Searcharxiv⌕ Search

arXiv subjects

Friedrich Hubalek

Publications and source records attributed to Friedrich Hubalek.

16 recordsLinked to original sources
arXiv2026

How the interpolation of life tables affects the decomposition of life insurance surplus

The surplus of a life insurance policy depends on both systematic changes in mortality risk and financial changes. We propose to decompose the surplus by the axiomatically justified IASU decomposition, which is a continuous time version of the Shapley value. However, life tables are not updated continuously, but rather, only once per year. In this yearly update cycle of the life tables, we apply different interpolation methods to perform the IASU decomposition and analyze the effects of these methods on the surplus decomposition. Our results show that Lee-Carter and linear interpolation yield almost identical decompositions, whereas constant approximations results in substantially different decompositions. As a consequence, reporting standards and regulators should clarify how to interpolate mortality risks.

q-fin.ST↗
arXiv2025

On the supremum of a quotient of power sums

We define a function of two real vectors by a certain homogeneous quotient involving power sums, and show that its supremum grows asymptotically linearly w.r.t. the dimension. From this, we deduce a condition under which a parametric set of real matrices satisfies a set of polynomial positivity constraints. This characterization finds an application in mathematical finance, in a recent study on price impact models.

math.CA↗
arXiv2025

Unparalleled instances of prolifickness, random walks, and square root boundaries

We revisit the problem of influencing the sex ratio of a population by subjecting reproduction of each family to some stopping rule. As an easy consequence of the strong law of large numbers, no such modification is possible in the sense that the ratio converges to 1 almost surely, for any stopping rule that is finite almost surely. We proceed to quantify the effects and provide limit distributions for the properly rescaled sex ratio. Besides the total ratio, which is predominantly considered in the pertinent literature, we also analyze the average sex ratio, which may converge to values different from 1. The first part of this note is largely expository, applying classical results and standard methods from the fluctuation theory of random walks. In the second part we apply tail asymptotics for the time at which a random walk hits a one-sided square root boundary, exhibit the differences to the corresponding two-sided problem, and use a limit law related to the empirical dispersion coefficient of a heavy-tailed distribution. Finally, we derive a large deviations result for a special stopping strategy, using saddle point asymptotics.

math.PR↗
arXiv2020

Convergence of Optimal Expected Utility for a Sequence of Binomial Models

We analyze the convergence of expected utility under the approximation of the Black-Scholes model by binomial models. In a recent paper by D. Kreps and W. Schachermayer a surprising and somewhat counter-intuitive example was given: such a convergence may, in general, fail to hold true. This counterexample is based on a binomial model where the i.i.d. logarithmic one-step increments have strictly positive third moments. This is the case, when the up-tick of the log-price is larger than the down-tick. In the paper by D. Kreps and W. Schachermayer it was left as an open question how things behave in the case when the down-tick is larger than the up-tick and -- most importantly -- in the case of the symmetric binomial model where the up-tick equals the down-tick. Is there a general positive result of convergence of expected utility in this setting? In the present note we provide a positive answer to this question. It is based on some rather fine estimates of the convergence arising in the Central Limit Theorem.

math.PR↗
arXiv2020

On binomial order avalanches

This paper introduces a discrete limit order book model where new orders are placed with a fixed displacement from the mid-price. Further, the trade event occurs whenever the mid-price hits the price level on which there is some volume. Therefore, the dynamics of the limit order book model leads to two trading mechanisms, namely Type I trade and Type II trade. A Type I trade takes place whenever the price maximum increases, while a Type II trade occurs if the price drops by $μ$ or more and then increases by $μ$ again. Our focus is mainly on the distribution of order avalanches length, and by an avalanche length we consider a series of order executions where the length of periods with no trade event cannot exceed $\varepsilon$.

math.PR↗
arXiv2020

The running maximum of the Cox-Ingersoll-Ross process with some properties of the Kummer function

We derive tail asymptotics for the running maximum of the Cox-Ingersoll-Ross process. The main result is proved by the saddle point method, where the tail estimate uses a new monotonicity property of the Kummer function. This auxiliary result is established by a computer algebra assisted proof. Moreover, we analyse the coefficients of the eigenfunction expansion of the running maximum distribution asymptotically.

math.PR↗
arXiv2017

Brownian trading excursions and avalanches

We study a parsimonious but non-trivial model of the latent limit order book where orders get placed with a fixed displacement from a center price process, i.e.\ some process in-between best bid and best ask, and get executed whenever this center price reaches their level. This mechanism corresponds to the fundamental solution of the stochastic heat equation with multiplicative noise for the relative order volume distribution. We classify various types of trades, and introduce the trading excursion process which is a Poisson point process. This allows to derive the Laplace transforms of the times to various trading events under the corresponding intensity measure. As a main application, we study the distribution of order avalanches, i.e.\ a series of order executions not interrupted by more than an $\varepsilon$-time interval, which moreover generalizes recent results about Parisian options.

q-fin.MF↗
arXiv2016

Multivariate Mixed Tempered Stable Distribution

The multivariate version of the Mixed Tempered Stable is proposed. It is a generalization of the Normal Variance Mean Mixtures. Characteristics of this new distribution and its capacity in fitting tails and capturing dependence structure between components are investigated. We discuss a random number generating procedure and introduce an estimation methodology based on the minimization of a distance between empirical and theoretical characteristic functions. Asymptotic tail behavior of the univariate Mixed Tempered Stable is exploited in the estimation procedure in order to obtain a better model fitting. Advantages of the multivariate Mixed Tempered Stable distribution are discussed and illustrated via simulation study.

q-fin.ST↗
arXiv2012

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 Lévy process. We prove that for almost all values of the index $α$ -- 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.

math.PR↗
arXiv2008

Probability measures, Lévy measures and analyticity in time

We investigate the relation of the semigroup probability density of an infinite activity Lévy process to the corresponding Lévy 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 Lévy measure and the third method uses the analytic continuation of the Lévy 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.

math.PR↗
arXiv2008

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 on the International Business Machines Corporation (IBM) stock from the New York Stock Exchange (NYSE) and the Microsoft Corporation (MSFT) stock from Nasdaq during a history of five years.

q-fin.ST↗
arXiv2008

Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models

We provide a simple explicit estimator for discretely observed Barndorff-Nielsen and Shephard models, 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. We develop in detail the martingale estimating function approach for a bivariate model, that is not a diffusion, but admits jumps. We do not use ergodicity arguments. We assume that both, logarithmic returns and instantaneous variance are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. As the instantaneous variance is not observable in practice, our results cannot be applied immediately. Our purpose is to provide a theoretical analysis as a starting point and benchmark for further developments concerning optimal martingale estimating functions, and for theoretical and empirical investigations, that replace the variance process with a substitute, such as number or volume of trades or implied variance from option data.

q-fin.ST↗
arXiv2008

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show, that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.

q-fin.CP↗
arXiv2006

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.

math.PR↗