A Betchov-type hydrodynamic formulation of the Ivancevic option-pricing equation

Abstract

We show that, under constant-coefficient assumptions, the Ivancevic option-pricing nonlinear Schrödinger equation admits a Betchov-type hydrodynamic formulation analogous to the one arising in the context of the vortex filament equation. Along with this structural correspondence, known Ivancevic solutions are interpreted in terms of density–velocity variables. As an illustration of the financial interpretation, we compare the normalized one-soliton density with the Black–Scholes risk-neutral density, obtaining an explicit relation between the Ivancevic coefficient ratio and standard market quantities. We further show that the probability current can be reconstructed from the temporal evolution of the density and demonstrate this reconstruction for an interacting two-soliton solution. Together, these results illustrate how the Betchov-type hydrodynamic formulation can be used to describe the evolution and transport of the Ivancevic density, while establishing a coefficient-explicit bridge between nonlinear wave models in mathematical finance and geometric fluid mechanics.