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.
Technical conditions are applied to the processes defining the short and forward interest rates as well as the money market account.
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.
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:
Here technical conditions are applied to parameters of the bond price process, thereby allowing the resulting bond price process to be well behaved.
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.
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:
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]. |
Assume Lemma 0.1 holds and define:
Then
Proof | |
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.
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: