2.10 Extending the model to allow time-dependent drift


2.10 Extending the model to allow time-dependent drift

Consider extending the model specification to allow for a time-dependent drift parameter. Hence the short- term interest rate dynamics become:

Due to the Markovian nature of the model, we assume that all information about past movements and expectations of future movements is contained in the current ( observed ) term structure. Therefore the functional form of the time-dependent parameter ( t ), may be determined from observed bond prices and the values of the constant parameters. No prior restrictions are placed on the functional form of ( t ) since it is determined so as to reflect the specific observed term structure.

Consider the conditional expectation of r ( s ) with the time-dependent parameter ( t ). Following the methodology of §(2.7.1) we have the integral form of the short-term interest rate process:

Taking expectations and differentiating with respect to s produces:

so, integrating over [ t, s ] gives the conditional expectation of r ( s )as:

The bond price takes the same functional form as specified in (2.27), with a modification to one of the parameters as depicted below:

where:

Given this formulation of the bond price and the observed term structure, (2.44) can be solved for ( s ) for all s ˆˆ [ t, T ] which could then be used in conjunction with (2.41) to determine future expectations of the short-term interest rate as specified by the current observed term structure.



2.11 Comparison of the Vasicek and CIR methods of derivation

The Vasicek and CIR models are very similar in their structure [23] and tractability, but their key difference lies in the derivation. Vasicek enforces the no arbitrage requirement between bonds but does not consider the existence of an underlying equilibrium economy consistent with the model. CIR begin with a specification of the equilibrium economy, from within which they obtain the valuation model. The following factors are contained within the CIR economy:

  • variables affecting the bond price,

  • endogenously determined stochastic properties driving the underlying variables,

  • the form of the factor risk premia.

Vasicek makes assumptions about the variables affecting the bond price and the stochastic factors driving these variables. These assumptions are exogenously specified and imposed directly on the relevant variables. Consider these assumptions in the framework of the CIR model:

  1. the bond price is assumed to be determined by the short- term interest rate only,

  2. the short-term interest rate r , is assumed to follow the stochastic process

    dr = ( * ˆ’ r ) dt + ƒ ˆ rdz

Application of Ito's Lemma and existence of the risk premium determines the excess expected return on a bond, that is ¼ ( t, T ) ˆ’ r = excess expected return = ( r, t, T )

If there exists an underlying equilibrium economy which supports (1) and (2), then this function ( r, t, T ) must exist. However, its dependence on the underlying variables is unspecified.

To preclude arbitrage must take on the following form:

where ( r, t ) is the required risk premium. Not all functions ( r, t, T ) will satisfy (2.45) and (2.46) and hence definite restrictions are placed on the functional form of the excess return.

However, this approach to the specification of a complete model of the term structure may lead to problems:

  1. Assumptions (1) and (2) do not guarantee a consistent underlying equilibrium economy;

  2. The no arbitrage approach does not guarantee the absence of arbitrage for every choice of ( r, t ).

The model specified by CIR does have a consistent underlying equilibrium economy and hence precludes arbitrage. Consider the following example which does not meet all the requirements specified by the CIR model and hence leads to disequilibrium in the underlying economy. Assuming ( r, t ) = + » r , (2.45) becomes

This is the same as (2.18) with = * ˆ’ , so the bond price takes the form:

where

The solution of the bond price equation (2.47) becomes:

and the bond price process may be specified as:

The linear form of the risk premium chosen above satisfies the no arbitrage condition and appears advantageous for empirical studies, but it can easily be shown that the resulting model is in fact not viable . Consider r = 0. Since the bond is instantaneously riskless, it should over the next instant, yield the corresponding risk-free rate. However, the bond price dynamics (2.48) reduce to:

and hence the instantaneous rate of return differs from the prevailing risk-free rate and the model guarantees arbitrage opportunities instead of precluding them. This model breaks down because there is no underlying economic equilibrium which is consistent with the chosen risk premium.

[23] They apply slightly different functional forms to the volatility of the short-term interest rate and the market price of risk.