Multisoliton Basis for Volatility Surface Reconstruction

Determine whether multisoliton solutions of the Financial Harry Dym equation can serve as a natural basis for reconstructing local volatility surfaces from option price data, and ascertain consistency of physical units within the Black–Scholes framework when performing such reconstructions.

Background

The paper derives a modified Harry Dym-type partial differential equation, termed the Financial Harry Dym (FHD) equation, via a zero-curvature deformation of the Black–Scholes operator with state- and time-dependent local volatility. The authors obtain travelling-wave solutions and observe qualitative similarities between these solutions and empirically reconstructed local volatility surfaces from commodity options.

Motivated by these similarities, the authors propose using coherent structures from integrable systems—specifically multisoliton solutions—as building blocks to represent and reconstruct volatility surfaces. They note that connecting these mathematical structures to financial reconstructions requires careful handling of units and scaling in the Black–Scholes framework.