Models of Self-Financing Hedging Strategies in Illiquid Markets: Symmetry Reductions and Exact Solutions

We study the general model of self-financing trading strategies in illiquid markets introduced by Schonbucher and Wilmott (SIAM J Appl Math 61(1):232-272, 2000). A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deeply connected to a marginal utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition, we show the way how to describe all types of functions g(alpha) for which the PDE admits an extended Lie group. Two of these special type functions correspond to the models introduced before by different authors, whereas one is new. We clarify the connection between these three special models and the general model for trading strategies in the illiquid markets. We also apply the Lie group analysis to the new special case of the PDE describing the self-financing strategies. For the general model, as well as for the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to ODEs. We provide explicit solutions to the new special model in all reduced cases. Moreover, in one of the cases the solutions describe power derivative products.