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

TL;DR

The paper develops a nonlinear evolution framework for a time-dependent local-volatility operator arising in the Black-Scholes setting via a zero-curvature deformation, yielding a Harry-Dym–type equation. It derives travelling-wave (soliton) solutions to the resulting Financial Harry Dym (FHD) equation $v_t = v^3(v_{xxx}-v_x)$ using a Lax-pair and Sagdeev-energy approach. The authors demonstrate qualitative parallels between these coherent structures and observed volatility surfaces, suggesting multisoliton solutions could serve as a natural basis for volatility-surface reconstructions. This work bridges integrable-systems methods with financial modeling and points to future research on complete integrability and market-calibrated, structure-based volatility representations.

Abstract

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 $L(x, {\partial_x}) := v^2(x,t) (\partial_x^2 -\partial_x)$ 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.

Figures (3)

• Figure 1: Plots of (a) the pseudopotential function, (b) phase portrait ($v_{\xi}$) and (c) the associated soliton form ($v$), for different values of $\Lambda$. Here, we have taken $v_0 = 1$ in the parametric analysis.
• Figure 2: (a) A 3D plot of v(x,t) is presented, in the $x-t$ plane. (b) A contour plot corresponding to the (numerically obtained) localized solution is shown, for a fixed value of $\Lambda=0.5$ and $v_0=1$.
• Figure 3: 3D plots of the local volatility surface $v(x,\tau)$ are presented, in the $x-\tau$ plane, where $x$ is the log-moneyness and $\tau$ is the time until expiration. The surfaces in (a) and (b) were reconstructed from call option prices of future contracts of Henry Hub gas and WTI oil, respectively.