Gambling and Pricing of Derivatives

We introduce a criterion how to price derivatives in incomplete markets, which is based on the theory of optimal strategies in repeated multiplicative games. Arguments are presented why such growth-optimal strategies should be relevant to the problem of pricing derivatives. Under the assumptions of no trading costs, and no restrictions on lending, we find an appropriate equivalent martingale measure that prices the underlying and the derivative securities. We compare our approach with other alternative pricing procedures in the literature. We also discuss the lognormal approximation to the compounded martingale measure over many time-steps, and its limits of applicability.

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