321 research outputs found
Surplus-invariant risk measures
No full textThis paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some special spaces of random variables. In this paper, we develop a theory of surplus invariance in its natural framework, namely, that of vector lattices. Besides providing a unifying perspective on the existing literature, we establish a variety of new results including dual representations and extensions of surplus-invariant risk measures and structural results for surplus-invariant acceptance sets. We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied
A continuous selection for optimal portfolios under convex risk measures does not always exist
No full textRisk control is one of the crucial problems in finance. One of the most common ways to mitigate risk of an investor's financial position is to set up a portfolio of hedging securities whose aim is to absorb unexpected losses and thus provide the investor with an acceptable level of security. In this respect, it is clear that investors will try to reach acceptability at the lowest possible cost. Mathematically, this problem leads naturally to considering set-valued maps that associate to each financial position the corresponding set of optimal hedging portfolios, i.e., of portfolios that ensure acceptability at the cheapest cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note, we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures
Risk measures beyond frictionless markets
No full textWe develop a general theory of risk measures to determine the optimal amount of capital to raise and invest in a portfolio of reference traded securities in order to meet a prespecified regulatory requirement. The distinguishing feature of our approach is that we embed portfolio constraints and transaction costs into the securities market. As a consequence, the property of translation invariance, which plays a key role in the classical theory, ceases to hold. We provide a comprehensive analysis of relevant properties, such as star shapedness, positive homogeneity, convexity, quasiconvexity, subadditivity, and lower semicontinuity. In addition, we establish dual representations for convex and quasiconvex risk measures. In the convex case, the absence of a special kind of arbitrage opportunity allows one to obtain dual representations in terms of pricing rules that respect market bid-ask spreads and assign a strictly positive price to each nonzero position in the regulator's acceptance set
Market-consistent prices: an introduction to arbitrage theory
No full textArbitrage Theory provides the foundation for the pricing of financial derivatives and has become indispensable in both financial theory and financial practice. This textbook offers a rigorous and comprehensive introduction to the mathematics of arbitrage pricing in a discrete-time, finite-state economy in which a finite number of securities are traded. In a first step, various versions of the Fundamental Theorem of Asset Pricing, i.e., characterizations of when a market does not admit arbitrage opportunities, are proved. The book then focuses on incomplete markets where the main concern is to obtain a precise description of the set of “market-consistent” prices for nontraded financial contracts, i.e. the set of prices at which such contracts could be transacted between rational agents. Both European-type and American-type contracts are considered. A distinguishing feature of this book is its emphasis on market-consistent prices and a systematic description of pricing rules, also at intermediate dates. The benefits of this approach are most evident in the treatment of American options, which is novel in terms of both the presentation and the scope, while also presenting new results. The focus on discrete-time, finite-state models makes it possible to cover all relevant topics while requiring only a moderate mathematical background on the part of the reader. The book will appeal to mathematical finance and financial economics students seeking an elementary but rigorous introduction to the subject; mathematics and physics students looking for an opportunity to get acquainted with a modern applied topic; and mathematicians, physicists and quantitatively inclined economists working or planning to work in the financial industry
Law-invariant functionals that collapse to the mean
Full text linkWe discuss when law-invariant convex functionals "collapse to the mean''. More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures. (C) 2021 The Author(s). Published by Elsevier B.V
Stability properties of Haezendonck–Goovaerts premium principles
No full textWe investigate a variety of stability properties of Haezendonck-Goovaerts premium principles on their natural domain, namely Orlicz spaces. We show that such principles always satisfy the Fatou property. This allows to establish a tractable dual representation without imposing any condition on the reference Orlicz function. In addition, we show that Haezendonck-Goovaerts principles satisfy the stronger Lebesgue property if and only if the reference Orlicz function fulfills the so-called Delta_2 condition. We also discuss (semi)continuity properties with respect to Phi-weak convergence of probability measures. In particular, we show that Haezendonck-Goovaerts principles, restricted to the corresponding Young class, are always lower semicontinuous with respect to the Phi-weak convergence. (C) 2020 Elsevier B.V. All rights reserved
Dual representations for systemic risk measures based on acceptance sets
No full textWe establish dual representations for systemic risk measures based on acceptance sets in a general setting. We deal with systemic risk measures of both "first allocate, then aggregate" and "first aggregate, then allocate" type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets and their support functions. The same approach delivers a simple and self-contained proof of the dual representation of utility-based risk measures for univariate positions
Robust portfolio selection under recovery average value at risk
No full textWe study mean-risk optimal portfolio problems where risk is measured by recovery average value at risk, a prominent example in the class of recovery risk measures. We establish existence results in the situation where the joint distribution of portfolio assets is known as well as in the situation where it is uncertain and only assumed to belong to a set of mixtures of benchmark distributions (mixture uncertainty) or to a cloud around a benchmark distribution (box uncertainty). The comparison with the classical average value at risk shows that portfolio selection under its recovery version enables financial institutions to exert better control on the recovery on liabilities while still allowing for tractable computations
Which eligible assets are compatible with comonotonic capital requirements?
No full textWithin the context of capital adequacy, we study comonotonicity of risk measures in terms of the primitives of the theory: acceptance sets and eligible, or reference, assets. We show that comonotonicity cannot be characterized by the properties of the acceptance set alone and heavily depends on the choice of the eligible asset. In fact, in many important cases, comonotonicity is only compatible with risk-free eligible assets. The incompatibility with risky eligible assets is systematic whenever the acceptability criterion is based on Value-at-Risk or any convex distortion risk measure such as Expected Shortfall. These findings qualify and arguably call for a critical appraisal of the meaning and the role of comonotonicity within a capital adequacy context. (C) 2018 Elsevier B.V. All rights reserved
Adjusted Expected Shortfall
Full text linkWe 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 is an element of [0, 1]. Through the choice of the benchmark risk profile gone 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
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