7.2 Extension of the Vasicek model


7.2 Extension of the Vasicek model

HW propose an extension to the Vasicek model of the form:

This is (7.2) with ² = 0. Assuming that the market price of interest rate risk has the functional form » ( t ), and is bounded on any time interval (0, ), we may apply Ito's Lemma to derive the general partial differential equation that must be satisfied by any interest rate contingent claim, g :

The market price of risk represents the excess return required above the risk-free rate. This relationship is denoted by:

and so

where ( t ) = a ( t ) b + ( t ) ˆ’ » ( t ) ƒ ( t ). Assume that the price of the contingent claim g , has the form:

with boundary condition g ( r, T, T ) = 1. Now:

Substituting into (7.5):

Therefore to solve (7.7) we must solve the system of simultaneous equations:

For ( t ), ƒ ( t ) and a ( t ) constant, (7.8a) and (7.8b) are solved to yield the Vasicek model where the bond price has the form assumed in (7.6) with [1] :

For the extended, time-dependent model, ƒ ( t ) should be chosen to reflect the current and future volatilities of the short- term interest rate. A (0, T ) and B (0, T ) are coefficients associated with the current term structure and are hence functions of the current interest rate term structure, current term structure of spot/forward interest rate volatilities, and ƒ (0) (the current volatility of the short-term interest rate). Since the current term structure is observable, we are able to determine A (0, T ), B (0, T ) and ƒ ( t ). Therefore, we must determine A ( t, T ), B ( t, T ), a ( t ) and ( t ) in terms of A (0, T ), B (0, T ) and ƒ ( t ).

First, differentiate (7.8a) and (7.8b) with respect to T . From (7.8a) we have:

Also, from (7.8a):

Hence:

with A ( T, T )= 1 and A (0, T ) = ¾ where ¾ is some known value. Similarly differentiating (7.8b) with respect to T yields:

From (7.8b) we have:

Therefore:

with B ( T, T ) = 0 and B (0, T ) = · where · is some known value. HW [ 28 ] solve (7.12) and (7.14) to yield:

where ( t, T ) = log A ( t, T )

We have solved for A ( t, T ) and B ( t, T ) in terms of the initial term structure.

Now solve for a ( t ) and ( t ). Differentiating (7.15) yields:

and so from (7.13) we solve for a ( t )as:

Differentiating (7.16) yields:

Since ( t, T ) = log A ( t, T ) we have:

Hence

Therefore substituting into (7.11) we solve for ( t ) in terms of the initial term structure:

Also, from (7.8b) and (7.17) we have

and

Hence

and we have specified all the required model parameters in terms of the initial yield curve.

[1] These are the same formulae as calculated for the Vasicek model in Chapter 1 equations (1.25) and (1.26) with the following notational substitutions:



7.3 Pricing contingent claims within the extended Vasicek framework

Let P ( r, t, T ) be the time t price of a discount bond maturing at time T .Since this is an interest rate contingent claim, it may be written in the functional form specified in (7.6):

By Ito's Lemma we have:

Hence the price process of the discount bond is described by the stochastic equation:

The relative volatility of P ( r, t, T ) is B ( t, T ) ƒ ( t ). Since it is independent of r , the distribution of the bond price at any time t *, conditional on its value at an earlier time , must be lognormally distributed.

Consider a European option on this discount bond. This option has the following characteristics:

X

-

exercise price,

T

-

option expiry time,

s

-

bond maturity time,

t

-

current (valuation) time, where t T s .

This option may be viewed as being equivalent to an option to exchange X units of a discount bond maturing at time T for one unit of a discount bond maturing at time s . Define:

± 1 ( )

-

time volatility of the price of a discount bond with maturity s ,

± 2 ( )

-

time volatility of the price of a discount bond with maturity T ,

( )

-

instantaneous correlation between the bond prices,

Then the price of a European call option may be written as [2] :

One of the characteristics of a one factor model is that instantaneous returns on bonds of all maturities are perfectly correlated. Hence ( ) = 1 for all . Also, from the equation of the general bond price process (7.19) we may write the volatilities of the two bonds as:

7.3.1 Time-dependent parameters.

Given the functional form of the volatilities above, calculate the volatility required for option valuation as:

From equation (7.15) we have and so:

Equations (7.20), (7.21) and (7.24) give analytical formulae for the price of a European call option on a discount bond. The corresponding European put option price may be obtained via put-call parity.

This formulation of the pricing formula is very attractive since a ( t ) and ƒ ( t ) may be chosen in such a way that a whole set of cap or swaption prices observed at time 0 can be exactly replicated. However, this full level of precision results in undesirable side effects [ 45 ], [ 29 ]. Examine the process for the short- term interest rate as considered thus far:

Here ( t ) is chosen such that the prices of all discount bonds at the initial time are reproduced, i.e. the initial term structure is matched, a ( t ) and ƒ ( t ) provide another two degrees of freedom which allow matching of the initial volatility term structure and volatilities of the short-term interest rate in the future. Initial volatilities of rates depend on ƒ (0) and a ( t ), hence a ( t ) defines the relative volatilities of long and short-term interest rates. Finally, ƒ ( t ) determines the future volatilities of the short-term interest rate.

It is appealing to take advantage of all these degrees of freedom since all the available initial market data will be incorporated into the model. Unfortunately this results in non-stationarity of the volatility term structure which could result in the mispricing of instruments contingent on the future, rather than simply the current, volatility structure.

Fitting all the model parameters to initial option prices results in a model which exactly reflects the initial term structure, but also introduces assumptions about the future evolution of the volatility structure. Making use of all the degrees of freedom produces an over-parameterisation of the model. Hull and White [ 29 ] recommend keeping the parameters a and ƒ constant. Within this simplified model, observed cap and swaption prices will only be approximated, but the model evolution can be more directly controlled. A stationary volatility term structure is achieved, resulting in robust pricing of more exotic interest rate options.

7.3.2 Constant parameters.

If we allow the volatility of the short-term interest rate and the rate of interest rate reversion to be constant, i.e. ƒ ( t ) ƒ and a ( t ) a , we have [3] :

The corresponding function A ( t, T ) is obtained from (7.16), setting ƒ ( t ) ƒ in the integral [ 29 ]. Making use of (7.15) and (7.25) we have:

Now, to calculate the volatility required for the option pricing, make use of (7.25) to give:

and so:

and

Substituting into (7.22), find the appropriate volatility ƒ P f ( T,s ) as [4] :

Hence:

where

[2] Here, the appropriate volatility to use is that of the forward bond price, i.e. the volatility of the time T price of the bond maturing at time s which may be expressed as . Use Ito's Lemma to determine this volatility:

Hence, the instantaneous volatility of the forward bond price is ± 1 ( t ) ˆ’ ± 2 ( t ) and so the square of forward price volatility over the life of the option is:

[3] This is the value of B ( , ·) obtained in Chapter 1.

[4] 4 Alternatively we may substitute the appropriate value of B (0, ·) into (7.24) to calculate ƒ P f ( T,s ) as: