A study of maximum and minimum operators with applications to piecewise linear payoff functions
- Authors: Seedat, Ebrahim
- Date: 2013
- Subjects: Options (Finance) Piecewise linear topology Geometry, Affine Riesz spaces Lattice theory Algebra, Boolean Pricing , Max and min operators
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:931 , http://hdl.handle.net/10962/d1001457
- Description: The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
- Full Text:
- Date Issued: 2013
- Authors: Seedat, Ebrahim
- Date: 2013
- Subjects: Options (Finance) Piecewise linear topology Geometry, Affine Riesz spaces Lattice theory Algebra, Boolean Pricing , Max and min operators
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:931 , http://hdl.handle.net/10962/d1001457
- Description: The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
- Full Text:
- Date Issued: 2013
Fixed points of single-valued and multi-valued mappings with applications
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
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