×

Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. (English) Zbl 1132.91426

Summary: In the general framework of a semimartingale financial model and a utility function \(U\) defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a “small” number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases:
1. for any utility function \(U\), if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance;
2. for any financial model, if and only if \(U\) is a power utility function (\(U\) is an exponential utility function if it is defined on the whole real line).

MSC:

91B16 Utility theory
90C26 Nonconvex programming, global optimization
91B30 Risk theory, insurance (MSC2010)

References:

[1] Davis, M. H. A. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities (M. A. H. Dempster and S. R. Pliska, eds.) 216–226. Cambridge Univ. Press. · Zbl 0914.90017
[2] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520. · Zbl 0865.90014 · doi:10.1007/BF01450498
[3] Delbaen, F. and Schachermayer, W. (1997). The Banach space of workable contingent claims in arbitrage theory. Ann. Inst. H. Poincaré Probab. Statist. 33 113–144. · Zbl 0872.90008 · doi:10.1016/S0246-0203(97)80118-5
[4] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250. · Zbl 0917.60048 · doi:10.1007/s002080050220
[5] Föllmer, H. and Schied, A. (2004). Stochastic Finance : An Introduction in Discrete Time , 2nd ed. de Gruyter, Berlin. · Zbl 1126.91028
[6] Henderson, V. (2002). Valuation of claims on nontraded assets using utility maximization. Math. Finance 12 351–373. · Zbl 1049.91072 · doi:10.1111/j.1467-9965.2002.tb00129.x
[7] Henderson, V. and Hobson, D. G. (2002). Real options with constant relative risk aversion. J. Econom. Dynam. Control 27 329–355. · Zbl 1027.91039 · doi:10.1016/S0165-1889(01)00052-5
[8] Hugonnier, J. and Kramkov, D. (2004). Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14 845–864. · Zbl 1086.91030 · doi:10.1214/105051604000000134
[9] Hugonnier, J., Kramkov, D. and Schachermayer, W. (2005). On utility-based pricing of contingent claims in incomplete markets. Math. Finance 15 203–212. · Zbl 1124.91338 · doi:10.1111/j.0960-1627.2005.00217.x
[10] Ilhan, A., Jonsson, M. and Sircar, R. (2005). Optimal investment with derivative securities. Finance Stoch. 9 585–595. · Zbl 1092.91018 · doi:10.1007/s00780-005-0154-y
[11] Jacka, S. D. (1992). A martingale representation result and an application to incomplete financial markets. Math. Finance 2 239–250. · Zbl 0900.90044 · doi:10.1111/j.1467-9965.1992.tb00031.x
[12] Kallsen, J. (2002). Derivative pricing based on local utility maximization. Finance Stoch. 6 115–140. · Zbl 1007.91020 · doi:10.1007/s780-002-8403-x
[13] 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 · doi:10.1214/aoap/1029962818
[14] Kramkov, D. and Sîrbu, M. (2006). On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16 1352–1384. · Zbl 1149.91035 · doi:10.1214/105051606000000259
[15] Musiela, M. and Zariphopoulou, T. (2004). An example of indifference prices under exponential preferences. Finance Stoch. 8 229–239. · Zbl 1062.93048 · doi:10.1007/s00780-003-0112-5
[16] Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options. Bell J. Econom. 7 407–425.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.