Class Of Convex Risk Measures Research Articles

Adjusted higher-order expected shortfall

How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, referred to as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) higher-order Expected Shortfall is also given.

A Reverse ES (CVaR) Optimization Formula

The celebrated Expected Shortfall (ES, also known as tail Value at Risk or conditional Value at Risk) optimization formula implies that ES at a fixed probability level is the minimum of a linear real function plus a scaled mean excess function. We establish a reverse ES optimization formula that says that a mean excess function at any fixed threshold is the maximum of an ES curve minus a linear function. Despite being a simple result, this formula reveals elegant symmetries between the mean excess function and the ES curve, as well as their optimizers. The reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. We analyze worst-case values of the mean excess function under two popular settings of model uncertainty to illustrate the usefulness of the reverse ES optimization formula, and this is further demonstrated with an application using insurance datasets.

Deep Equal Risk Pricing of Financial Derivatives with Non-Translation Invariant Risk Measures

The objective is to study the use of non-translation invariant risk measures within the equal risk pricing (ERP) methodology for the valuation of financial derivatives. The ability to move beyond the class of convex risk measures considered in several prior studies provides more flexibility within the pricing scheme. In particular, suitable choices for the risk measure embedded in the ERP framework, such as the semi-mean-square-error (SMSE), are shown herein to alleviate the price inflation phenomenon observed under the tail value at risk-based ERP as documented in previous work. The numerical implementation of non-translation invariant ERP is performed through deep reinforcement learning, where a slight modification is applied to the conventional deep hedging training algorithm so as to enable obtaining a price through a single training run for the two neural networks associated with the respective long and short hedging strategies. The accuracy of the neural network training procedure is shown in simulation experiments not to be materially impacted by such modification of the training algorithm.

Adjusted Rényi entropic Value-at-Risk

Entropy is a measure of self information or uncertainty. Using different concepts of entropy, we may get different risk measures by dual representation. In this paper, we introduce and study the main properties of a class of convex risk measures, called as adjusted Rényi entropic Value-at-Risk (VaR). The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its Rényi entropic-VaR does not exceed a pre-specified threshold for every probability level. When p∈[1,∞), the adjusted Rényi entropic-VaR of order p is intimately linked to the (p+1)-increasing convex order by choosing the risk threshold to be the Rényi entropic-VaR of a benchmark random loss.

Adjusted Expected Shortfall

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ES p ( X ) does not exceed a pre-specified threshold g ( p ) for every probability level p ∈ [ 0 , 1 ] . Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.

A Reverse Expected Shortfall Optimization Formula

The celebrated Expected Shortfall (ES) optimization formula implies that ES at a fixed probability level is the minimum of a linear real function plus a scaled mean excess function. We establish a reverse ES optimization formula, which says that a mean excess function at any fixed threshold is the maximum of an ES curve minus a linear function. Despite being a simple result, this formula reveals elegant symmetries between the mean excess function and the ES curve, as well as their optimizers. The reverse ES optimization formula is closely related to the Fenchel-Legendre transforms, and our formulas are generalized from ES to optimized certainty equivalents, a popular class of convex risk measures. We analyze worst-case values of the mean excess function under two popular settings of model uncertainty to illustrate the usefulness of the reverse ES optimization formula, and this is further demonstrated with an application using insurance datasets.

Adjusted Expected Shortfall

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp(X) does not exceed a pre-specified threshold g(p) for every probability level p\in[0,1]. Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.

Utility‐based shortfall risk: Efficient computations via Monte Carlo

AbstractWith the development of financial risk management, the notion of convex risk measures has been proposed and has gained increasing attentions. Utility‐based shortfall risk (SR), as a specific and important class of convex risk measures, has become popular in recent years. In this paper we focus on the computational aspects of SR, which are significantly understudied but fundamental for risk assessment and management. We discuss efficient estimation, optimization, and sensitivity analysis of SR, based on Monte Carlo techniques and stochastic optimization methods. We also conduct extensive numerical studies on the proposed approaches. The numerical results further demonstrate the effectiveness of these approaches.

Convex risk measures for the aggregation of multiple information sources and applications in insurance

We propose a novel class of convex risk measures, based on the concept of the Fréchet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty.

Convex Risk Measures for the Aggregation of Multiple Information Sources and Applications in Insurance

We consider the concept of convex risk measures under the presence of ambiguity regarding the probability model of the risk. A number of plausible probability models, possibly diverging, are provided to the decision maker by various information sources or experts, however it is not a priori known which of these sources are valid. We employ the concept of Frechet mean in order to define a new class of convex risk measures, the Frechet risk measures, which are appropriate for this framework, and provide closed form expressions for them which clarify the interaction between risk and uncertainty. We also investigate the robustness of these measures after applying them in some standard applications, such as (a) the total claim amount estimation and (b) the selection of the static hedging strategy for contingent claims, both of these applications considered under the concept of uncertainty. The proposed risk measures are shown to be very effective in aggregating models in the set of priors, thus filtering out the effects of uncertainty and leading to robust decision making.

Convex Risk Measures: Efficient Computations Via Monte Carlo

With the development of financial risk management, the notion of convex risk measures has been proposed and has gained more and more attentions. Utility-based shortfall risk (SR), as a specific and important class of convex risk measures, has become popular in recent years. In this paper we focus on computational aspects of SR, which are significantly understudied but fundamental for risk assessment and management. We discuss efficient estimation of SR, sensitivity analysis of SR, as well as optimization of SR, based on Monte Carlo techniques and stochastic optimization methods. We also conduct extensive numerical study on the proposed approaches. The numerical results further demonstrate the effectiveness of these approaches.

Shortfall Risk Minimization in Discrete Time Financial Market Models

In this paper, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined on $L^{p}(\mathbb{P})$ in terms of shortfall risk. Under mild assumptions, namely, the absence of arbitrage opportunity and the nondegeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. In a Markovian framework, the shortfall risk and optimal dynamic strategies are estimated using three main tools: Newton--Raphson Monte Carlo--based procedure, stochastic approximation algorithm, and Markovian quantization scheme. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets.

The restricted convex risk measures in actuarial solvency

In this article, we propose a class of convex risk measures defined on appropriate wedges of a space of financial positions which denote the cumulative surplus variables created by undertaking risks by either an insurance or a reinsurance company. The form of the wedge which is the domain of such a risk measure expresses the form of the company, and it is a subspace in the case of reinsurance companies and a cone in the case of the insurance companies. The value of such a risk measure on an insurance position denotes the capital that the corresponding company has to receive or to keep in advance so that it will not be exposed to risk due to this position. We prove some dual representation and continuity results being similar to the unrestricted case. Finally, we contribute to a decision theory related to the choice of a numeraire asset when the space in which the positions lie in is reflexive.

Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled (‘tilted’) one-period convex risk measures, using a d -dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete ‘driver’ characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.

Utility maximization under a shortfall risk constraint

The article analyzes optimal portfolio choice of utility maximizing agents in a general continuous-time financial market model under a joint budget and downside risk constraint. The risk constraint is given in terms of a class of convex risk measures. We do not impose any specific assumptions on the price processes of the underlying assets. We analyze under which circumstances the risk constraint is binding. We provide a closed-form solution to the optimization problem in a general semimartingale framework. For a complete market, the wealth maximization problem is equivalent to a dynamic portfolio optimization problem.

Large deviations bounds for estimating conditional value-at-risk

In this paper, we prove an exponential rate of convergence result for a common estimator of conditional value-at-risk for bounded random variables. The bound on optimistic deviations is tighter while the bound on pessimistic deviations is more general and applies to a broader class of convex risk measures.