Pub Date : 2006-01-25DOI: 10.1524/STND.2006.24.1.109
D. Hernández-Hernández, A. Schied
SUMMARY We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with negative risk aversion and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.
{"title":"Robust utility maximization in a stochastic factor model","authors":"D. Hernández-Hernández, A. Schied","doi":"10.1524/STND.2006.24.1.109","DOIUrl":"https://doi.org/10.1524/STND.2006.24.1.109","url":null,"abstract":"SUMMARY We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with negative risk aversion and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"24 1","pages":"109 - 125"},"PeriodicalIF":1.5,"publicationDate":"2006-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/STND.2006.24.1.109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66892785","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}
Pub Date : 2005-01-01DOI: 10.1524/stnd.2005.23.1.1
Christian Burgert, L. Rüschendorf
Summary In this paper we consider the problem of finding optimal consumption strategies in an incomplete semimartingale market model under model uncertainty. The quality of a consumption strategy is measured by not only one probability measure but as common in risk theory by a class of scenario measures. We formulate a dual version of the optimization problem and prove the existence of a saddle point and give a characterization of an optimal consumption strategy in terms of solutions of the dual problem. This generalizes results of Karatzas and Zitkovic (2003) for the optimal consumption problem under a fixed probability measure.
{"title":"Optimal consumption strategies under model uncertainty","authors":"Christian Burgert, L. Rüschendorf","doi":"10.1524/stnd.2005.23.1.1","DOIUrl":"https://doi.org/10.1524/stnd.2005.23.1.1","url":null,"abstract":"Summary In this paper we consider the problem of finding optimal consumption strategies in an incomplete semimartingale market model under model uncertainty. The quality of a consumption strategy is measured by not only one probability measure but as common in risk theory by a class of scenario measures. We formulate a dual version of the optimization problem and prove the existence of a saddle point and give a characterization of an optimal consumption strategy in terms of solutions of the dual problem. This generalizes results of Karatzas and Zitkovic (2003) for the optimal consumption problem under a fixed probability measure.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"23 1","pages":"1 - 14"},"PeriodicalIF":1.5,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/stnd.2005.23.1.1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66892230","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}
Pub Date : 2005-01-01DOI: 10.1524/stnd.2005.23.1.67
G. Szűcs
Summary First we polish an argument of Rémillard and Theodorescu for the weak convergence of the empirical probability generating process. Then we prove a general inequality between probability generating processes and the corresponding empirical processes, which readily implies a rate of convergence and trivializes the problem of weak convergence: whenever the empirical process or its non-parametric bootstrap version, or the parametric estimated empirical process or its bootstrap version converges, so does the corresponding probability generating process. Derivatives of the generating process are also considered.
{"title":"Approximations of empirical probability generating processes","authors":"G. Szűcs","doi":"10.1524/stnd.2005.23.1.67","DOIUrl":"https://doi.org/10.1524/stnd.2005.23.1.67","url":null,"abstract":"Summary First we polish an argument of Rémillard and Theodorescu for the weak convergence of the empirical probability generating process. Then we prove a general inequality between probability generating processes and the corresponding empirical processes, which readily implies a rate of convergence and trivializes the problem of weak convergence: whenever the empirical process or its non-parametric bootstrap version, or the parametric estimated empirical process or its bootstrap version converges, so does the corresponding probability generating process. Derivatives of the generating process are also considered.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"23 1","pages":"67 - 80"},"PeriodicalIF":1.5,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/stnd.2005.23.1.67","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66891994","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}
Pub Date : 2005-01-01DOI: 10.1524/STND.2005.23.1.33
P. Grigoriev
Summary The well-known Harrison–Plisse theorem claims that in the classical discrete time model of the financial market with finite Ω there is no arbitrage iff there exists an equivalent martingale measure. The famous Dalang–Morton–Willinger theorem extends this result for an arbitrary Ω. Kabanov and Stricker [KS01] generalized the Harrison–Pliska theorem for the case of the market with proportional transaction costs. Nevertheless the corresponding extension of the Kabanov and Stricker result to the case of non-finite Ω fails, the corresponding counter-example with 4 assets was constructed by Schachermayer [S04]. The main result of this paper is that in the special case of 2 assets the Kabanov and Stricker theorem can be extended for an arbitrary Ω. This is quite a surprising result since the corresponding cone of hedgeable claims ÂT is not necessarily closed.
{"title":"On low dimensional case in the fundamental asset pricing theorem with transaction costs","authors":"P. Grigoriev","doi":"10.1524/STND.2005.23.1.33","DOIUrl":"https://doi.org/10.1524/STND.2005.23.1.33","url":null,"abstract":"Summary The well-known Harrison–Plisse theorem claims that in the classical discrete time model of the financial market with finite Ω there is no arbitrage iff there exists an equivalent martingale measure. The famous Dalang–Morton–Willinger theorem extends this result for an arbitrary Ω. Kabanov and Stricker [KS01] generalized the Harrison–Pliska theorem for the case of the market with proportional transaction costs. Nevertheless the corresponding extension of the Kabanov and Stricker result to the case of non-finite Ω fails, the corresponding counter-example with 4 assets was constructed by Schachermayer [S04]. The main result of this paper is that in the special case of 2 assets the Kabanov and Stricker theorem can be extended for an arbitrary Ω. This is quite a surprising result since the corresponding cone of hedgeable claims ÂT is not necessarily closed.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"23 1","pages":"33 - 48"},"PeriodicalIF":1.5,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/STND.2005.23.1.33","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66892217","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}
Pub Date : 2005-01-01DOI: 10.1524/stnd.2005.23.1.49
Johannes Leitner
Summary We study reward over penalty for risk ratios E[u(V)]/E[ρ(V)], V ∈ V, where V ⊆ L1(P) describes a linear space of attainable returns in an arbitrage-free market, u is concave and ρ ≥ 0 is convex. It turns out that maximizing such reward over penalty ratios is essentially equivalent to maximizing the ratio α(V) := E[V]/E[V−] or the expected profit over expected loss ratio E[V+]/E[V−]. The lowest upper bound α– := supV ∈ Vα(V) can be determined by solving an appropriate dual problem over the set of bounded equivalent martingale measures for V. This observation leads to the definition of shortfall risk optimal equivalent martingale measures.
{"title":"Optimal portfolios with expected loss constraints and shortfall risk optimal martingale measures","authors":"Johannes Leitner","doi":"10.1524/stnd.2005.23.1.49","DOIUrl":"https://doi.org/10.1524/stnd.2005.23.1.49","url":null,"abstract":"Summary We study reward over penalty for risk ratios E[u(V)]/E[ρ(V)], V ∈ V, where V ⊆ L1(P) describes a linear space of attainable returns in an arbitrage-free market, u is concave and ρ ≥ 0 is convex. It turns out that maximizing such reward over penalty ratios is essentially equivalent to maximizing the ratio α(V) := E[V]/E[V−] or the expected profit over expected loss ratio E[V+]/E[V−]. The lowest upper bound α– := supV ∈ Vα(V) can be determined by solving an appropriate dual problem over the set of bounded equivalent martingale measures for V. This observation leads to the definition of shortfall risk optimal equivalent martingale measures.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"23 1","pages":"49 - 66"},"PeriodicalIF":1.5,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/stnd.2005.23.1.49","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66892411","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}
Pub Date : 2005-01-01DOI: 10.1524/STND.2005.23.1.15
P. Gapeev, C. Kühn
Summary A convertible (callable) bond is a security that the holder can convert into a specified number of underlying shares. In addition, the issuer can recall the bond, paying some compensation, or force the holder to convert it immediately. We give an explicit solution to the corresponding optimal stopping game in the context of a reduced form model driven by a Brownian motion and a compound Poisson process with exponential jumps. It turns out that the occurrence of jumps leads to optimal stopping strategies whose structure differs from the results for continuous models.
{"title":"Perpetual convertible bonds in jump-diffusion models","authors":"P. Gapeev, C. Kühn","doi":"10.1524/STND.2005.23.1.15","DOIUrl":"https://doi.org/10.1524/STND.2005.23.1.15","url":null,"abstract":"Summary A convertible (callable) bond is a security that the holder can convert into a specified number of underlying shares. In addition, the issuer can recall the bond, paying some compensation, or force the holder to convert it immediately. We give an explicit solution to the corresponding optimal stopping game in the context of a reduced form model driven by a Brownian motion and a compound Poisson process with exponential jumps. It turns out that the occurrence of jumps leads to optimal stopping strategies whose structure differs from the results for continuous models.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":"23 1","pages":"15 - 31"},"PeriodicalIF":1.5,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/STND.2005.23.1.15","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66892008","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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