In this paper we identify those shifts (continuous functions) of the term structure of interest r... more In this paper we identify those shifts (continuous functions) of the term structure of interest rates against which a given bond portfolio (BP) is immunized. The set of such shifts (IMMU) happens to be an (m − 1)-dimensional linear subspace in an m-dimensional linear space of all admissible shifts. In the proof we use triangular (Lagrange) functions by means of which we build a base for IMMU. How this IMMU space varies in response to changes in the cash flow generated by bond portfolio, BP, is also discussed in the last section of the paper.
In the following, we offer a theoretical approach that attempts to explain (Comments 1-3) why and... more In the following, we offer a theoretical approach that attempts to explain (Comments 1-3) why and when the Macaulay duration concept happens to be a good approximation of a bond's price sensitivity. We are concerned with the basic immunization problem with a single liability to be discharged at a future time q. Our idea is to divide the class K of all shifts a(t) of a term structure of interest rates s(t) into many classes and then to find a sufficient and necessary condition a given bond portfolio, dependent on a class of shifts, must satisfy to secure immunization at time q against all shifts a(t) from that class. For this purpose, we introduce the notions of dedicated duration and dedicated convexity. For each class of shifts, we show how to choose from a bond market under consideration a portfolio with maximal dedicated convexity among all immunizing portfolios. We demonstrate that the portfolio yields the maximal unanticipated rate of return and appears to be uniquely determined as a barbell strategy (portfolio) built up with 2 zero-coupon bearing bonds with maximal and respective minimal dedicated durations. Finally, an open problem addressed to researchers performing empirical studies is formulated.
In this paper our main goal is to demonstrate how listed companies can successfully defend themse... more In this paper our main goal is to demonstrate how listed companies can successfully defend themselves against falling share prices. We present a one-period model of the Polish financial market from the view point of KGHM. The ideas, notions, and tools presented in this article have high potential to be useful also to investment funds because the so-called replicating portfolios, when properly chosen, have negative prices and generate positive or zero income in all scenarios (states of the market). Financial instruments are represented here by vectors, while financial markets, by matrices. Having in mind that the stock price of KGHM declined from 126 PLN on April 15, 2015 to 57.50 PLN on January 15, 2016 and stayed at this level until May 15, 2016, we show how KGHM could create a financial instrument (with negative cost) which would fully compensate big potential declines of its share prices.
Journal of Optimization Theory and Applications, 1986
A two-person, zero-sum differential game of survival with general type phase constraints is inves... more A two-person, zero-sum differential game of survival with general type phase constraints is investigated. The dynamics of both players is governed by a system of differential inclusions. Player II can choose any strategy in the Varaiya-Lin sense, while player I can select any lower H-strategy (Ref. 1, p. 400). The existence of a value and an optimal strategy for player II is proved under the assumptions that the set of all player II's trajectories is compact in the Banach space of all continuous mappings and that some capturability condition is fulfilled.
Journal of Optimization Theory and Applications, 1982
A differential game of prescribed duration with generaltype phase constraints is investigated, Th... more A differential game of prescribed duration with generaltype phase constraints is investigated, The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.
Journal of Optimization Theory and Applications, 1980
Almost all results referring to the problem of the existence of a value in differential games con... more Almost all results referring to the problem of the existence of a value in differential games concern games without restricted phase coordinates. In this paper, we introduce a concept of value for differential games of pursuit and evasion and prove, under some general assumption, the existence of it. The players are required to satisfy some general phase constraints. The arguments employed in this paper are based to some extent on Krasovskii's method of extremal construction. We also show that the lower value in the Friedman sense is a generalization of our value. In a special linear case, the equivalence between pursuit differential games and time-optimal control problems is established.
In this paper we identify those shifts (continuous functions) of the term structure of interest r... more In this paper we identify those shifts (continuous functions) of the term structure of interest rates against which a given bond portfolio (BP) is immunized. The set of such shifts (IMMU) happens to be an (m − 1)-dimensional linear subspace in an m-dimensional linear space of all admissible shifts. In the proof we use triangular (Lagrange) functions by means of which we build a base for IMMU. How this IMMU space varies in response to changes in the cash flow generated by bond portfolio, BP, is also discussed in the last section of the paper.
In the following, we offer a theoretical approach that attempts to explain (Comments 1-3) why and... more In the following, we offer a theoretical approach that attempts to explain (Comments 1-3) why and when the Macaulay duration concept happens to be a good approximation of a bond's price sensitivity. We are concerned with the basic immunization problem with a single liability to be discharged at a future time q. Our idea is to divide the class K of all shifts a(t) of a term structure of interest rates s(t) into many classes and then to find a sufficient and necessary condition a given bond portfolio, dependent on a class of shifts, must satisfy to secure immunization at time q against all shifts a(t) from that class. For this purpose, we introduce the notions of dedicated duration and dedicated convexity. For each class of shifts, we show how to choose from a bond market under consideration a portfolio with maximal dedicated convexity among all immunizing portfolios. We demonstrate that the portfolio yields the maximal unanticipated rate of return and appears to be uniquely determined as a barbell strategy (portfolio) built up with 2 zero-coupon bearing bonds with maximal and respective minimal dedicated durations. Finally, an open problem addressed to researchers performing empirical studies is formulated.
In this paper our main goal is to demonstrate how listed companies can successfully defend themse... more In this paper our main goal is to demonstrate how listed companies can successfully defend themselves against falling share prices. We present a one-period model of the Polish financial market from the view point of KGHM. The ideas, notions, and tools presented in this article have high potential to be useful also to investment funds because the so-called replicating portfolios, when properly chosen, have negative prices and generate positive or zero income in all scenarios (states of the market). Financial instruments are represented here by vectors, while financial markets, by matrices. Having in mind that the stock price of KGHM declined from 126 PLN on April 15, 2015 to 57.50 PLN on January 15, 2016 and stayed at this level until May 15, 2016, we show how KGHM could create a financial instrument (with negative cost) which would fully compensate big potential declines of its share prices.
Journal of Optimization Theory and Applications, 1986
A two-person, zero-sum differential game of survival with general type phase constraints is inves... more A two-person, zero-sum differential game of survival with general type phase constraints is investigated. The dynamics of both players is governed by a system of differential inclusions. Player II can choose any strategy in the Varaiya-Lin sense, while player I can select any lower H-strategy (Ref. 1, p. 400). The existence of a value and an optimal strategy for player II is proved under the assumptions that the set of all player II's trajectories is compact in the Banach space of all continuous mappings and that some capturability condition is fulfilled.
Journal of Optimization Theory and Applications, 1982
A differential game of prescribed duration with generaltype phase constraints is investigated, Th... more A differential game of prescribed duration with generaltype phase constraints is investigated, The existence of a value in the Varaiya-Lin sense and an optimal strategy for one of the players is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. These results are next widened on more general games, examined earlier by Varaiya.
Journal of Optimization Theory and Applications, 1980
Almost all results referring to the problem of the existence of a value in differential games con... more Almost all results referring to the problem of the existence of a value in differential games concern games without restricted phase coordinates. In this paper, we introduce a concept of value for differential games of pursuit and evasion and prove, under some general assumption, the existence of it. The players are required to satisfy some general phase constraints. The arguments employed in this paper are based to some extent on Krasovskii's method of extremal construction. We also show that the lower value in the Friedman sense is a generalization of our value. In a special linear case, the equivalence between pursuit differential games and time-optimal control problems is established.
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Papers by Leszek Zaremba