  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 It\^o processes, we study how
restrictive this family is. In particular, in the special case of the standard
Homlstr\"om-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.
