Valuing American Continuous-Installment Options

Installment options are weakly path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividend-paying assets. The setup is a standard Black-Scholes-Merton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping and early exercise boundaries. Abelian theorems of Laplace transforms enable us to obtain a concise result for the perpetual case. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the boundaries.