A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. May 30, 2016.In this dissertation, we consider a number of modi ed Black-Scholes equations
being either non-linear or given in higher dimensions. In particular we focus
on the non-linear Black-Scholes equation describing option pricing with hedging
strategies in one case, and two dimensional models in the other. Classical
Lie point symmetry techniques are employed in an attempt to construct exact
solutions. Some large symmetry algebras are admitted. We proceeded by
determining the one dimensional optimal systems of sub-algebras for the admitted
Lie algebras. The elements of the optimal systems are used to reduce
the number of variables by one. In some cases, exact solutions are constructed.
For the cases for which exact solutions are di cult to construct, we employed
the numerical solutions. Some simulations are observed and interpretedMT201
