Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature. / South Africa

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

No description available.

Carefree handling refers to the ability of a pilot to operate an aircraft without the need to continuously monitor aircraft operating limits. At the heart of all carefree handling or maneuvering systems, also referred to as envelope protection systems, are algorithms and methods for predicting future limit violations. Recently, envelope protection methods that have gained more acceptance, translate limit proximity information to its equivalent in the control channel. Envelope protection algorithms either use very small prediction horizon or are static methods with no capability to adapt to changes in system configurations. Adaptive approaches maximizing prediction horizon such as dynamic trim, are only applicable to steady-state-response critical limit parameters. In this thesis, a new adaptive envelope protection method is developed that is applicable to steady-state and transient response critical limit parameters. The approach is based upon devising the most aggressive optimal control profile to the limit boundary and using it to compute control limits. Pilot-in-the-loop evaluations of the proposed approach are conducted at the Georgia Tech Carefree Maneuver lab for transient longitudinal hub moment limit protection. Carefree maneuvering is the dual of carefree handling in the realm of autonomous Uninhabited Aerial Vehicles (UAVs). Designing a flight control system to fully and effectively utilize the operational flight envelope is very difficult. With the increasing role and demands for extreme maneuverability there is a need for developing envelope protection methods for autonomous UAVs. In this thesis, a full-authority automatic envelope protection method is proposed for limit protection in UAVs. The approach uses adaptive estimate of limit parameter dynamics and finite-time horizon predictions to detect impending limit boundary violations. Limit violations are prevented by treating the limit boundary as an obstacle and by correcting nominal control/command inputs to track a limit parameter safe-response profile near the limit boundary. The method is evaluated using software-in-the-loop and flight evaluations on the Georgia Tech unmanned rotorcraft platform- GTMax. The thesis also develops and evaluates an extension for calculating control margins based on restricting limit parameter response aggressiveness near the limit boundary.