Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization
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Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization Birgit Rudloff1
· Firdevs Ulus2
Received: 20 April 2019 / Accepted: 26 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For incomplete preference relations that are represented by multiple priors and/or multiple— possibly multivariate—utility functions, we define a certainty equivalent as well as the utility indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduce the notion of a weak and a strong certainty equivalent. We will show that our definitions contain as special cases some definitions found in the literature so far on complete or special incomplete preferences. We prove monotonicity and convexity properties of utility buy and sell prices that hold in total analogy to the properties of the scalar indifference prices for complete preferences. We show how the (weak and strong) set-valued certainty equivalent as well as the indifference price bounds can be computed or approximated by solving convex vector optimization problems. Numerical examples and their economic interpretations are given for the univariate as well as for the multivariate case. Keywords Utility maximization · Indifference price bounds · Certainty equivalent · Incomplete preferences · Convex vector optimization JEL Classification D81 · C61 · G13
1 Introduction The certainty equivalent of a random payoff is a guaranteed return that a decision maker would accept now as it is equally desirable as the uncertain return that will be received in the future. Indifference pricing can be seen as a similar concept adapted to a dynamic setting. It plays an important role in pricing in incomplete markets as it typically yields a more narrow
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Firdevs Ulus [email protected] Birgit Rudloff [email protected]
1
Institute for Statistics and Mathematics, Vienna University of Economics and Business, 1020 Vienna, Austria
2
Department of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey
123
Mathematics and Financial Economics
pricing interval compared to the often very wide no-arbitrage pricing interval, see for instance [17]. The certainty equivalent and utility indifference pricing are well studied for complete preference relations that can be represented by a single univariate utility function and there are also some extensions for complete preferences represented by a single multivariate utility function. However, the completeness assumption of the preference relation is restrictive as it ignores the typical ‘indecisiveness’ that individuals face. This concern was stated already by von Neumann and Morgenstern in their 1947 paper [33] as “It is conceivable -and may even in a way be more realistic- to allow for cases where the individual is neither able to state which of two alternatives he prefers nor that they are equally desirable.” As Aumann [2] and many researchers agreed afterwards, it is natural and indeed more realistic to ex
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