Overview
This study presents a Betchov-type hydrodynamic formulation for the Ivancevic option-pricing nonlinear Schrödinger equation. This formulation is shown to be analogous to that found in the context of the vortex filament equation. The research identifies the continuity equation and a momentum-type conservation law that apply to the density-velocity pair within this framework. The formulation is illustrated using known Ivancevic-type soliton solutions.
Research Context
The Ivancevic option-pricing equation is positioned within the domain of mathematical finance, while the Betchov-type hydrodynamic formulation originates from geometric fluid mechanics. The investigation aims to establish a structural and model-dependent connection between these two distinct areas. Specifically, it explores whether a hydrodynamic interpretation, characteristic of certain physical systems, can be applied to a financial modeling equation.
Approach
The core approach involved analyzing the Ivancevic option-pricing nonlinear Schrödinger equation under specific conditions: constant-coefficient assumptions. Within this setup, the researchers sought to derive and identify a hydrodynamic formulation akin to the Betchov formulation. This entailed determining the mathematical expressions for a continuity equation and a momentum-type conservation law that govern a density-velocity pair associated with the option-pricing equation. The validity or applicability of this formulation was subsequently illustrated using existing Ivancevic-type soliton solutions.
Findings
- Under constant-coefficient assumptions, the Ivancevic option-pricing nonlinear Schrödinger equation admits a Betchov-type hydrodynamic formulation.
- This hydrodynamic formulation is analogous to the one observed in the context of the vortex filament equation.
- The corresponding continuity equation and a momentum-type conservation law for the density-velocity pair were identified.
- The formulation was illustrated using known Ivancevic-type soliton solutions.
- The resulting interpretation is structural and dependent on the specific model used.
Why This Matters
This work suggests a bridge between nonlinear wave formulations in mathematical finance and concepts from geometric fluid mechanics. The identified hydrodynamic interpretation provides a structural and model-dependent perspective on the Ivancevic option-pricing equation.