Investment system specific option pricing intervals in incomplete markets

Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [p(C),p(W)] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the interval. More specifically, buying call options with a price less than p(C) or writing covered call options with a price greater than p(W) will improve the investment system. Bounds for these thresholds are established. This investment system specific option pricing interval is compatible with option valuation methods of using a replicating portfolio or a risk neutrality argument.