Abstract The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minimizing setting is ruled out. The existence and concrete forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests are proven. The quantile hedging method applied in [Finance Stoch. 3 (1999), 251–273; Finance Stoch. 4 (2000), 117–146] as well as the classical Neyman–Pearson lemma are generalized. Optimal hedging strategies with shortfall constraints in the Black–Scholes and exponential Poisson model are explicitly determined.
{"title":"On the shortfall risk control: A refinement of the quantile hedging method","authors":"M. Barski","doi":"10.1515/strm-2014-1169","DOIUrl":"https://doi.org/10.1515/strm-2014-1169","url":null,"abstract":"Abstract The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minimizing setting is ruled out. The existence and concrete forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests are proven. The quantile hedging method applied in [Finance Stoch. 3 (1999), 251–273; Finance Stoch. 4 (2000), 117–146] as well as the classical Neyman–Pearson lemma are generalized. Optimal hedging strategies with shortfall constraints in the Black–Scholes and exponential Poisson model are explicitly determined.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-1169","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Nonparametric estimation of tail dependence can be based on a standardization of the marginals if their cumulative distribution functions are known. In this paper it is shown to be asymptotically more efficient if the additional knowledge of the marginals is ignored and estimators are based on ranks. The discrepancy between the two estimators is shown to be substantial for the popular Clayton and Gumbel–Hougaard models. A brief simulation study indicates that the asymptotic conclusions transfer to finite samples.
{"title":"A note on nonparametric estimation of bivariate tail dependence","authors":"Axel Bücher","doi":"10.1515/strm-2013-1143","DOIUrl":"https://doi.org/10.1515/strm-2013-1143","url":null,"abstract":"Abstract Nonparametric estimation of tail dependence can be based on a standardization of the marginals if their cumulative distribution functions are known. In this paper it is shown to be asymptotically more efficient if the additional knowledge of the marginals is ignored and estimators are based on ranks. The discrepancy between the two estimators is shown to be substantial for the popular Clayton and Gumbel–Hougaard models. A brief simulation study indicates that the asymptotic conclusions transfer to finite samples.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is devoted to the study of asymptotic properties of the regression function kernel estimate in the setting of continuous time stationary and ergodic data. More precisely, considering the Nadaraya–Watson type estimator, say m̂T(x), of the l-indexed regression function m(x) =𝔼 (l(Y)|X = x) built upon continuous time stationary and ergodic data (Xt, Yt)0≤t≤T, we establish its pointwise and uniform, over a dilative compact set, convergences with rates. Notice that the ergodic setting covers and completes various situations as compared to the mixing case and stands as more convenient to use in practice.
摘要本文研究了连续时间平稳遍历数据下回归函数核估计的渐近性质。更准确地说，考虑到建立在连续时间平稳和遍历数据(Xt, Yt)0≤T≤T上的l-索引回归函数m(x) = (l(Y)| x = x)的nadarya - watson型估计量m³T(x)，我们建立了它在一个扩张性紧集上的点性和均匀性，具有速率收敛性。请注意，与混合箱相比，遍历式设置涵盖并完成了各种情况，并且在实践中更方便使用。
{"title":"Asymptotic results for the regression function estimate on continuous time stationary and ergodic data","authors":"S. Didi, D. Louani","doi":"10.1515/strm-2012-1134","DOIUrl":"https://doi.org/10.1515/strm-2012-1134","url":null,"abstract":"Abstract This paper is devoted to the study of asymptotic properties of the regression function kernel estimate in the setting of continuous time stationary and ergodic data. More precisely, considering the Nadaraya–Watson type estimator, say m̂T(x), of the l-indexed regression function m(x) =𝔼 (l(Y)|X = x) built upon continuous time stationary and ergodic data (Xt, Yt)0≤t≤T, we establish its pointwise and uniform, over a dilative compact set, convergences with rates. Notice that the ergodic setting covers and completes various situations as compared to the mixing case and stands as more convenient to use in practice.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1134","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67311932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We characterize when a convex risk measure associated to a law-invariant acceptance set in L∞ can be extended to Lp, 1≤p<∞$1le p
摘要刻画了L∞上与一个律不变接受集相关联的凸风险测度可以扩展到Lp, 1≤p<∞$1le p
{"title":"Law-invariant risk measures: Extension properties and qualitative robustness","authors":"Pablo Koch-Medina, Cosimo Munari","doi":"10.1515/strm-2014-0002","DOIUrl":"https://doi.org/10.1515/strm-2014-0002","url":null,"abstract":"Abstract We characterize when a convex risk measure associated to a law-invariant acceptance set in L∞ can be extended to Lp, 1≤p<∞$1le p<infty $ , preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concrete examples including risk measures based on expected utility, max-correlation risk measures, and distortion risk measures.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen “tangent space”. We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability.
{"title":"Quasi-Hadamard differentiability of general risk functionals and its application","authors":"Volker Krätschmer, A. Schied, Henryk Zähle","doi":"10.1515/strm-2014-1174","DOIUrl":"https://doi.org/10.1515/strm-2014-1174","url":null,"abstract":"Abstract We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen “tangent space”. We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-1174","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study an inference problem for the regression coefcients in some multivariate regression models with multiple change-points occurring at unknown times, when the regression coe cients may satisfy some restrictions. The hypothesized restriction is more general than that given in recent literature. Under a weaker assumption than that given in recent literature, we derive the joint asymptotic normality of the restricted and unrestricted estimators. Finally, we construct a test for the hypothesized restriction and derive its asymptotic power.
{"title":"Constrained inference in multiple regression with structural changes","authors":"Fuqi Chen, S. Nkurunziza","doi":"10.1515/strm-2012-1154","DOIUrl":"https://doi.org/10.1515/strm-2012-1154","url":null,"abstract":"In this paper, we study an inference problem for the regression coefcients in some multivariate regression models with multiple change-points occurring at unknown times, when the regression coe cients may satisfy some restrictions. The hypothesized restriction is more general than that given in recent literature. Under a weaker assumption than that given in recent literature, we derive the joint asymptotic normality of the restricted and unrestricted estimators. Finally, we construct a test for the hypothesized restriction and derive its asymptotic power.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1154","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider convex risk measures in a spatial setting, where the outcome of a financial position depends on the states at different nodes of a network. In analogy to the theory of Gibbs measures in Statistical Mechanics, we discuss the local specification of a global risk measure in terms of conditional local risk measures for the single nodes of the network, given their environment. Under a condition of local law invariance, we show that a consistent local specification must be of entropic form. Even in that case, a global risk measure may not be uniquely determined by the local specification, and this can be seen as a source of “systemic risk”, in analogy to the appearance of phase transitions in the theory of Gibbs measures
{"title":"Spatial risk measures and their local specification: The locally law-invariant case","authors":"H. Föllmer","doi":"10.1515/strm-2013-5001","DOIUrl":"https://doi.org/10.1515/strm-2013-5001","url":null,"abstract":"Abstract We consider convex risk measures in a spatial setting, where the outcome of a financial position depends on the states at different nodes of a network. In analogy to the theory of Gibbs measures in Statistical Mechanics, we discuss the local specification of a global risk measure in terms of conditional local risk measures for the single nodes of the network, given their environment. Under a condition of local law invariance, we show that a consistent local specification must be of entropic form. Even in that case, a global risk measure may not be uniquely determined by the local specification, and this can be seen as a source of “systemic risk”, in analogy to the appearance of phase transitions in the theory of Gibbs measures","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-5001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.
{"title":"Stochastic dominance with respect to a capacity and risk measures","authors":"Miryana Grigorova","doi":"10.1515/strm-2014-1167","DOIUrl":"https://doi.org/10.1515/strm-2014-1167","url":null,"abstract":"Abstract In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2014-1167","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we formulate the classical optimal risk allocation problem for convex risk functionals defined on products of real Banach spaces as risk domains. This generality includes in particular the classical case of Lp risks but also allows to describe the influence of dependence in the risk allocation problem. We characterize optimal allocations and complete known existence and uniqueness results from the literature. We discuss in detail an application to expected risk functionals. This case can be dealt with by the Banach space approach applied to Orlicz hearts associated to the risk functionals. We give a detailed discussion of the necessary continuity and differentiability properties. Based on ordering results for Orlicz hearts we obtain extensions of the optimal allocation results to different Orlicz hearts as domain of risk functionals and establish a general form of the classical Borch theorem. In some numerical examples, optimal redistributions are determined for the expected risk case and the precision of the numerical calculation is checked.
{"title":"Optimal risk allocation for convex risk functionals in general risk domains","authors":"S. Kiesel, L. Rüschendorf","doi":"10.1515/strm-2012-1156","DOIUrl":"https://doi.org/10.1515/strm-2012-1156","url":null,"abstract":"Abstract In this paper, we formulate the classical optimal risk allocation problem for convex risk functionals defined on products of real Banach spaces as risk domains. This generality includes in particular the classical case of Lp risks but also allows to describe the influence of dependence in the risk allocation problem. We characterize optimal allocations and complete known existence and uniqueness results from the literature. We discuss in detail an application to expected risk functionals. This case can be dealt with by the Banach space approach applied to Orlicz hearts associated to the risk functionals. We give a detailed discussion of the necessary continuity and differentiability properties. Based on ordering results for Orlicz hearts we obtain extensions of the optimal allocation results to different Orlicz hearts as domain of risk functionals and establish a general form of the classical Borch theorem. In some numerical examples, optimal redistributions are determined for the expected risk case and the precision of the numerical calculation is checked.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1156","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the change point test for the tail index of scaleshifted processes. To this task, we propose two tests. The rst is designed via examining the discrepancy between the two Hill estimators obtained from the observations before and after a preliminary change point estimate. The second is a modi ed recursive test which uses scale-adjusted observations. Both methods produce a tail index estimator that outperforms the Hill estimator. A simulation study and real data analysis are provided for illustration.
{"title":"Change point test for tail index of scale-shifted processes","authors":"Moosup Kim, Sangyeol Lee","doi":"10.1515/strm-2012-1147","DOIUrl":"https://doi.org/10.1515/strm-2012-1147","url":null,"abstract":"In this paper, we study the change point test for the tail index of scaleshifted processes. To this task, we propose two tests. The rst is designed via examining the discrepancy between the two Hill estimators obtained from the observations before and after a preliminary change point estimate. The second is a modi ed recursive test which uses scale-adjusted observations. Both methods produce a tail index estimator that outperforms the Hill estimator. A simulation study and real data analysis are provided for illustration.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1147","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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