The short-term interest rate at each node in the tree is found such that the term structure produced by the model matches the current observed term structure. European-style contingent claims may then be priced. The value at a node is the discounted expected value one time period in the future. Since the binomial tree is calibrated to the market- observed risk-free rate, the contingent claim is priced in a risk-neutral environment, where the probabilities of an up and down move are equal. Hence our expectation of the price of the contingent claim after one period is:
where S u and S d are the prices of the contingent claim after an up and down move respectively. Discounting by the current one period interest rate r ,the current price of the contingent claim S ,is:
This method may be used to determine the price at any node in the tree, from the prices one step in the future. Iterative application of (8.1) allows valuation of contingent claims of any duration, as long as the tree of short-term interest rates extends sufficiently far into the future.
To value bond options, the tree must first be used to determine the bond value at every node according to the interest rate associated with that node.
Then, making use of the known option value at expiry [1] , and working backwards through time, the bond option value may be found at every node prior to expiry.
[1] This is the option payoff at expiry, hence for a call option the payoff is spot less the strike, while for a put option it is the strike less the spot.