1.2 Term structure equation


1.2 Term structure equation

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 , ).



1.3 Risk-neutral valuation

The mean rate of return on a bond can be written as a function of its variance, the risk-free interest rate and market price of risk. From (1.12) we have:

Hence the bond price dynamics (1.9), may be written in terms of the market price of risk as [1] :

Let d = ˆ’ qdt + dz and the above equation becomes:

where is the Wiener process in the risk-neutral world being governed by the probability measure. Since dz = qdt + , the equation describing the dynamics of the short- term rate, (1.4) may be written in terms of as:

Using the dynamics of the bond price in (1.15) and the PDE of the bond price (1.13) with boundary condition P ( T , T ) = 1, the Feynman-Kac theorem [2] may be applied to yield the valuation:

Here we take the expectation with respect to which corresponds to the risk-neutral world. corresponds to the equivalent probability measure which utilises risk-neutral probabilities. (As opposed to the utility dependent probability measure, Q , which represents investor specific probabilities.) By introducing the market price of risk q we are able to transform the probability measure from a utility-dependent to a risk-neutral one. The Girsanov Theorem [3] defines this transformation. First consider the Wiener process:

Let t t * T and define

as the Radon-Nikodym derivative used to define the new probability measure, that is:

Also, expectation with respect to is calculated as:

for any random variable Y . Hence the expected bond price (1.17) may be expressed in terms of the utility-dependent measure as:

[1] To lighten the notation, the functional dependence of r , q , v , s and ƒ on r , t and T is suppressed.

[2] The discounted Feynman-Kac theorem is applicable in this case. This theorem defines the relationship between a stochastic differential equation (SDE) and the corresponding partial differential equation (PDE). Considering the SDE:

Let 0 t T where T > 0 is fixed, and let h ( y ) be some function. Define:

Then the corresponding PDE is:

See [ 45 ] for more details.

[3] For more details about the application of Girsanov's Theorem and the Radon-Nikodym derivative in the change of measure see [ 45 ] and [ 41 ].