Dynamic Risk Measures Research Articles

Quantile Diffusions

We develop a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: the first, which we largely focus on, features a dynamic random quantile level and allows for direct interpretation of the resulting quantile process characteristics such as location, scale, skewness and kurtosis, in terms of the model parameters. The second type are function-valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. By the proposed innovative and simple -- yet powerful -- construction method, quantile processes are obtained by transforming the marginals of a diffusion process under a composite map consisting of a distribution and a quantile function. Such maps, analogous to rank transmutation maps, produce the marginals of the resulting quantile process. We discuss the relationship and differences between our approach and existing methods and characterisations of quantile processes in discrete and continuous time. As an example of an application of quantile diffusions, we show how probability measure distortions, a form of dynamic tilting, can be induced. Though particularly useful in financial mathematics and actuarial science, examples of which are given in this work, measure distortions feature prominently across multiple research areas. For instance, dynamic distributional approximations (statistics), non-parametric and asymptotic analysis (mathematical statistics), dynamic risk measures (econometrics), behavioural economics, decision making (operations research), signal processing (information theory), and not least in general risk theory including applications thereof, for example in the context of climate change.

Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

For an $\cF_T$-measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an $\cF_T$-measurable stochastic process (called a position process, not necessarily $\dbF$-adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in $Z(t,s)$). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed.

Mean Field BSDEs and Global Dynamic Risk Measures

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.

Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Let (Ω,F,F,P) be a filtered probability space with a filtration F=(Ft)t∈[0,T] satisfying the usual conditions and T a finite time, L0(F0) the algebra of equivalence classes of F0–measurable real–valued random variables on Ω, Lp(FT,Rd) the usual function space and LF0p(FT,Rd) the L0(F0)–module generated by Lp(FT,Rd). The usual backward stochastic equations (BSEs) are studied for their terminal conditions ξ in Lp(FT,Rd). Motivated by the study of continuous–time conditional mean–conditional convex risk measure portfolio selection, this paper, for the first time, formulates and studies a more general class of BSEs with their terminal conditions in LF0p(FT,Rd). The main results of this paper are to prove two fixed point theorems in complete random normed modules, which are respectively the random generalizations of Banach contraction mapping principle and Browder–Kirk fixed point theorem, and give their applications to the general class of BSEs.

Efficient Nested Simulation for Conditional Tail Expectation of Variable Annuities

Monte Carlo simulations—in particular, nested Monte Carlo simulations—are commonly used in variable annuity (VA) risk modeling. However, the computational burden associated with nested simulations is substantial. We propose an Importance-Allocated Nested Simulation (IANS) method to reduce the computational burden, using a two-stage process. The first stage uses a low-cost analytic proxy to identify the tail scenarios most likely to contribute to the Conditional Tail Expectation risk measure. In the second stage we allocate the entire inner simulation computational budget to the scenarios identified in the first stage. Our numerical experiments show that, in the VA context, IANS can be up to 30 times more efficient than a standard Monte Carlo experiment, measured by relative mean squared errors, when both are given the same computational budget.

The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization

The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. So far, it has only been applied to set-valued dynamic risk measures within a stochastic framework, but not to deterministic calculus of variations and optimal control problems. In this paper, a multi-objective calculus of variations problem is considered which is turned into a set-valued problem by a straightforward extension. A new set-valued value function is introduced, for which a Bellman's optimality principle holds. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time and a directional derivative with respect to the state variable of the value function are defined. The value function is proved to be a solution of a corresponding Hamilton-Jacobi equation.

Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures

In this paper we extend to a dynamic setting the robust Orlicz premia and Haezendonck–Goovaerts risk measures introduced in Bellini, Laeven and Rosazza Gianin (2018). We extensively analyze the properties of the resulting dynamic risk measures. Furthermore, we characterize dynamic Orlicz premia that are time-consistent, and establish some relations between the time-consistency properties of dynamic robust Orlicz premia and the corresponding dynamic robust Haezendonck–Goovaerts risk measures.

Pricing under dynamic risk measures

Abstract In this paper, we study the discrete-time super-replication problem of contingent claims with respect to an acceptable terminal discounted cash flow. Based on the concept of Immediate Profit, i.e., a negative price which super-replicates the zero contingent claim, we establish a weak version of the fundamental theorem of asset pricing. Moreover, time consistency is discussed and we obtain a representation formula for the minimal super-hedging prices of bounded contingent claims.

Optimal execution with dynamic risk adjustment

This article considers the problem of optimal liquidation of a position in a risky security quoted in a financial market, where price evolution are risky and trades have an impact on price as well as uncertainty in the filling orders. The problem is formulated as a continuous time stochastic optimal control problem aiming at maximising a generalised risk-adjusted profit and loss function. The expression of the risk adjustment is derived from the general theory of dynamic risk measures and is selected in the class of g-conditional risk measures. The resulting theoretical framework is nonclassical since the target function depends on backward components. We show that, under a quadratic specification of the driver of a backward stochastic differential equation, it is possible to find a closed form solution and an explicit expression of the optimal liquidation policies. In this way, it is immediate to quantify the impact of risk adjustment on the profit and loss and on the expression of the optimal liquidation policies.

$$l_1$$-Regularization for multi-period portfolio selection

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.

A Framework for Time-Consistent, Risk-Sensitive Model Predictive Control: Theory and Algorithms

In this paper, we present a framework for risk-sensitive model predictive control (MPC) of linear systems affected by stochastic multiplicative uncertainty. Our key innovation is to consider a time-consistent, dynamic risk evaluation of the cumulative cost as the objective function to be minimized. This framework is axiomatically justified in terms of time-consistency of risk assessments, is amenable to dynamic optimization, and is unifying in the sense that it captures a full range of risk preferences from risk neutral (i.e., expectation) to worst case. Within this framework, we propose and analyze an online risk-sensitive MPC algorithm that is provably stabilizing. Furthermore, by exploiting the dual representation of time-consistent, dynamic risk measures, we cast the computation of the MPC control law as a convex optimization problem amenable to real-time implementation. Simulation results are presented and discussed.

Conditional submodular Choquet expected values and conditional coherent risk measures

We provide an axiomatic definition of conditional submodular capacity that allows conditioning on “null” events and is the basis for the notions of consistency and of consistent extension of a partial assessment. The same definition gives rise to an axiomatic definition of conditional submodular Choquet expected value, which is a conditional functional defined on conditional gambles, that can be expressed as the Choquet integral with respect to its restriction on conditional indicators. Finally, the notion of conditional submodular Choquet expected value is used to provide a definition of conditional submodular coherent risk measure that, locally on every conditioning event, has an upper expected loss interpretation.

Conditional excess risk measures and multivariate regular variation

Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.

Asymptotics of multivariate conditional risk measures for Gaussian risks

This paper investigates accurate approximations of marginal moment excess, marginal conditional tail moment and marginal moment shortfall for multivariate Gaussian system risks. Based on the dimension reduction property via the quadratic programming problem, the super-exponential and polynomial convergence speeds are specified. Two interesting questions involved in risk management are well addressed, namely the minimal additional risk capital injection to avoid infinite risk contagion and a sufficient and necessary condition to alternate the convergence speeds. Numerical study and typical examples are given to illustrate the efficiency of our findings. Due to the flexible moment order, additional applications may involve in risk management, including tail mean–variance portfolio and multivariate conditional risk measures of tail covariance, tail skewness with dependence and extremal risk contagion under consideration.

The Impact of Outliers on Computing Conditional Risk Measures for Crude Oil and Natural Gas Commodity Futures Prices

In this study, we aim to build better risk models for energy commodities by employing statistical procedures to identify outliers in the prices for all crude oil and natural gas futures contracts traded on the CME over the period of December 2003 through March 2017. Our results show that it is important to investigate and control for potential outlier effects when performing parametric estimation of risk parameters because outliers can have a large impact on the estimation of value at risk. We illustrate for actual crude oil and natural gas contract how the use of value at risk metrics based on raw data can lead to higher than expected actual losses. Our research demonstrates that it is crucial to include intervention parameters to address outlier impacts in order to obtain robust risk metrics.

Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.

Dynamic Systemic Risk Measures for Bounded Discrete-Time Processes

The question of measuring and managing systemic risk - especially in view of the recent financial crises - 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. (2013) and extended in Kromer et al. (2014). The aim of this paper is to generalize the static approach in Kromer et al. (2014) and analyze systemic risk measures in a dynamic setting. We work in the framework of Cheridito et al. (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.

Dynamic systemic risk measures for bounded discrete time processes

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.

Establishment of safety structure theory

In order to reveal the essence of “safety”, to realize the active identification of hazard and the active prevention and control of accidents, the safety structure theory (SST) is established in this paper. A new definition of “safety” is given from the view of the interaction between activities and environment. And the definition of the instability domain of safety events is given based on the definition of connotation in logic. According to the factor space theory (FST), factors are used to express activities and environment, then the environment factors space cane (EFSC) is developed to “bearing” the states of environment and safety events, and the atomic activity factor space (AAFS) is developed to “bearing” the states of activities. The mapping from EFSC × AAFS to EFSC is established to describe the interaction between the atomic activity factor (AAF) and atomic environment factor (AEF), then the safety structure consisting of AAF, AEF, and the interaction between the two is developed. The evolution mechanism and evolution process of safety issues are analyzed according to the changes of states of safety structures and the relationships between the states of simple environment factors (SEFs) and instability domain of safety events. Then the definition of hazard based on safety structures and the active identification method are given, and the guiding ideology for establishing dynamic risk measurement and prediction models of safety events is given. According to the risk assessment and prediction, a three-stage echelon active safety control mechanism for safety events consisting of advanced feedforward control, feedforward control and feedback control is established. SST is a new safety science theory, which can provide a new analysis method for revealing the essence of “safety” and the evolution mechanism of safety issues, and open up a new way for the development of active safety control.

Dynamic risk measures for processes via backward stochastic differential equations

We provide some time-consistent dynamic convex (resp. coherent) risk measures for processes via backward stochastic differential equations (BSDEs for short), and establish the one-to-one correspondence between the generators of BSDEs and the associated dynamic convex (resp. coherent) risk measures for processes. Furthermore, we show that the dynamic convex (resp. coherent) risk measures for processes via BSDEs coincide with the classical dynamic convex (resp. coherent) risk measures under the framework of Peng’s g-expectations.