The required calibration procedure is a simpler version of that outlined in Chapter 8, §8.5. Again we make use of forward induction and state or Arrow-Debreu prices. Define Q i,j as the time 0 value of security paying 1 if node ( i, j ) is reached and 0 otherwise . By definition Q 0,0 = 1.
These Arrow-Debreu prices may be seen as discounted probabilities and hence may be used to specify the value of any instrument. Specifically, the time 0 price of a discount bond maturing at time ( i + 1) ” t may be written as:
where d i,j is the one period discount factor at node ( i, j ) expressed as:
At each node ( i, j ) of the tree, the state prices are determined as functions of the state prices at time ( i ˆ’ 1) using:
Initial step: Initialise variables at time i = 0 as follows :
Now, for i > 0, we assume the following are known for all states j at time step ( i ˆ’ 1): Q i ˆ’ 1, j , u ( i ˆ’ 1), r i ˆ’ 1, j and d i ˆ’ 1, j . Hence, the values of Q i,j , u ( i ), r i,j and d i,j may be found for all states j at time i as follows:
Step 1: Make use of (14.5) to generate Q i,j as follows:
Step 2: A numerical search technique such as Newton-Raphson (e.g. see [ 13 ]) is used to find u ( i ) such that the following is satisfied [2] :
Step 3: Using the u ( i )s calculated in Step 2, the short- term interest rates r i,j , and corresponding discount factors d i,j , are updated for each state j at time step i as:
Steps 1-3 are repeated for all i = 1, , N where N ” t is the longest maturity discount bond.
[2] Here the summation is over all states j at time step i , hence j = ˆ’ i , ˆ’ i +2, , i ˆ’ 2, i .