arXiv subjects

Alois Pichler

Publications and source records attributed to Alois Pichler.

36 records · Page 2Linked to original sources

arXiv2021

Quantification of Risk in Classical Models of Finance

This paper enhances the pricing of derivatives as well as optimal control problems to a level comprising risk. We employ nested risk measures to quantify risk, investigate the limiting behavior of nested risk measures within the classical models in finance and characterize existence of the risk-averse limit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen risk measure. Within the classical models risk aversion gives rise to a stream of risk premiums, comparable to dividend payments. In this context we connect coherent risk measures with the Sharpe ratio from modern portfolio theory and extract the Z-spread -- a widely accepted quantity in economics to hedge risk. The results for European option pricing are then extended to risk-averse American options, where we study the impact of risk on the price as well as the optimal time to exercise the option. We also extend Merton's optimal consumption problem to the risk-averse setting.

q-fin.MF↗

arXiv2021

Nested Sinkhorn Divergence To Compute The Nested Distance

The nested distance builds on the Wasserstein distance to quantify the difference of stochastic processes, including also the information modelled by filtrations. The Sinkhorn divergence is a relaxation of the Wasserstein distance, which can be computed considerably faster. For this reason we employ the Sinkhorn divergence and take advantage of the related (fixed point) iteration algorithm. Furthermore, we investigate the transition of the entropy throughout the stages of the stochastic process and provide an entropy-regularized nested distance formulation, including a characterization of its dual. Numerical experiments affirm the computational advantage and supremacy.

math.OC↗

arXiv2020

New Algorithms And Fast Implementations To Approximate Stochastic Processes

We present new algorithms and fast implementations to find efficient approximations for modelling stochastic processes. For many numerical computations it is essential to develop finite approximations for stochastic processes. While the goal is always to find a finite model, which represents a given knowledge about the real data process as accurate as possible, the ways of estimating the discrete approximating model may be quite different: (i) if the stochastic model is known as a solution of a stochastic differential equation, e.g., one may generate the scenario tree directly from the specified model; (ii) if a simulation algorithm is available, which allows simulating trajectories from all conditional distributions, a scenario tree can be generated by stochastic approximation; (iii) if only some observed trajectories of the scenario process are available, the construction of the approximating process can be based on non-parametric conditional density estimates.

math.OC↗

arXiv2020

Convex Risk Measures based on Divergence

Risk measures connect probability theory or statistics to optimization, particularly to convex optimization. They are nowadays standard in applications of finance and in insurance involving risk aversion. This paper investigates a wide class of risk measures on Orlicz spaces. The characterizing function describes the decision maker's risk assessment towards increasing losses. We link the risk measures to a crucial formula developed by Rockafellar for the Average Value-at-Risk based on convex duality, which is fundamental in corresponding optimization problems. We characterize the dual and provide complementary representations.

q-fin.RM↗

arXiv2019

Risk averse stochastic programming: time consistency and optimal stopping

Bellman formulated a vague principle for optimization over time, which characterizes optimal policies by stating that a decision maker should not regret previous decisions retrospectively. This paper addresses time consistency in stochastic optimization. The problem is stated in generality first. The paper discusses time consistent decision-making by addressing risk measures which are recursive, nested, dynamically or time consistent and introduces stopping time risk measures. It turns out that the paradigm of time consistency is in conflict with various desirable, classical properties of general risk measures.

math.OC↗

arXiv2019

Fractional Risk Process in Insurance

Important models in insurance, for example the Carm{é}r--Lundberg theory and the Sparre Andersen model, essentially rely on the Poisson process. The process is used to model arrival times of insurance claims. This paper extends the classical framework for ruin probabilities by proposing and involving the fractional Poisson process as a counting process and addresses fields of applications in insurance. The interdependence of the fractional Poisson process is an important feature of the process, which leads to initial stress of the surplus process. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process.

math.ST↗

arXiv2019

Incorporating statistical model error into the calculation of acceptability prices of contingent claims

The determination of acceptability prices of contingent claims requires the choice of a stochastic model for the underlying asset price dynamics. Given this model, optimal bid and ask prices can be found by stochastic optimization. However, the model for the underlying asset price process is typically based on data and found by a statistical estimation procedure. We define a confidence set of possible estimated models by a nonparametric neighborhood of a baseline model. This neighborhood serves as ambiguity set for a multi-stage stochastic optimization problem under model uncertainty. We obtain distributionally robust solutions of the acceptability pricing problem and derive the dual problem formulation. Moreover, we prove a general large deviations result for the nested distance, which allows to relate the bid and ask prices under model ambiguity to the quality of the observed data.

q-fin.PR↗

arXiv2018

Martingale Characterizations of Risk-Averse Stochastic Optimization Problems

This paper addresses risk awareness of stochastic optimization problems. Nested risk measures appear naturally in this context, as they allow beneficial reformulations for algorithmic treatments. The reformulations presented extend usual Hamilton-Jacobi-Bellman equations in dynamic optimization by involving risk awareness in the problem formulation. Nested risk measures are built on risk measures, which originate by conditioning on the history of a stochastic process. We derive martingale properties of these risk measures and use them to prove continuity. It is demonstrated that stochastic optimization problems, which incorporate risk awareness via nesting risk measures, are continuous with respect to the natural distance governing these optimization problems, the nested distance.

math.OC↗

arXiv2018

Fundamental Properties of Process Distances

Information is an inherent component of stochastic processes and to measure the distance between different stochastic processes it is not sufficient to consider the distance between their laws. Instead, the information which accumulates over time and which is mathematically encoded by filtrations has to be accounted for as well. The nested distance/bicausal Wasserstein distance addresses this challenge by incorporating the filtration. It is of emerging importance due to its applications in stochastic analysis, stochastic programming, mathematical economics and other disciplines. This article establishes a number of fundamental properties of the nested distance. In particular we prove that the nested distance of processes generates a Polish topology but is itself not a complete metric. We identify its completion to be the set of nested distributions, which are a form of generalized stochastic processes. We also characterize the extreme points of the set of couplings which participate in the definition of the nested distance, proving that they can be identified with adapted deterministic maps. Finally, we compare the nested distance to an alternative metric, which could possibly be easier to compute in practical situations.

math.OC↗

arXiv2018

Entropy Based Risk Measures

Entropy is a measure of self-information which is used to quantify losses. Entropy was developed in thermodynamics, but is also used to compare probabilities based on their deviating information content. Corresponding model uncertainty is of particular interest in stochastic programming and its applications like mathematical finance, as complete information is not accessible or manageable in general. This paper extends and generalizes the Entropic Value-at-Risk by involving Rényi entropies. We provide explicit relations of different entropic risk measures, we elaborate their dual representations and present their relations explicitly. We consider the largest spaces which allow studying the impact of information in detail and it is demonstrated that these do not depend on the information loss. The dual norms and Hahn-Banach functionals are characterized explicitly.

math.PR↗

arXiv2017

Geometry Of The Expected Value Set And The Set-Valued Sample Mean Process

The law of large numbers extends to random sets by employing Minkowski addition. Above that, a central limit theorem is available for set-valued random variables. The existing results use abstract isometries to describe convergence of the sample mean process towards the limit, the expected value set. These statements do not reveal the local geometry and the relations of the sample mean and the expected value set, so these descriptions are not entirely satisfactory in understanding the limiting behavior of the sample mean process. This paper addresses and describes the fluctuations of the sample average mean on the boundary of the expectation set.

math.PR↗

arXiv2017

On Banach spaces of vector-valued random variables and their duals motivated by risk measures

We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces and it is shown that these functionals are Lipschitz continuous. The risk functionals cannot be defined on strictly larger spaces of random variables which creates a particular interest for the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.

math.FA↗

arXiv2013

Premiums And Reserves, Adjusted By Distortions

The net-premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, an insurance company, applying the net-premium principle, goes bankrupt with probability one in the long run, even if the company covers its entire costs by collecting the respective fees from its customers. It is therefore an intrinsic necessity for the insurance industry to apply premium principles, which guarantee at least further existence of the company itself; otherwise, the company naturally could not insure its clients to cover their potential, future claims. Beside this intriguing fact the underlying loss distribution typically is not known precisely. Hence alternative premium principles have been developed. A simple principle, ensuring risk-adjusted credibility premiums, is the distorted premium principle. This principle is convenient in insurance companies, as the actuary does not have to change his or her tools to compute the premiums or reserves. This paper addresses the distorted premium principle from various angles. First, dual characterizations are developed. Next, distorted premiums are typically computed by under-weighting or ignoring low, but over-weighting high losses. It is demonstrated here that there is an alternative, opposite point of view, which consists in leaving the probability measure unchanged, but increasing the outcomes instead. It turns out that this new point of view is natural in actuarial practice, as it can be used for premium calculations, as well as to determine the reserves of subsequent years in a time consistent way.

q-fin.RM↗

arXiv2013

The Natural Banach Space for Version Independent Risk Measures

Risk measures, or coherent measures of risk are often considered on the space L^\infty, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure---the Average Value-at-Risk---are well defined on the larger space L^1 and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some L^p space. But in many situations this is possibly unnatural, because any L^p with p>p_0, say, is suitable to define the spectral risk measure as well. In addition to that risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is? Abstract This paper introduces a norm, which is built from the risk measure, and a Banach space, which carries the risk measure in a natural way. It is often strictly larger than its original domain, and obeys the key property that the risk measure is finite valued and continuous on that space in an elementary and natural way.

math.FA↗

arXiv2013

Uniqueness of Kusuoka Representations

This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka representation is derived from this initial result. Further, stochastic order relations are employed to identify the minimal Kusuoka representation. It is shown that measures in the minimal representation are extremal with respect to the order relations. The tools are finally employed to provide the minimal representation for important practical examples. Although the Kusuoka representation is usually given only for nonatomic probability spaces, this presentation closes the gap to spaces with atoms.

math.ST↗

arXiv2012

Spectral Risk Measures, With Adaptions For Stochastic Optimization

Stochastic optimization problems often involve the expectation in its objective. When risk is incorporated in the problem description as well, then risk measures have to be involved in addition to quantify the acceptable risk, often in the objective. For this purpose it is important to have an adjusted, adapted and efficient evaluation scheme for the risk measure available. In this article different representations of an important class of risk measures, the spectral risk measures, are elaborated. The results allow concise problem formulations, they are particularly adapted for stochastic optimization problems. Efficient evaluation algorithms can be built on these new results, which finally make optimization problems involving spectral risk measures eligible for stochastic optimization.

math.ST↗

arXiv2012

On A Rapidly Converging Series For The Riemann Zeta Function

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann's hypothesis to similar properties on polynomials. And interestingly, Laguerre's polynomials form a kind of leitmotif through all sections.

math.NT↗

arXiv2012

Strategies To Evaluate The Riemann Zeta Function

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta function. The results originate from attempts to extend the zeta function by classical means on the complex plane. This is particularly of interest for representations which converge rapidly in a given area of the complex plane, or for the purpose to make error bounds available.

math.NT↗