Time consistent dynamic risk processes

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.

Different approaches to defining dynamic market risk measures are available in the literature. Most are focused or derived from probability theory, economic behavior or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representation. The proposed approach is to be implementable and to inherit properties from static market 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.

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.

In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

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.

ABSTRACTWe provide a representation for strong-weak continuous dynamic risk measures from Lp into Lpt spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.

Asymptotic analysis of a dynamic systemic risk measure in a renewal risk model

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.

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 .

Článek se zabývá problematikou dynamických rizikových měr, konkrétně odhadem dynamického VaR a dynamického ES. Po obecném úvodu do problematiky řízení rizik popisujeme v naší práci metody, které jsou klíčové pro samotný výpočetní a odhadovací postup. Nejprve je čtenářům popsána metoda ARMA-GARCH modelů z oboru časových řad a jsou komentovány předpoklady modelu, které se ukazují v praxi jako nereálné. Díky této skutečnosti je navrženo použití metodologie EVT, konkrétně metoda POT. Model POT je schopen aproximovat odlehlé konce námi pozorovaného rozdělení, což je zásadní vlastnost pro odhadování kvantilových rizikových měr. V předposledním odstavci je prezentován přímý výpočet odhadovaných rizikových měr. Poslední odstavec se zabývá některými nedostatky diskutovaných rizikových měr pro delší předpovědní období a také dává podněty pro další výzkumnou činnost této problematiky.

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.

On Dynamic Measures of Risk

We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the “stopper”) who chooses the termination time of the game, and an agent (the “controller”, or “nature”) who selects the probability measure.

We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.