DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES

The (subjective) indifference value of a payoff in an incomplete financial market is that monetary amount which leaves an agent indifferent between buying or not buying the payoff when she always optimally exploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, that is, minus a dynamic convex risk measure. For that purpose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional convex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we construct a dynamic risk measure (or utility functional) associated to superreplication in a market with trading constraints and prove that it is time-consistent. By combining these results, we deduce that the corresponding indifference valuation functional is again time-consistent. As an auxiliary tool, we establish a variant of the representation theorem for conditional convex risk measures in terms of equivalent probability measures.     

ASYMPTOTIC EQUIVALENCE OF RISK MEASURES UNDER DEPENDENCE UNCERTAINTY

In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst‐case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst‐case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.     

SCHUR CONVEX FUNCTIONALS: FATOU PROPERTY AND REPRESENTATION

The Fatou property for every Schur convex lower semicontinuous (l.s.c.) functional on a general probability space is established. As a result, the existing quantile representations for Schur convex l.s.c. positively homogeneous convex functionals, established on for either p= 1 or p=∞ and with the requirement of the Fatou property, are generalized for , with no requirement of the Fatou property. In particular, the existing quantile representations for law invariant coherent risk measures and law invariant deviation measures on an atomless probability space are extended for a general probability space.     

AN OLD-NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and puts it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the φ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.     

MODEL UNCERTAINTY AND SCENARIO AGGREGATION

This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value‐at‐risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method.     

RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES

In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

DYNAMIC COHERENT ACCEPTABILITY INDICES AND THEIR APPLICATIONS TO FINANCE

In this paper, we present a theoretical framework for studying coherent acceptability indices (CAIs) in a dynamic setup. We study dynamic CAIs (DCAIs) and dynamic coherent risk measures (DCRMs), and we establish a duality between them. We derive a representation theorem for DCRMs in terms of a so‐called dynamically consistent sequence of sets of probability measures. Based on these results, we give a specific construction of DCAIs. We also provide examples of DCAIs, both abstract and also some that generalize selected classical financial measures of portfolio performance.     

DISTRIBUTION-INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY

In the first part of the paper, we characterize distribution-invariant risk measures with convex acceptance and rejection sets on the level of distributions. It is shown that these risk measures are closely related to utility-based shortfall risk.
In the second part of the paper, we provide an axiomatic characterization for distribution-invariant dynamic risk measures of terminal payments. We prove a representation theorem and investigate the relation to static risk measures. A key insight of the paper is that dynamic consistency and the notion of "measure convex sets of probability measures" are intimately related. This result implies that under weak conditions dynamically consistent dynamic risk measures can be represented by static utility-based shortfall risk.     

MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."     

COHERENCE AND ELICITABILITY

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile‐based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law‐invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.     

CHOQUET INSURANCE PRICING: A CAVEAT

We show that, if prices in a market are Choquet expectations, the existence of one frictionless asset may force the whole market to be frictionless. Any risky asset will cause this collapse if prices depend only on the distribution with respect to a given nonatomic probability measure; the frictionless asset has to be fully revealing if such dependence is not assumed. Similar considerations apply to law-invariant coherent risk measures.     

CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY

A new class of risk measures called cash subadditive risk measures is introduced to assess the risk of future financial, nonfinancial, and insurance positions. The debated cash additive axiom is relaxed into the cash subadditive axiom to preserve the original difference between the numéraire of the current reserve amounts and future positions. Consequently, cash subadditive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented, and in such contexts cash additive risk measures cannot be used. Several representations of the cash subadditive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sublinear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash subadditive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash subadditive risk measure.     

COMONOTONIC MEASURES OF MULTIVARIATE RISKS

We propose a multivariate extension of a well‐known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invariance, subadditivity, and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we reformulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.     

COHERENT ACCEPTABILITY MEASURES IN MULTIPERIOD MODELS

The framework of coherent risk measures has been introduced by Artzner et al. (1999; Math. Finance 9, 203–228) in a single-period setting. Here, we investigate a similar framework in a multiperiod context. We add an axiom of dynamic consistency to the standard coherence axioms, and obtain a representation theorem in terms of collections of multiperiod probability measures that satisfy a certain product property. This theorem is similar to results obtained by Epstein and Schneider (2003; J. Econ. Theor. 113, 1–31) and Wang (2003; J. Econ. Theor. 108, 286–321) in a different axiomatic framework. We then apply our representation result to the pricing of derivatives in incomplete markets, extending results by Carr, Geman, and Madan (2001; J. Financial Econ. 32, 131–167) to the multiperiod case. We present recursive formulas for the computation of price bounds and corresponding optimal hedges. When no shortselling constraints are present, we obtain a recursive formula for price bounds in terms of martingale measures.     

EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL‐REPRESENTATION FORMULA

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·) . Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual‐representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex‐duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρ^L(·) which is continuous from below, and if the European option H is essentially bounded.     

Fundamental Theorems of Asset Pricing for Good Deal Bounds

We prove fundamental theorems of asset pricing for good deal bounds in incomplete markets. These theorems relate arbitrage-freedom and uniqueness of prices for over-the-counter derivatives to existence and uniqueness of a pricing kernel that is consistent with market prices and the acceptance set of good deals. They are proved using duality of convex optimization in locally convex linear topological spaces. The concepts investigated are closely related to convex and coherent risk measures, exact functionals, and coherent lower previsions in the theory of imprecise probabilities.     

MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

This paper deals with multidimensional dynamic risk measures induced by conditional g‐expectations. A notion of multidimensional g‐expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional g‐expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g‐risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi‐monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g‐risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for ‐tolerant g‐risk measures, and risk contribution for coherent g‐risk measures are investigated. Insurance g‐risk measure and other ways to induce g‐risk measures are also studied at the end of the paper.     

SHAREHOLDER RISK MEASURES

The aim of this paper is to put forward a new family of risk measures that could guide investment decisions of private companies. But at the difference of the classical approach of Artzner, Delbaen, Eber, and Heath and the subsequent extensions of this model, our risk measures are built to reflect the risk perception of shareholders rather than regulators. Instead of an axiomatic approach, we derive risk measures from the optimal policies of a shareholder value‐maximizing company. We study these optimal policies and the related risk measures that we call shareholder risk measures. We emphasize the fact that due to the specific corporate environment, in particular the limited shareholders' liability and the possibility to pay out dividends from cash reserves, these risk measures are not convex. Also, they depend on the specific economic situation of the firm, in particular its current cash level, and thus they are not translation invariant. This paper bridges the gap between two important branches of mathematical finance: risk measures and optimal dividends.     

动态货币政策、投资水平与企业价值波动

我国货币政策松紧交替进行,但鲜有文献探讨宽松与紧缩交替的动态货币政策对企业风险、业绩等产出指标的共同影响作用。本文选取2001年至2012年A股上市公司为样本,将此期间划分为宽松与紧缩四个交替的货币政策时期。研究发现：货币政策宽松期（紧缩期）投资水平越高的企业,在货币政策紧缩期（宽松期）经营业绩下降越大（小）,陷入财务危机的可能性越高（低）,企业价值越低（高）。