Time consistent pricing of options with embedded decisions

Many financial contracts are equipped with exercise rights or other features enabling the parties to actively shape the contract’s payoff. These decisions pose a great challenge for the pricing and hedging of such contracts. The existing literature deals with these decisions by providing methods for specific contracts that are not easily transferable to other models. In this paper we present a framework that allows us to separate the treatment of the decisions from the pricing problem and derive a general pricing principle for the price of an option with decisions by both parties. To accomplish this, we present a general version of the duality between acceptance sets and pricing functions, and use it to translate the pricing problem into the language of acceptance. Expressing certain aspects of economic behavior in this language is sufficient to fully eliminate the decisions from the problem. Further, we demonstrate why time consistent pricing functions are crucial when dealing with options with embedded decisions and how the pricing functions used in many contributions can be derived if time consistency is added to our minimal set of assumptions.