On non-uniqueness in the option valuation problem

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 $x=\infty$.