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.
- Full Text:
- 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.
- Full Text:
Dynamic Mathematical Modeling in Chemical Reaction Networks
- Authors: Gurajena, Simba
- Date: 2018
- Subjects: Chemical reaction
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/11961 , vital:39122
- Description: Many students are familiar with the idea of ecological, financial modeling and modeling in other fields, but modeling in engineering fields is still an area to be looked at. In this discussion the researcher will deal with chemical reaction networks. This will cover areas in chemical reaction, interaction diagrams and the associated models. The discussion will also cover dynamic behaviour of reaction networks. The law of mass action and examples of simple networks will be dealt with. Differential equation models of biochemical and genetic systems are invariantly nonlinear, and as such numerical simulation is used to solve such models. The use of numerical simulation packages will be discussed and separation of timescale and model reduction will form part of the discussion.
- Full Text:
- Authors: Gurajena, Simba
- Date: 2018
- Subjects: Chemical reaction
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10353/11961 , vital:39122
- Description: Many students are familiar with the idea of ecological, financial modeling and modeling in other fields, but modeling in engineering fields is still an area to be looked at. In this discussion the researcher will deal with chemical reaction networks. This will cover areas in chemical reaction, interaction diagrams and the associated models. The discussion will also cover dynamic behaviour of reaction networks. The law of mass action and examples of simple networks will be dealt with. Differential equation models of biochemical and genetic systems are invariantly nonlinear, and as such numerical simulation is used to solve such models. The use of numerical simulation packages will be discussed and separation of timescale and model reduction will form part of the discussion.
- Full Text:
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