2.3 The Long-Term Spot Rate
• l (t )  lim R(t , T ) the long term spot rate.
T 
• Empirical research (Cairns 1998) suggests that
l(t) fluctuates substantially over long periods
of time.
• None of the models we will examine later in
this book allow l(t) to decrease over time.
• Almost all arbitrage-free models result in a
constant value for l(t) over time.
• This suggest that a fluctuating l(t) is not
consistent with no arbitrage.
• Theorem2.6 (Dybvig-Ingersoll-Ross Theorem)
Suppose that the dynamics of term structure are
arbitrage free. Then l(t) in non-decreasing
almost surely.
proof
• At time 0, we invest an amount 1/[T(T+1)] in
the bond maturing at time T.

•
1
V (0)  
T 1
T (T  1)
1
Dybvig-Ingersoll-Ross Theorem
• Assume l (1)  l (0)
• Goal: check V(1).
• let  (l (0)  l (1)) / 3  0 , there exists
such that T  T0 | l (0)  R(0, T ) | 
 l (0)  R(0, T )   or R(0, T )  l (0)  
  R(0, T )  (l (0)   )
T0  0
Dybvig-Ingersoll-Ross Theorem
 R ( 0,T )T
 ( l ( 0 )  )T
P
(
0
,
T
)

e

e
•
as T  T0 .
• With similar argument, we can get
as T  T
P(1, T )  e  R (1,T )T  e (l (1) )T
0


1
1 
• V (0) 
1


1
 T (T  1)   T
T 1
•
T 1

T 1
T0 1

P(1, T )
P(1, T )
1
e  (l (1) )T
V (1)  


 ( l ( 0 )  )T
T 1 T (T  1) P (0, T )
T 1 T (T  1) P (0, T )
T T0 T (T  1) e

T0 1

P(1, T )
1


eT  
T 1 T (T  1) P (0, T )
T T0 T (T  1)
L' Hopital 
Dybvig-Ingersoll-Ross Theorem
• Since dynamics are arbitrage free, there exists
an equivalent martingale measure, Q , such
that V(1)/B(1) is a martingale (Theorem 2.2)
i.e. E V (1)    V (0)  1  
Q

 B(1)
0


B(0)
t
r ( s ) ds

0
where B(t )  e
 V (1) / B(1)
is the cash account.
is a.e. real-valued.
 Q(V (1) / B(1))    0
Dybvig-Ingersoll-Ross Theorem
 QV (1)    0
 Ql (1)  l (0)  QV (1)    0
 Pl (1)  l (0)  0
(equivalent measure)
• l (t ) is non-decreasing almost surely under the
real world measure P.
• What the D-I-R Theorem tell us is that we will
not be able to construct an arbitrage-free
model for the term structure that allows the
long-term rate l(t) to go down.
Example 2.7
• Suppose under the equivalent martingale
measure that
0.05

r (t )  0.04

0.06
for 0  t  1
for
t 1
with probabilit y 0.5
for
t 1
with probabilit y 0.5
Example 2.7
Then, for T  1

P (0, T )  e 0.05  0.5e 0.04(T 1)  0.5e 0.06(T 1)

 0.5e 0.010.04T 1  e 0.020.02T



 1

l (0)  lim R (0, T )  lim  log P (0, T )  P (t , T )  e (T t ) R ( t ,T )
T 
T 
 T

1
1

 0.01
0.02 0.02T 
 lim 0.04  log 0.5e
 log 1  e

T 
T
T


 0.04





Example 2.7
At time 1, P(1, T ) is equal to e 0.04(T 1) or e 0.06(T 1) with equal probabilit y
 1

  0.04(T  1) 
 l (1)  lim  log P(1, T )  lim 
 0.04 (or 0.06)

T 
T


T
 T



 l (t ) is constant or increasing , as indicated by the DIR theorem.
• This example is included here to demonstrate that we
can construct models under which l(t) may increase
over time.
• In practice, many models we consider have a recurrent
stochastic structure which ensures that l(t) is constant.
In other models l(t) is infinite for all t > 0.