14.3 Forward Induction: making use of state prices

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 .