Invariant solutions for nonlinear models of illiquid markets

Abstract

A general nonlinear model of illiquid markets with feedback effects is considered. This equation with 2 free functional parameters contains as partial cases the classical Black-Scholes equation, Schönbucher-Wilmott equation, andSircar-Papanicolaou equation of option pricing. We obtain here the complete group classification of the equation, and for every parameters specification we obtain the principal Lie algebra and its optimal system of 1-dimensional subalgerbras. For every such subalgebra we calculate the invariant submodel and invariant solution, when it is possible. Thus, the series of invariant submodels and invariant solutions are derived for the considered nonlinear model.

Comparing of some sensitivities for nonlinear models Comparing of some sensitivities (Greeks) for nonlinear models of option pricing with market illiquidity

“…Using the group classification of equation ( 4), obtained in the works [7][8][9][10], authors single out several cases of the free functional parameter v, which are nonequivalent in the group structure sense. These cases correspond to the all widest symmetry groups of the equation.…”

Symmetry Analysis of a Model of Option Pricing and Hedging

“…In the last decade, in the works of Bordag [24,25], of Dyshaev and Fedorov [26][27][28][29][30][31][32] group properties of various nonlinear Black-Scholes type models were studied, and their invariant solutions and submodels were calculated. In the papers of Dyshaev and Fedorov, group classifications for various classes of nonlinear Black-Scholes type models were obtained.…”

Invariant analytical solutions for the motion of an elastic string with electric current in a static magnetic field

“…The modern group theory analysis of differential equations is a powerful tool to find analytical solutions for complicated systems of differential equations that describe natural phenomena. As recent advances in this field, we refer to the works on gas dynamics, 1 fluid mechanics, 2,3 epidemiology, 4 economy sciences, 5,6 plasticity, 7 nonlinear equations of Korteweg‐de Vries type, 8,9 variable‐coefficient Burgers equations, 10 and generalized Benney system 11 …”