Paper: arXiv 2412.19020

Authors: Jorge P. Zubelli, Kuldeep Singh, Vinicius Albani, Ioannis Kourakis

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation linking zero-curvature conditions and Harry Dym equations to the Black-Scholes/Dupire local volatility operator. However, it lacks quantitative empirical validation or backtesting, offering only a brief qualitative visual comparison to reconstructed volatility surfaces.

Research Flowchart

  flowchart TD
    A[Research Goal: Find non-linear evolution eq for Black-Scholes operator with Dupire local volatility] --> B[Input: Second-order differential operator L with variable volatility v]
    B --> C[Methodology: Apply time-dependent deformation to operator L]
    C --> D[Methodology: Derive Zero-Curvature Condition]
    D --> E[Outcome: Variant of Harry Dym non-linear evolution equation]
    E --> F[Computation: Apply Soliton Theory to find travelling wave solutions]
    F --> G[Key Finding: Family of travelling wave solutions & application of coherent structures to market volatility]

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0 of 79 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.