On the multiplicity of option prices under CEV with positive elasticity of variance

The discounted stock price under the Constant Elasticity of Variance model is not a martingale when the elasticity of variance is positive. Two expressions for the European call price then arise, namely the price for which put-call parity holds and the price that represents the lowest cost of replicating the call option’s payoffs. The greeks of European put and call prices are derived and it is shown that the greeks of the risk-neutral call can substantially differ from standard results. For instance, the relation between the call price and variance may become non-monotonic. Such unfamiliar behavior then might yield option-based tests for the potential presence of a bubble in the underlying stock price.