Travelling 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 $L(x,{\mathrm{\partial }}_x):=v^2(x,t)({\mathrm{\partial }}_x^2-{\mathrm{\partial }}_x)$ that carries a variable volatility term $v(x,t)$ and which is dependent 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 travelling wave solutions. This brings in extensive machinery from soliton theory and integrable systems. As a by-product, it opens up the way to the use of coherent structures in financial-market volatility studies.