Time Consistent Pricing of Options with Embedded Decisions

Abstract

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. Yet, the literature lacks a consistent way of dealing with these decisions, and instead only provides methods for specific contracts and not 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 ad-hoc pricing functions used in many contributions can be derived if time consistency is added to our minimal set of assumptions.

Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions

“…Applying the methods of Gerer and Dorfleitner (2016) to the problem of hedging options with decisions allows us to derive a general hedging principle in a rigorous but straight forward manner, starting from a small set of clearly stated assumptions. This principle is then further specialized to a formula for realistically hedging American options; a formula that is not conjectured, but formally derived and proved in nonpreexisting fashion.…”

“…Our approach is motivated by the understanding of a "price" as an intrinsically one-dimensional quantity, which does not leave much conceptual freedom. The results from Gerer and Dorfleitner (2016) imply, that under mild assumptions, the decisions can be formally eliminated from the problem in a consequent manner without the need to resort to external concepts and without any further motivating argument. Our analysis differs from others in that we seek to calculate the agent's indifference or reservation price in an uncompromising fashion.…”

“…Time consistency (requirement 8), which is neglected in many contributions, is an important property, especially in the context of pricing options with complex decisions. Results obtained by time-inconsistent methods will either assume suboptimal future choices or do not give the optimal solution from today's perspective (see Gerer and Dorfleitner, 2016, for more details on the relation of decisions and time consistency in option pricing).…”

“…In Section 2, we summarize the formalism and results from Gerer and Dorfleitner (2016) used in this paper. In Section 3, a general duality between pricing functions and acceptance sets for payoffs with decisions is applied to derive a general pricing and hedging principle for options with decision by both parties.…”