Moral Hazard Under Ambiguity

Abstract

In this paper, we extend the Holmström and Milgrom problem [47] by adding uncertainty about the volatility of the output for both the Agent and the Principal. We study more precisely the impact of the "Nature" playing against the Agent and the Principal by choosing the worst possible volatility of the output. We solve the first-best and the second-best problems associated with this framework and we show that optimal contracts are in a class of contracts similar to [14,15], linear with respect to the output and its quadratic variation. We compare our results with the classical problem in [47].to designing optimal incentives, and are therefore present in a very large number of situations. Beyond the obvious application to the optimal remuneration of an employee, one can for instance think on how regulators with imperfect information and limited policy instruments can motivate banks to operate entirely in the social interest, on how a company can optimally compensate its executives, on how banks achieve optimal securitisation of mortgage loans or on how investors should pay their portfolio managers (see Bolton and Dewatripont [6] or Laffont and Martimort [49] for many more examples).Early studies of the risk-sharing problem can be found, among others, in Borch [7], Wilson [99] or Ross [78]. Since then, a large literature has emerged, solving very general risk-sharing problems, for instance in a framework with several Agents and recursive utilities (see Duffie et al. [25] or Dumas et al. [26], or for studying optimal compensation of portfolio managers (see or Cadenillas et al. [11]). From the mathematical point of view, these problems can usually be tackled using either their dual formulation or the so-called stochastic maximum principle, which can characterize the optimal choices of the Principal and the Agent through coupled systems of Forward Backward Stochastic Differential Equations (FBSDEs in the sequel) (see the very nice monograph [20] by Cvitanić and Zhang for a systematic presentation). One of the main findings in almost all of these works, is that one can find an optimal contract which is linear in the terminal value of the output managed by the Agent (a result already obtained in [78]) and possibly some benchmark to which his performance is compared. In specific cases, one can even have Markovian optimal contracts which are given as a (possibly linear) functional of the terminal value of the output (see in particular [11] for details).Concerning the so-called moral hazard problem, the first paper on continuous-time Principal-Agent problems is the seminal paper by Holmström and Milgrom [47]. They consider a Principal and an Agent with exponential utility functions and find that the optimal contract is linear. Their work was generalized by Schättler and Sung [83,84], Sung [90,91], Müller [54,55], and Hellwig and Schmidt [46], using a dynamic programming and martingales approach, which is classical in stochastic control theory (see also the survey paper by Sung [92] for more references). The papers by Wi...

Contract Theory in a VUCA World

“…An example of estimate sets of volatility which satisfy Assumption 2.1 is the learning model presented in [31].…”

“…We study a generalization of both the classical problem of Holmström and Milgrom [24] and the problem of moral hazard under volatility uncertainty studied in [31,44]. In our model, the Agent is hired by the Principal to control the drift of the outcome process X, but none of them have certainty about what is the volatility of the project.…”

“…In this section we study the Agent's problem (2.6). We follow both the study made in Section 4.1 of [31] by extending it to a more general framework, and [14] by adding uncertainty on the volatility. We mention also that another approach which does not use the theory of 2BSDEs has been proposed in [44].…”

“…In this section, we aim at solving the contracting problem (2.7). This corresponds to an extension of both [14] to the uncontrolled volatility case and [31] in a more general model, without assuming that a dynamic programming principle holds for the value function of the Principal. We follow the ideas of [1, 2, 39].…”

“…As a consequence, both the Principal and the Agent play zero-sum stochastic differential games against the Nature. This work is an extension of the models proposed by Mastrolia and Possamaï [31] and Sung [44] to more general frameworks, by dropping several explicit and implicit assumptions made in these papers. We consider a more general framework by not considering only exponential utilities and by not restricting a priori the class of admissible contracts.…”