixed and floating strike under continuous monitoring, when the underlying stock price process exhibits both stochastic volatility and jumps. More precisely, we shall work in the Barndorff-Nielsen and Shephard (BNS) model framework. We shall provide some numerical illustrations of the results obtained.

QUADRATIC HEDGING FOR THE BATES MODEL

In the present paper we give some preliminary results for option pricing and hedging in the framework of the Bates model based on quadratic risk minimization. We provide an explicit expression of the mean-variance hedging strategy in the martingale case and study the Minimal Martingale measure in the general case.

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 closed form solutions for some relevant affine model classes.

On the variance of the internal path length of generalized digital trees — the Mellin convolution approach

Abstract This paper studies a generalization of the internal path length of a random digital search tree to bucket trees, which are capable of storing up to b records per node. We show that under the assumption of the symmetric Bernoulli probabilistic model the expected path length of a tree built from N records is asymptotically N log 2 N+(A+δ 1 ( log 2 N))N and the variance is (C+e 1 ( log 2 N))N , where A and C depend on b only. The continuous functions δ 1 and e 1 are periodic with mean 0 and period 1. The proofs are analytical and make use of generating functions, harmonic sums and the Mellin integral transform. An important and very general tool for the analysis is Mellins convolution integral. These results and techniques are motivated by a paper by Flajolet and Richmond and by Kirschenhofer, Prodinger, and Szpankowski.