Coherent Risk Measures Research Articles

On the link between monetary and star-shaped risk measures

Recently, Castagnoli et al. (2021) introduced the class of star-shaped risk measures as a generalization of convex and coherent ones, proving that there is a representation as the pointwise minimum of some family composed by convex risk measures. Concomitantly, Jia et al. (2020) prove a similar representation result for monetary risk measures, which are more general than star-shaped ones. Then, there is a question on how both classes are connected. In this letter, we provide an answer by casting light on the importance of the acceptability of 0, which is linked to the property of normalization. We then show that under mild conditions, a monetary risk measure is only a translation away from star-shapedness.

Quantification of risk in classical models of finance

This paper treats optimal control problems and derivative pricing with regard to fixed levels of risk. We employ nested risk measures to quantify risk, investigate the limiting behavior of nested risk measures within the classical models in finance and characterize existence of the risk-averse limit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen risk measure. Within the classical models, risk aversion gives rise to a stream of risk premiums comparable to dividend payments. In this context we connect coherent risk measures with the Sharpe ratio from modern portfolio theory and extract the Z-spread— a widely accepted quantity in economics for hedging risk. The results for European option pricing are extended to risk-averse American options, to study the impact of risk on the price and optimal time to exercise such options. We also extend Merton's optimal consumption problem to the risk-averse setting.

Acceptability maximization

<p style='text-indent:20px;'>The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and the tail-value-at-risk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem, similar to the classical mean-variance criteria. To overcome this form of time-inconsistency, we adapt to our setup the set-valued Bellman's principle recently proposed in [<xref ref-type="bibr" rid="b23">23</xref>] applied to two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate mean-risk efficient frontiers.</p>

Laws of Large Numbers for Dynamic Coherent Risk Measures

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

Robust policy selection and harvest risk quantification for natural resources management under model uncertainty

<p style='text-indent:20px;'>In this work the problem of optimal harvesting policy selection for natural resources management under model uncertainty is investigated. Under the framework of the neoclassical growth model dynamics, the associated optimal control problem is investigated by introducing the concept of model uncertainty on the initial conditions of the operational procedure. At this stage, the notion of convex risk measures, and in particular the class of Fréchet risk measures, is employed in order to quantify the total operational and marginal risk, whereas simultaneously obtaining robust to model uncertainty harvesting strategies.</p>

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.

From Regularization to Risk-Sensitivity–and Back Again

We explore (h,f)-divergence regularized RL objectives (as a generalization of KL regularized and maximum entropy objectives) through the lens of risk sensitivity and offer two iterative schemes for approaching such regularized objectives. This allows for (I) understanding of RL algorithms through the lens of risk optimization which in turn provides a more cohesive view of well-known RL algorithms and (II) introduction of theoretically well-motivated regularization terms that lead to risk-sensitive RL algorithms. We offer two iterative frameworks for using (h,f)-divergence-based Convex risk measures to facilitate further algorithmic development in risk-sensitive RL.

A Game Theoretic Approach on an Optimal Investment-Consumption-Insurance Strategy

We find the possible risk minimizing portfolio strategies in a two dimensional market consisting of a risk asset and risk-less asset. The investor in the market is subjected to consumption, purchasing of life insurance and stochastic income with inflation risk. The problem is formulated as zero sum game problem between the market and the investor. The strategies are determined for the different generations of the life of an investor, that is before the investor dies and after the investor dies. We used the concept of convex risk measures and monetary utility maximizing problem-concept studied before finding the risk minimizing portfolios which was solved using the game theoretic approach to obtain the strategies explicitly given in the propositions in the study.

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.

Optimal Auctions: Non-expected Utility and Constant Risk Aversion

Abstract We study auction design for bidders equipped with non-expected utility preferences that exhibit constant risk aversion (CRA). The CRA class is large and includes loss-averse, disappointment-averse, mean-dispersion, and Yaari’s dual preferences as well as coherent and convex risk measures. Any preference in this class displays first-order risk aversion, contrasting the standard expected utility case which displays second-order risk aversion. The optimal mechanism offers “ full-insurance” in the sense that each agent’s utility is independent of other agents’ reports. The seller excludes less types than under risk neutrality and awards the object randomly to intermediate types. Subjecting intermediate types to a risky allocation while compensating them when losing allows the seller to collect larger payments from higher types. Relatively high types are willing to pay more, and their allocation is efficient.

Existence of solutions for a class of bilevel stochastic linear programs

We study bilevel stochastic linear programs where the upper and lower level goal functions as well as the right-hand side of the follower’s constraint system are affected by randomness. Invoking a robust representation result, we prove that any objective functional derived from a convex risk measure is lower semicontinuous, even if the underlying distribution is not absolutely continuous with respect to the Lebesgue measure. Moreover, we provide a continuity result that also applies to pessimistic models and show that the existence of solutions can be guaranteed under standard compactness assumptions.

Risk-theoretic optimal design of output-feedback controllers via iterative convex relaxations

This work presents a framework to address the MIMO (multiple-input multiple-output) control design problem under model uncertainty. The plant is considered to be LTI (linear and time-invariant) and that it is parametrized by a random variable whose distribution may be unknown but can be sampled from. The objective is to design a dynamic LTI MIMO controller optimizing the closed-loop H2-performance. To account for the uncertainty in the plant, we employ measures of risk on the performance and discuss how coherent measures of risk are suitable to address this problem. In particular, we introduce a measure called Conditional Value-at-Risk (CVaR) and analyze its advantages and disadvantages regarding other traditional measures of risk. A simulation study complements the discussion.

Equal risk pricing and hedging of financial derivatives with convex risk measures

In this paper, we consider the problem of equal risk pricing and hedging in which the fair price of an option is the price that exposes both sides of the contract to the same level of risk. Focusing for the first time on the context where risk is measured according to convex risk measures, we establish that the problem reduces to solving independently the writer and the buyer's hedging problems with zero initial capital. By further imposing that the risk measures decompose in a way that satisfies a Markovian property, we provide dynamic programming equations that can be used to solve the hedging problems for both European and American options. All of our results are general enough to accommodate situations where the risk is measured according to a worst-case risk measure, as is typically done in robust optimization. Our numerical study illustrates the advantages of equal risk pricing over schemes that only account for a single party, pricing based on quadratic hedging (i.e. ϵ-arbitrage pricing), or pricing based on a fixed equivalent martingale measure (i.e. Black–Scholes pricing). In particular, the numerical results confirm that when employing an equal risk price both the writer and the buyer end up being exposed to risks that are more similar and on average smaller than what they would experience with the other approaches.

A pessimistic bilevel stochastic problem for elastic shape optimization

We consider pessimistic bilevel stochastic programs in which the follower maximizes over a fixed compact convex set a strictly convex quadratic function, whose Hessian depends on the leader’s decision. This results in a random upper level outcome which is evaluated by a convex risk measure. Under assumptions including real analyticity of the lower-level goal function, we prove the existence of optimal solutions. We discuss an alternate model, where the leader hedges against optimal lower-level solutions, and show that solvability can be guaranteed under weaker conditions in both, a deterministic and a stochastic setting. The approach is applied to a mechanical shape optimization problem in which the leader decides on an optimal material distribution to minimize a tracking-type cost functional, whereas the follower chooses forces from an admissible set to maximize a compliance objective. The material distribution is considered to be stochastically perturbed in the actual construction phase. Computational results illustrate the bilevel optimization concept and demonstrate the interplay of follower and leader in shape design and testing.

Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations.

Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.

Distributionally Robust Markov Decision Processes and Their Connection to Risk Measures

We consider robust Markov decision processes with Borel state and action spaces, unbounded cost, and finite time horizon. Our formulation leads to a Stackelberg game against nature. Under integrability, continuity, and compactness assumptions, we derive a robust cost iteration for a fixed policy of the decision maker and a value iteration for the robust optimization problem. Moreover, we show the existence of deterministic optimal policies for both players. This is in contrast to classical zero-sum games. In case the state space is the real line, we show under some convexity assumptions that the interchange of supremum and infimum is possible with the help of Sion’s minimax theorem. Further, we consider the problem with special ambiguity sets. In particular, we are able to derive some cases where the robust optimization problem coincides with the minimization of a coherent risk measure. In the final section, we discuss two applications: a robust linear-quadratic problem and a robust problem for managing regenerative energy.

Portfolio optimization with optimal expected utility risk measures

The purpose of this article is to evaluate optimal expected utility risk measures (OEU) in a risk-constrained portfolio optimization context where the expected portfolio return is maximized. We compare the portfolio optimization with OEU constraint to a portfolio selection model using value at risk as constraint. The former is a coherent risk measure for utility functions with constant relative risk aversion and allows individual specifications to the investor’s risk attitude and time preference. In a case study with three indices, we investigate how these theoretical differences influence the performance of the portfolio selection strategies. A copula approach with univariate ARMA-GARCH models is used in a rolling forecast to simulate monthly future returns and calculate the derived measures for the optimization. The results of this study illustrate that both optimization strategies perform considerably better than an equally weighted portfolio and a buy and hold portfolio. Moreover, our results illustrate that portfolio optimization with OEU constraint experiences individualized effects, e.g., less risk-averse investors lose more portfolio value in the financial crises but outperform their more risk-averse counterparts in bull markets.

Robustness in the Optimization of Risk Measures

We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk-measurement-related optimization problem is robust, which we call “robustness against optimization.” The new notion is studied for various classes of risk measures and expected utility and loss functions. Motivated by practical issues from !nancial regulation, special attention is given to the twomost widely used riskmeasures in the industry, Valueat-Risk (VaR) and Expected Shortfall (ES).We establish that for a class of general optimization problems, VaR leads to nonrobust optimizers, whereas convex riskmeasures generally lead to robust ones. Our results offer extra insight on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the !eld of robust optimization, to which some interesting links are derived.

Risk-informed participation in T&D markets

Power producers can exhibit strategic behavior in electricity markets to maximize their profits. This behavior is more pronounced with the deregulation of distribution markets, which offers an opportunity for profit arbitrage between transmission and distribution (T&D) markets. However, the temporally distinct nature of these two markets introduces a significant risk in profit for such producers. This paper finds its motivation from the perspective of a strategic producer and develops a Single Leader Multi-Follower (SLMF) game for deriving its participation strategies in T&D markets, while accounting for different T&D coordination schemes based on the individual market Gate Closure Times (GCT). We compare and contrast joint and sequential market-clearing models with regulated and deregulated distribution environments and evaluate the risk of producer by leveraging consistent and coherent risk measures. SLMF game is reformulated as a Mathematical Program with Equilibrium Constraints (MPEC) and is solved using the seminal Scholtes’s relaxation scheme. We validate the efficacy of our model and solution approach via the case study carried out on the 11-zone New York ISO, and 7-bus Manhattan power networks, used as T&D markets, respectively.

Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.