Once the short-term interest rate tree has been constructed such that the short- term interest rate and associated discount factor are known at each node of the tree, any interest rate contingent claim may be priced by a simple backward induction procedure.
Consider a coupon bond maturing at time [3] T = N T ” t and paying coupons at discrete time steps { t 1 , t 2 , , t m } where m is the number of coupons due until maturity. If c is the amount payable at each coupon time and the last coupon payment coincides with bond maturity, i.e. t m = N T , the maturity value of the bond may be initialised at time i = N T as:
where P c i,j is the value of the coupon paying bond at node ( i, j ).
Now for i < N T , the value of the coupon bond is equal to the discounted expected value of the bond at the next time step ( i + 1). Since the risk-neutral probability associated with each branch of the binomial tree is ½, the value of the coupon bond for all i < N T is determined as [4] follows .
If i ˆˆ { t 1 , t 2 , , t m } then
else
Once we have determined the value of the coupon paying bond at each node of the binomial tree, we may price claims contingent on this coupon bond. Consider a European call option on the above coupon paying bond with:
expiry date s = N s ” t ,
strike price X .
Knowing the value of the coupon bond in all states at option expiry time i = N s , we determine the option payoff as:
where C i,j is the value of the European call option at node ( i, j ). For each i < N s the value of the European call is determined as the discounted expectation of option values at time ( i + 1); hence:
More directly, utilise the discounted probabilities or state prices Q i,j to determine the time i = 0 European option value directly from the expiry condition (14.6) as:
[3] Hence, this bond may be seen to mature N T time steps from initial time i = 0.
[4] Here j represents each node at time step i , hence j = ˆ’ i, ˆ’ i +2, , i ˆ’ 2, i .