Publications:Models of self-financing hedging strategies in illiquid markets : Symmetry reductions and exact solutions

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Title Models of self-financing hedging strategies in illiquid markets : Symmetry reductions and exact solutions
Author Ljudmila A. Bordag and Anna Mikaelyan
Year 2011
PublicationType Journal Paper
Journal Letters in Mathematical Physics
HostPublication
Conference
DOI http://dx.doi.org/10.1007/s11005-011-0463-3
Diva url http://hh.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:346570
Abstract We study the general model of self-financing trading strategies inilliquid markets introduced by Schoenbucher and Wilmott, 2000.A hedging strategy in the framework of this model satisfies anonlinear partial differential equation (PDE) which contains somefunction g(alpha). This function is deep connected to anutility function.We describe the Lie symmetry algebra of this PDE and provide acomplete set of reductions of the PDE to ordinary differentialequations (ODEs). In addition we are able to describe all types offunctions g(alpha) for which the PDE admits an extended Liegroup. Two of three special type functions lead to modelsintroduced before by different authors, one is new. We clarify theconnection between these three special models and the generalmodel for trading strategies in illiquid markets. We study withthe Lie group analysis the new special case of the PDE describingthe self-financing strategies. In both, the general model and thenew special model, we provide the optimal systems of subalgebrasand study the complete set of reductions of the PDEs to differentODEs. In all cases we are able to provide explicit solutions tothe new special model. In one of the cases the solutions describepower derivative products.