American Options with Discontinuous Two-Level Caps

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This paper examines the valuation of American capped call options with two-level caps. The structure of the immediate exercise region is significantly more complex than in the classical case with constant cap. When the cap grows over time, making extensive use of probabilistic arguments and local time, we show that the exercise region can be the union of two disconnected set. Alternatively, it can consist of two sets connected by a line. The problem then reduces to the characterization of the upper boundary of the first set, which is shown to satisfy a recursive integral equation. When the cap decreases over time, the boundary of the exercise region has piecewise constant segments alternating with non-increasing segments. General representation formulas for the option price, involving the exercise boundaries and the local time of the underlying price process, are derived. An efficient algorithm is developed and numerical results are provided.