State-space and time

In the last chapter we developed a one period binomial model for the contingent claim prices. This can be extended in two directions: One can increase the number of periods in question, and/or one can increase the number of the possible states. We define as the state-space the set of all possible security prices. The next few chapters will investigate models with a discrete state-space --like the binomial trees of the previous discussion, and models where the state-space is continuous, allowing the possible values of the security prices to span a whole compact set of nonnegative real numbers. In addition, the following chapters will consider models that allow trading to occur at discrete points in time, and models where trading is considered to be continuous. In fact there are three possibilities:

• models in discrete state-space and discrete time --trees;

• models in continuous state-space and discrete time --semi-continuous; and

• models in continuous state-space and continuous time --continuous.

These notes are going to deal extensively with the two extreme cases, namely the first and the third one. The intermediate [second] case will be also, briefly, considered.