Chapter 7 - Question 1

The instalment option has the same payoff as a vanilla call of put option; it may be European or American. Its unusual feature is that, as well as paying the initial premium, the holder must pay 'instalments' during the life of the option. The instalments may be paid either continuously or discretely. The holder can choose at any time to stop paying the instalments, at which point the contract is cancelled and the option ceases to exist. When instalments are paid continuously at a rate L(t) per unit time, derive the differential equation satisfied by the option price. What new constraint must it satisfy? Formulate a free boundary problem for its value.