American Continuous-Installment Options: Valuation and Premium Decomposition

Abstract

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 continuous-installment 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 the Laplace transform of the initial premium in an explicit form, which is decomposed into the value of the associated European vanilla option with the same payoff plus the premiums of early exercise and halfway cancellation. We also obtain a pair of nonlinear equations for the Laplace transforms of the boundaries. An Abelian theorem of Laplace transforms enables 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.

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