Abstract
The aim of this paper is to describe globally the behavior and preferences of heterogeneous agents. Our starting point is the aggregate wealth of a given economy, with a given repartition of the wealth among investors, which is not necessarily Pareto optimal. We propose a construction of an aggregate forward utility, market consistent, that aggregates the marginal utility of the heterogeneous agents. This construction is based on the aggregation of the pricing kernels of each investor. As an application we analyze the impact of the heterogeneity and of the wealth market on the yield curve.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
\(\kappa _t= \sigma _t.\kappa _t = \sigma _{t}^{T}\pi _{t}\), with \(\pi \) being the fraction of wealth invested in the risky assets, and \(\sigma \) being the volatility process, and .\(^{T}\) denotes the transpose operator.
- 2.
That is \(\mathbf U\) is a \({\mathcal{K}}^{1,\delta }_{loc}\cap {{\mathcal C}}^{2}\)-semimartingale, see the Appendix and Theorem 5.1(iv).
- 3.
See the Appendix for the definition of these classes of regularity.
- 4.
Actually, one may choose any deterministic initial utility, as soon as it satisfies sufficient integrability conditions, as the ones required in Theorem 2.11.
- 5.
That is \(\phi \) is m-times continuously differentiable with \(\phi ^{(m)}\) being \(\delta \)-Hölder.
References
Bank, P., Kramkov, D.: A model for a large investor trading at market indifference prices. ii: continuous-time case. Ann. Appl. Probab. 25(5), 2708–2742 (2015)
Berrier, F.P., Rogers, L.G.C., Tehranchi, M.R.: A characterization of forward utility functions. Preprint (2009)
Berrier, F.P., Tehranchi, M.R.: Forward utility of investment and consumption. Preprint (2011)
Bjork, T.: Equilibrium theory in continuous time. In: Lecture Notes (2012)
Chan, Y.L., Kogan, L.: Catching up with the joneses: heterogeneous preferences and the dynamics of asset prices. J. Polit. Econ. 110, 1255–1285 (2002)
Cox, J.C., Ingersoll, J.C., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53(2), 385–403 (1985)
Cvitanić, J., Jouini, E., Malamud, S., Napp, C.: Financial markets equilibrium with heterogeneous agents. Rev. Financ. 16(1), 285–321 (2011)
Davis, M.H.A.: Option pricing in incomplete markets. In Pliska, S.R. (ed.) Mathematics of Derivative Securities, pp. 216–226. M.A.H. Dempster and S.R. Pliska, Cambridge University Press edition (1998)
Dumas, B.: Two-person dynamic equilibrium in the capital market. Rev. Financ. Stud. 2(2), 157–188 (1989)
Dybvig, P.H., Rogers, L.C.G.: Recovery of preferences from observed wealth in a single realization. Rev. Financ. Stud. 10(1), 151–174 (1997)
El Karoui, N., Hillairet, C., Mrad, M.: Affine long term yield curves: an application of the Ramsey rule with progressive utility. J. Financ. Eng. 1(1) (2014)
El Karoui, N., Hillairet, C., Mrad, M.: Consistent utility of investment and consumption: a forward/backward spde viewpoint. Stochastics (2017)
El Karoui, N., Hillairet, C., Mrad, M.: Ramsey rule with forward/backward utility for long term yield curves modeling. Preprint (2017)
El Karoui, N., Mrad, M.: Mixture of consistent stochastic utilities, and a priori randomness. Preprint (2016)
El Karoui, N., Mrad, M.: Recover dynamic utility from monotonic characteristic/extremal processes (2019), in revision. https://hal.archives-ouvertes.fr/hal-01966312
El Karoui, N., Mrad, M.: An exact connection between two solvable SDEs and a non linear utility stochastic PDEs. SIAM J. Financ. Math. 4(1), 697–736 (2013)
El Karoui, N., Geman, H., Rochet, J.C.: Changes of numeraire, changes of probability measure and option pricing. J. Appl. Probab. 32(2), 443–458 (1995)
Filipovic, D., Platen, E.: Consistent market extensions under the benchmark approach. Math. Financ. 19(1), 41–52 (2009)
Geman, H., El Karoui, N., Rochet, J.-C.: Changes of numeraire, changes of probability measure and option pricing. J. Appl. Probab. 32(2), 443–458 (1995)
Henderson, V., Hobson, D.: Indifference Pricing: Theory and Application (Ed. R. Carmona). Princeton University Press, Princeton (2009)
Ikefuji, M., Laeven, R.J.A., Magnus, J.R., Muris, C.: Pareto utility. Theor. Decis. 75(1), 43–57 (2013)
Jouini, E., Napp, C.: Unbiased disagreement in financial markets, waves of pessimism and the risk-return trade-off. Rev. Financ. 15(3), 575–601 (2010)
Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer (2001)
Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Optimal portfolio and consumption decisions for a small investor on a finite horizon. SIAM J. Control Optim. 25, 1557–1586 (1987)
Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)
Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13(4), 1504–1516 (2003)
Kunita, H.: Stochastic flows and stochastic differential equations. In: Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press, Cambridge (1997). Reprint of the 1990 original
Musiela, M., Zariphopoulou, T.: Backward and forward utilities and the associated pricing systems: the case study of the binomial model. In: Indifference Pricing: Theory and Application, pp. 3–44. Princeton University Press (2009)
Musiela, M., Zariphopoulou, T.: Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. In: Advances in Mathematical Finance, pp. 303–334. Birkhäuser, Boston (2007)
Musiela, M., Zariphopoulou, T.: Stochastic partial differential equations in portfolio choice. In: Chiarella, C., Novikov, A. (eds.) Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, pp. 161–170 (2010)
Musiela, M., Zariphopoulou, T.: Portfolio choice under space-time monotone performance criteria. SIAM J. Financ. Math. 1(1), 326–365 (2010)
Piazzesi, M.: Affine term structure models. Handbook Financ. Econ. 1, 691–766 (2010)
Platen, E., Heath, D.: A benchmark approach to quantitative finance. In: Springer Finance. Springer-Verlag, Berlin (2006)
Rogers, L.C.G.: Duality in constrained optimal investment and consumption problems: a synthesis. In: Paris-Princeton Lectures on Mathematical Finance 2002, pp. 95–131. Springer (2003)
Vasicek, O.: An equilibrium characterization of the term structure. J. Financ. Econ. 5(2), 177–188 (1977)
Wang, J.: The term structure of interest rates in a pure exchange economy with heterogeneous investors. J. Financ. Econ. 41(1), 75–110 (1996)
Yan, H.: Is noise trading cancelled out by aggregation? Manag. Sci. 56(7), 1047–1059 (2010)
Zitkovic, G.: A dual characterization of self-generation and log-affine forward performances. Ann. Appl. Probab. 19(6), 2176–2210 (2009)
Acknowledgements
The authors thank the financial supports of Chaire “Risques Financiers”, of Labex Ecodec and of Labex MME-DII.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
El Karoui, N., Hillairet, C., Mrad, M. (2019). Construction of an Aggregate Consistent Utility, Without Pareto Optimality. Application to Long-Term Yield Curve Modeling. In: Cohen, S., Gyöngy, I., dos Reis, G., Siska, D., Szpruch, Ł. (eds) Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications. BSDE-SPDE 2017. Springer Proceedings in Mathematics & Statistics, vol 289. Springer, Cham. https://doi.org/10.1007/978-3-030-22285-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-22285-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-22284-0
Online ISBN: 978-3-030-22285-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)