Zahid Hussain; Muhammad Sulaiman; Edward K. E. Sackey 2009:07, pp. 69. ING/School of Engineering, 2009.
The main purpose of this thesis is to use modern goal-oriented
adaptive methods of Lie group analysis to construct the optimal sys-
tem of Black-Scholes equation. We will show in this thesis how to
obtain all invariant solutions by constructing what has now become so
popular, optimal system of sub-algebras, the main Lie algebra admit-
ted by the Black-Scholes equation. First, we obtain the commutator
table of already calculated symmetries of the Black-Scholes equation.
We then followed with the calculations of transformation of the gen-
erators with the Lie algebra L6 which provides one-parameter group
of linear transformations for the operators. Here we make use of the
method of Lie equations to solve the partial di®erential equations.
Next, we consider the construction of optimal systems of the Black-
Scholes equation where the method requires a simpli¯cation of a vector
to a general form to each of the transformations of the generators.
Further, we construct the invariant solutions for each of the op-
timal system. This study is motivated by the analysis of Lie groups
which is being taken to another level by ALGA here in Blekinge In-
stitute Technology, Sweden. We give a practical and in-depth steps
and explanation of how to construct the commutator table, the calcu-
lation of the transformation of the generators and the construction of
the optimal system as well as their invariant solutions.
Black-Scholes Equation, commutators, commutator table, Lie equa-
tions, invariant solution, optimal system, generators, Airy equation,
It was an accolade for us to work with Professor Nail.H. Ibrgimov.