Continuous-Time Principal-Agent Problem with Drift and Stochastic Volatility Control

We introduce a continuous-time principal-agent model where the agent can privately influence both the drift and diffusion of the cash flows. The total diffusion is the product of the agent's volatility choice and a stochastic volatility process that is unobservable to the principal. This model is applicable to delegated portfolio management problems where the risky asset has stochastic volatility, and to corporate project management problems where both idiosyncratic project risks and exogenous macro risks determine total project cash flow risk. We find: (i) as long as the stochastic volatility cannot behave too erratically and that there are reasonable bounds to the agent's volatility choice, it is still optimal for the principal to make the agent's compensation variable to unexpected cash flow shocks, as in the case of classical drift-only control models; (ii) the lump-sum payment condition to the agent is random, and in particular it is a moving boundary dependent on the volatility level; and (iii) because of possible substitutability or complementarity trade-offs between effort and volatility for both the principal and the agent, an agent exerting higher effort needs not have higher compensation variability to cash flow shocks