Valuation of American Continuous-Installment Options

We present three approaches to value American continuous-installment calls and puts and compare their computational precision. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black-Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponen∗We are grateful to Manfred Gilli, Henri Loubergé and Evis Këllezi for encouragements, suggestions and remarks.