On non-uniqueness of solutions to degenerate parabolic equations in the context of option pricing in the Heston model

Paper: arXiv 2511.11288

Authors: Ruslan R. Boyko

Abstract

It is known that the price of call options in the Heston model is determined in a non-unique way. In this paper, this problem is analyzed from the point of view of the existing mathematical theory of uniqueness classes for degenerate parabolic equations. For the special case of degeneracy, a new example is constructed demonstrating the accuracy of the uniqueness theorem for a solution in the class of functions with sublinear growth at infinity.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of degenerate parabolic partial differential equations (PDEs) and Fichera boundary conditions in the Heston option pricing framework. While it effectively constructs a counterexample demonstrating non-uniqueness and sublinear growth boundaries, it is purely theoretical with zero empirical or backtesting components.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze non-uniqueness of Heston call option prices via degenerate parabolic equation theory] --> B[Input: Heston Model PDE & Boundary Conditions]
    B --> C[Methodology: Apply Uniqueness Class Theory for Degenerate Parabolic Equations]
    C --> D[Analysis: Evaluate Degeneracy Special Case at Volatility Boundary]
    D --> E[Construction: Build Counterexample with Sublinear Growth at Infinity]
    E --> F[Outcome: Demonstrate Sharpness of Uniqueness Theorem for Sublinear Growth Solutions]