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Erschienen in: Mathematics and Financial Economics 2/2021

10.10.2020

Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization

verfasst von: Birgit Rudloff, Firdevs Ulus

Erschienen in: Mathematics and Financial Economics | Ausgabe 2/2021

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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.

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Fußnoten
1
There are alternative approaches to define the indifference buy and sell prices in the literature. Indeed, there is a recent discussion stating that the indifference prices provided above satisfy the so called “complementary symmetry property”, see for instance [10, 22]; and there are experiments showing that this property is systematically violated [6]. Accordingly, it is possible to define, for instance, the utility indifference sell price as a solution of
$$\begin{aligned} \sup _{V_T \in \mathcal {A}(x_0)}{\mathbb {E}}u(V_T+C_T) = \sup _{V_T \in \mathcal {A}(x_0+p^s)} {\mathbb {E}}u(V_T), \end{aligned}$$
(3)
which accounts for the situation that one owns \(C_T\) in order to sell it. Thus, the agent’s initial pre-trade position is \((x_0, C_T)\), that is, \(x_0\) at time zero, and \(C_T\) initial wealth at time T. This would lead to an alternative description for \(P^s\). The definition in (4) corresponds to the situation, where the agent’s initial pre-trade position is \((x_0, 0)\), that is, \(x_0\) at time zero, and zero initial wealth at time T, see also [17]. This could also be interpreted as leading to the indifference short-selling price, with (3) as the indifference sell price. However, when we discuss the extensions of these concepts in Sect. 4, we keep the usual terminology and the sets as in (4), since they are quite standard in Financial Mathematics, see for instance [3, 8, 9, 17, 18].
 
2
Note that it is also possible to consider the slighly more general preference relation in [26], where there is a set of probability measure and utility pairs, say, \(\mathcal {UQ}\) and
$$\begin{aligned} Y \succsim Z \iff \,\, \forall (u,Q)\in \mathcal {UQ}: \,\,\, {\mathbb {E}}_Q u(Y) \ge {\mathbb {E}}_Q u(Z). \end{aligned}$$
In this case, we would assume that there exists finitely many pairs in \(\mathcal {UQ}\) instead of what is stated in Assumption 2.8a. However, keeping the representation as in Definition 2.5 will be useful in simplifying some expressions throughout.
 
3
In [7], Campi and Owen define a multivariate utility function in a similar way. Different from Definition 2.7, they require \(C_u:=\mathrm{cl \,}(\mathrm{dom\,}u)\) to be a convex cone such that \(\mathbb {R}^d_+\subseteq C_u \ne \mathbb {R}^d\) and u to be increasing with respect to the partial order \(\le _{C_u}\). Note that as \(C_u\supseteq \mathbb {R}^d_+\), our definition is more general.
 
4
If we consider a representation given by a set of probability measures paired with utility functions as in [26], we would list all the pairs in order to obtain \(U(\cdot )\) and all the results of this section would remain the same, see also Footnote 2.
 
5
Following the remark given in Footnote 1, an alternative definition for the set-valued sell price of \(C_T\) would be \({\tilde{P}}^s(C_T) =\{p^s \in \mathbb {R}^d |\;V(x_0 + p^s, 0) \supseteq V(x_0, C_T)\}\). With this definition, Remark 4.6 is not correct anymore. Hence, one needs to check the rest of the results in Sect. 4 separately for \({\tilde{P}}^s(C_T)\). It is straightforward to see that Propositions 4.7-1., 4.7-2. and 4.9 hold correct for this definition. Moreover, both the statements and the proofs of Propositions 4.11 and 4.12 can be modified accordingly. However, the steps followed to prove Propositions 4.7-3. and 4.8 can not be applied directly to the alternative definition. Note that as Proposition 4.7-1. holds correct, the computations of \({\tilde{P}}^s(C_T)\) can be done by applying similar techniques as described in Sect. 5.
 
6
Note also the relationship to the definition of the certainty equivalent, in particular between \(\mathrm{C^{up} \,}(C_T)\) and \(P^s(C_T)\). A certain amount \(c\in \mathrm{bd \,}\mathrm{C^{up} \,}(C_T)= \mathrm{C^{w} \,}(C_T)\) is preferred to \(C_T\) (\(c \succsim C_T\)), but for any \(\varepsilon \in \mathrm{int \,}\mathbb {R}^d_+\), \(c-\varepsilon \) is not anymore preferred to it (\(c-\varepsilon \not \succsim C_T\)). Similarly, for a price \(p\in \mathrm{bd \,}P^s(C_T)\), the decision maker would prefer selling the claim at that price rather than not taking any action, but for any \(\varepsilon \in \mathrm{int \,}\mathbb {R}^d_+\), \(p-\varepsilon \) is not anymore a sell price for him/her (see Propositions 3.5-1. and 4.11-2.). Moreover both sets are convex upper sets. The situation is somehow different when we consider \(\mathrm{C^{lo} \,}(C_T)\). This set is not necessarily convex (unlike \(P^b(C_T)\)). Moreover, for any \(c\in \mathrm{C^{lo} \,}(C_T)\), \(C_T\) is preferred to c (not the other way around), and for any \(\varepsilon \in \mathrm{int \,}\mathbb {R}^d_+\), \(C_T\) is not anymore preferred to \(c+\varepsilon \).
 
Literatur
1.
Zurück zum Zitat Armbruster, B., Delage, E.: Decision making under uncertainty when preference information is incomplete. Manag. Sci. 61, 111–128 (2015)CrossRef Armbruster, B., Delage, E.: Decision making under uncertainty when preference information is incomplete. Manag. Sci. 61, 111–128 (2015)CrossRef
2.
Zurück zum Zitat Aumann, R.: Utility theory without the completeness axiom. Econometrica 30, 445–462 (1962)CrossRef Aumann, R.: Utility theory without the completeness axiom. Econometrica 30, 445–462 (1962)CrossRef
3.
Zurück zum Zitat Benedetti, G., Campi, O.: Multivariate utility maximization with proportional transaction costs and random endowment. SIAM J. Control Optim. 50(3), 1283–1308 (2012)MathSciNetCrossRef Benedetti, G., Campi, O.: Multivariate utility maximization with proportional transaction costs and random endowment. SIAM J. Control Optim. 50(3), 1283–1308 (2012)MathSciNetCrossRef
4.
Zurück zum Zitat Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13, 1–24 (1998)MathSciNetCrossRef Benson, H.P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J. Glob. Optim. 13, 1–24 (1998)MathSciNetCrossRef
6.
Zurück zum Zitat Birnbaum, M.H., Yeary, S., Luce, R.D., Zhao, L.: Empirical evaluation of four models of buying and selling prices of gambles. J. Math. Psychol. 75, 183–193 (2016)MathSciNetCrossRef Birnbaum, M.H., Yeary, S., Luce, R.D., Zhao, L.: Empirical evaluation of four models of buying and selling prices of gambles. J. Math. Psychol. 75, 183–193 (2016)MathSciNetCrossRef
7.
Zurück zum Zitat Campi, L., Owen, M.P.: Multivariate utility maximization with proportional transaction costs. Finance Stoch. 15(3), 461–499 (2011)MathSciNetCrossRef Campi, L., Owen, M.P.: Multivariate utility maximization with proportional transaction costs. Finance Stoch. 15(3), 461–499 (2011)MathSciNetCrossRef
8.
Zurück zum Zitat Carmona, R.: Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2008)CrossRef Carmona, R.: Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2008)CrossRef
9.
Zurück zum Zitat Cheridito, P., Kupper, M.: Recursiveness of indifference prices and translation-invariant preferences. Math. Financ. Econ. 2(3), 173–188 (2009)MathSciNetCrossRef Cheridito, P., Kupper, M.: Recursiveness of indifference prices and translation-invariant preferences. Math. Financ. Econ. 2(3), 173–188 (2009)MathSciNetCrossRef
10.
Zurück zum Zitat Chudziak, J.: On complementary symmetry under cumulative prospect theory. J. Math. Psychol. 95, 102312 (2020)MathSciNetCrossRef Chudziak, J.: On complementary symmetry under cumulative prospect theory. J. Math. Psychol. 95, 102312 (2020)MathSciNetCrossRef
11.
Zurück zum Zitat Dubra, J., Maccheroni, F., Ok, E.: Expected utility theory without the completeness axiom. J. Econ. Theory 115, 118–133 (2004)MathSciNetCrossRef Dubra, J., Maccheroni, F., Ok, E.: Expected utility theory without the completeness axiom. J. Econ. Theory 115, 118–133 (2004)MathSciNetCrossRef
12.
Zurück zum Zitat Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Glob. Optim. 50(3), 397–416 (2011)MathSciNetCrossRef Ehrgott, M., Shao, L., Schöbel, A.: An approximation algorithm for convex multi-objective programming problems. J. Glob. Optim. 50(3), 397–416 (2011)MathSciNetCrossRef
13.
Zurück zum Zitat Eichfelder, G.: Adaptive scalarization methods in multiobjective optimization. Springer, Berlin (2008)CrossRef Eichfelder, G.: Adaptive scalarization methods in multiobjective optimization. Springer, Berlin (2008)CrossRef
14.
Zurück zum Zitat Galaabaatar, T., Karni, E.: Subjective expected utility with incomplete preferences. Econometrica 81(1), 255–284 (2013)MathSciNetCrossRef Galaabaatar, T., Karni, E.: Subjective expected utility with incomplete preferences. Econometrica 81(1), 255–284 (2013)MathSciNetCrossRef
15.
Zurück zum Zitat Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7(1), 19–37 (2006)MathSciNetMATH Hamel, A., Löhne, A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlinear Convex Anal. 7(1), 19–37 (2006)MathSciNetMATH
16.
Zurück zum Zitat Hamel, A., Wang, S.Q.: A set optimization approach to utility maximization under transaction costs. Decis. Econ. Finance 40(1–2), 257–275 (2017)MathSciNetCrossRef Hamel, A., Wang, S.Q.: A set optimization approach to utility maximization under transaction costs. Decis. Econ. Finance 40(1–2), 257–275 (2017)MathSciNetCrossRef
17.
Zurück zum Zitat Henderson, V., Hobson, D.: Indifference Pricing: Theory and Applications, Chapter Utility Indifference Pricing—An Overview, pp. 44–74. Princeton University Press, Princeton (2009)MATH Henderson, V., Hobson, D.: Indifference Pricing: Theory and Applications, Chapter Utility Indifference Pricing—An Overview, pp. 44–74. Princeton University Press, Princeton (2009)MATH
18.
Zurück zum Zitat Henderson, V., Liang, G.: A multidimensional exponential utility indifference pricing model with applications to counterparty risk. SIAM J. Control Optim. 54(2), 690–717 (2016)MathSciNetCrossRef Henderson, V., Liang, G.: A multidimensional exponential utility indifference pricing model with applications to counterparty risk. SIAM J. Control Optim. 54(2), 690–717 (2016)MathSciNetCrossRef
19.
Zurück zum Zitat Jahn, J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2004)MATH Jahn, J.: Vector Optimization—Theory, Applications, and Extensions. Springer, Berlin (2004)MATH
20.
Zurück zum Zitat Kabanov, Y., Rásonyi, M., Stricker, C.: No-arbitrage criteria for financial markets with efficient friction. Finance Stoch. 6, 371–382 (2002)MathSciNetCrossRef Kabanov, Y., Rásonyi, M., Stricker, C.: No-arbitrage criteria for financial markets with efficient friction. Finance Stoch. 6, 371–382 (2002)MathSciNetCrossRef
21.
Zurück zum Zitat Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. Theory Methods Appl. 30(3), 1487–1496 (1997)MathSciNetCrossRef Kuroiwa, D., Tanaka, T., Ha, T.X.D.: On cone convexity of set-valued maps. Nonlinear Anal. Theory Methods Appl. 30(3), 1487–1496 (1997)MathSciNetCrossRef
22.
Zurück zum Zitat Lewandowski, M.: Complementary symmetry in cumulative prospect theory with random reference. J. Math. Psychol. 82, 52–55 (2018)MathSciNetCrossRef Lewandowski, M.: Complementary symmetry in cumulative prospect theory with random reference. J. Math. Psychol. 82, 52–55 (2018)MathSciNetCrossRef
23.
Zurück zum Zitat Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Finance 17(2), 1450012 (2014)MathSciNetCrossRef Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Finance 17(2), 1450012 (2014)MathSciNetCrossRef
24.
Zurück zum Zitat Löhne, A., Rudloff, B., Ulus, F.: Primal and dual approximation algorithms for convex vector optimization problems. J. Glob. Optim. 60(4), 713–736 (2014)MathSciNetCrossRef Löhne, A., Rudloff, B., Ulus, F.: Primal and dual approximation algorithms for convex vector optimization problems. J. Glob. Optim. 60(4), 713–736 (2014)MathSciNetCrossRef
25.
Zurück zum Zitat Luc, D.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer (1989) Luc, D.: Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer (1989)
27.
Zurück zum Zitat Nobakhtian, S., Shafiei, N.: A Benson type algorithm for nonconvex multiobjective programming problems. TOP 25(2), 271–287 (2017)MathSciNetCrossRef Nobakhtian, S., Shafiei, N.: A Benson type algorithm for nonconvex multiobjective programming problems. TOP 25(2), 271–287 (2017)MathSciNetCrossRef
28.
29.
Zurück zum Zitat Ok, E., Ortoleva, P., Riella, G.: Incomplete preferences under uncertainty: indecisiveness in beliefs versus tastes. Econometrica 80(4), 1791–1808 (2012)MathSciNetCrossRef Ok, E., Ortoleva, P., Riella, G.: Incomplete preferences under uncertainty: indecisiveness in beliefs versus tastes. Econometrica 80(4), 1791–1808 (2012)MathSciNetCrossRef
30.
Zurück zum Zitat Pagani, E.: Certainty equivalent: many meanings of a mean. Working Paper Series Department of Economics, University of Verona (2015) Pagani, E.: Certainty equivalent: many meanings of a mean. Working Paper Series Department of Economics, University of Verona (2015)
31.
Zurück zum Zitat Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42(4), 499–524 (1984)MathSciNetCrossRef Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42(4), 499–524 (1984)MathSciNetCrossRef
32.
Zurück zum Zitat Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126(3), 473–501 (2005)MathSciNetCrossRef Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126(3), 473–501 (2005)MathSciNetCrossRef
33.
Zurück zum Zitat Von Neumann, N., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947)MATH Von Neumann, N., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947)MATH
34.
Zurück zum Zitat Wang, S.Q.: Utility. Junior Project. Princeton University, Princeton (2010) Wang, S.Q.: Utility. Junior Project. Princeton University, Princeton (2010)
Metadaten
Titel
Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization
verfasst von
Birgit Rudloff
Firdevs Ulus
Publikationsdatum
10.10.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 2/2021
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-020-00282-x

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