Equation (1.5) implies that the bond price * P * = * P * ( * r * ( * t * ), * t * , * T * ) is a function of the short rate. Applying Ito's Lemma to the bond price and using (1.4) we derive the stochastic differential equation for the bond price:

Set

and hence

where ¼ ( * t * , * T * ) and ƒ ^{ 2 } ( * t * , * T * ) are the time * t * mean and variance of the instantaneous rate of return of a * T * -maturity zero coupon bond.

Since a single state variable is used to determine all bond prices, the instantaneous returns on bonds of varying maturities are perfectly correlated. Hence a portfolio of positions in two bonds with different maturity dates can be made instantaneously risk-free. This means that the instantaneous return on the portfolio will be the risk-free rate of interest. Consider a time * t * portfolio of a short position in * V * _{ 1 } bonds with maturity * T * _{ 1 } and a long position in * V * _{ 2 } bonds with maturity * T * _{ 2 } . The change over time in the value of the portfolio, * V * = * V * _{ 2 } ˆ’ * V * _{ 1 } , is obtained from (1.9):

Choosing * V * _{ 1 } and * V * _{ 2 } such that the coefficient of the Wiener coefficient in (1.10) reduces to zero will result in a portfolio with a strictly deterministic instantaneous return. Hence we require:

Similarly:

and hence (1.10) becomes:

Invoking assumption 3, that no riskless arbitrage is possible, the instantaneous return on the portfolio must be the risk-free rate, * r * ( * t * ). That is:

Rearranging the terms in the equation, we have:

and since this equality is independent of the bond maturity dates, * T * _{ 1 } and * T * _{ 2 } , we can define:

where * q * ( * r * , * t * ) is independent of * T * . * q * ( * r * , * t * ) measures the increase in expected instantaneous return on a bond, for a unit increase in risk, and is referred to as the market price of risk. Substituting the formulae for ¼ and ƒ from (1.7) and (1.8), we derive a partial differential equation for the bond price:

This equation, referred to as the term structure equation, is a general zero coupon bond pricing equation in a market characterised by assumptions 1, 2 and 3. To solve (1.13), we need to specify the parameters of the short-term interest rate process defined by (1.4), the market price of risk * q * ( * r * , * t * ), and apply the boundary condition:

Using equation (1.1) we can evaluate the entire term structure, * R * ( * t * , ).

Interest Rate Modelling (Finance and Capital Markets Series)

ISBN: 1403934703

EAN: 2147483647

EAN: 2147483647

Year: 2004

Pages: 132

Pages: 132

Authors: Simona Svoboda

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