Consider a project that is supposed to work for a long period of time, with. initial value V(0). A partition of this period is considered characterizing a sequence of points in time t = 0,1,2. In order to characterize the value of the project, an underlying uncertainty structure over time is required. For simplicity, the value of the project will be described as following what is known as a binomial tree. Assume that at each point in time t, future uncertainty is characterized by the sample space A= {w+ , w_ }, reflecting the existence of only two states of nature at time t+1. The probabilities are taken as P(w+)=p and P(w_) =1-p.
For sake of interpretation, in one of the states the value of the project is assumed to increase,and in the other state to decrease. Moreover, the value of the project at any point in time is assumed to be determined only by the number of times that its value increased. A tree is formally defined as a pair (B,A), where a is B set of nodes and A is a set of arcs linking such nodes. Under this setting, a node is defined as the pair (j, t), with 0 < j < t. More specifically, a node (j,t) represents a possible state of nature at time t. The arcs are defined under this binomial setting by the pairs of nodes of the form [(j, t); (j, t+1)} or [(j, t); (j+1, t+1)].The initial node is (0,0)
Seen from t=0, at t=1 there are two possible states denoted as (1,1) ·and (0,1), depending on whether the value of the project increased or decreased with respect to its initial value.At time t=2 there are in principle four possible states, two for each of the possible states at t=1. In fact, from state (1,1), the value of the pro ject may increase, leading to state (2,2), but may also decrease, leading to (1,2). Similarly, from state (0,1), one can achieve either state (1, 2) or. (0, 2). Notice that two of these four states collapse into the same one, leaving only three distinct states at t=2.One such tree is said to be recombining.this property can be rephrased as follows:
Given a binomial tree (B,A) with a node (j,t)∈A with 0<j<t, this tree is said to be recombining if both [(j,t-1); (j,t)]∈A and [(j-1,t-1); (j,t)]∈A.Figure 1.5 depicts this property.Finally,markets are taken as complete at any point in time in the sense that there is always a one-period risk-free asset besides the project.
Let j be the number of times up to date t that the value of the project increased. At each nodethe value of the project is denoted by Vi(t), and the payoff matrix for the next period by CC with R=1+r reflecting the discount rate of the risk-free project. Under the absence of arbitrage opportunities, there exists ψ(j, t)∈R2 such that DD
fully characterizing the value of the project at any node.
Figure 1.6 shows the tree for the time evolution of the project's value. The binomial model is a further simplification of this structure. It establishes the existence of rates U and D such that D and EE.Under the assumption that U＞D, absence of arbitrage opportunities . implies that U>R >D. The particular feature of this model that the state-price vector ψ(j, t) is independent of j and t. In fact, equatlon (1.19) becomes simply FF.Solving for ψ, it is easily seen that the martingale measure is characterized by R - D
and the value of a project can simply be. written as a function of its future payoffs after n periods as FF. Of course, this value could be written with the real probabilities p and 1-p.
