Interest Rate Modelling (Finance and Capital Markets Series)
13.1 Using information from the observed term structure of interest rates and volatilities
Let [0, T *] be some trading interval. Data representing the term structure consists of the following:
| r ( t ) | - | the instantaneous, continuously compounded short-term interest rate at time t , |
| R ( r, t, T ) | - | the interest rate term structure, that is continuously compounded rates for a series of maturity dates, T , T ˆˆ [0, T *], |
| ƒ r ( r, t ) | - | instantaneous short-term interest rate volatility, |
| ƒ R ( r, t, T ) | ˆ’ | term structure of interest rate volatilities with maturities corresponding to those of the term structure of interest rates. |
The price of a zero coupon bond takes the functional form:
where B ( t, T ) may be found from the time t term structure as (refer to equation (7.37) Chapter 7):
Letting t = 0 denote the current time, the term structure of B (0, ·) is expressed in terms of the current volatility and interest rate term structures as:
Additionally, the initial term structure of interest rates R ( r, 0, ·) may be used to determine the term structure of zero coupon bond prices or discount factors as:
This term structure of zero coupon bond prices, together with the term structure of B (0, ·) allows the term structure of A (0, ·) to be found as: