Moral Hazard in Dynamic Risk Management

Abstract

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent's action is explicitly characterized, and, using the recent theory of singular changes of measures for Itô processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstrom-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.

Contract Theory in a VUCA World

“…As explained in [38], then in [13,14], the problem (4.7) coincides with the weak formulation of a (non standard) zero-sum stochastic differential game with the following characteristics • state variables: the output process X x,Θ and the continuation utility of the Agent Y y,Θ , with dynamic given for any t ≤ s ≤ T , P − a.s., by…”

“…We now fix an arbitrary Y 0 ∈ Y 0 and turn to the procedure to solve (4.7). The main issue is that the class of controls K Y 0 is too general since, as explained in [31, Section 4.3.2] and [13,14], the non-decreasing process K impacts the dynamic of Y Y 0 ,Z,K only throught the minimality condition (3.3) and more information on this process is required to solve the problem. As emphasized in [39,Remark 3.4], we need to consider piecewise controls and restrict our investigation on elementary strategies.…”

Regulation of Renewable Resource Exploitation

“…Taking into account this optimal reaction of the Agent, the goal is to compute the optimal compensation. As emphasized in [25] and then in [8,9], this problem remains to a (classical) stochastic control problem with the wealth of the Principal and the continuation utility value of the Agent as state variables.…”

Dynamic optimal contract under parameter uncertainty with risk-averse agent and principal

“…A nice survey of the literature was provided by Sung in [19]. Recently, [3] considered a general formulation of the principal-agent problem with a lump-sum payment on a finite horizon, where the agent influences both the agent and the volatility of the output, where the proofs use techniques based on a backward stochastic differential equations approach to non-Markovian stochastic control.…”