On non-uniqueness in the option valuation problem

TL;DR

This paper explains the non-uniqueness in call option valuation under constant elasticity processes with elasticity parameter exceeding one, using boundary condition theory for degenerate parabolic equations.

Contribution

It applies existing mathematical boundary condition theory to clarify the causes of non-uniqueness in option pricing models with specific elasticity parameters.

Findings

Non-uniqueness occurs for 1<α≤3/2 due to initial data outside Täcklind class.

For α>3/2, non-uniqueness is caused by missing boundary conditions at infinity.

Theoretical explanation aligns with known issues in degenerate parabolic PDEs.

Abstract

It is known that the value of a call option in the case of constant elasticity processes (CEV) with the indicator $\alpha$ exceeding the critical $\alpha=1$ 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 $1<\alpha\le \frac32$ the non-uniqueness is due to the fact that the initial data of the call option are outside the T\"acklind class, and for $\alpha> \frac32$ it is due to the absence boundary condition for $x=\infty$.