The Premium Reduction of European, American, and Perpetual Log Return Options

Traditional plain vanilla options may be regarded as contingent claims whose value depends upon the simple returns of an underlying asset. These options have convex payoffs, and as a consequence of Jensen's inequality, their prices increase as a function of maturity in the absence of interest rates. This results in long-dated call option premia being excessively expensive in relation to the fraction of a corresponding insured portfolio. We show that replacing the simple return payoff with the log return call option payoff leads to substantial premium savings while providing the similar insurance protection. Call options on log returns have favorable prices for very long maturities on the scale of decades. This property enables them to be attractive securities for long-term investors, such as pension funds.