Continuously controlled options: derivatives with added flexibility

Abstract

The paper introduces special options such that the holder selects dynamically a continuous time process controlling the distribution of the payments (benefits) over time. For instance, the holder can select dynamically the quantity of a commodity purchased or sold by a fixed price given constraints on the cumulative quantity. In a modification of the Asian option, the control process can represent the averaging kernel describing the distribution of the purchases. The pricing of these options requires to solve special stochastic control problems with constraints for the cumulative control similar to a knapsack problem. Some existence results and pricing rules are obtained via modifications of parabolic Bellman equations.