Options and Bubbles

The Black-Scholes-Merton option valuation method involves derivingand solving a partial differential equation (PDE). But thismethod can generate multiple values for an option. We providenew solutions for the Cox-Ingersoll-Ross (CIR) term structuremodel, the constant elasticity of variance (CEV) model, andthe Heston stochastic volatility model. Multiple solutions reflectasset pricing bubbles, dominated investments, and (possiblyinfeasible) arbitrages. We provide conditions to rule out bubbleson underlying prices. If they are not satisfied, put-call paritymight not hold, American calls have no optimal exercise policy,and lookback calls have infinite value. We clarify a longstandingconjecture of Cox, Ingersoll, and Ross. (JEL G12 and G13)