11.2 Specifications of the various processes

We present a family of stochastic processes describing the evolution of forward rates and hence uniquely determining the short- term interest rate and bond price processes. A series of conditions is presented ensuring the processes are bounded and well behaved.

11.2.1 Forward and short-term interest rate processes.

Technical conditions are applied to the processes defining the short and forward interest rates as well as the money market account.

Condition 1

A family of forward rate processes. Define a family of forward rate processes f ( t, T ), for fixed T ˆˆ [0, ]:

where ^[2] :

{ f (0, T ): T ˆˆ [0, ] } is a fixed, non-random initial forward rate curve, measurable as a mapping f (0, ·):([0, ], [0, ]) ’ ( R , ) where [0, ] is a Borel ƒ -algebra restricted to [0, ].
± : {( t, s ):0 t s T } ’ R is a family of drift functions jointly measurable from F {( t, s ):0 t s T } ’ , adapted and having
ƒ _i : {( t, s ): 0 t s T } ’ R are volatilities, jointly measurable from

F {( t, s ): 0 t s T } ’ , adapted and with

Starting from the initial fixed forward rate curve { f (0, T ): T ˆˆ [0, ]}, the n independent Brownian motions determine the stochastic evolution of the whole forward curve through time. The sensitivity of the change in a given maturity forward rate to each Brownian motion, is specified by the volatility coefficients. The only restrictions imposed on the forward rate process that have economic implications are:

time is continuous and
stochastic movement is specified by a finite number of random shocks.

Condition 2

Regularity of the money market account. Given the forward rate process in (11.4), the dynamics of the short-term interest rate may be expressed as:

Now, define an accumulation factor or money market account B ( t ), as:

with initial condition B (0) = 1. The value of this money market account must satisfy :

To guarantee that this condition is satisfied, we require:

11.2.2 Bond price process.

Here technical conditions are applied to parameters of the bond price process, thereby allowing the resulting bond price process to be well behaved.

Condition 3

Regularity of the bond price process. To ensure a well-behaved bond price process, the following regularity conditions are imposed:

and

Given Conditions 2 and 3 and using the lemma and two corollaries below, we determine the bond price process.

Lemma 0.1

This is a generalised form of the Fubini theorem for stochastic integrals. Given the following:

( , F,Q )	ˆ’	probability space,
{ F _t }	ˆ’	filtration generated by a Brownian motion { z ( t ): t ˆˆ [0, ]}

let { ( , t, a ):( t, a ) ˆˆ [0, ] [0, ]} be a family of real random variables such that

(( , t ), a ) ˆˆ {( [0, ]) [0, ]} ’ ( , t, a ) is L B [0, ] measurable ^[3] ,
ˆ« ^t ² ( , s, a ) ds < + ˆ a.e. ˆ t ˆˆ [0, ];
ˆ« ( ˆ« ( , s, a ) da ) ² ds < + ˆ a.e. ˆ t ˆˆ [0, ].

If t ’ ˆ« ( ˆ« ( , s, a ) dz ( s )) da is continuous a.e. then:

Proof

Let _A and _B be characteristic functions such that:

and

where A is a set { t : t ˆˆ [ s , )} and B ˆˆ F _s . Now we have:

where » is the Lebesque measure and Q the measure associated with filtration F . Also:

Therefore:

Let be a class of functions h ( t , ):[0, ˆ ] ’ R such that:

( t , ) ’ h ( t , ) is F ˆ’ measurable ^[4] ,
for each t 0 the function ’ h ( t , ) is F ^t -measurable and
[ ˆ« _S ^T h ² ( t , ) dt ] < ˆ

Now an elementary function ˆ ˆˆ may be defined as a sum of characteristic function as ^[5] :

Since ˆˆ , each function e _j must be F _{t _j} -measurable. Hence we may define:

for some 0 S T . Therefore the Ito Integral of some function h ˆˆ may be written as:

where the limit is taken in L ² ( P ) and { ˆ _n } is a sequence of elementary functions such that:

We have shown that the integral of any function h ˆˆ may be written as the limit of the integral of a sequence of elementary functions. The elementary functions may be expressed as sums of characteristic functions. A similar results (used in the proof of the standard Fubini Theorem, e.g. [ 16 ]) exists for the purely deterministic case. Hence we conclude that since relationship (11.7) holds for characteristic functions, it holds for any function ( , t, a ).

For an alternative description of this proof see [ 33 , Chapter 3, Problem 6.12].

Corollary 0.1.1

Assume Lemma 0.1 holds and define:

Then

Proof

Corollary 0.1.2

Assume Lemma 0.1 holds and define:

Then

Proof

Now consider the bond price (11.2):

Substituting (11.4) we have ^[6] :

Now, apply the standard Fubini theorem to the double integral on ± ( v,y ) and Corollary 0.1.1 to the double integral on ƒ _i ( v,y ) toget:

Applying Corollary 0.1.2 to the last two terms of (11.9) gives:

and from (11.2) we know:

Hence (11.9) becomes:

However, from (11.5) we have ^[7] :

and so the dynamics of the bond price process are:

Let:

and (11.10) becomes:

which may be expressed in differential form as:

Now applying Ito's Lemma, we find the differential equation satisfied by the bond price P ( t, T ), to be:

where

and so:

Since in (11.13) both the drift term r ( t )+ b ( , t, T ) and the volatility coefficients

a _i ( , t, T ), i = 1, , n , may depend on the history of the Brownian motions, the bond price process is non-Markovian.

11.2.3 Relative bond price process.

Let T ˆˆ [0, ], t ˆˆ [0, T ] be the time t relative price of a T -maturity bond. Here, the bond price is expressed in terms of the money market account, so its drift with respect to the short-term interest rate is removed. Make use of Ito's Lemma to determine the dynamics of the relative bond price as ^[8] :

Also:

Hence the integral form of the relative bond price process is:

Again, the relative bond price is non-Markovian since the drift and volatility coefficients may depend on the history of the Brownian motions through the cumulative forward rate drift and volatility terms b ( , ·, T ) and a _i ( , ·, T ), i = 1, , n .

^[2] Here, and in subsequent formulae denotes the possible dependence on the history of the Brownian motions.

^[3] L is the smallest ƒ -field on [0, ] such that all left-continuous F _t -adapted processes Y :( , t ) ˆˆ [0, ] ’ Y ( , t ) ˆˆ R ^d are measurable.

^[4] As before represents the Borel ƒ -algebra on [0, ˆ ).

^[5] The following has been adapted from [ 43 ].

^[6] Here to improve readability, we suppress the notational dependence on .

^[7] Directly integrating (11.5) yields:

However, by definition of ± ( v, y ) and ƒ _i ( v, y ) as drift and volatility parameters of the forward rate process, we require v y for all v, y ˆˆ [0, ] and hence the upper limit on the inner integrals must become min ( y, t ) y ˆ§ t .

^[8] Dynamics of the money market account are easily found from (11.6) as: