Installment Options : A Closed-Form Solution and the Limiting Case

In Foreign Exchange Markets Compound options (options on options) are traded frequently. Installment options generalize the concept of Compound options as they allow the holder to prolong a Vanilla Call or Put option by paying installments of a discrete payment plan. We derive a closed-form solution to the value of such an option in the Black-Scholes model and prove that the limiting case of an Installment option with a continuous payment plan is equivalent to a portfolio consisting of a European Vanilla option and an American Put on this Vanilla option with a time-dependent strike