HJM develop their model within a continuous trading economy, with trading interval [0, ], > 0 fixed. Uncertainty within the economy is represented by the probability space ( , F,Q ), where represents the state space, F the ƒ -algebra representing all measurable events and Q the probability measure. Information becomes available over the trading period according to the filtration { F t : t ˆˆ [0, ]} which is generated by n independent Brownian motions { z 1 ( t ), , z n ( t ): t ˆˆ [0, ]} with n ‰ 1.
Assume there exist default-free zero coupon bonds with maturities on each trading day T , T ˆˆ [0, ]. If P ( t, T ) represents the time t price of a T -maturity bond, where T ˆˆ [0, ] and t ˆˆ [0, T ], then the following must be true:
Define the time t instantaneous forward rate for time T , T > t as:
Solving this differential equation for the bond price yields:
The short- term interest rate at time t is the instantaneous forward rate for time t , hence:
Alternatively, expressed in terms of the bond price [1] :
Hence the short-term interest rate may be interpreted as the rate of return on an instantaneously maturing bond.
[1] Here, make use of the Taylor series expansion of the natural logarithm of a number:
Consider:
By definition h is small, so P ( t, T ) is only slightly greater than P ( t, T + h ), and is only slightly larger than 1; hence:
Therefore applying this expansion:
since the higher order terms are negligibly small by the definition of h .