Traveling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework

The Black-Scholes framework is crucial in pricing a vast number of financial instruments that permeate the complex dynamics of world markets. Associated with this framework, we consider a second-order differential operator L(x, ∂_x):=v²(x,t) ( ∂_x² − ∂_x) that carries a variable volatility term v ( x , t ) and which depends on the underlying log-price x and a time parameter t motivated by the celebrated Dupire local volatility model. In this context, we ask and answer the question of whether one can find a non-linear evolution equation derived from a zero-curvature condition for a time-dependent deformation of the operator L. The result is a variant of the Harry Dym equation for which we can then find a family of traveling wave solutions. This brings in extensive machinery from soliton theory and integrable systems. As a by-product, it opens the way to the use of coherent structures in financial-market volatility studies.