11.1 Initial specifications


11.1 Initial specifications

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 .




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net