Optimal System of Subalgebras and Invariant Solutions for the Black-Scholes Equation
Independent thesis Advanced level (degree of Master (Two Years))Student thesis
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. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant,
Place, publisher, year, edition, pages
2009. , 69 p.
Keywords:Black-Scholes Equation, commutators, commutator table, Lie equainvariant solution, optimal system, generators, Airy equation, structure constant
Mathematics Mathematical Analysis Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:bth-2817Local ID: oai:bth.se:arkivexBFAC08FDFEDF00FDC12576810045ED21OAI: oai:DiVA.org:bth-2817DiVA: diva2:830112
UppsokPhysics, Chemistry, Mathematics
IBRAGIMOV, PROF. NAIL H.
It was an accolade for us to work with Professor Nail.H. Ibrgimov. +467626009532015-04-222009-12-032015-06-30Bibliographically approved