OPTIMAL PORTFOLIOS WITH LOWER PARTIAL MOMENT CONSTRAINTS AND LPM-RISK-OPTIMAL MARTINGALE MEASURES

We optimize the ratio     over an (arbitrage-free) linear sub-space     of attainable returns in an incomplete market model. If a solution exists for  1 < r < ∞  , then the 1st order optimality condition allows to construct an equivalent martingale measure for     , which is shown to be the solution of an appropriate dual minimization problem over the set of all equivalent martingale measures for     . The dual minimization problem admits a solution iff there exists an equivalent martingale measure for     and its optimal value     equals the lowest upper bound     of all α-ratios over     . This new type of non-concave duality also provides an indifference pricing method. The duality result can be extended to the case     and leads to a new no (approximate) arbitrage condition: no great expectations with vanishing risk.     

RISK MEASURES ON ORLICZ HEARTS

Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on L^p -spaces for  1 ≤ p < ∞  and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given.     

RISK MEASURES FOR NON-INTEGRABLE RANDOM VARIABLES

We show that when a real-valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables. As a result we see that a risk measure defined for, say, Cauchy-distributed random variable, must take infinite values for some of the random variables.     

MARTINGALE MEASURES FOR DISCRETE-TIME PROCESSES WITH INFINITE HORIZON

Let ( S_t ) _tεI be an R^d-valued adapted stochastic process on (Ω, , ( _t ) _tεI , P ). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on  equivalent to P such that ( S_t ) _tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics. We introduce the intermediate concept of "no free lunch with bounded risk." This is a somewhat more precise version of the notion of "no free lunch." It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of "no free lunch." We give an argument as to why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process ( S_t ) _tεI of a securities market. We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process ( S_t ) _{t εR+} is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be "almost a martingale" in order to allow an equivalent martingale measure.     

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.     

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.     

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.     

INDIFFERENCE PRICE WITH GENERAL SEMIMARTINGALES

Using duality methods, we prove several key properties of the indifference price π for contingent claims. The underlying market model is very general and the mathematical formulation is based on a duality naturally induced by the problem. In particular, the indifference price π turns out to be a convex risk measure on the Orlicz space induced by the utility function.     

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.     

A SHORT NOTE ON SECOND-ORDER STOCHASTIC DOMINANCE PRESERVING COHERENT RISK MEASURES

We improve results on law invariant coherent risk measures satisfying the Fatou property due to Kusuoka (2001; Adv. Math. Econ . 3, 83–95) by considering risk measures which are in addition second order stochastic dominance preserving. In particular, we derive a representation result for such risk measures.     

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.     

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.     

RISK MEASURES: RATIONALITY AND DIVERSIFICATION

When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash‐additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash‐subadditivity for example, the equivalence between convexity and the diversification principle no longer holds. In fact, this principle only implies (and it is implied by) quasiconvexity. For this reason, in this paper quasiconvex risk measures are studied. We provide a dual characterization of quasiconvex cash‐subadditive risk measures and we establish necessary and sufficient conditions for their law invariance. As a byproduct, we obtain an alternative characterization of the actuarial mean value premium principle.     

OVERLAPPING SETS OF PRIORS AND THE EXISTENCE OF EFFICIENT ALLOCATIONS AND EQUILIBRIA FOR RISK MEASURES

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short‐selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no‐arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow–Debreu equilibria.     

CLOSED-FORM SOLUTIONS FOR OPTIMAL PORTFOLIO SELECTION WITH STOCHASTIC INTEREST RATE AND INVESTMENT CONSTRAINTS

We examine the portfolio choice problem of an investor with constant relative risk aversion in a financial market with partially hedgeable interest rate risk. The individual shadow price of the portfolio constraint is characterized as the solution of a new backward equation involving Malliavin derivatives. A generalization of this equation is studied and solved in explicit form. This result, applied to our financial model, yields closed-form solutions for the shadow price and the optimal portfolio. The effects of parameters such as risk aversion, interest rate volatility, investment horizon, and tightness of the constraint are examined. Applications of our method to a monetary economy with inflation risk and to an international setting with currency risk are also provided.     

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.     

金融监管新模式下合理风险度量的条件与结构框架

在回顾近年来风险度量的公理化研究及进展的基础上,本文提出和论证了计量资本要求的合理风险度量应满足的几个条件,并从风险度量的经济理论基础出发,引入一个可以生成满足这些条件的风险度量模型的结构框架。