×

zbMATH — the first resource for mathematics

Exponential utility maximization under model uncertainty for unbounded endowments. (English) Zbl 1419.91285
Summary: We consider the robust exponential utility maximization problem in discrete time: An investor maximizes the worst case expected exponential utility with respect to a family of nondominated probabilistic models of her endowment by dynamically investing in a financial market, and statically in available options.
We show that, for any measurable random endowment (regardless of whether the problem is finite or not) an optimal strategy exists, a dual representation in terms of (calibrated) martingale measures holds true, and that the problem satisfies the dynamic programming principle (in case of no options). Further, it is shown that the value of the utility maximization problem converges to the robust superhedging price as the risk aversion parameter gets large, and examples of nondominated probabilistic models are discussed.

MSC:
91B16 Utility theory
49L20 Dynamic programming in optimal control and differential games
60G42 Martingales with discrete parameter
91G80 Financial applications of other theories
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Acciaio, B., Beiglböck, M., Penkner, F. and Schachermayer, W. (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance26 233–251. · Zbl 1378.91129
[2] Backhoff Veraguas, J. D. and Fontbona, J. (2016). Robust utility maximization without model compactness. SIAM J. Financial Math.7 70–103. · Zbl 1368.91163
[3] Bartl, D. (2016). Conditional nonlinear expectations. Preprint. Available at arXiv:1612.09103v2.
[4] Bartl, D., Drapeau, S. and Tangpi, L. (2017). Computational aspects of robust optimized cer-tainty equivalents and option pricing. Math. Finance. To appear.
[5] Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-independent bounds for option prices—A mass transport approach. Finance Stoch.17 477–501. · Zbl 1277.91162
[6] Beiglböck, M., Nutz, M. and Touzi, N. (2017). Complete duality for martingale optimal transport on the line. Ann. Probab.45 3038–3074. · Zbl 1417.60032
[7] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering139. Academic Press, New York. · Zbl 0471.93002
[8] Blanchard, R. and Carassus, L. (2017). Convergence of utility indifference prices to the superreplication price in a multiple-priors framework. Preprint. Available at arXiv:1709.09465.
[9] Blanchard, R. and Carassus, L. (2018). Multiple-priors optimal investment in discrete time for unbounded utility function. Ann. Appl. Probab.28 1856–1892. · Zbl 1411.91484
[10] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab.25 823–859. · Zbl 1322.60045
[11] Burzoni, M., Frittelli, M., Hou, Z., Maggis, M. and Obłój, J. (2016). Pointwise arbitrage pricing theory in discrete time. Math. Oper. Res. To appear.
[12] Cheridito, P. and Kupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. Int. J. Theor. Appl. Finance14 137–162. · Zbl 1211.91147
[13] Cheridito, P., Kupper, M. and Tangpi, L. (2015). Representation of increasing convex functionals with countably additive measures. Preprint. Available at arXiv:1502.05763. · Zbl 06667688
[14] Cheridito, P., Kupper, M. and Tangpi, L. (2017). Duality formulas for robust pricing and hedging in discrete time. SIAM J. Financial Math.8 738–765. · Zbl 1407.91243
[15] Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance12 99–123. · Zbl 1072.91019
[16] Dellacherie, C. and Meyer, P.-A. (1988). Probabilities and Potential. C: Potential Theory for Discrete and Continuous Semigroups. North-Holland, Amsterdam. · Zbl 0716.60001
[17] Deng, S., Tan, X. and Yu, X. (2018). Utility maximization with proportional transaction costs under model uncertainty. Preprint. Available at arXiv:1805.06498.
[18] Denis, L. and Kervarec, M. (2013). Optimal investment under model uncertainty in nondominated models. SIAM J. Control Optim.51 1803–1822. · Zbl 1331.60076
[19] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York. · Zbl 0904.60001
[20] Föllmer, H. and Leukert, P. (1999). Quantile hedging. Finance Stoch.3 251–273. · Zbl 0977.91019
[21] Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, 3rd revised and extended ed. de Gruyter, Berlin.
[22] Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance10 39–52. · Zbl 1013.60026
[23] Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with nonunique prior. J. Math. Econom.18 141–153. · Zbl 0675.90012
[24] Gundel, A. (2005). Robust utility maximization for complete and incomplete market models. Finance Stoch.9 151–176. · Zbl 1106.91027
[25] Hobson, D. G. (1998). Robust hedging of the lookback option. Finance Stoch.2 329–347. · Zbl 0907.90023
[26] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab.15 1691–1712. · Zbl 1083.60048
[27] Jacod, J. and Shiryaev, A. N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch.2 259–273. · Zbl 0903.60036
[28] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab.9 904–950. · Zbl 0967.91017
[29] Lacker, D. (2018). Liquidity, risk measures, and concentration of measure. Mathematics of Operations43. · Zbl 1396.65013
[30] Leese, S. J. (1978). Measurable selections and the uniformization of Souslin sets. Amer. J. Math.100 19–41. · Zbl 0384.28005
[31] Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica74 1447–1498. · Zbl 1187.91066
[32] Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab.15 2113–2143. · Zbl 1134.91449
[33] Matoussi, A., Possamaï, D. and Zhou, C. (2015). Robust utility maximization in nondominated models with 2BSDE: The uncertain volatility model. Math. Finance25 258–287. · Zbl 1335.91067
[34] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory3 373–413. · Zbl 1011.91502
[35] Neufeld, A. and Nutz, M. (2018). Robust utility maximization with Lévy processes. Math. Finance28 82–105. · Zbl 1403.91321
[36] Neufeld, A. and Sikic, M. (2017). Nonconcave robust optimization under knightian uncertainty. Preprint. Available at arXiv:1711.03875.
[37] Neufeld, A. and Šikić, M. (2018). Robust utility maximization in discrete-time markets with friction. SIAM J. Control Optim.56 1912–1937. · Zbl 1387.93189
[38] Nutz, M. (2016). Utility maximization under model uncertainty in discrete time. Math. Finance26 252–268. · Zbl 1378.91114
[39] Nutz, M. and van Handel, R. (2013). Constructing sublinear expectations on path space. Stochastic Process. Appl.123 3100–3121. · Zbl 1285.93104
[40] Owari, K. (2011). Robust utility maximization with unbounded random endowment. In Advances in Mathematical Economics. Adv. Math. Econ.14 147–181. Springer, Tokyo. · Zbl 1236.91150
[41] Peng, S. (2007). \(G\)-expectation, \(G\)-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications. Abel Symp.2 541–567. Springer, Berlin. · Zbl 1131.60057
[42] Quenez, M.-C. (2004). Optimal portfolio in a multiple-priors model. In Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability58 291–321. Birkhäuser, Basel. · Zbl 1061.93100
[43] Schied, A. (2006). Risk measures and robust optimization problems. Stoch. Models22 753–831. · Zbl 1211.91151
[44] Sion, M. (1958). On general minimax theorems. Pacific J. Math.8 171–176. · Zbl 0081.11502
[45] Villani, C.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.