Option Pricing When the Underlying Asset Earns a Below-Equilibrium Rate of Return: A Note

IN THEIR CLASSIC PAPER, Black and Scholes [2] present two derivations of the European call option pricing formula. The first, which we will refer to as the "replication derivation," relies on the fact that the returns from the option can be replicated by holding a continuously revised position in the underlying asset and bonds. This implies that if an option is not priced according to the BlackScholes formula, there is a sure profit to be made by some combination of either short or long sales of the option and underlying asset. This argument is seemingly independent of considerations relating to capital market equilibrium. The second derivation, which we will call the "equilibrium derivation," evolves from the requirement that the option earn an expected rate of return commensurate with the risk involved in holding the option as an asset. This note considers the pricing of a call option on an asset which earns an expected rate of return lower than that necessary to induce investors to hold it. We show in this case that even if the replication strategy is feasible, it is inefficient. A consequence of this is that the correct option pricing formula is different than the Black-Scholes formula. Unlike the Black-Scholes formula, our formula requires as an input the difference between the expected rate of return on the underlying asset and that expected rate of return which would be required to induce holding of the asset. Other papers have made a similar point. Constantinides [3] and Cox et al. [4] have argued that in equilibrium the prices of traded contingent claims must satisfy a partial differential equation which is identical to the Black-Scholes partial differential equation only when the underlying asset pays no dividends and earns a rate of return sufficient to induce holding of the asset. More recently, Ingersoll [5] has noted that commodities often pay a nonpecuniary service yield, the value of which must be taken into account in deriving the option price. None of these authors, however, explain why the Black-Scholes formula is sometimes inappropriate. The purpose of this note is to demonstrate why the Black-Scholes formula needs modification even if replication is feasible. We provide an intuitive derivation of the correct formula and discuss why divergences of the equilibrium price from the Black-Scholes formula do not result in arbitrage opportunities.