Paper: arXiv 2501.18721

Authors: Ekaterina A. Ladykova, Olga S. Rozanova

Abstract

It is known that the value of a call option in the case of constant elasticity processes (CEV) with the indicator $α$ exceeding the critical $α=1$ is determined in a non-unique way. We show how, based on an already existing mathematical theory concerning the correctness of boundary conditions for degenerate parabolic equations on the semi-axis $[0,\infty)$, this phenomenon can be explained. Namely, for $1<α\le \frac32$ the non-uniqueness is due to the fact that the initial data of the call option are outside the Täcklind class, and for $α> \frac32$ it is due to the absence boundary condition for $x=\infty$.

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 partial differential equation (PDE) theory perspective to explain non-uniqueness in option pricing under the CEV model, utilizing Fichera theory and Täcklind classes. It is a purely theoretical paper with no empirical backtesting or real market data validation, but it contributes strong analytical insights to mathematical finance.

Research Flowchart

  flowchart TD
    A[Research Goal: Explain non-uniqueness of Call Option Pricing in CEV Model for α > 1] --> B[Inputs: CEV Process Equation & Boundary Conditions on 0, ∞]
    B --> C[Methodology: Apply theory of degenerate parabolic equations]
    C --> D{Evaluate α Parameter Range}
    D -- "1 < α ≤ 3/2" --> E[Outcome 1: Non-uniqueness caused by Initial Data falling outside the Täcklind Class]
    D -- "α > 3/2" --> F[Outcome 2: Non-uniqueness caused by Absence of Boundary Condition at x = ∞]