American Options by Perturbing Closed Form Solutions

The pricing and hedging of American options generally requires the numerical solution of optimal stopping problems.  This paper reveals analytical solutions for a class of optimal stopping problems, specifically maximising... [ view full abstract ]

The pricing and hedging of American options generally requires the numerical solution of optimal stopping problems.  This paper reveals analytical solutions for a class of optimal stopping problems, specifically maximising the expected product of a Wiener process and the survival function of a generalised Pareto distribution (GPD). The solution emerges because of a self-similarity property of the GPD, which is a consequence of the Pickands–Balkema–de Haan theorem. The property allows us to reduce the partial differential equation with movable boundary, to an ordinary differential equation. The same approach also works when the Wiener process is reflected at zero.