Instalment options are a simple extension of a plain vanilla contract with the added touch of being able to pay the premium of the option over a period of time. One can look at it as a series of compound options as the holder of the option is also able to cancel the instalment option at any of the premium payment periods.

Pricing:

The pricing of this option can be seen as string of n-1 compound options where n is the number of instalment dates. For example if the instalment option has 2 instalment dates, t = 0 and t = 1 then one can view this as a compound option initiated at t = 0 where the underlying is a call or put at t = 1. Compared to vanilla options, the potential profits and losses are less for an instalment option.

We can price the option under a standard Black-Scholes framework by considering the underlying asset to follow a Wiener process and closed form solutions can be attained.

Ben-Ameur, Breton & Fraincois (???) derive an approximation for instalment options by solving a dynamic programming equation through piecewise linear interpolation. Wystup & Griebsch (2003) note that the method derived by Ben-Ameur, Breton & Fraincois (???) works well when the instalment dates are spaced equally apart.

Lattice Methods

Lattice methods such as a binomial or trinomial trees do not give particularly good results - especially in cases where the number of instalment periods is large. The accuracy and speed of convergence of these tree methods is poor compared to other methods - even against the approximation outlined above

Other Known Names / Variants:

Compound Options
Continuation Options
Continuous Intalment Options
Pay-as-you-go Options

Additional/Useful List of resources

Papers:

Wystup, U. & Griebsche, S. "Instalment Options", Presentation (2003)