Numéraire-invariant preferences in financial modeling

Numéraire-invariant preferences in financial modeling

Constantinos Kardaras

Source: Ann. Appl. Probab. Volume 20, Number 5 (2010), 1697-1728.

Abstract

We provide an axiomatic foundation for the representation of numéraire-invariant preferences of economic agents acting in a financial market. In a static environment, the simple axioms turn out to be equivalent to the following choice rule: the agent prefers one outcome over another if and only if the expected (under the agent’s subjective probability) relative rate of return of the latter outcome with respect to the former is nonpositive. With the addition of a transitivity requirement, this last preference relation has an extension that can be numerically represented by expected logarithmic utility. We also treat the case of a dynamic environment where consumption streams are the objects of choice. There, a novel result concerning a canonical representation of unit-mass optional measures enables us to explicitly solve the investment–consumption problem by separating the two aspects of investment and consumption. Finally, we give an application to the problem of optimal numéraire investment with a random time-horizon.

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Primary Subjects: 60G07, 91B08, 91B16, 91B28

Keywords: Preferences; choice rules; numéraire-invariance; optional measures; investment–consumption problem; random time-horizon utility maximization

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1282747398
Digital Object Identifier: doi:10.1214/09-AAP669
Mathematical Reviews number (MathSciNet): MR2724418

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References

[1] Ankirchner, S., Dereich, S. and Imkeller, P. (2006). The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34 743–778.

Mathematical Reviews (MathSciNet): MR2223957

Zentralblatt MATH: 1098.60065

Digital Object Identifier: doi:10.1214/009117905000000648

Project Euclid: euclid.aop/1147179988

[2] Ansel, J.-P. and Stricker, C. (1994). Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré Probab. Statist. 30 303–315.

Mathematical Reviews (MathSciNet): MR1277002

Zentralblatt MATH: 0796.60056

[3] Becherer, D. (2001). The numeraire portfolio for unbounded semimartingales. Finance Stoch. 5 327–341.

Mathematical Reviews (MathSciNet): MR1849424

Digital Object Identifier: doi:10.1007/PL00013535

[4] Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk. Econometrica 22 23–36.

Mathematical Reviews (MathSciNet): MR59834

Digital Object Identifier: doi:10.2307/1909829

JSTOR: links.jstor.org

[5] Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M. and Martellini, L. (2008). Optimal investment decisions when time-horizon is uncertain. J. Math. Econom. 44 1100–1113.

Mathematical Reviews (MathSciNet): MR2456470

Zentralblatt MATH: 1153.91018

Digital Object Identifier: doi:10.1016/j.jmateco.2007.09.004

[6] Bouchard, B. and Pham, H. (2004). Wealth-path dependent utility maximization in incomplete markets. Finance Stoch. 8 579–603.

Mathematical Reviews (MathSciNet): MR2212119

Zentralblatt MATH: 1063.91029

[7] Brannath, W. and Schachermayer, W. (1999). A bipolar theorem for L⁰₊(Ω, ℱ, P). In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 349–354. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1767997

[8] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.

Mathematical Reviews (MathSciNet): MR1304434

Zentralblatt MATH: 0865.90014

Digital Object Identifier: doi:10.1007/BF01450498

[9] Elworthy, K. D., Li, X. M. and Yor, M. (1997). On the tails of the supremum and the quadratic variation of strictly local martingales. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 113–125. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1478722

Zentralblatt MATH: 0886.60035

[10] Goll, T. and Kallsen, J. (2003). A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13 774–799.

Mathematical Reviews (MathSciNet): MR1970286

Zentralblatt MATH: 1034.60047

Digital Object Identifier: doi:10.1214/aoap/1050689603

Project Euclid: euclid.aoap/1050689603

[11] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing.

Mathematical Reviews (MathSciNet): MR1219534

Zentralblatt MATH: 0781.60002

[12] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1943877

[13] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 447–493.

Mathematical Reviews (MathSciNet): MR2335830

Digital Object Identifier: doi:10.1007/s00780-007-0047-3

[14] Kelly, J. L. Jr. (1956). A new interpretation of information rate. Bell. System Tech. J. 35 917–926.

Mathematical Reviews (MathSciNet): MR90494

[15] Latane, H. A. (1959). Criteria for choice among risky ventures. Journal of Political Economy 67 144–155.

[16] Long, J. B. Jr. (1990). The numéraire portfolio. Journal of Financial Economics 26 29–69.

[17] Mas-Colell, A., Whinston, M. D. and Green, J. R. (1995). Microeconomic Theory. Oxford Univ. Press, Oxford.

[18] Nikeghbali, A. and Yor, M. (2006). Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois J. Math. 50 791–814.

Mathematical Reviews (MathSciNet): MR2247846

Zentralblatt MATH: 1101.60059

Project Euclid: euclid.ijm/1258059492

[19] Platen, E. and Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2267213

Zentralblatt MATH: 1104.91041

[20] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Cambridge Mathematical Library 2. Cambridge Univ. Press, Cambridge.

[21] Samuelson, P. A. (1971). The “fallacy” of maximizing the geometric mean in long sequences of investing or gambling. Proc. Natl. Acad. Sci. USA 68 2493–2496.

Mathematical Reviews (MathSciNet): MR295739

Zentralblatt MATH: 0226.62111

Digital Object Identifier: doi:10.1073/pnas.68.10.2493

JSTOR: links.jstor.org

[22] Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. Journal of Banking and Finance 3 305–307.

[23] Savage, L. J. (1972). The Foundations of Statistics, revised ed. Dover, New York.

Mathematical Reviews (MathSciNet): MR348870

[24] von Neumann, J. and Morgenstern, O. (2007). Theory of Games and Economic Behavior, anniversary ed. Princeton Univ. Press, Princeton, NJ.

Mathematical Reviews (MathSciNet): MR2316805

[25] Williams, J. B. (1936). Speculation and the carryover. Quarterly Journal of Economics 50 436–455.

[26] Žitković, G. (2005). Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Probab. 15 748–777.

Mathematical Reviews (MathSciNet): MR2114989

Zentralblatt MATH: 1108.91032

Digital Object Identifier: doi:10.1214/105051604000000738

Project Euclid: euclid.aoap/1107271667

[27] Žitković, G. (2008). Convex-compactness and its applications. Math. Financ. Econ. To appear.

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