Stochastic Finance

Föllmer, Hans; Schied, Alexander

doi:10.1515/9783110198065

Cited by 438 publications

(455 citation statements)

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“…For the evaluation of risks, the notion of risk measures-in particular of coherent and convex risk measures-has been introduced in an axiomatic way for real risks in Artzner et al (1999), Delbaen (2002), Föllmer and Schied (2002) and has been extended to vector risks in Jouini et al (2004), Burgert and Rüschendorf (2006), and many others. This notion leads to the comparison of two risks X, Y (resp., distributions Q, P) by ρ(X ) − ρ(Y ) (resp., ρ(P) − ρ(Q)).…”

Section: Motivationmentioning

confidence: 99%

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Risk excess measures induced by hemi-metrics

Faugeras

Rüschendorf

2018

Probab Uncertain Quant Risk

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which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. AbstractThe main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties, and to describe some basic construction methods. To compare the risk excess of one distribution Q w.r.t. a given risk distribution P, we apply the concept of hemi-metrics on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of Q and P by the values ρ(Q), ρ(P) of a risk measure ρ. We argue that the difference ρ(Q) − ρ(P) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

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Section: Motivationmentioning

confidence: 99%

“…where X ∼ Q, A is a class of scenario measures and α(ν) is a penalization term, see Föllmer and Schied (2002). This representation suggests to consider for a class F of real functions on E the following hemi-metric…”

Section: Motivation and Definitionmentioning

confidence: 99%

Risk excess measures induced by hemi-metrics

Faugeras

Rüschendorf

2018

Probab Uncertain Quant Risk

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“…Various different approaches have been proposed to measure risks (see, for instance, Föllmer and Schied (2002), Chapter 4, for an introduction). For simplicity, we collect several important properties of risk measures in the following three definitions.…”

Section: Admissible Convex Risk Measuresmentioning

confidence: 99%

A General Framework for Portfolio Theory. Part II: Drawdown Risk Measures

Maier-Paape

Zhu

2018

Risks

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Abstract:The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures related to the "current" drawdown of the portfolio equity. In contrast to references Chekhlov, Zabarankin (2003, 2005), Goldberg and Mahmoud (2017), and Zabarankin, Pavlikov, and Uryasev (2014), who used the absolute drawdown, our risk measure is based on the relative drawdown process. Combined with the results of Part I, Maier-Paape and Zhu (2018), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint.

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“…Prices of primitive assets will thereafter be posted along that tree. As in [11], time is discrete. That choice facilitates both analysis and computation.…”

Section: The Scenario Tree and The Assetsmentioning

confidence: 99%

“…It may well be random [29]. 3 See for instance the excellent text [11]. 4 Customary but weaker definitions of arbitrage require that θ be self-financing in that G n (θ) = 0 for all n = 0 (or for all n); see e.g.…”

Section: Arbitragementioning

confidence: 99%

Option pricing by mathematical programming†

Flåm

2008

Optimization

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ForewordStandard financial problems of pricing, portfolio selection and hedging often arise in integrated modeling of socio-economic and environmental processes. For example, long-term investments in mitigations of potential disasters can be considered with such financial instruments as real option, catastrophe bounds, and contingent credits. On such occasions, the exposed agent should evaluate and price the risks and finally choose to either carry or sell.For a rational choice some techniques developed in mathematical finance are helpful. Such techniques support estimation of price ranges for an insurance policy or a financial security, by hedging or replicating the resulting contingent claim of assets (contract) has been traded in the market. Of particular importance are options, real or financial, that incorporate the corresponding exercise date. A standard mathematical programming representation of such models include integer decision variables, therefore such problems are often difficult to solve. This paper shows that the reformulation of the standard financial problems as optimization models leads to problems that are often easier to be solved. For example, integer choice solutions may come automatically from solving a linear programming model with continuous decision variables. The discussed links between finance and optimization problems allows explanation, in a unified manner, of such important concepts as arbitrage, the fundamental theorem of asset pricing, martingale pricing, key ideas of complete markets, American-like options. The proposed approach is computationally efficient, direct, and simple. All considered standard financial problems are stated as mathematical programs, often linear. Therefore the results summarized in this paper offer an efficient approach for analysis of a class of problems related to integrated risk management.This report also describes a part of the research done by Sjur D. Flåm when he was a visiting scholar with the Integrated Modeling Environment Project.-iii - AbstractFinancial options typically incorporate times of exercise. Alternatively, they embody setup costs or indivisibilities. Such features lead to planning problems with integer decision variables. Provided the sample space be finite, it is shown here that integrality constraints can often be relaxed. In fact, simple mathematical programming, aimed at arbitrage or replication, may find optimal exercise, and bound or identify option prices. When the asset market is incomplete, the bounds system from nonlinear pricing functionals.

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Stochastic Finance

Cited by 438 publications

References 0 publications

Risk excess measures induced by hemi-metrics

Risk excess measures induced by hemi-metrics

A General Framework for Portfolio Theory. Part II: Drawdown Risk Measures

Option pricing by mathematical programming†

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