Pricing contingent claims in incomplete markets when the holder can choose among different payoffs

We suggest a valuation principle to price general claims giving the holder the right to choose (in a predefined way) among several random payoffs in an incomplete financial market. Examples are so-called “chooser options” and American options with finitely many possible exertion times but also some life insurance contracts. Our premium is defined by the minimal amount the writer must receive at time zero such that for all possible decision functions of the holder, the writer’s utility is at least as big as the utility he would have if he did not offer this contingent claim. The valuation principle is consistent with no-arbitrage and can be interpreted as a generalization of Schweizer’s indifference principle (Schweizer [Insurance: Mathematics and Economics 28 (2001) 31–47]). We show that in a complete financial market or, in general, if the writer has an exponential utility function, our premium is the supremum over all “utility-indifference premiums” related to all fixed random payoffs we get by fixing the decision function of the holder. For every other utility function our premium can be even larger than this supremum.