Abstract
In the context of model uncertainty, we study the optimal design and the pricing of financial instruments aiming to hedge some of non-tradable risks. For the existence of model uncertainty, the preference can be represented by the robust expected utility (also called maxmin expected utility) which can be put in the framework of sublinear expectation. The problem of maximizing the issuer’s robust expected utility under the constraint imposed by the buyer can be transformed to the problem of minimizing the issuer’s convex measure under the corresponding constraint. And here the convex measure measures not only the risks but also the model uncertainties.
Similar content being viewed by others
References
Ellsberg, D.: Risk, ambiguity, and the Savage axioms. Q. J. Econ., 75, 643–669 (1961)
Knight, F.: Risk, Uncertainty and Profit, Houghton Mifflin, Boston, 1921
Mehra, R., Prescott, E. C.: The equity premium: a puzzle. Journal of Monetary Economics, 15, 145–161 (1985)
Epstein, L. G., Wang, T.: Uncertainty, risk-neutral measures and security price booms and crashes. Journal of Economic Theory, 67, 40–80 (1995)
Routledge, B. R., Zin, S.: Model uncertainty and liquidity. NBER working Paper No. 8683, (2001)
Chen, Z., Epstein, L. G.: Ambiguity, risk and asset returns in continuous time. Econometrica, 70(4), 1403–1443 (2002)
Hansen, L., Sargent, T., Tallarini, T.: Robust permanent income and pricing. Rev. Econ. Stud., 66, 872–907 (1999)
Hansen, L., Sargent, T., Turluhambetova, G. A., Williams, N.: Robustness and uncertainty aversion. Working Paper (2002)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ., 18, 141–153 (1989)
Epstein, L. G.: A definition of uncertainty aversion. Rev. Econ. Stud., 65, 579–608 (1999)
El Karoui, N., Quenez, M. C.: Dynamic programming and the pricing of contingent claims in incomplete markets. SIAM J. Control Optim., 33(1), 29–66 (1995)
Avellaneda, M., Levy, A., Paras, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance, 2, 73–88 (1995)
Avellaneda, M., Levy, A., Paras, A.: Managing the volatility risk of portfolios of derivative securities: the lagrangian uncertain volatility model. Appl. Math. Finance, 3, 31–52 (1996)
Lyons, T. J.: Uncertain volatility and the risk free synthesis of derivatives. Appl. Math. Finance, 2, 117–133 (1995)
Epstein, L. G., Wang, T.: Intertemporal asset pricing under knightian uncertainty. Econometrica, 62, 283–322 (1994)
Artzner, P., Delbaen, F., Eber, J. M., Heath, D.: Thinking coherently. Risk, 10, 68–71 (1997)
Artzner, P., Delbaen, F., Eber, J. M., Heath, D.: Coherent measures of risk. Math. Finance, 9(3), 203–228 (1999)
Delbaen, F.: Coherent risk measures. Lectures given at the Cattedra Galileiana at the Scuola Normale Superiore di Pisa, March 2000, Published by the Scuola Normale Superiore di Pisa, 2002
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch., 6(4), 429–447 (2002)
Peng, S.: G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In: Stochastic Analysis and Applications, The Abel Symposium 2005, Abel Symposia 2, New York, Springer-Verlag, 2006, 541–567
Peng, S.: Law of large numbers and central limit theorem under nonlinear expectations. ar**v: math.PR/07-02358v1 13 Feb 2007
Peng, S.: G-Brownian motion and dynamic risk measure under volatility uncertainty. ar**v: 0711.2834v1 [math.PR] 19 Nov 2007
Peng, S.: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Processes Appl., 118(12), 2223–2253 (2008)
Peng, S.: A new central limit theorem under sublinear expectations. ar**v: 0803.2656v1 [math.PR] 18 Mar 2008
Peng, S.: Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China, Ser. A, 52(7), 1391–1411 (2009)
El Karoui, N., Rouge, R.: Pricing via utility maximization and entropy. Math. Finance, 10, 259–276 (2000)
Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance: Math. Econ., 33, 1–28 (2003)
Davis, M.: Option pricing in incomplete markets. In: Mathematics of Derivative Securities (Dempster, M. A. H. et al. eds.), Cambridge, Cambridge University Press, 1997, 227–254
Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential preferences. Finance Stochast., 8, 229–239 (2004)
Barrieu, P., El Karoui, N.: Inf-convolution of risk measures and optimal risk transfer. Finance Stochast., 9, 269–298 (2005)
Borch, K.: Equilibrium in a reinsurance market. Econometrica, 30, 424–444 (1962)
Föllmer, H., Schied, A.: Stochastic finance: An introduction in discrete time (De Gruyter Studies in Mathematics, 27), Berlin, New York, De Gruyter, 2004
Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Analysis, 34(2), 139–161 (2011)
Komlós, J.: A generalisation of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar., 18, 217–229 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Bei**g Natural Science Foundation (Grant No. 1112009)
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Fan, Y.L. The risk transfer of non-tradable risks under model uncertainty. Acta. Math. Sin.-English Ser. 28, 1597–1614 (2012). https://doi.org/10.1007/s10114-011-0281-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-011-0281-7