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 instanta-neous 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 Louberge ́ and Evis Këllezi for encouragements, suggestions and remarks. 1 tial functions. This process is compared to the finite-difference method to solve the inhomogeneous Black-Scholes PDE and a Monte Carlo approach.