Rizikové míry a dynamické systémy

Assessing systemic risk presents a significant challenge in finance and insurance, where conditional risk measures are essential for capturing contagion effects. This paper introduces two novel systemic risk measures – conditional interval value-at-risk (CoIVaR) and conditional interval expected shortfall (CoIES) – which extend traditional metrics by incorporating interval-based uncertainty. A formal theoretical framework is developed for both measures, offering a detailed characterization of their key properties and risk contributions. We then propose a comprehensive comparison methodology for systemic risk assessment, leveraging stochastic orders, dependence structures, and marginal distributions to establish conditions for ranking risk vectors. Finally, through numerical experiments and real-world stock market applications, we demonstrate the practical utility of CoIVaR and CoIES in quantifying systemic risk under uncertainty. The findings provide valuable insights into systemic risk propagation and establish a robust foundation for risk management in interconnected financial systems.

Risk-consistent conditional systemic risk measures

In this paper, we introduce two kinds of time consistent properties for set-valued dynamic risk measures for discrete-time processes that are adapted to a given filtration, named time consistency and multi-portfolio time consistency. Equivalent characterizations of multi-portfolio time consistency are deduced for normalized dynamic risk measures. In the normalized case, multi-portfolio time consistency is equivalent to the recursive form for risk measures as well as a decomposition property for the acceptance sets. The relations between time consistency and multi-portfolio time consistency are addressed. We also provide a way to construct multi-portfolio time consistent versions of any dynamic risk measure. Finally, we investigate the relationship about time consistency and multi-portfolio time consistency between risk measures for processes and risk measures for random vectors on some product space.

Tail risk of hedge funds: an extreme value application

This paper mainly investigates long term equity return in the Taiwan stock market. We attempted to find the optimal model among and perform diagnostic analysis of the stochastic asset models, such as the independent lognormal model, the ARCH-type model, the Regime-Switching Lognormal model, and the Stochastic Log-Volatility model, all of which have been proposed by the American Academy of Actuaries. In addition, the most suitable long term equity model was determined. In the empirical study of the Taiwan stock market, the parameter estimations were carried out using maximum likelihood estimate and efficient moment of method. Following, we compared log-likelihood, Akaike Information Criterion and Schwartz Bayesian Criterion and drew Quantile-Quantile plots to test tails, to determine the appropriateness of models. Finally, Conditional Tail Expectation risk measures of stochastic equity models were estimated from simulation of the liability of a guaranteed minimum accumulation benefit contract. The results indicate that the generalized autoregressive conditional heteroskedastic and regime switching log normal models perform much better than other considered models for analyzing the Taiwan financial market.

The question of measuring and managing systemic risk—especially in view of the recent financial crisis—became more and more important. We study systemic risk by taking the perspective of a financial regulator and considering the axiomatic approach originally introduced in Chen et al. (Manag Sci 59(6):1373–1388, 2013) and extended in Kromer et al. (Math Methods Oper Res 84:323–357, 2016). The aim of this paper is to generalize the static approach in Kromer et al. (2016) and analyze systemic risk measures in a dynamic setting. We work in the framework of Cheridito et al. (Electron J Probab 11:57–106, 2006) who consider risk measures for bounded discrete-time processes. Apart from the possibility to consider the “evolution of financial values”, another important advantage of the dynamic approach is the possibility to incorporate information in the risk measurement and management process. In context of this dynamic setting we also discuss the arising question of time-consistency for our dynamic systemic risk measures.

The article aims to survey recent advancements in risk management field. First a popular quantile-based risk measure Value-at-Risk (VaR), nowadays widely used to asses exposure to market and credit risk, is presented. Four different approaches are introduced, implemented and backtested on PSE index PX-50 time series. A class of so called coherent risk measures satisfying four qualities highly desired for a risk measure is propounded. As a response to VaR deficiencies several "improved" variants of VaR, some of them satisfying coherence axioms, are proposed. In the last section the motion of coherent risk measures is adapted to the multiperiod framework.

In this paper, we study the asymptotic behavior of dynamic coherent risk measures in general settings regardless of specific representations of the risk measures. In particular, we develop three different types laws of large numbers (LLN) for the average values of portfolios. These LLNs capture the limiting behavior of time-consistent dynamic coherent risk measures under appropriate conditions. Our results apply to general probability spaces with a sequence of financial returns characterized by a set of probability measures. We show that the limit of these averages will generally be multivalued within an identified set. We give examples to illustrate the potential applicability of our results and derive asymptotic results on estimation for the risk of returns of financial assets using a time-consistent dynamic coherent risk measure induced by a class of g-expectations.

In this paper, we study the effect of network structure between agents and objects on measures for systemic risk. We model the influence of sharing large exogeneous losses to the financial or (re)insurance market by a bipartite graph. Using Pareto-tailed losses and multivariate regular variation, we obtain asymptotic results for conditional risk measures based on the Value-at-Risk and the Conditional Tail Expectation. These results allow us to assess the influence of an individual institution on the systemic or market risk and vice versa through a collection of conditional risk measures. For large markets, Poisson approximations of the relevant constants are provided. Differences of the conditional risk measures for an underlying homogeneous and inhomogeneous random graph are illustrated by simulations.

Time consistent dynamic risk processes

In this paper, we consider a finite-horizon Markov decision process (MDP) for which the objective at each stage is to minimize a quantile-based risk measure (QBRM) of the sequence of future costs; we call the overall objective a dynamic quantile-based risk measure (DQBRM). In particular, we consider optimizing dynamic risk measures where the one-step risk measures are QBRMs, a class of risk measures that includes the popular value at risk (VaR) and the conditional value at risk (CVaR). Although there is considerable theoretical development of risk-averse MDPs in the literature, the computational challenges have not been explored as thoroughly. We propose data-driven and simulation-based approximate dynamic programming (ADP) algorithms to solve the risk-averse sequential decision problem. We address the issue of inefficient sampling for risk applications in simulated settings and present a procedure, based on importance sampling, to direct samples toward the “risky region” as the ADP algorithm progresses. Finally, we show numerical results of our algorithms in the context of an application involving risk-averse bidding for energy storage.The online appendix is available at https://doi.org/10.1287/moor.2017.0872 .

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

On Dynamic Measures of Risk

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

Considerably much work has been done on Backward Stochastic Differential Equations (BSDEs) in continuous-time with deterministic terminal horizon or stopping times. Various new models have been introduced in these years in order to generalize BSDEs to solve new practical financial problems.
\nOne strand is focused on discrete-time models. Backward Stochastic Difference Equations (also called BSDEs if no ambiguity) on discrete-time finite-state space were introduced by Cohen and Elliott (2010a). The associated theory required only weak assumptions. In the first topic of this thesis, properties of non-linear expectations defined using the discrete-time finite-state BSDEs were studied. A converse comparison theorem was established. Properties of risk measures defined by non-linear expectations, especially the representation theorems, were given. Then the theory of BSDEs was applied to optimal design of dynamic risk measures. Another strand is about a general random terminal time, which is not necessarily a stopping time. The motivation of this model is a financial problem of hedging of defaultable contingent claims, where BSDEs with stopping times are not applicable. In the second topic of this thesis, discrete-time finite-state BSDEs under progressively enlarged filtration were considered. Martingale representation
\ntheorem, existence and uniqueness theorem and comparison theorem were established. Application to nonlinear expectations was also explored. Using the theory of BSDEs, the explicit solution for optimal design of dynamic default risk measures was obtained.
\nIn recent work on continuous-time BSDEs under progressively enlarged filtration, the reference filtration is generated by Brownian motions. In order to deal with cases with jumps, in the third topic of this thesis, a general reference filtration with predictable representation property and an initial time with immersion property were considered. The martingale representation theorem for square-integrable martingales under progressively enlarged filtration was established. Then the existence and uniqueness theorem of BSDEs under enlarged filtration using Lipschitz continuity of the driver was proved. Conditions for a comparison theorem were also presented. Finally applications to nonlinear expectations and hedging of defaultable contingent claims on Brownian-Poisson setting were explored.