Symmetry Reductions and Solutions for Hedging in Illiquid Markets

Summary

This mathematical study analyzes a general model of self-financing hedging in illiquid markets, originally introduced by Schoenbucher and Wilmott. In the model, hedging strategies satisfy a nonlinear partial differential equation whose coefficient function is linked to an investor utility function. The authors use Lie symmetry analysis to characterize the equation's symmetries and reduce it to ordinary differential equations.

The paper classifies coefficient functions that allow additional symmetry, relates three resulting special cases to earlier models or a new case, and derives optimal subalgebra systems and reductions for both the general and new special models. It also provides explicit solutions for the new special case, including solutions describing power derivative products. These are analytical results about model structure and solvability; the supplied description does not report empirical tests, market data, or evidence that the solutions improve practical hedging outcomes. The account is therefore most useful for understanding the theoretical framework and its reductions, with real-world applicability left unassessed.

Key ideas

• Self-financing hedging strategies in the model satisfy a nonlinear PDE with a coefficient tied to utility.
• Lie symmetry analysis is used to describe symmetries and reduce the PDE to ODEs.
• The paper classifies coefficient functions that permit extended symmetry and connects special cases to prior models.
• Explicit solutions are derived for a new special model, including solutions for power derivative products.
• The described contribution is mathematical; no empirical evaluation is reported.

Tags

# Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions


# 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 Schoenbucher and Wilmott, 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 deep connected to an 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 are able to describe all types of functions g(alpha) for which the PDE admits an extended Lie group. Two of three special type functions lead to models introduced before by different authors, one is new. We clarify the connection between these three special models and the general model for trading strategies in illiquid markets. We study with the Lie group analysis the new special case of the PDE describing the self-financing strategies. In both, the general model and the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to different ODEs. In all cases we are able to provide explicit solutions to the new special model. In one of the cases the solutions describe power derivative products.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.