760 results on '"dynamic risk measure"'
Search Results
2. Set-valued dynamic risk measures for processes and for vectors.
- Author
-
Chen, Yanhong and Feinstein, Zachary
- Subjects
MEASUREMENT ,DEFINITIONS - Abstract
The relationship between set-valued risk measures for processes and vectors on the optional filtration is investigated. The equivalence of risk measures for processes and vectors and the equivalence of their penalty function formulations are provided. In contrast to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations.
- Author
-
Mabitsela, Lesedi, Guambe, Calisto, and Kufakunesu, Rodwell
- Subjects
- *
STOCHASTIC differential equations - Abstract
We derive a representation for dynamic capital allocation when the underlying asset price process includes extreme random price movements. Moreover, we consider the representation of dynamic risk measures defined under Backward Stochastic Differential Equations (BSDE) with generators that grow quadratic-exponentially in the control variables. Dynamic capital allocation is derived from the differentiability of BSDEs with jumps. The results are illustrated by deriving a capital allocation representation for dynamic entropic risk measure and static coherent risk measure. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. Set-valued risk measures as backward stochastic difference inclusions and equations.
- Author
-
Ararat, Çağın and Feinstein, Zachary
- Subjects
STOCHASTIC difference equations ,STOCHASTIC differential equations ,DIFFERENCE equations ,DIFFERENTIAL inclusions - Abstract
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
5. Equivalence between time consistency and nested formula.
- Author
-
Gérard, Henri, De Lara, Michel, and Chancelier, Jean-Philippe
- Subjects
- *
MATHEMATICAL equivalence , *STOCHASTIC processes , *DEFINITIONS , *LITERATURE reviews - Abstract
Figure out a situation where, at the beginning of every week, one has to rank every pair of stochastic processes starting from that week up to the horizon. Suppose that two processes are equal at the beginning of the week. The ranking procedure is time consistent if the ranking does not change between this week and the next one. In this paper, we propose a minimalist definition of time consistency (TC) between two (assessment) mappings. With very few assumptions, we are able to prove an equivalence between time consistency and a nested formula (NF) between the two mappings. Thus, in a sense, two assessments are consistent if and only if one is factored into the other. We review the literature and observe that the various definitions of TC (or of NF) are special cases of ours, as they always include additional assumptions. By stripping off these additional assumptions, we present an overview of the literature where the specific contributions of authors are enlightened. Moreover, we present two classes of mappings, translation invariant mappings and Fenchel–Moreau conjugates, that display time consistency under suitable assumptions. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
6. De-risking Fleet Replacement Decisions
- Author
-
Liret, Anne, Ansaripoor, Amir H., Oliveira, Fernando S., Bramer, Max, editor, and Petridis, Miltos, editor
- Published
- 2014
- Full Text
- View/download PDF
7. Solvency requirement for long term guarantee: risk measure versus probability of ruin.
- Author
-
Devolder, Pierre
- Abstract
Solvency requirements are based on the idea that risk can be accepted if enough capital is present. The determination of this minimum level of capital depends on the way to consider and measure the underlying risk. Apart from the kind of risk measure used, an important factor is the way to integrate time in the process. This topic is particularly important for long term liabilities such as life insurance or pension benefits. In this paper we study the market risk of a life insurer offering a fixed guaranteed rate on a certain time horizon and investing the premium in a risky fund. We develop and compare various risk measurements based either on a single point analysis or on a continuous time test. Dynamic risk measures are also considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
8. A DYNAMIC MODEL OF CENTRAL COUNTERPARTY RISK.
- Author
-
BIELECKI, TOMASZ R., CIALENCO, IGOR, and FENG, SHIBI
- Subjects
DYNAMIC models ,COUNTERPARTY risk ,COUNTERPARTIES (Finance) ,ALLOCATION (Accounting) ,PORTFOLIO management (Investments) - Abstract
We introduce a dynamic model of the default waterfall of derivatives central counterparties and propose a risk sensitive method for sizing the initial margin, and the default fund and its allocation among clearing members. Using a Markovian structure model of joint credit migrations, our evaluation of the default fund takes into account the joint credit quality of clearing members as they evolve over time. Another important aspect of the proposed methodology is the use of the time consistent dynamic risk measures for computation of the initial margin and the default fund. We carry out a comprehensive numerical study, where, in particular, we analyze the advantages of the proposed methodology and its comparison with the currently prevailing methods used in industry. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
9. Dynamic Risk Measures
- Author
-
Delong, Łukasz and Delong, Łukasz
- Published
- 2013
- Full Text
- View/download PDF
10. A Unified Approach to Time Consistency of Dynamic Risk Measures and Dynamic Performance Measures in Discrete Time.
- Author
-
Bielecki, Tomasz R., Cialenco, Igor, and Pitera, Marcin
- Subjects
DISCRETE-time systems ,MATHEMATICAL analysis ,DECISION making ,COMPARATIVE studies ,NUMERICAL analysis - Abstract
In this paper, we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures (also known as acceptability indices). The proposed framework not only integrates existing forms of time consistency, but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making. In particular, it allows for in-depth comparative analysis of (most of) the existing types of time consistency--a feat that has not been possible before and, which is done in the companion paper by the authors. In our approach, the time consistency is studied for a large class of maps that are postulated to satisfy only two properties--monotonicity and locality. We call these maps LM-measures. The time consistency is defined in terms of an update rule. The form of the update rule introduced here is novel, and is perfectly suited for developing the unifying framework that is worked out in this paper. As an illustration of the applicability of our approach, we show how to recover almost all concepts of weak time consistency by means of constructing appropriate update rules. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
11. Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading.
- Author
-
Redeker, Imke and Wunderlich, Ralf
- Subjects
DISCRETE time filters ,INVESTMENTS ,CONSUMPTION (Economics) ,RISK assessment ,MARKOV processes - Abstract
We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
12. DIFFERENTIABILITY OF BSVIEs AND DYNAMIC CAPITAL ALLOCATIONS.
- Author
-
KROMER, EDUARD and OVERBECK, LUDGER
- Subjects
CAPITAL ,RISK management in business ,CONTINUOUS time systems ,STOCHASTIC analysis ,INTEGRAL equations - Abstract
Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios for the first time in a continuous-time dynamic context. For this purpose, we introduce a differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
13. On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation.
- Author
-
Madan, D., Pistorius, M., and Stadje, M.
- Subjects
INVESTMENTS ,FINANCIAL risk ,LIMIT theorems ,STOCHASTIC differential equations ,RANDOM walks ,LATTICE theory - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
14. Existence and uniqueness of M-solutions for backward stochastic Volterra integral equations
- Author
-
Wenxue Li, Ruihua Wu, and Ke Wang
- Subjects
Backward stochastic Volterra integral equations ,existence ,uniqueness ,dynamic risk measure ,Mathematics ,QA1-939 - Abstract
In this article, we study general backward stochastic Volterra integral equations (BSVIEs). Combining the contractive-mapping principle, step-by-step iteration method and mathematical induction, we establish the existence and uniqueness theorem of M-solution for the BSVIEs. This theorem could be applied directly to many models, for example, using the result to a kind of financial models provides a new and easy method to discuss the existence of dynamic risk measure.
- Published
- 2014
15. Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs.
- Author
-
Grigorova, Miryana and Quenez, Marie-Claire
- Subjects
- *
OPTIMAL stopping (Mathematical statistics) , *GAME theory , *DISCRETE-time systems , *STOCHASTIC differential equations , *NASH equilibrium - Abstract
We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear)g-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed byg-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamadène and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz drivergwithout any additional assumption on the driver besides that ensuring the monotonicity of the correspondingg-expectation. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
- View/download PDF
16. A note on representation of BSDE-based dynamic risk measures and dynamic capital allocations
- Author
-
Rodwell Kufakunesu, Lesedi Mabitsela, and Calisto Guambe
- Subjects
Statistics and Probability ,Dynamic risk measure ,Process (engineering) ,Capital (economics) ,Representation (systemics) ,Econometrics ,Asset (economics) ,Mathematics ,Capital allocation line - Abstract
We derive a representation for dynamic capital allocation when the underlying asset price process includes extreme random price movements. Moreover, we consider the representation of dynamic risk m...
- Published
- 2020
17. Optimal stopping with f-expectations: The irregular case
- Author
-
Miryana Grigorova, Youssef Ouknine, Peter Imkeller, and Marie-Claire Quenez
- Subjects
Statistics and Probability ,Comparison theorem ,Applied Mathematics ,Infinitesimal ,010102 general mathematics ,Optional stopping theorem ,01 natural sciences ,Dynamic risk measure ,010104 statistics & probability ,Modeling and Simulation ,Snell envelope ,Filtration (mathematics) ,Applied mathematics ,Optimal stopping ,0101 mathematics ,Nonlinear expectation ,Mathematical economics ,Mathematics - Abstract
We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.
- Published
- 2020
18. Time-consistency of risk measures with GARCH volatilities and their estimation.
- Author
-
Klüppelberg, Claudia and Zhang, Jianing
- Subjects
RISK assessment ,EXTREME value theory ,PARETO distribution ,VALUE at risk ,STATISTICS - Abstract
In this paper we study time-consistent risk measures for returns that are given by a GARCH(1,1) model. We present a construction of risk measures based on their static counterparts that overcomes the lack of time-consistency. We then study in detail our construction for the risk measures Value-at-Risk (VaR) and Average Value-at-Risk (AVaR). While in the VaR case we can derive an analytical formula for its time-consistent counterpart, in the AVaR case we derive lower and upper bounds to its time-consistent version. Furthermore, we incorporate techniques from extreme value theory (EVT) to allow for a more tail-geared statistical analysis of the corresponding risk measures. We conclude with an application of our results to a data set of stock prices. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
19. Dynamic risk measure for BSVIE with jumps and semimartingale issues
- Author
-
Nacira Agram
- Subjects
Statistics and Probability ,Actuarial science ,Applied Mathematics ,Risk measure ,010102 general mathematics ,Mathematics::Optimization and Control ,60H07, 60H20, 60H30, 45D05, 45R05 ,01 natural sciences ,Dynamic risk measure ,010104 statistics & probability ,Semimartingale ,Optimization and Control (math.OC) ,Life insurance ,FOS: Mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Optimization and Control ,Insurance industry ,Mathematics - Abstract
Risk measure is a fundamental concept in finance and in the insurance industry, it is used to adjust life insurance rates. In this current paper, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues., Comment: 11 pages
- Published
- 2019
20. Set-valued risk measures as backward stochastic difference inclusions and equations
- Author
-
Zachary Feinstein, Çağın Ararat, and Ararat, Çağın
- Subjects
Statistics and Probability ,Difference inclusion ,Time-consistency ,Set-valued difference equation ,Mathematical finance ,Univariate ,Scalar (physics) ,Set-valued risk measure ,Dynamic risk measure ,Stochastic differential equation ,Differential inclusion ,Time consistency ,26E25, 28B20, 34A60, 39A50, 46A20, 91B30 ,Applied mathematics ,Statistics, Probability and Uncertainty ,Finance ,Differential (mathematics) ,Mathematics - Probability ,Mathematics ,Quantitative Finance - Risk Management - Abstract
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time., Comment: 27 pages
- Published
- 2021
21. Risk arbitrage and hedging to acceptability under transaction costs
- Author
-
Emmanuel Lépinette and Ilya Molchanov
- Subjects
Statistics and Probability ,Computer science ,01 natural sciences ,FOS: Economics and business ,Dynamic risk measure ,010104 statistics & probability ,510 Mathematics ,Computer Science::Computational Engineering, Finance, and Science ,0502 economics and business ,FOS: Mathematics ,Econometrics ,0101 mathematics ,050208 finance ,91G20, 60D05, 60G42 ,Mathematical finance ,Risk measure ,Probability (math.PR) ,05 social sciences ,Mathematical Finance (q-fin.MF) ,310 Statistics ,Quantitative Finance - Mathematical Finance ,Fixed asset ,Portfolio ,Risk arbitrage ,Arbitrage ,Statistics, Probability and Uncertainty ,Acceptance set ,Mathematics - Probability ,Finance - Abstract
The classical discrete time model of proportional transaction costs relies on the assumption that a feasible portfolio process has solvent increments at each step. We extend this setting in two directions, allowing for convex transaction costs and assuming that increments of the portfolio process belong to the sum of a solvency set and a family of multivariate acceptable positions, e.g. with respect to a dynamic risk measure. We describe the sets of superhedging prices, formulate several no (risk) arbitrage conditions and explore connections between them. In the special case when multivariate positions are converted into a single fixed asset, our framework turns into the no good deals setting. However, in general, the possibilities of assessing the risk with respect to any asset or a basket of the assets lead to a decrease of superhedging prices and the no arbitrage conditions become stronger. The mathematical technique relies on results for unbounded and possibly non-closed random sets in Euclidean space., 31 page. Accepted for publication in Finance and Stochastics
- Published
- 2021
- Full Text
- View/download PDF
22. Convexity and sublinearity of [formula omitted]-expectations.
- Author
-
Ji, Ronglin, Shi, Xuejun, Wang, Shijie, and Zhou, Jinming
- Subjects
- *
SUBDIFFERENTIALS , *STOCHASTIC differential equations - Abstract
Under the basic assumptions of g -expectations defined in Chen and Wang (2000), we establish the one-to-one correspondence between generators of backward stochastic differential equations (BSDEs for short) and the convexity (resp. conditional convexity, F t -convexity) of g -expectations, respectively. We also obtain the relationship between generators of BSDEs and the sublinearity (resp. conditional sublinearity, F t -sublinearity) of g -expectations, respectively. Moreover, we provide the reasonable assumptions on generators of BSDEs for the further study on dynamic risk measures by applying the theory of g -expectations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective
- Author
-
Bielecki, Tomasz R., Cialenco, Igor, and Pitera, Marcin
- Published
- 2017
- Full Text
- View/download PDF
24. REPRESENTATION OF BSDE-BASED DYNAMIC RISK MEASURES AND DYNAMIC CAPITAL ALLOCATIONS.
- Author
-
KROMER, EDUARD and OVERBECK, LUDGER
- Subjects
STOCHASTIC difference equations ,CAPITAL ,ALLOCATION (Accounting) ,ECONOMIC research ,MEASURE theory - Abstract
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 entropie 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 Gâteauxdifferentiable. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
25. VECTOR-VALUED COHERENT RISK MEASURE PROCESSES.
- Author
-
TAHAR, IMEN BEN and LÉPINETTE, EMMANUEL
- Subjects
VECTOR analysis ,FINANCIAL risk management ,ECONOMIC models ,VALUE at risk ,DYNAMICAL systems - Abstract
Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
26. Optimal stopping with f-expectations: The irregular case
- Author
-
Grigorova, Miryana, Imkeller, Peter, Ouknine, Youssef, Quenez, Marie-Claire, Universität Bielefeld, Institut für Mathematik [Berlin], Technische Universität Berlin (TU), Faculté des Sciences Semlalia Marrakech, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
nonlinear expectation ,strong $\mathcal{E}^f$ -supermartingale ,Probability (math.PR) ,aggregation ,backward stochastic differential equation ,f-expectation ,Computational Finance (q-fin.CP) ,dynamic risk measure ,Snell envelope ,[QFIN.CP]Quantitative Finance [q-fin]/Computational Finance [q-fin.CP] ,General filtration ,FOS: Economics and business ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Quantitative Finance - Computational Finance ,reflected backward stochastic differential equation ,optimal stopping ,Optimization and Control (math.OC) ,FOS: Mathematics ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,American option ,comparison theorem ,Tanaka-type formula ,Mathematics - Optimization and Control ,Mathematics - Probability - Abstract
We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.
- Published
- 2020
27. l1 -Regularization for multi-period portfolio selection
- Author
-
Francesca Perla, Zelda Marino, Stefania Corsaro, Valentina De Simone, Corsaro, Stefania, De Simone, Valentina, Marino, Zelda, and Perla, Francesca
- Subjects
Mathematical optimization ,021103 operations research ,Portfolio optimization, Time consistency, l1 norm, Constrained optimization ,Computer science ,91G10, 90C30, 65K05 ,Multi period ,Portfolio optimization ,0211 other engineering and technologies ,Holding cost ,General Decision Sciences ,l1 norm ,02 engineering and technology ,Portfolio optimization · Time consistency · l1 norm · Constrained optimization ,Management Science and Operations Research ,Regularization (mathematics) ,Dynamic risk measure ,Time consistency ,Optimization and Control (math.OC) ,FOS: Mathematics ,Portfolio ,Constrained optimization ,Mathematics - Optimization and Control - Abstract
In this work we present a model for the solution of the multi-period portfolio selection problem. The model is based on a time consistent dynamic risk measure. We apply $$l_1$$ -regularization to stabilize the solution process and to obtain sparse solutions, which allow one to reduce holding costs. The core problem is a nonsmooth optimization one, with equality constraints. We present an iterative procedure based on a modified Bregman iteration, that adaptively sets the value of the regularization parameter in order to produce solutions with desired financial properties. We validate the approach showing results of tests performed on real data.
- Published
- 2020
28. Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic
- Author
-
Gloria Pérez, María Merino, and Unai Aldasoro
- Subjects
Mathematical optimization ,021103 operations research ,Computation ,0211 other engineering and technologies ,General Decision Sciences ,02 engineering and technology ,Variance (accounting) ,Management Science and Operations Research ,Dynamic risk measure ,Production planning ,Theory of computation ,Benchmark (computing) ,Quadratic programming ,Linear combination ,Mathematics - Abstract
In this paper, we present a multistage time consistent Expected Conditional Risk Measure for minimizing a linear combination of the expected mean and the expected variance, so-called Expected Mean-Variance. The model is formulated as a multistage stochastic mixed-integer quadratic programming problem combining risk-sensitive cost and scenario analysis approaches. The proposed problem is solved by a matheuristic based on the Branch-and-Fix Coordination method. The multistage scenario cluster primal decomposition framework is extended to deal with large-scale quadratic optimization by means of stage-wise reformulation techniques. A specific case study in risk-sensitive production planning is used to illustrate that a remarkable decrease in the expected variance (risk cost) is obtained. A competitive behavior on the part of our methodology in terms of solution quality and computation time is shown when comparing with plain use of CPLEX in 150 benchmark instances, ranging up to 711,845 constraints and 193,000 binary variables.
- Published
- 2018
29. Conditional expectiles, time consistency and mixture convexity properties
- Author
-
Fabio Bellini, Valeria Bignozzi, Giovanni Puccetti, Bellini, F, Bignozzi, V, and Puccetti, G
- Subjects
Mixture concavity ,Statistics and Probability ,Economics and Econometrics ,050208 finance ,Sequential consistency ,05 social sciences ,Conditional expectation ,01 natural sciences ,Minimisation (clinical trials) ,Convexity ,Conditional expectile ,010104 statistics & probability ,Time consistency ,Quadratic equation ,Dynamic risk measure ,0502 economics and business ,Applied mathematics ,First derivative test ,Supermartingale property ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
We study conditional expectiles, defined as a natural generalisation of conditional expectations by means of the minimisation of an asymmetric quadratic loss function. We show that conditional expectiles can be equivalently characterised by a conditional first order condition and we derive their main properties. For possible applications as dynamic risk measures, we discuss their time consistency properties.
- Published
- 2018
30. Mean field BSDEs and global dynamic risk measures
- Author
-
Agnès Sulem, Rui Chen, Andreea Minca, Roxana Dumitrescu, Mathematical Risk Handling (MATHRISK), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), King‘s College London, Cornell University [New York], Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Random graph ,History ,050208 finance ,Polymers and Plastics ,Risk measure ,05 social sciences ,Dual representation ,Industrial and Manufacturing Engineering ,BSDEs ,Dynamic risk measure ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Operator (computer programming) ,Mean field theory ,0502 economics and business ,Systemic risk ,Applied mathematics ,050207 economics ,Business and International Management ,Intensity (heat transfer) ,Mathematics ,Systemic risk measures - Abstract
We study Mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher order interactions such as those occurring on an inhomogeneous random graph. We provide comparison and strict comparison results. Based on these, we interpret the BSDE solution as a global dynamic risk measure that can account for the intensity of system interactions and therefore incorporate systemic risk. Using Fenchel-Legendre transforms, we establish a dual representation for the risk measure, and in particular we exhibit its dependence on the mean-field operator.
- Published
- 2019
31. On the properties of the Lambda value at risk: robustness, elicitability and consistency
- Author
-
Matteo Burzoni, Ilaria Peri, and Chiara Maria Ruffo
- Subjects
Deviation risk measure ,050208 finance ,Risk measure ,05 social sciences ,ems ,Entropic value at risk ,01 natural sciences ,FOS: Economics and business ,Dynamic risk measure ,010104 statistics & probability ,Spectral risk measure ,Time consistency ,Risk Management (q-fin.RM) ,0502 economics and business ,Coherent risk measure ,Econometrics ,0101 mathematics ,General Economics, Econometrics and Finance ,Finance ,Value at risk ,Quantitative Finance - Risk Management ,Mathematics - Abstract
© 2017 Informa UK Limited, trading as Taylor & Francis Group. Recently, the financial industry and regulators have enhanced the debate on the good properties of a risk measure. A fundamental issue is the evaluation of the quality of a risk estimation. On the one hand, a backtesting procedure is desirable for assessing the accuracy of such an estimation and this can be naturally achieved by elicitable risk measures. For the same objective, an alternative approach has been introduced by Davis [Stat. Risk Model. Appl. Finance Insurance, 2016, 33, 67–93] through the so-called consistency property. On the other hand, a risk estimation should be less sensitive with respect to small changes in the available data-set and exhibit qualitative robustness. A new risk measure, the Lambda value at risk (Λ V a R), has been recently proposed by Frittelli et al. [Math. Finance, 2014, 24, 442–463], as a generalization of VaR with the ability to discriminate the risk among P&L; distributions with different tail behaviour. In this article, we show that Λ V a R also satisfies the properties of robustness, elicitability and consistency under some conditions.
- Published
- 2019
32. Dynamic risk measures for stochastic asset processes from ruin theory
- Author
-
Shuji Tanaka and Yasutaka Shimizu
- Subjects
Statistics and Probability ,Economics and Econometrics ,Solvency ,050208 finance ,Stochastic process ,Computer science ,Risk measure ,05 social sciences ,Ruin theory ,01 natural sciences ,Measure (mathematics) ,Dynamic risk measure ,010104 statistics & probability ,0502 economics and business ,Econometrics ,Penalty method ,Asset (economics) ,0101 mathematics ,Statistics, Probability and Uncertainty - Abstract
This article considers a dynamic version of risk measures for stochastic asset processes and gives a mathematical benchmark for required capital in a solvency regulation framework. Some dynamic risk measures, based on the expected discounted penalty function launched by Gerber and Shiu, are proposed to measure solvency risk from the company’s going-concern point of view. This study proposes a novel mathematical justification of a risk measure for stochastic processes as a map on a functional path space of future loss processes.
- Published
- 2018
33. Strongly consistent multivariate conditional risk measures
- Author
-
Thilo Meyer-Brandis, Gregor Svindland, and Hannes Hoffmann
- Subjects
Statistics and Probability ,Multivariate statistics ,050208 finance ,Mathematical finance ,05 social sciences ,Univariate ,Strong consistency ,Function (mathematics) ,Conditional probability distribution ,Mathematical Finance (q-fin.MF) ,01 natural sciences ,FOS: Economics and business ,Dynamic risk measure ,010104 statistics & probability ,Quantitative Finance - Mathematical Finance ,0502 economics and business ,Statistics ,Econometrics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Conditional variance ,Finance ,Mathematics - Abstract
We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (Stoch Process Appl 126(7):2014–2037, 2016). Further, in analogy to the univariate case in Follmer (Stat Risk Model 31(1):79–103, 2014), we prove that under law-invariance strong consistency implies that multivariate conditional risk measures are necessarily multivariate conditional certainty equivalents.
- Published
- 2018
34. Entropic risk measures and their comparative statics in portfolio selection: Coherence vs. convexity
- Author
-
Robert Rischau, Mario Brandtner, and Wolfgang Kürsten
- Subjects
050208 finance ,Information Systems and Management ,Actuarial science ,General Computer Science ,05 social sciences ,Diversification (finance) ,Management Science and Operations Research ,Entropic value at risk ,Industrial and Manufacturing Engineering ,Dynamic risk measure ,Expected shortfall ,Spectral risk measure ,Modeling and Simulation ,0502 economics and business ,Coherent risk measure ,Economics ,Portfolio optimization ,050205 econometrics ,Entropic risk measure - Abstract
We conduct a decision-theoretic analysis of optimal portfolio choices and, in particular, their comparative statics under two types of entropic risk measures, the coherent entropic risk measure (CERM) and the convex entropic risk measure (ERM). Starting with the portfolio selection between a risky and a risk free asset (framework of Arrow (1965) and Pratt (1964)), we find a restrictive all-or-nothing investment decision under the CERM, while the ERM yields diversification. We then address a portfolio problem with two risky assets, and provide comparative statics with respect to the investor’s risk aversion (framework of Ross (1981)). Here, both the CERM and the ERM exhibit closely interrelated inconsistencies with respect to the interpretation of their risk parameters as a measure of risk aversion: for any two investors with different risk parameters, it may happen that the investor with the higher risk parameter invests more in the riskier one of the two assets. Finally, we analyze the portfolio problem “risky vs. risk free” in the presence of an independent background risk, and analyze the effect of changes in this background risk (framework of Gollier and Pratt (1996)). Again, we find questionable predictions: under the CERM, the optimal risky investment is always increasing instead of decreasing when a background risk is introduced, while under the ERM it remains unaffected.
- Published
- 2018
35. Minimizing Risk Exposure When the Choice of a Risk Measure Is Ambiguous
- Author
-
Jonathan Yu-Meng Li and Erick Delage
- Subjects
050208 finance ,021103 operations research ,Actuarial science ,Strategy and Management ,Risk measure ,05 social sciences ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Entropic value at risk ,Dynamic risk measure ,Expected shortfall ,Spectral risk measure ,Time consistency ,0502 economics and business ,Coherent risk measure ,Economics ,Distortion risk measure - Abstract
Since the financial crisis of 2007–2009, there has been a renewed interest in quantifying more appropriately the risks involved in financial positions. Popular risk measures such as variance and value-at-risk have been found inadequate because we now give more importance to properties such as monotonicity, convexity, translation invariance, positive homogeneity, and law invariance. Unfortunately, the challenge remains that it is unclear how to choose a risk measure that faithfully represents a decision maker’s true risk attitude. In this work, we show that one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions. We assume that the decision maker can commit to a subset of the above properties (the use of a law invariant convex risk measure for example) and that he can provide a series of assessments comparing pairs of potential risky payoffs. Given this information, we propose to seek financial positions that perform best with respect to the most pessimistic estimation of the level of risk potentially perceived by the decision maker. We present how this preference robust risk minimization problem can be solved numerically by formulating convex optimization problems of reasonable size. Numerical experiments on a portfolio selection problem, where the problem reduces to a linear program, will illustrate the advantages of accounting for the fact that the choice of a risk measure is ambiguous. This paper was accepted by Yinyu Ye, optimization.
- Published
- 2018
36. Set-valued risk statistics with scenario analysis
- Author
-
Yijun Hu and Yanhong Chen
- Subjects
Statistics and Probability ,050208 finance ,05 social sciences ,Nonparametric statistics ,Regular polygon ,Extension (predicate logic) ,Entropic value at risk ,01 natural sciences ,Convexity ,Dynamic risk measure ,Set (abstract data type) ,010104 statistics & probability ,0502 economics and business ,Statistics ,Econometrics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Abstract
In this paper, we introduce two new classes of risk statistics, named set-valued coherent and convex risk statistics. These new risk statistics can be considered as a kind of set-valued extension of risk statistics introduced by Kou, Peng and Heyde (2013), and also empirical versions of set-valued coherent and convex risk measures introduced by Jouini, Meddeb and Touzi (2004) and Hamel (2009), respectively. Representation results for these new introduced risk statistics are provided. Finally, we also provide some examples of set-valued coherent and convex risk statistics.
- Published
- 2017
37. Time consistency for set-valued dynamic risk measures for bounded discrete-time processes
- Author
-
Yanhong Chen and Yijun Hu
- Subjects
Statistics and Probability ,Mathematical optimization ,050208 finance ,Mathematical finance ,05 social sciences ,01 natural sciences ,Dynamic risk measure ,Set (abstract data type) ,010104 statistics & probability ,Discrete time and continuous time ,Time consistency ,Bounded function ,0502 economics and business ,Filtration (mathematics) ,Product topology ,0101 mathematics ,Statistics, Probability and Uncertainty ,Finance ,Mathematics - Abstract
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.
- Published
- 2017
38. Measuring systemic risk of the US banking sector in time-frequency domain
- Author
-
Ivana Kvapilikova and Petr Teply
- Subjects
Economics and Econometrics ,050208 finance ,Actuarial science ,05 social sciences ,Tail dependence ,Dynamic risk measure ,Expected shortfall ,Wavelet ,0502 economics and business ,Economics ,Systemic risk ,050207 economics ,Volatility (finance) ,Finance ,Stock (geology) ,Panel data - Abstract
To estimate short-term, medium-term, and long-term financial connectedness, we propose a frequency-based approach and measure the contribution of individual financial institutions to overall systemic risk. We derive Wavelet Conditional Value at Risk (WCoVaR) – a robust market-based measure of systemic risk across financial cycles of differing length. We evaluate the systemic importance of financial institutions based on their stock returns and use wavelet framework to analyze returns in a time-frequency domain. Empirical analysis on US banking sector data between 2004 and 2013 demonstrates that wavelet decomposition can improve the forecast power of the CoVaR measure. We use panel regression to explain systemic importance of individual banks, using their objectively measurable characteristics and conclude that size, volatility and value-at-risk are the most robust determinants of systemic risk.
- Published
- 2017
39. Dynamic no-good-deal pricing measures and extension theorems for linear operators on L.
- Author
-
Bion-Nadal, Jocelyne and Nunno, Giulia
- Subjects
LINEAR operators ,PRICING ,MATHEMATICS theorems ,DISCRETE-time systems ,MATHEMATICAL bounds ,ECONOMIC models - Abstract
In an L-framework, we present majorant-preserving and sandwich-preserving extension theorems for linear operators. These results are then applied to price systems derived by a reasonable restriction of the class of applicable equivalent martingale measures. Our results prove the existence of a no-good-deal pricing measure for price systems consistent with bounds on the Sharpe ratio. We treat both discrete- and continuous-time market models. Within this study we present definitions of no-good-deal pricing measures that are equivalent to the existing ones and extend them to discrete-time models. We introduce the corresponding version of dynamic no-good-deal pricing measures in the continuous-time setting. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
40. Dynamically consistent nonlinear evaluations with their generating functions in L.
- Author
-
Hu, Feng
- Subjects
- *
NONLINEAR analysis , *GENERATING functions , *PRESUPPOSITION (Logic) , *EXISTENCE theorems , *MATHEMATICAL analysis , *FUNCTIONAL analysis - Abstract
In this paper, we study dynamically consistent nonlinear evaluations in L (1 < p < 2). One of our aim is to obtain the following result: under a domination condition, an F-consistent evaluation is an E-evaluation in L. Furthermore, without the assumption that the generating function g ( t, ω, y, z) is continuous with respect to t, we provide some useful characterizations of an E-evaluation by g and give some applications. These results include and extend some existing results. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
41. Backward Stochastic Difference Equations for a Single Jump Process.
- Author
-
Shen, Leo and Elliott, Robert
- Subjects
STOCHASTIC difference equations ,JUMP processes ,MATHEMATICAL proofs ,UNIQUENESS (Mathematics) ,EXISTENCE theorems ,NONLINEAR theories ,MARKOV processes - Abstract
We define Backward Stochastic Difference Equations related to a discrete finite time single jump process. We prove the existence and uniqueness of solutions under some assumptions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated. In this paper the single jump process takes values in a general measurable space where as previous work has considered the situation where the noise is a finite state Markov chain, so the state space is finite. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
42. How to value risk
- Author
-
Shen, Leo and Elliott, Robert J.
- Subjects
- *
VALUE at risk , *RISK management in business , *STOCHASTIC difference equations , *JUMP processes , *NUMERICAL analysis - Abstract
Abstract: We review various risk measures which have been introduced. By considering backward stochastic difference equations related to a single jump process, we define some risk measures related to the solutions. Some simple numerical examples are given. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
43. Backward Stochastic Differential Equations for a Single Jump Process.
- Author
-
Shen, Leo and Elliott, Robert J.
- Subjects
- *
NUMERICAL solutions to stochastic differential equations , *NONLINEAR theories , *MATHEMATICAL proofs , *LIPSCHITZ spaces , *FINITE fields , *EXISTENCE theorems - Abstract
We consider backward stochastic differential equations (BSDEs) related to a finite continuous time single jump process. We prove the existence and uniqueness of solutions when the coefficients satisfy Lipschitz continuity conditions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
44. VALUATIONS AND DYNAMIC CONVEX RISK MEASURES.
- Author
-
Jobert, A. and Rogers, L. C. G.
- Subjects
CONCAVE functions ,VALUATION ,AXIOMS ,RISK ,SUBSIDIARY corporations ,SPREAD (Finance) - Abstract
This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-consistency properties for a firm seeking to spread its risk across a group of subsidiaries. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
45. Continuous-time dynamic risk measures by backward stochastic Volterra integral equations.
- Author
-
Jiongmin Yong
- Subjects
- *
STOCHASTIC processes , *VOLTERRA equations , *INTEGRAL equations , *FUNCTIONAL equations , *FUNCTIONAL differential equations - Abstract
Continuous-time dynamic convex and coherent risk measures are introduced. To obtain existence of such risk measures, backward stochastic Volterra integral equations (BSVIEs, for short) are studied. For such equations, notion of adapted M-solution is introduced, well-posedness is established, duality principles and comparison theorems are presented. Then a class of dynamic convex and coherent risk measures are identified as a component of the adapted M-solutions to certain BSVIEs. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
46. The Analysis of Portfolio Risk Management using VAR Approach Based on Investor Risk Preference
- Author
-
Putu Anom Mahadwartha and Agus Suwarno
- Subjects
Dynamic risk measure ,Expected shortfall ,Actuarial science ,Spectral risk measure ,Single-index model ,Risk premium ,Financial risk ,Portfolio ,Business ,Value at risk - Abstract
Ackert and Deaves (2010) said that most people have tendency to being risk averse, but with appropriate amount of compensation, people may take more risk. Understanding those circumstances, this research trying to figure risk involved in a Mean-Variance Model. This model has taken consideration about investor risk preference in composed VAR model. VAR define as a measure of the risk of investments, which in this research focuses on risk preferences. This research also conducts comparison between optimum portfolio model known as Single Index Model and Mean-Variance Mode. Robustness test taken too analyze the outcomes from different data input. Research showed that risk preference has an impact on generating portfolio based on Mean-Variance Mode (MVM). Meanwhile, Single Index Model (SIM) found to given a similar result as MVM in high risk preference. This has shown that SIM may not adequate for those who have low risk preference. Research also show that risk taker investor get more gain and endure more risk than risk averse investor. But, based on robustness test, we found that the lowest risk an investor bear is on the highest risk preference. Thus, we make a conclusion that variance is not the only factor that might cause VaR increased, data dispersion has became more major factor.Keywords: Value at risk, Single Index Model, Optimum Portfolio.
- Published
- 2017
47. Statistical Inference for a Relative Risk Measure
- Author
-
Jiliang Sheng, Yi He, Yanxi Hou, and Liang Peng
- Subjects
Statistics and Probability ,Deviation risk measure ,Economics and Econometrics ,05 social sciences ,Nonparametric statistics ,Asymptotic distribution ,01 natural sciences ,Copula (probability theory) ,Dynamic risk measure ,010104 statistics & probability ,Expected shortfall ,0502 economics and business ,Statistics ,Econometrics ,Statistical inference ,Systemic risk ,0101 mathematics ,Statistics, Probability and Uncertainty ,Social Sciences (miscellaneous) ,050205 econometrics ,Mathematics - Abstract
For monitoring systemic risk from regulators’ point of view, this article proposes a relative risk measure, which is sensitive to the market comovement. The asymptotic normality of a nonparametric ...
- Published
- 2017
48. An application of extreme value theory in estimating liquidity risk
- Author
-
C Martin, Sonia Benito Muela, and Raquel Arguedas Sanz
- Subjects
Economics and Econometrics ,Strategy and Management ,Risk-adjusted return on capital ,lcsh:Business ,Dynamic risk measure ,Value-at-risk ,ddc:650 ,0502 economics and business ,Economics ,G32 ,C14 ,Basel capital accord ,Business and International Management ,C53 ,050205 econometrics ,Marketing ,050208 finance ,Actuarial science ,05 social sciences ,Extreme value theory ,Liquidity crisis ,Financial risk management ,Liquidity risk ,Liquidity premium ,Expected shortfall ,lcsh:HF5001-6182 ,C22 ,Value at risk - Abstract
The last global financial crisis (2007–2008) has highlighted the weaknesses of value at risk (VaR) as a measure of market risk, as this metric by itself does not take liquidity risk into account. To address this problem, the academic literature has proposed incorporating liquidity risk into estimations of market risk by adding the VaR of the spread to the risk price. The parametric model is the standard approach used to estimate liquidity risk. As this approach does not generate reliable VaR estimates, we propose estimating liquidity risk using more sophisticated models based on extreme value theory (EVT). We find that the approach based on conditional extreme value theory outperforms the standard approach in terms of accurate VaR estimates and the market risk capital requirements of the Basel Capital Accord.
- Published
- 2017
49. On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation
- Author
-
Martijn Pistorius, Dilip Madan, and Mitja Stadje
- Subjects
ASSET RETURNS ,Social Sciences ,01 natural sciences ,Dynamic risk measure ,010104 statistics & probability ,Spectral risk measure ,JUMPS ,Business & Economics ,0102 Applied Mathematics ,CONVERGENCE ,Limit of a sequence ,Mathematics ,UTILITY ,050208 finance ,CONTINUOUS-TIME ,0104 Statistics ,05 social sciences ,Poisson random measure ,Random walk ,Expected shortfall ,Iterated function ,Risk Management (q-fin.RM) ,Physical Sciences ,60H10, 60Fxx ,Limit theorem ,Dynamic portfolio optimisation ,Statistics, Probability and Uncertainty ,Mathematical Methods In Social Sciences ,Mathematics - Probability ,Quantitative Finance - Risk Management ,Mathematics, Interdisciplinary Applications ,Statistics and Probability ,Mathematical optimization ,Statistics & Probability ,Choquet expectation ,g-expectation ,NONLINEAR EXPECTATIONS ,STOCHASTIC DIFFERENCE-EQUATIONS ,CONSISTENCY ,FOS: Economics and business ,0502 economics and business ,FOS: Mathematics ,COHERENT RISK ,0101 mathematics ,(Strong) Time-consistency ,Science & Technology ,DISCRETE-TIME ,Mathematical finance ,Probability (math.PR) ,Distortion ,Social Sciences, Mathematical Methods ,Business, Finance ,Finance - Abstract
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 time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR., Comment: To appear in Finance and Stochastics
- Published
- 2017
50. On Coherent Risk Measures Induced by Convex Risk Measures
- Author
-
Zhiping Chen and Qianhui Hu
- Subjects
Statistics and Probability ,Mathematical optimization ,050208 finance ,021103 operations research ,CVAR ,General Mathematics ,05 social sciences ,0211 other engineering and technologies ,02 engineering and technology ,Entropic value at risk ,Measure (mathematics) ,Dynamic risk measure ,Spectral risk measure ,0502 economics and business ,Coherent risk measure ,Distortion risk measure ,Mathematics ,Entropic risk measure - Abstract
We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.
- Published
- 2017
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.