2009/01/19 by Maxim Bobrov, Bobrov, Maxim
Economics, Econometrics and Finance · #Computational Finance (q-fin.CP) #FOS: Economics and business #q-fin.CP
paper · pdf · doi:10.48550/arxiv.0901.2826
arxiv created 2009/01/19 · arxiv updated 2009/12/01
The main object of our study is a four dimensional Lie algebra which describes the symmetry properties of a nonlinear Black-Scholes model. This model implements a feedback effect which is typical for an illiquid market. The structure of the Lie algebra depends on one parameter, i.e. we have to do with a one-parametric family of algebras. We provide a classification of these algebras using Patera--Winternitz method. Optimal systems of one-, two- and three- dimensional subalgebras are described for the family of symmetry algebras of the nonlinear Black-Scholes equation. The optimal systems give us the possibility to describe a complete set of invariant solutions to the equation.