This paper proposes a new continuous-time first best contract framework that has the ability to explicitly limit the agent's risks. This limitation of risk is achieved by formulating the contract design problem as a mean-variance constrained stochastic optimal control problem. To obtain a globally optimal contract even when system dynamics are nonlinear, we develop a dynamic programming-based solution approach. The major theoretical challenge arises from the variance inequality constraint. To overcome this difficulty, we track and limit the agent's risk using a new stochastic system, whose state value can be interpreted as the agent's remaining budget for risks. We also propose an approximately decoupled contract design approach for multiple agents to resolve the scalability issue inherent in dynamic programming. The procedures in this contract design for multiple agents can be completely parallelized. We show that this approximate contract satisfies each agent's individual rationality condition and has a provable suboptimality bound. The performance and usefulness of the proposed contract method and its application to demand response are demonstrated using data on the electric energy consumption of customers in Austin, Texas, and the locational marginal price data from the Electricity Reliability Council of Texas.