### Symmetry methods and conservation laws applied to the Black-Scholes partial differential equation

**Authors:**McDonald, Ruth Leigh**Date:**2012-07-03**Subjects:**Applied mathematics , Symmetry (Mathematics) , Conservation laws (Mathematics) , Differential equations , Differential equations, Partial**Type:**Thesis**Identifier:**uj:8783 , http://hdl.handle.net/10210/5141**Description:**M.Sc. , The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.**Full Text:**

### Linear optical quantum computing, associated Hamilton operators and computer algebra implementations

**Authors:**Le Roux, Jaco**Date:**2012-06-07**Subjects:**Linear optical quantum computing , Applied mathematics , Hamilton operators , SymbolicC++ , Probabilistic gates , Optical data processing , Quantum computers , Hamiltonian system , Algebra - Data processing**Type:**Thesis**Identifier:**uj:8644 , http://hdl.handle.net/10210/5000**Description:**M.Sc. , In this thesis we study the techniques used to calculate the Hamilton operators related to linear optical quantum computing. We also discuss the basic building blocks of linear optical quantum computing (LOQC) by looking at the logic gates and the physical instruments of which they are made.**Full Text:**