13.1 Using information from the observed term structure of interest rates and volatilities


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:




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

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