The Black-Scholes-Merton option valuation method involves deriving and solving a partial differential equation (PDE). But this method can generate multiple values for an option. We provide new solutions for the Cox-Ingersoll-Ross (CIR) term structure model, the constant elasticity of variance (CEV) model, and the Heston stochastic volatility model. Multiple solutions reflect asset pricing bubbles, dominated investments, and (possibly infeasible) arbitrages. We provide conditions to rule out bubbles on underlying prices. If they are not satisfied, put-call parity might not hold, American calls have no optimal exercise policy, and lookback calls have infinite value. We clarify a longstanding conjecture of Cox, Ingersoll, and Ross.

Standard literature concludes that transaction costs only have a second‐order effect on liquidity premia. We show that this conclusion depends crucially on the assumption of a constant investment opportunity set. In a regime‐switching model in which the investment opportunity set varies over time, we explicitly characterize the optimal consumption and investment strategy. In contrast to the standard literature, we find that transaction costs can have a first‐order effect on liquidity premia. However, with reasonably calibrated parameters, the presence of transaction costs still cannot fully explain the equity premium puzzle.