A systematic approach for valuing American-style installment options with continuous payment plan

In this paper we present an integral equation approach for the valuation of American-style installment derivatives when the premium payments, made continuously throughout the contract’s life, are assumed to be a function of the asset price and time. The contribution of this study is three-fold. First, we show that in the Black-Scholes framework the option pricing problem can be formulated as a free boundary problem under very general conditions on payo structure and payment plan. Second, by applying a Fourier transform-based solution technique, we derive two recursive integral equations for the free boundaries along with an analytic representation for the option price. Third, within this systematic treatment of the American installment options, we propose a unified and easily applicable method to deal with a wide range of monotonic payo functions and continuous payment plans. Finally, by using the illustrative example of American vanilla installment call options, an explicit pricing formula is obtained for payment schedules depending on the time only.