A derivation of the the black-scholes equation using martingales
- Authors: Nyarko , Ebenezer Narh
- Date: 2018
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14572 , vital:40016
- Description: This work focuses on the application of stochastic differential equations, with martingales, in finance. The emphasis is on the derivation of the Black-Scholes model for the valuation of options. A theoretical framework in stochastic analysis, together with Itô calculus (Kiyoshi Itô), is explored. The Girsanov Theorem is applied in order to transform a modelled stochastic equation based, on predetermined stock and bond prices, into equivalent martingale measures. A replication strategy is then adopted to solve the two equations analytically, by finding the natural logarithm of the expectation of the solution to the stochastic models. We finally compute the resulting solution based on a standard, normal distribution to get the desired outcome of the Black-Scholes model.
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- Date Issued: 2018
Phase Plane Analysis of Linear Systems in Dynamic Mathematical Models
- Authors: Marange, Simukai Daniel
- Date: 2019
- Subjects: Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/14715 , vital:40064
- Description: A plethora of dynamic mathematical models exist and to understand and master all of them would be a gargantuan task. The author had, nonetheless, attempted to outline some of the methods used to analyse linear systems in modeling. Systems techniques are fundamental to current research in molecular cell-biology. The systems-approach stands in stark contrast to the historically, reductionist paradigm of molecular biology. Field work can be very dangerous. The main purpose of this study was to come up with the best analysis that would be used without going to the real field and thus saving time, money and risks associated with remote field localities. This research showed that the best analysis depends on the nature of the objectives intended to be solved by the model. Phase plane analysis on linear systems assisted in gaining deeper knowledge on the characteristics of such systems. This work analysed some dynamic models looking at phase planes, bifurcation, sensitivity and stability. The research provided a qualitative analysis of the processes not a numerical analysis.
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- Date Issued: 2019