Controlled options: derivatives with added flexibility

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The paper introduces a limit version of multiple stopping options such that the holder selects dynamically a weight function that control the distribution of the payments (benefits) over time. In applications for commodities and energy trading, a control process can represent the quantity that can be purchased by a fixed price at current time. In another example, the control represents the weight of the integral in a modification of the Asian option. The pricing for these options requires to solve a stochastic control problem. Some existence results and pricing rules are obtained via modifications of parabolic Bellman equations.