In many cases, demand for electricity can be viewed as demand for energy over a time horizon and not instantaneous demand for power (i.e., energy rate). Demand for energy translates to more flexibility in energy delivery. Assuming availability of proper information, which is promised by smart grids, this flexibility can be utilized to reduce costs for the consumers alongside providing other benefits. In this paper, we propose a stochastic model for delayaverse flexible demands subject to real-time stochastic spot prices. Based on this model, we obtain the optimal consumption policy and discuss its computational efficiency under different assumptions. Using this optimal scheme we quantify the value of time flexibility (i.e., delay tolerance) in terms of the reduction in the expected cost of satisfying consumer demand as well as the cost-delay trade-off. Finally, through simulations, we analyze the collective behavior of such opportunistic loads and their effects on the power system. Beyond obtaining computationally efficient algorithms for optimal behavior of delay-averse flexible loads, our model provides insights into the value of time flexibility and exhibits cost-delay trade-offs. Furthermore, we show that opportunism on the demand side in real-time pricing environments can result in undesired effects in terms of the aggregate power profile of the loads.