Models studied in the previous chapters specify the movement of the short- term interest rate and thereby endogenously determine the form of term structure (including its initial value). Ho and Lee (HL) [ 27 ] developed a model which takes as input, the initial interest rate term structure and derives its subsequent stochastic evolution. Hence the theoretical zero coupon bond prices (that is, those produced by the model) will be exactly consistent with those observed in the market.
HL use all information within the current observed term structure to price contingent claims in such as way as to ensure that profitable arbitrage is precluded.
The assumptions made by HL are the standard assumptions for a perfect capital market in a discrete time framework.
The market is frictionless, i.e. there are no taxes or transaction costs and securities are perfectly divisible.
In a discrete time framework each time period is taken to be one unit of time. Hence a zero coupon bond with term to maturity T pays $1 at the end of the T th time period (taken from valuation time).
The bond market is complete, with a bond maturing at the end of each time period n , n =0, 1, 2, .
At each time period n , there are a finite number of possible states of the world. At time n , state i , denote the equilibrium price of a T -maturity zero coupon bond as P ( n ) i ( T ). This function is termed a discount function. At any time n , state i , the interest rate term structure is fully described by a series of discount functions.
By its definition as a discount function, P ( n ) i ( ·) must satisfy certain conditions. That is: