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

Ruslan R. Boyko

TL;DR

The paper studies non-uniqueness in option pricing under the degenerate parabolic PDE of the Heston model. It employs boundary-value theory for degenerate equations, leveraging the Fichera function and boundary classification to identify where boundary data must be prescribed. By embedding generalized Tikhonov–Täcklind growth classes, it derives explicit uniqueness conditions: under the Feller condition, the solution is unique in the class with growth as $S\to\infty$ and $v\to\infty$, prescribed growth as $S\to0$, and sublinear growth $V=o(1/v)$ as $v\to0$; the paper also constructs non-uniqueness examples for certain power-law degeneracies (e.g., alpha = 2 and alpha = 3/2). These results clarify how boundary data and far-field growth constrain pricing solutions, with implications for arbitrage-free option pricing in degenerate models and guidance for boundary specification in practice.

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.

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

TL;DR

The paper studies non-uniqueness in option pricing under the degenerate parabolic PDE of the Heston model. It employs boundary-value theory for degenerate equations, leveraging the Fichera function and boundary classification to identify where boundary data must be prescribed. By embedding generalized Tikhonov–Täcklind growth classes, it derives explicit uniqueness conditions: under the Feller condition, the solution is unique in the class with growth as and , prescribed growth as , and sublinear growth as ; the paper also constructs non-uniqueness examples for certain power-law degeneracies (e.g., alpha = 2 and alpha = 3/2). These results clarify how boundary data and far-field growth constrain pricing solutions, with implications for arbitrage-free option pricing in degenerate models and guidance for boundary specification in practice.

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.

Paper Structure

This paper contains 9 sections, 1 theorem, 25 equations.

Table of Contents

  1. Option price in the Heston model
  2. Correct setting of boundary conditions
  3. The uniqueness problem in option pricing in the Heston model under the Feller condition
  4. Classes of uniqueness in the problem of option valuation in the Heston model
  5. Tikhonov--Täcklind classes ($S\to\infty$, $v\to\infty$).
  6. Uniqueness class for \ref{['eq:H1']} as $S\to 0$.
  7. Uniqueness class for \ref{['eq:H1']} as $v\to 0$.
  8. Main theorem
  9. Conclusion and discussion

Key Result

Theorem 1

Assume that the parameters of model eq:Heston1 are such that the Feller condition Feller is satisfied. Then the classical solution to problem eq:H1, eq:Heston_fund_cond is unique in the class of functions having, as $S\to +\infty$, $v\to +\infty$, $S\to 0$, growth determined by condition V, and belo

Theorems & Definitions (1)

  • Theorem 1