Alternative Characterizations of the European Continuous-Installment Option Valuation Problem

This paper is concerned with the pricing of European continuous-installment options where the aim is to determine the initial premium given the installment payments schedule. The particular feature of this pricing problem is the determination, along with the initial premium, of an optimal stopping boundary since the option holder has the right to stop paying the installments at any time before maturity. Given that installments are paid continuously, it can be possible to derive the Black-Scholes differential equation satisfied by the initial value of the option. Using this key result, we obtain two alternative characterizations of the European continuous-installment option valuation problem, for which no closed-form solution is available. First, we formulate the pricing problem as a free boundary value problem and using the integral representation method we obtain integral expressions for both the initial premium and the optimal stopping boundary. Second, we use the variational inequality formulation of the pricing problem for determining the initial value and the early stopping curve implicitly. To solve the system of discretized linear complementarity problems we adopt a Newton method