4 The Evolution of the Term Structure of Interest
Rates
A Motivation
This chapter introduces the stochastic evolution
for the zero-coupon bond price curve.
The
stochastic
structure
is
introduced
sequentially, starting with a one-factor model,
then presenting a two-factor model, and so forth.
This section motivates the discrete time processes
constructed in this chapter.
Consider the historical forward rate curve
evolution.
1
Figure 4.1: Forward Rate Curve Evolutions over January 1973 - March 1997
2
From this evolution, we can generate histograms
for changes in a particular maturity forward rate.
3
Figure 4.2: Histogram of Monthly Changes in Forward Rates
from January 1973 - March 1997.
3 Months Forward Rate
6 Months Forward Rate
0.25
0.35
0.3
0.2
0.25
0.15
0.2
0.15
0.1
0.1
0.05
0.05
0
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0
-0.1
-0.08
-0.06
-0.04
-0.02
Df
0
0.02
0.04
0.06
0.08
Df
1 Year Forward Rate
3 Years Forward Rate
0.35
0.16
0.3
0.14
0.12
0.25
0.1
0.2
0.08
0.15
0.06
0.1
0.04
0.05
0
-0.08
0.02
-0.06
-0.04
-0.02
0
Df
0.02
0.04
0.06
0
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Df
4
Figure 4.2 (continued): Histogram of Monthly Changes in Forward Rates
from January 1973 - March 1997.
5 Years Forward Rate
7 Years Forward Rate
0.2
0.25
0.18
0.16
0.2
0.14
0.12
0.15
0.1
0.08
0.1
0.06
0.04
0.05
0.02
0
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0
-0.08
-0.06
-0.04
-0.02
0
Df
0.02
0.04
0.06
0.08
Df
10 Years Forward Rate
20 Years Forward Rate
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
-0.1
-0.08
-0.06
-0.04
-0.02
0
Df
0.02
0.04
0.06
0.08
0
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Df
5
What we want to do in this chapter is to build a model for the
evolution of forward rates such that the model provides a
reasonable approximation to the true underlying joint
distribution whose marginals are given in Figure 4.2.
The binomial (and its generalization – the multinomial) model
provides such an approach.
This process is illustrated in Figure 4.3.
6
.05
1/2
Probability

1/2 .05
Df
.1
.1
1/2
.05
Probability
1/2
1/2
1/2
.075

1/2 1/4 .05
.1
.025
1/2
1/2
.075
Df
.1
Probability
1/2
.05
1/2
1/2
.075

3/8 1/8 .025
.05
.075
.1
Df
.1
1/2
1/2 1/2
...
True
Distribution
0
..
.3

Df
Figure 4.3: Example of A Binomial Approximation to the True Distribution--A Normal 7
Distribution
B The One-Factor Economy
1 The State Space Process
EXAMPLE:
PROCESS
ONE-FACTOR STATE SPACE
8
uuu
3/4
1/4
uu
uud
3/4
u
1/4
3/4
3/4
ud
udu
1/4
udd
3/4
1/4
3/4
du
duu
1/4
dud
d
1/4
ddu
3/4
dd
time
0
1
2
1/4
ddd
3
Figure 4.4: An Example of a One-Factor State Space Tree Diagram
9
At this time, the up state and the down state have
no economic interpretation. They are just used as
“place-holders” for an economic state, e.g. “good”
or “bad”.
The state space process in Fig. 4.4 is called a onefactor model because at each node in the tree, only
one of two possibilities can happen (up or down).
Each branch also occurs with strictly positive
probability. One can conceptualize the tree's being
constructed by tossing one coin (one-factor).
If, instead, at each node of the tree there were
three branches, each with strictly positive
probability, it would be called a two-factor model.
It would be a two-factor model because it would
take two coins to construct the tree.
10

uuu
q  1 (uu)
q1 (u)
1
uu
uu
1  q 1 (uu)
u
q0
q t (st )
1  q1 (u) ud
q1 (d)
1  q0
1  q t (st )
du
st u
st
• ••
u…ud
•
•
•
• ••
st d
d
d…du
q  1 (dd)
1
d…d
1  q1 (d) dd

d…dd
1  q 1 (dd)
time
0
1
2
• ••
t
t+1
• ••

Figure 4.5: One Factor State Space Tree Diagram

11
EXAMPLE: ONE-FACTOR
BOND CURVE EVOLUTION.
ZERO-COUPON
The underlying state space process is that
contained in Figure 4.4.
12
3/4
.967826
.984222
1
3/4
.947497
.965127
.982699
1
3/4
P(0,4)
P(0,3)
P(0,2)
P(0,1)
P(0,0)
=
.923845
.942322
.961169
.980392
1
.960529
.980015
1
.937148
.957211
.978085
1
.962414
.981169
1
3/4
time
0
1
2
1
.982456
1
1
.977778
1
1
.983134
1
1
.978637
1
1
1/4
1/4
.953877
.976147
1
.981381
1
1/4
3/4
1/4
1
1/4
1/4
3/4
.985301
1
3/4
.979870
1
1
.974502
1
1
1/4
3
4
Figure 4.6: An Example of a One-Factor Bond Price Curve Evolution.
Actual Probabilities Along Each Branch of the Tree
13
We can now give the “up” and “down” state an economic
interpretation. The “up” state refers to the fact that bond
prices are greater in state “u” then they are in state “d”.
This is true for the entire tree.
We see that as each period occurs, the shortest maturity
bond matures and then disappears from the tree
The last step in the tree is a residual, not used for analysis.
14
q1( u)
 P(1, ; u)   u(0, )P(0, ) 
P(1,   1; u) u(0,   1)P(0,   1)

  

 P(1, 2; u)   u(0, 2)P(0, 2) 


r(0)P(0, 1)
P(1,1; u)  1
 


1  q1( u)
q0
 P(2, ; uu)   u(1, ; u)P(1, ; u) 
P(2,   1; uu) u(1,   1; u)P(1,   1; u)

  

 P(2, 3; uu)   u(1, 3; u)P(1, 3; u) 


r(1; u)P(1, 2; u)
P(2, 2; uu)  1
 


 P(2, ; ud)   d(1, ; u)P(1, ; u) 
P(2,   1; ud) d(1,   1; u)P(1,   1; u)

  

 P(2, 3; ud)   d(1, 3; u)P(1, 3; u) 


r(1; u)P(1, 2; u)
P(2, 2; ud)  1
 


 P(0, ) 
P(0,   1)


 P(0, 2) 
 P(0,1) 

P(0, 0)  1

1q0
q1(d)
 P(1, ;d) 
P(1,   1;d)


 P(1, 2;d) 

P(1,1;d)  1

 d(0, )P(0, ) 
d(0,   1)P(0,   1)


 d(0, 2)P(0, 2) 


r(0)P(0, 1)


1  q1(d)
time
0
1
 P(2, ; du) 
P(2,   1;du)

 
 P(2, 3;du) 

P(2, 2;du)  1

 u(1, ;d)P(1, ; d) 
u(1,   1;d)P(1,   1;d)


 u(1, 3;d)P(1, 3;d) 


r(1;d)P(1, 2;d)


 P(2, ; dd) 
P(2,   1;dd)

 
 P(2, 3;dd) 

P(2, 2;dd)  1

 d(1, ;d)P(1, ;d) 
d(1,   1;d)P(1,   1; d)


 d(1, 3;d)P(1, 3;d) 


r(1; d)P(1, 2;d)


2
Figure 4.7: One Factor Bond Price Curve Evolution
15
 Pt  1, ;st u   ut, ;st Pt, ;st  
 P t  1,   1;s u  u t,   1;s P t,   1; s 
t    
t 
t 
 

 

P
t

1,
t

2;s
u



t
 ut, t  2;st Pt, t  2; st 
Pt  1, t  1; st u 1  rt;st Pt, t  1;st  

 

q 1( u
1  q1( u
uu)  1
u)
P(, ; u
ud)  1
...
1 qt st 
q 1(d
 Pt  1, ;st d   dt, ; st Pt, ; st  
 P t  1,   1;s d  d t,   1; s P t,   1;s 
t    
t 
t 
 

 

P
t

1,
t

2;s
d
d
t,
t

2;
s
P
t,
t

2;s







t
 
t
t 
Pt  1, t  1; st d  1  rt;st Pt, t  1;st  

 

t
P(, ; u
P  1, ; u u



P  1,   1; u u  1
qt st 
 Pt, ;st  
P t,   1; s 
t 
 


Pt, t  1; st 
Pt, t;st   1


u)
t+1
d)
P(, ; d
du)  1
P(, ; d
dd)  1
P  1, ;d d



P  1,   1;d d  1
1  q1(d
d)
-1
Figure 4.7: One Factor Bond Price Curve Evolution (continued)

16
We can summarize the evolution of the one-factor
zero-coupon bond price curve analytically as in
expression (4.3):


u( t ,T ; s ) P ( t ,T ; s ) if

t
t

P (t  1,T ; s )  
with qt ( st )  0
t 1 
d ( t ,T ; s ) P ( t ,T ; s ) if
t
t


with 1  qt ( st )  0

s
s u
t 1 t
(4.3)
s
s d
t 1 t
where
u(t ,T ; st )  d (t ,T ; st )
for t  T  1
and
u(t , t  1; st ) P (t , t  1; st )  d (t , t  1; st ) P (t , t  1; st )
 r (t; st ) P (t , t  1; st )  1.
17
For simplicity of presentation, we have constructed
the economy so that at each trading date, a
zero-coupon bond matures and it is removed from
trading.
In addition, no new zero-coupon bonds are issued.
It is an easy adjustment to introduce a newly issued
-maturity zero-coupon bond at each trading date.
18
3
The Forward Rate Process
This evolution can be derived from the evolution
of the zero-coupon bond price curve, given the
definition of the forward rate.
EXAMPLE: ONE-FACTOR FORWARD RATE
CURVE EVOLUTION.
19
1.014918
3/4
1.016941
1.016031
1/4
3/4
1.018607
1.018207
1.017606
1.018972
1/4
3/4
f(0,3)
f(0,2)
=
f(0,1)
f(0,0)
1.017857
3/4
1.020286
1.020393
1/4
1.022727
1.02
1.02
1.02
1.02
1/4
3/4
1.021408
1.021808
1.022406
1.019487
1.019193
3/4
1.017155
1/4
1.021830
1/4
3/4
1.023347
1.024436
1.020543
1/4
1.026165
time
0
1
2
Figure 4.8: An Example of a One-Factor Forward Rate Curve. Actual
Probabilities Along Each Branch of the Tree
3
20
 f 1,  1; u 


f 1,1;u r 1;u

 

0,   1f 0,  1


 0,1f 0,1 



 f 1,   1;d 


f 1,1;d r1;d

 

0,   1f 0,   1


0,1f 0,1 



q0
 f 0,  1 


 f 0,1 

f 0,0 r 0

1  q0
time
0
1
Figure 4.9: One Factor Forward Rate Curve Evolution
21
f t  1,   1; s t u 

 t,   1; s t f t,   1; s t 





f t  1, t  1; s t u   r t  1; s t u   t, t  1; s t f t, t  1; s t 
q 1( u
u)
N/A
f  1,   1; u u r  1; u u
q t s t 
1  q1( u
u)
N/A
 f t,   1; s t  



f t, t; s t   rt; s t 
1 q t s t 
q 1(d
f t  1,   1; s t d 

 t,   1; s t f t,   1; s t 





f t  1, t  1; s t d   rt  1; s t d   t, t  1; s t f t, t  1; s t 
d)
N/A
f  1,   1;d d r  1; d d
1  q1(d
d)
N/A
t
t+1
-1
Figure 4.9: One Factor Forward Rate Curve Evolution
22
(continued)

We can summarize this evolution for an arbitrary
time t as in expression (4.5):
 ( t ,T ; s ) f ( t ,T ; s )
t
t

 with q ( s )  0
t t
f (t  1,T ; s )  
t 1   (t ,T ; s ) f (t ,T ; s )
t
t

 with 1  q ( s )  0
t t

where
if s
s u
t 1 t
if s
s d
t 1 t
(4.5)
  1  T  t  1.
23
4
The Spot Rate Process
The stochastic process for the spot rate can be
deduced from the zero-coupon bond price
process's evolution in Figure 4.7 or read off the
forward rate curve evolution in Figure 4.9.
EXAMPLE:
ONE-FACTOR
PROCESS EVOLUTION.
SPOT
RATE
24
1.014918
3/4
1.016031
3/4
1/4
1.018972
1.017606
1/4
3/4
1.017857
3/4
1.020393
1/4
1.022727
r(0) = 1.02
1.017155
3/4
1/4
1.019193
3/4
1/4
1.021830
1.022406
1/4
3/4
1.024436
1.020543
1/4
1.026165
time
0
1
2
Figure 4.10: An Example of a One-Factor Spot Rate Process. Actual
Probabilities Along Each Branch of the Tree.
3
25
r(2;uu)  u(2,3;uu)
q1(u)
r(1;u)  u(1,2;u)
q0
1 q1(u)
r(2;ud)  d(2,3;ud)
r(0)
r(2;du)  u(2,3;du)
q1(d)
1 q0
r(1;d)  d(1,2;d)
1 q1(d)
time
0
1
r(2;dd)  d(2,3;dd)
2
Figure 4.11: One Factor Spot Rate Process
26
rt 1;st u ut 1,t  2;st u
q 2(u
r   2;u
q t st 
u)
r   1;u
uu u  1,;u
uu
r   1;u
ud d  1,;u
ud
r  1; d
du u  1, ; d
du
r  1;d
dd d  1, ;d
dd
u
1 q2(u
u)
r t;st 
1 qt st 
q  2(d
rt 1;st d dt 1,t  2;std
r  2; d
d
1  q 2(d
t
t+1
d)
d)
-2
Figure 4.11: One Factor Spot Rate Process (continued)
-1
27


u t  1, t  2; s u
t

 with probability q ( s )  0
t t




r  t  1; s   
t 1  

d t  1, t  2; st d

 with probability 1  qt ( st )  0


(4.11)
for all st and t+1  T-1.
28
Question: Is it possible to generate the zero-coupon
bond prices from the spot rate process evolution?
Answer: No, not without additional information.
Try it!
The extension from a one-factor economy to a
two-factor economy is straightforward. It just
corresponds to adding an additional branch on
every node in the appropriate tree (or analytic
expression). This procedure for extending the
economy to three factors, four factors, and so on is
similar.
29
1
E Consistency with Equilibrium
The preceding material exogenously imposes a
stochastic structure on the evolution of the
zero-coupon bond price curve.
The evolution, except for the number of factors, is
almost completely unrestricted.
But a moment's reflection reveals that this cannot be
the case.
30
1
A T-maturity zero-coupon bond is, of course, a close
substitute (for investment purposes) to a T-1 or a T+1
maturity zero-coupon bond.
Therefore, in an
economic equilibrium, the returns on these similar
maturity zero-coupon bonds cannot be too different.
Furthermore, the entire zero-coupon bond curve is
pairwise linked in this manner.
Consequently, to be consistent with an economic
equilibrium, there must be some additional explicit
structure required on the parameters.
Chapter 7 introduces these restrictions by studying the
meaning and existence of arbitrage opportunities.
31