<p>This master project is dedicated to the analysis of one of the nancialmarket models in an illiquid market. This is a nonlinear model. Using analytical methods we studied the symmetry properties of theequation which described the given model. We called this equation aSchonbucher-Wilmott equation or the main equation. We have foundinnitesimal generators of the Lie algebra, containing the informationabout the symmetry group admitted by the main equation. We foundthat there could be dierent types of the unknown function g, whichwas located in the main equation, in particular four types which admitsricher symmetry group. According to the type of the function gthe equation was split up into four PDEs with the dierent Lie algebrasin each case. Using the generators we studied the structure ofthe Lie algebras and found optimal systems of subalgebras. Then weused the optimal systems for dierent reductions of the PDE equationsto some ODEs. Obtained ODEs were easier to solve than the correspondingPDE. Thereafter we proceeded to the solution of the desiredSchonbucher-Wilmott equation. In the project we were guided by thepapers of Bank, Baum [1] and Schonbucher, Wilmott [2]. In these twopapers authors introduced distinct approaches of the analysis of thenonlinear model - stochastic and dierential ones. Both approaches leadunder some additional assumptions to the same nonlinear equation - the main equation.</p>

This master project is dedicated to the analysis of one of the nancialmarket models in an illiquid market. This is a nonlinear model. Using analytical methods we studied the symmetry properties of theequation which described the given model. We called this equation aSchonbucher-Wilmott equation or the main equation. We have foundinnitesimal generators of the Lie algebra, containing the informationabout the symmetry group admitted by the main equation. We foundthat there could be dierent types of the unknown function g, whichwas located in the main equation, in particular four types which admitsricher symmetry group. According to the type of the function gthe equation was split up into four PDEs with the dierent Lie algebrasin each case. Using the generators we studied the structure ofthe Lie algebras and found optimal systems of subalgebras. Then weused the optimal systems for dierent reductions of the PDE equationsto some ODEs. Obtained ODEs were easier to solve than the correspondingPDE. Thereafter we proceeded to the solution of the desiredSchonbucher-Wilmott equation. In the project we were guided by thepapers of Bank, Baum [1] and Schonbucher, Wilmott [2]. In these twopapers authors introduced distinct approaches of the analysis of thenonlinear model - stochastic and dierential ones. Both approaches leadunder some additional assumptions to the same nonlinear equation - the main equation.