Abstract
We examine a continuous-time principal-agent problem under mean-volatility joint ambiguity uncertainties. Both the principal and the agent exhibit Gilboa–Schmeidler’s extreme ambiguity aversion with exponential utilities. We distinguish between expost realized and exante perceived volatilities, and argue that the second-best contract necessarily consists of two sharing rules: one for realized outcome and the other for realized volatility. The outcome-sharing rule is for uncertainty sharing and work incentives, as usual, and the volatility-sharing rule is to align the agent’s worst prior with that of the principal. At optimum, their worst priors are symmetrized, and realized compensation is positively related to realized volatility. This theoretical positive relation can be consistent with popular managerial compensation practices such as restricted stock plus stock option grants. A closed-form solution to a linear-quadratic example is provided.
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References
Baker, G.P., Hall, B.J.: CEO incentives and firm size. J. Labor Econ. 22(4), 767–798 (2004)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wieley-interscience, Hoboken (2006)
Bouchard, B., Nutz, M.: Weak dynamic programming for generalized state constraints. SIAM J. Control Optim. 50(6), 3344–3373 (2102)
Chen, X., Sung, J.: Managerial Compensation and outcome volatility. Working paper (2018)
Chen, Z., Epstein, L.G.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70, 1403–1443 (2002)
Coles, J.L., Daniel, N.D., Naveen, L.: Managerial incentives and risk-taking. J. Financ. Econ. 76, 431–468 (2006)
Core, J., Guay, W.: The other side of the trade-off: the impact of risk on executive compensation: a revised comment. Working Paper, University of Pennsylvania (2002)
Cvitanić, J., Possamaï, D., Touzi, N.: Moral hazard in dynamic risk management. Manag. Sci. 63, 3328–3346 (2017a)
Cvitanić, J., Possamaï, D., Touzi, N.: Dynamic programming approach to principal agent problems. Finance Stochast. 22, 1–37 (2017b)
Davis, M.H.A.: On the existence of optimal policies in stochastic control. SIAM J. Control 11, 587–594 (1973)
Davis, M.H.A.: Martingale methods in stochastic control. In: Kohlmann, M., Vogel, W. (eds.) Stochastic Differential Systems. Lecture Notes in Control and Information Science, vol. 16, pp. 85–117. Springer Verlag, New York (1979)
Davis, M.H.A., Varaiya, P.: Dynamic programming conditions for partially observable stochastic systems. SIAM J. Control 11, 226–261 (1973)
Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827–852 (2006)
Ditttmann, I., Maug, E.: Lower salaries and no options? On the optimal structure of executive pay. J. Finance 62, 303–343 (2007)
Edmans, A., Gabaix, X.: The effect of risk on the CEO market. Rev. Financ. Stud. 24, 2822–2863 (2011)
Edmans, A., Gabaix, X., Jenter, D.: Executive compensation: a survey of theory and evidence. Working Paper (2017)
Epstein, L.G., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740–1786 (2013)
Epstein, L.G., Schneider, M.: Ambiguity, information quality and asset pricing. J. Finance 63, 197–228 (2008)
Epstein, L.G., Schneider, M.: Ambiguity, and asset markets. Annu. Rev. Financ. Econ. 2, 315–46 (2010)
Gilboa, I., Schmeidler, D.: Maximin expected utility with non-unique priors. J. Math. Econ. 18, 141–153 (1989)
Guay, W.: The sensitivity of CEO wealth to equity risk: and analysis of the magnitude and determinants. J. Financ. Econ. 53, 43–71 (1999)
Hall, B.J., Murphy, K.: Stock options for undiversified executives. J. Account. Econ. 33, 3–42 (2002)
Hellwig, M., Schmidt, K.M.: Discrete-time approximation of Holmström–Milgrom Brownian-motion model of intertemporal incentive provision. Econometrica 70, 2225–264 (2002)
Hirshleifer, D., Suh, R.: Risk, managerial effort, and project choice. J. Financ. Intermed. 2, 308–345 (1992)
Holmstrom, B., Milgrom, P.: Aggregation and linearity in the provision of intertemporal incentives. Econometrica 55, 303–328 (1987)
Klibanoff, P., Marinacci, M., Mukerji, S.: A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892 (2005)
Liu, Q., Liu, L., Sun, B.: Incentive contracting under ambiguity aversion. Econ. Theor. 66, 929–950 (2018). https://doi.org/10.1007/s00199-017-1073-9
Mastrolia, T., Possamaï, D.: Moral hazard under ambiguity. J. Optim. Theory Appl. 179(2), 452–500 (2018)
Miao, J., Rivera, A.: Robust contracts in continuous time. Econometrica 84, 1405–1440 (2016)
Murphy, K.J.: Executive compensation: where we are, and how we got there. In: Constantinides, G., Harris, M., Stulz, R. (eds.) Handbook of the Economics of Finance, pp. 211–356. Elsevier, Amsterdam (2013)
Neufeld, A., Nutz, M.: Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18(48), 1–14 (2013)
Nutz, M.: Pathwise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1–7 (2012a)
Nutz, M.: A quasi-sure approach to the control of non-Markovian stochastic differential equations. Electron. J. Probab. 17(23), 1–23 (2012b)
Nutz, M., Soner, H.M.: Superhedging and dynamic risk measures under volatility uncertainty. SIAM J. Control Optim. 50(4), 2065–2089 (2012)
Ou-Yang, H.: Optimal contracts in a continuous-time delegated portfolio management problem. Rev. Financ. Stud. 16, 173–208 (2003)
Pham, T., Zhang, J.: Two person zero-sum game in weak formulation and path dependent Bellman–Isaacs equation. SIAM J. Control Optim. 52(4), 2090–2121 (2014)
Rishel, R.: Necessary and sufficient dynamic programming conditions for continuous-time stochastic optimal control. SIAM J. Control 8, 559–571 (1970)
Sannikov, Y.: A continuous-time version of the principal-agent problem. Rev. Econ. Stud. 75, 957–984 (2008)
Schättler, H., Sung, J.: The first-order approach to continuous-time principal-agent problem with exponential utility. J. Econ. Theory 61, 331–371 (1993)
Schättler, H., Sung, J.: On optimal sharing rules in discrete- and continuous-time principal-agent problems with exponential utility. J. Econ. Dyn. Control 21, 551–574 (1997)
Soner, M., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis though aggregation. Electron. J. Probab. 16, 1844–1879 (2011)
Soner, M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Relat. Fields 153, 149–190 (2012)
Sung, J.: Linearity with project selection and controllable diffusion rate in continuous-time principal-agent problems. RAND J. Econ. 24, 720–743 (1995)
Szydlowski, M.: Ambiguity in dynamic contracts. Working Paper, University of Minnesota (2012)
Takayama, A.: Mathematical Economics, 2nd edn. Cambridge University Press, Cambridge (1985)
Weinschenk, P.: Moral hazard and ambiguity. Working Paper, Bonn Graduate School of Economics (2010)
Wu, Y., Yang, J., Zou, Z.: Ambiguity sharing and the lack of relative performance. Econ. Theor. 66, 141–157 (2018). https://doi.org/10.1007/s00199-017-1056-x
Zábojník, J.: Pay-performance sensitivity and production uncertainty. Econ. Lett. 53, 291–296 (1996)
Zhang, J.: Backwards Stochastic Differential Equations: From Linear to Fully Nonlinear Theory. Springer, New York (2017)
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I would like to thank for useful comments/discussions anonymous referees, Xiaoyan Chen, Zengjing Chen, Jaksa Cvitanić, Shige Peng, Jianfeng Zhang, and participants in seminars at Nanjing University of Science and Technology, Shanghai University of Finance and Economics, University of Southern California, 2017 Workshop on Mathematical Finance and Financial Data Processing at Qufu Normal University, China, and North American Summer Meeting of the Econometric Society 2018. All remaining errors of course are mine.
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Sung, J. Optimal contracting under mean-volatility joint ambiguity uncertainties. Econ Theory 74, 593–642 (2022). https://doi.org/10.1007/s00199-021-01362-9
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DOI: https://doi.org/10.1007/s00199-021-01362-9
Keywords
- Mean-volatility ambiguity
- Perception asymmetry
- Moral hazard
- Optimal contract
- Stock option
- Managerial compensation
- Volatility control