The Distribution of the Value of the Firm and
Stochastic Interest Rates: An Application to Structural
Models of Default Risk
by
S. Lakshmivarahan1, Shengguang Qian1 and Duane Stock2
December 2, 2007
1) School of Computer Science, University of Oklahoma, Norman, OK 73019
2) Division of Finance, Michael F. Price College of Business, University of Oklahoma,
Norman, OK 73019. Contact Author: Duane Stock, email: dstock@ou.edu
1
Abstract
The time evolution of the value of a firm is commonly modeled by the linear, scalar
stochastic differential equation (SDE) of the type d Vt = rV
t t d t + s v ( t )Vt d Wv where the
coefficient rt in the drift term denotes the (exogenous) stochastic short term interest rate
and s v ( t ) is the given volatility of the value process. In turn, the dynamics of the short
term interest rate, d rt , are modeled by a scalar SDE. We solve this pair of equations for a
variety of commonly used interest rate processes which in turn provide explicit expressions
for Vt . We show that Vt exhibits a lognormal distribution when rt is a normal/Gaussian
process defined by a common variety of narrow sense linear SDEs. While we also provide
explicit solutions for Vt when rt evolves according to general linear and some well
known nonlinear SDEs, the distribution of Vt is not explicitly known in many of these
other cases. The results can be applied to many financial situations where modeling value of
the firm is critical. For example, there is a large literature concerning structural models of
yields on corporate debt.
Some structural models assume a constant rate of interest while
others utilize, for example, interest rate models consistent with the popular Vasicek (1977)
model. Our solutions for the distribution of Vt readily permit detailed analysis of the term
structure of default risky yields and credit spreads. More specifically, we analyze how the
probability of default varies with maturity and, in turn, analyze the complex interactions of
maturity, drift in Vt , and variance of Vt upon default risky yields and credit spreads.
2
Introduction
Modeling the value of the firm is one of the more important research topics in
finance. The value of an unlevered firm is the value of expected future cash flows
discounted at a rate appropriate for an all-equity firm whereas the value of a levered
firm is commonly expressed as the value of an unlevered firm plus the gain from
leverage due to a tax shield provided by the debt. Including business disruption costs,
the optimal capital structure can then be characterized as a trade-off between the
interest tax shield and disruption costs. Recent analysis by Hackbarth, Hennessy and
Leland (2007) extends this line of research by examining an optimal mixture of debt;
that is, the optimal mixture of bank debt and market debt (bonds).
Leland and Toft (1996) develop an ambitious model of firm value that addresses
optimal capital structure, optimal debt maturity, and shape of credit spreads. They
describe alternative shapes of credit spread term structures dependent upon various
conditions. Generally, the shapes are either positively sloped throughout or humped.
Recently, Qi (2007) has modified the Leland and Toft (1996) by setting the lower
bankruptcy boundary to be a fraction of bond face value.
The importance of good structural models for credit spreads has been enhanced
with the growth of credit derivatives and the credit crisis of 2007.
More specifically,
notional amounts of credit derivatives grew by over 100% for every year from 2004
through 2006. At the end of 2006, there was 34.5 trillion outstanding. 1 The weakened
credit quality of many financial firms in 2007 caused high volatility in equity markets
and, also, large changes in the value of credit spreads and credit default swaps. For
example, spreads on Citigroup credit default swaps rose 30 basis points in the first
week of November where a basis point is worth $1,000 per year. Eom, Helwege, and
1
“Credit Derivatives Show Surge”, A. Saha-Bubna and E. Barrett, Wall Street Journal, April 28, 2007.
3
Huang (2004) strongly suggest there is a need to improve structural models of credit
spreads because their empirical tests reveal obviously large weaknesses.
Our purpose is two-fold. The first purpose is to derive alternative distributions of
Vt for commonly used processes of short term interest rates. This is desirable
because value of the firm, Vt , processes are strongly dependent on processes of short
term interest rates.
Alternative short term interest rate processes can be classified as
narrow sense linear functions of r, generally linear in r, or nonlinear in rt. The resulting
distributions of Vt can be easily compared to distributions where rt is assumed
constant.
The second purpose is to apply these Vt distributions to structural models of
credit spreads. We develop equations for default risky spot rates and credit spreads
utilizing Vasicek (1977) interest rate processes, a popular narrow sense linear model
very commonly used in structural models. Probability of default is determined by a first
passage approach where default occurs if value of the firm pierces certain lower
thresholds. Probability of default depends on all parameters of the Vasicek (1977)
model including mean reversion parameters. Probability of default may be expressed as
a function of expected Vt and variance of Vt . More specifically, greater maturity
increases both expected Vt and, also, variance of Vt. Thus, the impact of increasing
maturity upon spread is complex. Surprisingly, probability of default may or may not
increase with maturity.
The next section describes models of firm value and the following section
describes the framework for solutions of firm value.
Then we describe solutions in
the cases where interest rate processes are narrow sense linear. Finally, we describe an
application of our theoretical results to modeling credit spreads of corporate bonds as
4
dependent upon features of a popular interest rate process.
1. Models of Firm Value Dependent Upon Interest Rate Processes
The time evolution of the value, Vt , of a firm is routinely modeled by a linear,
scalar, stochastic differential equation (SDE)
dVt
= rt dt + s v ( t ) dWv , t
Vt
,
(1.1)
where the instantaneous drift rt denotes the (exogenous) stochastic short-term interest
rate process and s v ( t ) is the instantaneous volatility. See, for example, Acharya and
Carpenter (2002). The dynamics of rt are commonly modeled by a (scalar) SDE of the
type
drt = a ( rt , t )dt + s ( rt , t )dWr ,t ,
(1.2)
where the instantaneous drift, a ( rt , t ) and the volatility, s ( rt , t ) are smooth
functions. It is further assumed that the Wiener increment processes dWv and dWr
are correlated where
E éê( dWv , t
ë
with
)( dWr ,t ) ùûú= r ( t )dt
(1.3)
r ( t ) £ 1 . It is worth noting that in this framework the flow of information is
only one way: rt affects Vt and not vice versa. Our first purpose in this paper is to
solve for Vt and characterize the distribution of the process Vt for different choices
of the rt processes.
Models utilized for the rt process can be divided into linear and nonlinear models.
Following Arnold (1974), linear models can be further subdivided into two subclasses.
The single factor model in (1.2) is called a narrow sense linear model if
5
a (rt , t )= a1 (t )rt + a2 (t )
,
(1.4)
and
s ( rt , t )= s r (t )
(1.5)
That is, the drift term is linear in rt and the volatility term is not a function of rt .
In
contrast, the model is called a general linear model if a ( r, t ) is of the form (1.4) and
s ( rt , t )= b1 (t )rt + b2 (t ) ,
where ai (t ), bi (t ), i = 1, 2 and s r (t ) are smooth functions of time t .
(1.6)
Finally, the
model is called nonlinear if either a ( rt , t ) and/or s ( rt , t ) are nonlinear functions of
the short rate rt .2
Similarly, Black (1995) defines three classes of rt processes by considering “a
simple random process without worrying about the forces that influence the interest
rate”. Thus, if
drt = s (rt , t )dWr ,t ,
(1.7)
then, rt is (1) a normal/Gaussian process if s ( rt , t ) is independent of rt , (2) a
lognormal process if
s ( rt , t )= s (t )rt , and (3) a square root process if
1
s ( r , t )= s (t )rt 2 .
The above classifications have a lot in common as given in Tables 1a,b,c. The
narrow sense linear models of Merton (1973), Vasicek (1977), Ho-Lee (1986), Hull and
White (1990) and generalized Hull and White (2000) are special cases of the Heath,
Jarrow and Morton (1992) model and define normal/Gaussian processes.
The nonlinear models of Black, Derman and Toy (1990) and Black and Karasinki
(1991) are in fact narrow sense linear SDE’s in ht = ln rt implying ln rt is a normal
2
See Cairns (2004) for more on the classification of interest rate models.
6
process and hence rt is a lognormal process (Johnson et. al. (1994)). The general
linear models of Dothan (1978) and Brennan and Schwartz (1979) give rise to
lognormal processes. The nonlinear models of Cox, Ingersoll and Ross (1985) and
Pearson and Sun (1994) define the so called square root processes.
There are essentially two ways of solving the system (1.1) - (1.2). The first method
T
is to define a vector Markov process X t = (Vt , rt )
and express (1.1) - (1.2) in a
single equation
dX t = f ( X t , t )dt + s ( X t , t )dBt ,
where f ( X t , t ) =
T
( f1 ( X t , t ), f 2 ( X t , t ))
T
dBt = ( dB1,t , dB2,t )
, s ( X t ,t )
is
(1.8)
a
2´ 2
matrix
and
is a vector of two independent Wiener increment processes. It
can be verified that if (1.2) is linear, then so is (1.8). In this case we can solve (1.8)
explicitly using the methods in chapter 8 of Arnold (1974). However, since solving
even the simple linear vector equations can be very demanding, in this research we use
a simpler alternative approach that exploits the one way dependence of Vt on rt . We
first solve the scalar SDE (1.2) for rt and using the solution in (1.1), we then
recover Vt is a lognormal process when rt º r , a constant. However, it is very
unappealing to assume interest rates are constant in many cases. For example, the V t
process is critical in valuing corporate bonds whose value clearly depends on the level
of interest rates. We show that Vt also exhibits a lognormal distribution where rt is a
normal process defined by the narrow sense linear models in Table 1. The results are
quite different when rt is a lognormal process defined by general linear models or by
narrow sense linear models in ln rt .
In these cases, while we derive expressions for
the exact solution of Vt , its distribution is not lognormal. Similar conclusions apply
7
when rt evolves according to the nonlinear models. (See Table 1.)
To our knowledge this is the first attempt to quantify the sensitivity of the solution,
Vt of (1.1) and its distribution with respect to the various interest rate models. One
general result is that the distribution of Vt varies widely from lognormal to other
classes of distributions as we vary interest rate models across the narrow sense linear,
general linear and nonlinear models. Thus, two important open questions are: (1) What
is an appropriate model for the distribution of Vt and (2) in the framework given by
(1.1) - (1.2), what classes of interest rate models can give rise to a well defined Vt
process?
2.
The Framework for the Solution
In this section we develop a framework for solving (1.1) - (1.2). Setting
g t = ln Vt and applying Ito’s lemma, (1.1) becomes (Kloeden and Platen (1992))
æ
ö
1
dg t = çç rt - s v2 (t )÷
dt + s v (t )dWv ,t .
÷
÷
çè
ø
2
(2.1)
The following result can be easily verified.
Lemma 2.1
Let dW1,t and d W 2,t be two independent standard Wiener increment
processes. The two correlated processes d Wv ,t and d W r ,t in (1.3) can be expressed
as linear functions of dW1,t and d W 2,t given by
dWr ,t  dW1,t ,
(2.2a)
and
dWv ,t = r (t )dW1,t + 1- r 2 (t )dW2,t
(2.2b)
Substituting (2.2) in (2.1) and (1.2), we get
8
drt = a (rt , t )dt + s (rt , t )dW1,t ,
(2.3)
and
é
ù
1
dg t = êrt - s v2 (t )údt + s v (t )éêr (t )dW1,t + 1- r 2 (t )dW2,t ù
.
ú
êë
ú
ë
û
2
û
(2.4)
Integrating (2.4), it follows that
gt = g0 +
ò
t
0
rs ds -
1 t 2
s v ( s )ds +
2 ò0
ò
t
0
s v ( s )r ( s )dW1, s +
ò
t
0
s v ( s ) 1- r 2 ( s )dW2, s
(2.5)
The following two properties of the Wiener process are easily verified (Shiryaev
(1999), Oksendal (2003), Kuo (2006)).
Lemma 2.2. If Wt is a standard Wiener process, then, for c>0,
Wt =
1
W2 .
c ct
(2.6)
If c > 1 ( < 1 ), this is equivalent to time stretching (compression). Here t is called the
“physical time” and t = c 2t is called the “operation time”.
Let f : Â ® Â be a square integrable function on any finite interval such that
T (t ) =
ò
t
0
f 2 (t )dt < ¥
,
for all 0 < t < ¥
,
(2.7)
is a strictly increasing function with unique inverse.
Lemma 2.3 Relation between Ito integrals and Wiener process
The Ito integral
ò
0
t
f ( s )d Ws
9
with f(s) > 0 for 0 ≤ s ≤ t is equivalent to the standard Wiener process W t where
t = T (t ) =
ò
0
t
f 2 ( s )ds .
Combining Lemmas 2.2 and 2.3, and setting c 2 =
ò
0
t
f ( s )dWs = WT (t ) =
T (t )
t
, it follows that
T (t )
Wt
t
(2.8)
Applying (2.8) to the last two terms in (2.5) we obtain
ò
0
t
s v ( s )r ( s )dW1, s = W1,T1(t ) =
T1 (t )
W1,t ,
t
(2.9a)
where
T1 (t ) =
ò
t
0
s v2 ( s )r 2 ( s )ds .
(2.9b)
Also,
ò
t
0
s v ( s ) 1- r 2 ( s )dW2, s = W2,T2 (t ) =
T2 (t )
W2,t
t
,
(2.10a)
where
T2 (t ) =
ò
t
0
s v2 ( s )(1- r 2 ( s ))ds .
(2.10b)
Substituting (2.9) and (2.10) in (2.5) and simplifying, it follows that
é T (t )
ù
é 1 t
ù
T2 (t )
é t
ù
Vt = V0 exp êò rs ds úexp ê- ò s v2 ( s )ds úexp êê 1 W1,t +
W2,t úú . (2.11)
êë 2 0
ú
t
êë t
ú
14444êë420444443úû144444444
42 4444444443û1444444444444
42 44444444444443û
I
II
III
Since s v (t ) is assumed to be deterministic, the second factor, I I , in (2.11) is
deterministic. From the independence of W1,t and W2,t , it follows that the sum of the
10
two terms in the exponent of the third factor I I I in (2.11) is Gaussian from which we
infer that the third factor I I I in (2.11) is lognormal. Thus, the distribution of Vt in
(2.11) critically depends on the properties of the rt process in (2.3).
For later reference, we consider two special cases.
Case 1: s v (t )= s v , a constant and r (t )º r , a constant.
In this case,
T1 (t )= s v2r 2t and T2 (t ) = s v2 (1- r 2 )t .
(2.12)
Substituting these into (2.11) and simplifying we get
é t
ù
1
é
ù
Vt = V0 exp êò rs ds - s v2t úexp ês v r W1,t + 1- r 2 W2,t ú .
êë 0
ú
ë
û
2
û
{
Case 2: s v (t )= s v
,
}
(2.13)
r = 0 . In this case (2.13) reduces to
é
ù
1
é t
ù
Vt = V0 exp êò rs ds úexp ês vW2,t - s v2t ú ,
êë
2 3ú
û14444444
û
14444êë420444443ú
42 44444444
I
(2.14)
II
where the second factor I I in (2.14) is called a Brownian martingale or the stochastic
exponential (Shiryaev (1999), Kuo (2006)). In the following section we solve (2.3) for
rt for various choices of a ( rt , t ) and s ( rt , t ).
3. Narrow Sense Linear Solutions
Setting
a (rt , t )= q (t )- c (t )rt and s r (rt , t )= s r (t ) ,
(3.1)
in (2.3), we get a narrow sense (time varying) linear model known as the generalized
Hull and White (2000) model given by
drt = - c (t )rt dt + q (t )dt + s r (t )dW1,t
(3.2)
11
Since all the other narrow sense linear models in Table 1 are special cases of (3.2), we
first concentrate on solving (3.2).
Defining
ò
c (t ) =
t
0
c ( s )d s ,
(3.3)
from chapter 8 of Arnold (1974) and Gard (1988) we get
F ( t ) = e - c (t ) ,
(3.4)
as the fundamental solution of (3.2). Hence the solution of (3.2) is given by
rt = rt (det )+ rt (ran ) ,
(3.5a)
where
rt ( det ) = e
- c (t )
r0 +
ò
t
e
- éëc (t )- c ( s )ùû
q ( s )ds ,
0
(3.5b)
and
rt ( ran ) =
ò
t
e
- éëc (t )- c ( s )ùû
0
s r ( s )dW1, s .
(3.5c)
T3 (t )
W1,t ,
t
(3.6a)
By lemmas (2.2) and (2.3), we obtain
rt ( r a n ) = W1,T3 (t ) =
where
T3 (t ) =
ò
t
0
s r2 ( s )e
- 2 éëc (t )- c ( s )ùû
ds .
(3.6b)
Combining (3.5) and (3.6), it is immediate that
rt = rt ( det )+
T3 (t )
W1,t
t
~ N ( rt ( det ), T3 (t )) .
(3.7a)
(3.7b)
Hence
12
é T (t ) ù
é t
ù
é t
ù
exp êò rs ds ú= exp êò rs (det )ds úexp êê 3 W1,t ú
ú.
ú
êë 0
ú
t
ëê 0
û
û
êë
ú
û
(3.8)
Substituting (3.8) in (2.11) and simplifying we get an expression for Vt .
éæ T t
ù
ö
é t ïì
T3 (t ) ÷
T2 (t )
1 2 ïü ù
êçç 1 ( )
ú
÷
ê
ú
Vt = V0 exp ò í rs ( det )- s v (t )ý dt exp êç
+
W1,t +
W2,t ú. (3.9)
÷
÷
êë 0 ïîï
ú
ç
ï
2
t
t ÷
t
êçè
ú
ï 43û
þ
ø
1444444444444442 4444444444444
ë
1444444444444444444
42 44444444444444444443û
I
II
The first factor I in (3.9) is a deterministic function and the second factor I I in (3.9)
is a lognormal variate (Johnson et al. (1994)).
We summarize this result in the following:
Theorem 3.1: Let the interest rate rt evolve according to a narrow sense linear scalar
SDE of the type (3.2). Then, rt is a normal or Gaussian process and the solution Vt ,
called the value process in (1.1), is a lognormal process given by (3.9).
We now
enlist a number of corollaries as special cases.
Case 1: Let s v (t )º s v , and r (t )º r be constants. Then, using (2.12) in (3.9) we
get
éæ
é t
1 2 ù
ç
Vt = V0 exp êò rs ( det )ds - s v t úexp êêçç s v r +
0
ú
2
êççè
ëê
û
ë
ù
T3 (t ) ö
÷
2
÷
W1,t + s v (1- r )W2,t ú
÷
ú. (3.10)
÷
t ø
÷
ú
û
Case 2: Hull and White (1990) model: Setting c (t )º c and s r (t )º s r in (3.5) (3.6), it follows that
rt HW ( det ) = e - ct r0 +
rt
HW
( ran )=
ò
t
e
- c(t - s )
0
T3HW (t )
W1,t ,
t
q ( s )ds ,
(3.11)
13
HW
3
T
(t ) = s
2
r
ò
t
e
s r2 é
ds =
1- e ê
ë
2c
- 2 c(t - s )
0
2 ct
ù .
ú
û
Setting rt (det )= rt HW (det ) and T3 (t )= T3HW (t ) in (3.9), it follows that Vt H W is
lognormal.
Case 3: Ho-Lee (1986) model: Setting c (t )º 0 in (3.11), we obtain
rt HL ( det ) = r0 +
ò
0
t
q ( s )ds ,
rt HL (ran )= s rW1,t ,
(3.12)
T3HL (t )= s r2t .
Again by setting rt (det )= rt HL (det ) and T3 (t )= T3HL (t ) in (3.9), we see that
Vt H L is lognormal.
Setting c (t )º c , q (t )º q and s r (t )= s r in
Case 4: Vasicek (1977) model:
(3.11) we get
rtV ( det ) = e - ct r0 +
rtV ( ran )=
T3V (t ) =
qé
1- e - ct ù
ê
ú
ë
û,
c
T3V (t )
W1,t ,
t
s r2
(1- e2c
2 ct
)
(3.13)
.
Substituting these into (3.9), we readily see that VtV is lognormal.
Case 5: Merton (1973): Setting q (t )º q , c (t )º 0 and s r (t )º s r , we get
rt M (det )= r0 + qt ,
14
rt M (ran )= s rW1,t ,
(3.14)
T3M (t )= s r2t .
which on substitution into (3.9) implies that Vt M is lognormal.
Case 6: Let rt º r . Then s r (t )º 0 , T3 (t )º 0 , rt ( ran )º 0 , rt (det )º r . For this
choice, (3.9) implies that Vt is lognormal. If we further assume that s v (t )º s and
r t º r , then (2.13) and (3.9) it follows that
éæ 1 ö
ù
ú .
Vt = V0 exp êçç r - s v2 ÷
t
+
s
W
÷
v 2,t
êëçè
ú
ø
2 ÷
û
(3.15)
æV ö
éæ 1 ö 2 ù
ú
÷
ln ççç t ÷
~ N êçç r - s v2 ÷
÷
÷
÷t , s v t ú ,
êëçè
2 ø
è V0 ÷
ø
û
(3.16)
Since
the lognormal probability distribution of Vt is given by
æV ö
÷
pt ççç t ÷
÷=
è V0 ø÷
é
ê
ê
1
exp êêæVt ÷
ö
ê
2ps v t çç ÷
ê
÷
÷
çè V0 ø
êë
ù
ö æ s v2 ÷
ö ïü
ïìï çæVt ÷
ïý ú
çç r ÷
t
í ln çç ÷
÷
ú
÷
÷ çè
ïï è V0 ø
2 ÷
ø ïþ
ïú.
î
ú
2s v2t
ú
ú
ú
û
(3.17)
The mean and variance of Vt are given by Johnson et. al. (1994).
E Vt   V0ert and Var (Vt )= V02e 2rt éêe s v t ë
2
1ù
ú
û
(3.18)
Examples of plots of (3.18) are given in Figures 1 and 2. Figure 1 gives examples
of the Vt distribution when rt is constant and the probability densities of VT at
T = 5, 10, 15, 20 are generated from (3.17).
VT increases with T and moves the VT
distribution curve toward the right, while, at same time, the variance of VT also
15
increases with T.
Figure 2 gives examples of the Vt distribution when rt follows Vasicek’s
(1977) model. First, we build a Vasicek system in Figure 2a to be very similar to the
constant interest rate model illustrated in Figure 1; that is, we set
q
= r0 , r = 0 and
c
use a small s r . We compute and plot the VT distribution at T = 5, 10, 15, 20 in
Figure 2a.
As expected, the distributions in Figures 1 and Figure 2a are very similar.
In Figure 2b, we introduce positive correlation ( r = 0.9 ) where the VT distribution
now has a lower peak and fatter tails than in Figure 2a.
Then, in Figure 2c, we
introduce negative correlation ( r = - 0.9 ) and the VT distribution shows a higher
peak and less fat tails.
We furthermore consider general linear models of rt as given in Table 1.
see Appendix A.
Please
Generally, we cannot determine the distribution of Vt . Next
consider nonlinear models of rt such as Black and Karasinki (1991); Black, Derman
and Toy (1990); and Cox, Ingersoll, and Ross (1985). In these cases we also cannot
determine the distribution of Vt . Please see Appendix B for details.
The above results may be used to derive expressions for the distribution of Vt .
Assuming positive drift, E Vt  grows with time as does the variance around that
expected value. In cases below we refer to a specific time as T which is a terminal or
intermediate date for a stage in the life of the firm, such as maturity of the firm’s debt.
For brevity, we concentrate on narrow sense linear models in Table 2. As given in that
table, it is useful to represent log Vt V0 

2
as distributed N μ 't , σ 't

where μ 't is a
2
drift unique to the particular interest rate process and σ 't is a variance also unique to the
16
particular process.
For example, in the Vasicek model,
1  
  1


T'    r0  e cT   T   r0      v2T
c  c
c  2


and
 T' 2  2 v  T3V T  T  T3V T    v2T
.
Then, E Vt  , Var Vt  , and probability densities can be generally expressed as given
2
at the top of Table 2 where all depend upon particular μ 't and σ 't values.
The more complex drift and variance terms for the different rt processes can be
compared to the constant r case where the drift term is merely
variance merely  T'   v2T .
2
1 2

 r  v  T
2 

and the
In Merton (1973),  is needed to prevent arbitrage in
the drift and ρ is needed in the variance. Furthermore, Vasicek (1977) requires both θ
and mean reversion parameter c in the drift.
4. Application to Structural Models of Credit Spreads
4.1 Expressions for Risky Spot Rates and Credit Spreads
The first structural model of credit spreads was given by Merton (1974) where
he assumed a constant short term interest rate.
Leland (1994) and Leland and Toft
(1996) also assume a constant interest rate and thus, as in Merton (1974), there is no
correlation of stochastic changes in interest rates and stochastic changes in firm value.
Of course, it is unappealing to not allow interest rates to change in a model of bond
valuation. In contrast, Longstaff and Schwartz (1995) and Collin-Dufresne and
Goldstein (2001) models include both a Vt
process and, also, a rt process. Their
usage of dual processes is as we describe in Section 1. The rt process used by both is
17
a Vasicek (1977) model that is narrow sense linear.
Eom, Helwege, and Huang (2004)
provide a characterization of these and other popular structural models in an appendix.
The above structural models do not provide the distribution of Vt .
In
contrast, the distributions of Vt we have derived in earlier sections can now be used to
develop compact expressions for default risky spot rates, Rd T  , and the spot rate
spread over a case with no default risk, Rdf(T).
Here Rd T   Rdf T  is denoted as
S T  . Our solution of distributions permits spot rates and spreads to be concisely and
explicitly expressed as a function of probability of default. Of course, this combines the
two processes for rt and Vt where the level and volatility of Vt represents default
risk. Debt with no default risk and maturity T has present value at T of
PVdf T   PAR . For a bond with default risk, the present value at T is
PVd  t | t  T   PVdf T  1  Pd  K , D, T    PVdf T  Pd  K , D, T  RR  Pd  K , D, T   .
Here, Pd  K , D,T  is the first time (first passage) the firm value hits a level where
default occurs. Default occurs when the value of the firm falls below the face
value of the debt ( K ) at time T.
Additionally, as in Giescke (2004), default
occurs before maturity, t < T , when the value of the firm falls below a level, D .
That is, frequently creditors have a right to reorganize the firm if value falls below
D.
Various covenants in bank loans, bonds, and other debt may contain such a
condition. For our purposes, we assume, as given in Giescke (2004), that D is
below K . Thus, Pd  K , D,T  is the probability the first passage is at t or T .
RR is the recovery rate of principle in case the firm defaults. For our purposes,
18
we assume the recovery rate is a constant, 0.50.
For various short rate models, we have  v ,  r ,   t  , c ,  as parameters
in the model. By applying the above exact solutions, we can find the distribution
of firm value at any specific time.
thresholds
At a specific time, if we also know the default
 K , D  , then we easily can find the default probability
Pd  K , D, t  at
time t. With this, we can further derive the default risky bond price at maturity
T , PVd T  , from the default-free bond present value, PVdf T  , the PAR value of
the bond.
The risky spot rate and spread are derived from the default-free spot rate
Rdf T  and the default risky bond price at maturity, PVd T  .
PVd  0   PVd T  e
 Rdf T T
PVd  0  is the present value of the bond which may default at T. Of, course,
PVd  0  PVd f  0 due to the risk of default.
To define the risky spot rate, Rd T  , we easily construct a default risky bond,
which sells at PVd  0  at present, and is valued at PAR at maturity. So,
PVd  0  e
Rd T T
 PAR  PVd f T 
.
After substituting,
PVd T  e
 Rdf T T
e
Rd T T
 PAR  PVdf T 
.
Then, dividing both sides by PVd (T ) and simplifying we obtain
19
Rd T  
1
1
ln
 Rdf T 
T 1  1  RR  Pd  K , D, T    Pd  K , D, T 


and
S (T )  Rd T   Rdf T  
1
1
ln
T 1  1  RR  Pd  K , D, T    Pd  K , D, T 




As long as Pd  K , D, T   0 ,and RR Pd  K , D, T  1 ,
1  1  RR  Pd  K , D, T    Pd  K , D, T   1 , which implies that Rd T   Rdf T  . The
probability of default, Pd  K , D, T  , can be derived from the alternative distributions of Vt
prescribed above.
4.2 Computations of Risky Spot Rates and Credit Spreads
Given the above equations for Pd  K , D, T  , Rd T  , and S T  , we may now
apply the prior alternative solutions for Vt to analyze risky spot rate term structures
and credit spread term structures. Our examination has an analytical advantage
compared to prior research in that we can readily compute expected values and
variances from the above distributions. Assuming, for example, a popular Vasicek
(1977) model of short term interest rates, we can very easily plot E VT  and
Var VT  , and Pd  K , D, T  for any parameters of that particular interest rate process.
Then we analyze how complex interactions dictate various levels and shapes of Rd T 
and S T  term structures.
E Vt  values that grow with time obviously tend to reduce default risk
because, in the great majority of
cases, the value of the firm is assumed to drift
20
upward with the passing of time.
In the above cases, the growth rate in firm value is
obviously affected by the level of interest rates and the rt process in the alternative
interest rate models.
However, at the same time that E Vt  grows with time, the
variance of Vt also increases with time. At each future point in time, the probability of
default (first passage) can be computed where greater default probability obviously
increases Rd (T ) and S (T ).
A large rate of increase in probability of default due to time passage tends to
increase the relative slope of Rd T  and S (T ) even though the slopes may positive
or negative. However, we note that the impact of a first passage probability that grows
with time is complex. That is, first passage probability may grow with time but, S (T )
may or may not increase because the present value of the loss is diminished with
greater time.
Furthermore, it is quite interesting to demonstrate that the probability of
default may not necessarily increase with time.
Figure 3 is one example of how our analysis permits detailed analysis of default
spreads using the Vasicek (1977) model common to many structural models. Here we
assume an initial firm value of 150, θ of 0.03, and no correlation between firm value
and level of interest rates. Other Vasicek parameter values are given in Figure 3.
Expected value of the firm rises with maturity where it is 268 at a maturity of 10. The
square root of Var VT  in the second panel increases to 59 at maturity 10. The first
passage probability always grows with maturity which is shown in the third panel
( Pd  K , D,T  ) and fourth panel which displays the derivative of Pd with respect to T.
The last two panels display Rd T  and S(T). Note that Rd T  peaks at around
T=3
which is in contrast to R d f . Thus, even though the probability of default
21
consistently increases, its impact on the spot rate weakens because the present value of
the loss declines with time. Similarly, the spread peaks at around T=3 and then
declines.
Figure 4 is another set of graphs depicting default spreads where θ is 0.07 instead
of 0.03. Here expected value of the firm grows more rapidly but so does variance
around expected value. The net effect is to reduce probability of default relative to the
previous figure.
Interestingly, the probability of default peaks and then falls and thus
its derivative becomes negative. For this case, Rd T  always increases but S(T) peaks
at about T=2.0, a little bit earlier than our previous figure. The earlier peak
and
subsequent steep decline is due to the peak in the probability of default.
The peak in probability of default Pd is counterintuitive at first glance but can be
partially explained as follows. Assume the D default threshold is relatively low such
that default before T is low. Then, the likelihood of default is largely dependent upon
default potential at the higher threshold K at time T. The passage of time increases
expected value and also increases variance where the net effect on probability of
default is unclear. Greater E Vt  may dominate the increase in variance so that
default declines with T.
3
5. Conclusion
The value of the firm, Vt , is the sum of all claims upon it. In the simplest case,
Vt is just the sum of equity and zero coupon debt. More complex cases consider
secured versus unsecured debt, junior versus senior debt, convertible bonds and
preferred stock. A weakness of much of the research concerning Vt is the lack of an
3
See Giesecke (2004) for separate expressions of default before T and at T.
22
expression for its expected value and variance as time advances. That is, much of the
research generally describes expected firm value that grows with time where the
instantaneous variance of the process describes the volatility and riskiness of the firm.
There is a need for precision in the distribution of Vt .
One outstanding stream of research that uses the concept of Vt quite intensively
concerns structural models of credit risk. Strong growth in credit derivatives and the
recent credit crises have both increased the demand for improved models of credit risk.
We note that popular structural models commonly utilize simultaneous processes for
Vt and short term interest rates, rt , where changes in Vt are dependent upon the
particular rt process assumed.
to linearity in the rt
Recognizing different classes of models with respect
process, we show that the exact distribution of Vt depends on
the type of rt process assumed. In some cases, Vt is lognormally distributed. For
example, the Vasicek (1977) process commonly assumed in structural models leads to a
lognormal Vt.
A well defined Vt distribution permits us to easily analyze default
probabilities and term structures of credit risk in great detail. Greater passage of time
suggests greater expected firm value but, at the same time, greater variance of Vt so that
the impact of maturity on default probability is complex. The behavior of default
probability with passage of time has a clear impact on term structure of credit spreads
where a strong growth in probability of default increases the slope of the term structure
of credit spreads. It is interesting to note that probability of default may decrease with
greater maturity.
23
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Black, F. (1995) “Interest rate options”, The Journal of Finance, Vol. 50, 1371-1376.
Black, F., E. Derman and W. Toy (1990) “A one-factor model of interest rates and its
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Cairns, A.J.G. (2004) Interest Rate Models: An Introduction, Princeton University
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Eom, Y.H.; J. Helwege; and J. Huang (2004) “Structural Models of Corporate Bond
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Theory Explain Debt Structure?”, Review of Financial Studies, Vol. 20,
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Interest Rates: A New Methodology for Contingent Claims Valuation”,
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Contingent Claims”, Journal of Finance, Vol. 41, 1011-1029.
24
Hull, J.C. (2000) Options, Futures and other Derivatives, Prentice Hall, (4th
Edition), 698 pages.
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Johnson, N.L., S. Kotz and N.Balakrishnan (1994) Continuous univariate
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Kuo, H.H. (2006) Introduction to Stochastic Integration, Springer, New York, 278
pages.
Leland, H. E. and K. B. Toft (1996) “Optimal Capital Structure, Endogenous
Bankruptcy, and the Term Structure of Credit Spreads, “ Journal of Finance,
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Longstaff,, F. A. and E. S. Schwartz (1995) “ A Simple Approach to Valuing Risky
Fixed and Floating Rate Debt”, Journal of Finance, Vol. 50, 789-819.
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25
Appendix A:
General linear models for rt
Consider a general linear (time invariant) scalar SDE, known as the
Brennan-Schwartz (1979) model in Table 1
drt = - crt dt + qdt + s r rt dWr ,t .
(A.1)
From chapter 8 of Arnold (1974), it follows that
é æ s2ö
ù
÷
F (t ) = exp êê- çç c + r ÷
t + s rW1,t ú
÷
ú ,
2 ÷
ø
êë çè
ú
û
(A.2)
is the solution of
drt = - crt dt + s r rt dW1,t ,
and is the fundamental solution of (A.1). The solution of (4.1) is then given by
t
rt = F (t )r0 + q F (t )ò F - 1 (u )du .
1442 443 1444444442
0 444444443
I
(A.3)
II
Combining this with (2.11), it follows that
t
t
t
é
ù
Vt = V0 exp êr0 ò F ( s )ds + q ò F ( s )ò F - 1 (u )duds ú
0
êë 0
144444444444444444444
420 44444444444444444444
43úû
I
é T (t )
ù
é 1 t
ù
T2 (t )
exp ê- ò s v2 ( s )ds úexp êê 1 W1,t +
W2,t úú.
êë 2 0
ú
t
êë t
ú
144444444
42 4444444443û1444444444444
42 44444444444443û
II
(A.4)
III
To our knowledge, all we can claim at this stage is that F (t ) in (A.2) is lognormal
and so is F -
1
(t ). Since the distribution of the second term, I I , in (A.3) is not known,
we do not know the distribution rt in (A.3).
Similarly, we know that the second term, I I , in (A.4) is deterministic and third
term, I I I , in (A.4) is lognormal. Since the first term, I , in (A.4) involves the integral of
rt , its distribution is not known. Hence, while we know the exact form of the solution
26
Vt in (A.4), its distribution is not known. We conjecture that it is not lognormal.
In the special case when q º 0 and the sign of c in (A.1) is changed, we get
Dothan’s (1978) model. In this case
rt = F (t )r0 ,
(A.5)
and
é T (t )
ù
t
é 1 t
ù
T2 (t )
é
ù
Vt = V0 exp êr0 ò F ( s )ds úexp ê- ò s v2 ( s )ds úexp êê 1 W1,t +
W2,t úú.
êë 2 0
ú
êë 420 444444443úû
t
êë t
ú
14444444
144444444
42 4444444443û1444444444444
42
4444444444444
3û
I
II
III
(A.6)
In this case, while F (t ) and hence rt are lognormal, we don’t know the distribution
of the first term, I , which is correlated with the first term,
T1 (t )
W1,t in the
t
exponent of the third term, I I I , in (A.6). Again, we have an explicit expression for Vt ,
but don’t know its distribution.
27
Appendix B: Nonlinear Models for rt
In this section, we consider the class of nonlinear models due to Black and
Karasinki (1991). In this model ht = ln rt evolves according to a scalar, narrow sense
(time varying) linear SDE
d ht = - c (t )ht dt + q (t )dt + s r (t )dW1,t .
(B.1)
By exploiting the similarity between (B.1) and (3.2), we readily obtain
ht = ht (det )+ ht (ran ) ,
h t ( det ) = e
- c (t )
r0 +
ò
t
e
0
(B.2)
- éëc (t )- c ( s )ùû
q ( s )ds ,
and
h t ( ran ) = W1,T3 (t ) =
T3 (t )
W1,t .
t
where c (t ) and T3 (t ) are defined in (3.3) and (3.6b) respectively. Hence
h t = h t ( det )+
T3 (t )
W1,t ~ N éëh t ( det ), T3 (t )ù
û,
t
(B.3)
and
é
rt = exp êêht (det )+
êë
T3 (t ) ù
W1,t ú
ú.
t
ú
û
(B.4)
That is, rt has a lognormal distribution. Combining this with (2.11), we get
é t
ïì
ïü ùú
T3 ( s )
ê
Vt = V0 exp êò exp ïí h s ( det )+
W1, s ïý ds ú
ïï
ïï ú
s
ê 0
îï
þï 43û
ë
144444444444444444
42 44444444444444444
I
é T (t )
ù
é 1 t
ù
T2 (t )
exp ê- ò s v2 ( s )ds úexp êê 1 W1,t +
W2,t úú.
t
ëê 2 420 4444444443ûú
êë t
ú
144444444
1444444444444
42 44444444444443û
II
(B.5)
III
The second factor, I I , in (B.5) is deterministic and the third factor, I I I , in (B.5) is
28
lognormal. The distribution of the first factor, I , in (5.5) is not known. In view of the
similarity between the Black and Karasinki (1991) and Black, Derman and Toy (1990)
model, the above analysis and conclusion carry over to the latter class of models as
well.
Also, we observe that if rt evolves according to the Cox, Ingersoll and Ross
(1985) model, then it can be shown (Chapter 4, Cairns (2004)) that rt has a non-central
chi-squared distribution. Referring to (2.11), in this case, since the first factor, I ,
involves the exponential of the integral of a non-central chi-squared process rt , its
distribution and hence that of Vt are unknown.
29
Table 1: Alternative Models of rt According to Linearity in rt
Table 1a: Narrow sense linear stochastic interest rate models
Merton (1973)
drt = qdt + s r dWr
rt - Gaussian
Vasicek (1977)
drt = (q - crt )dt + s r dWr
rt - Gaussian
Ho-Lee (1986)
drt = q (t )dt + s r dWr
rt - Gaussian
Hull and White (1990)
drt = (q (t )- crt )dt + s r dWr
rt - Gaussian
Generalized Hull and White (2000)
drt = (q (t )- c (t )rt )dt + s r (t )dWr
rt - Gaussian
Table 1b:
General linear stochastic interest rate models
Dothan (1978)
drt = qrt dt + s r rt dWr
rt - lognormal
Brennan-Schwartz (1979)
drt = (q - crt )dt + s r rt dWr
rt - lognormal
Table 1c:
Nonlinear stochastic interest rate models
Cox-Ingersoll-Ross
(1985)
drt = q (m- rt )dt + s r rt dWr
rt
Pearson-Sun (1994)
drt = q (m- rt )dt + s r rt - b dWr
- Non-centered
chi-squared
distribution
rt - modified
Black, Derman and Toy
(1990)
ö
s '(t )
drt æ
1
÷
= ççç q (t )+ s 2 (t )+
ln rt ÷
dt + s r (t )dWr
÷
÷
çè
rt
2
s (t )
ø
or
with
t  ln rt
'
æ
s (t ) ö
÷
d h t = ççç q (t )+
ht ÷
dt + s r (t )dWr
÷
÷
çè
s (t ) ø
square root
process
rt - lognormal
,
30
TABLE 2 Distribution of Value of Firm: Narrow Sense Linear Interest Rates Models
éæ 1
 Vt 
2
2
'
'2
Vt = V0 e x pêçç r- s
ln    a1  bW
1 1  b2W2 ~ N  a1 , b1 t  b2 t   N t ,  t
êëçè
2
 V0 

E éëV (T )ù
û= V0e
mT' +
2
s T'
2
,
Var V T   V02e
2
 T'  2 T'
Model
e 1
2
 T'
V
P T
 V0

1
e
V
'
T

2 T
V0
 VT
' 
 ln  T 
V0


2
2 T'
,
2


ö
÷
+t s
÷
÷
ø
v
ù
ú
W
2 , t
ú
û
 VT
ln  T'

V  1 1
V0
cdf  T    erf 
2 T'

 V0  2 2


2
T' and  T'
rt Process
rt º r
Constant r
,

2
v






Processes


1
T'   r   v2  T
2
 T'   v2T
2
Merton (1973)
drt = qdt + s r dWr
1


1


T'   T 2   r0   v2  T
2
2
 T'  2 v  rT   r2T   v2T
2
Vasicek (1977)
drt = (q - crt )dt + s r dWr
1  
  1


T'    r0  e cT   T   r0      v2T
c  c
c  2


 T'  2 v 
Ho-Lee (1986)
drt = q (t )dt + s r dWr
 r2
2
1  e 2 cT  T  r 1  e 2 cT    v2T
2c
2c
b1t
b2t
Assume q (t ) = a1e + a2 e , where a1 , a2 , b1 , b2 are constants
2
mT' =
ö
a1 b1T a2 b2T æ
a
a
a
a
1 ÷
e + 2 e + ççç r0 - 1 - 2 - s v2 ÷
T - 12 - 22
÷
÷
b1 b2 2 ø
b12
b2
b1 b2
è
 T'  2 v  rT   r2T   v2T
2
Hull and
(1990)
White
d rt = (q (t )- crt )d t + s r d Wr
Assume q (t ) = a1e b1t + a 2e b2t , where a1, a2, b1, b3 are constants
mT' =
a1
a2 ö÷
a1
a2
a1 a2 ö÷
1 - cT æ
1 2
1æ
b1T
b2T
ç
÷
e çç- r0 +
+
÷+ cb + b 2 e + cb + b 2 e - 2 s v T - c çç- r0 + b + b ÷
÷
÷
çè
c
c + b1 c + b2 ø÷
è
1
2 ø
1
1
2
2
 T'  2 v 
2
 r2
2
1  e 2 cT  T  r 1  e 2 cT    v2T
2c
2c
31
Figure 1
Probability Density of VT for Constant r
Model: constant r t=r
Parameters: r = 0.1 V0 = 150
 = 0.2
v
1
T
T
T
T
0.9
Probability density of V
T
0.8
=
=
=
=
5
10
15
20
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 100 200
400
600
800 1000 1200
Value of firm V T
1400
1600
1800
2000
32
Figure 2a
Probability Density of VT for Vasicek with σr = 0.1, ρ = 0
Model: Vasicek:
Parameters: r 0 = 0.1  = 0.05 c = 0.5 r = 0.1
 = 0.2  = 0 V = 150
0
v
1
T
T
T
T
0.9
Probability density of V
T
0.8
=
=
=
=
5
10
15
20
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 100 200
400
600
800 1000 1200
Value of firm V T
1400
1600
1800
2000
33
Figure 2b
Probability Density of VT for Vasicek with σr = 0.1, ρ = 0.9
Model: Vasicek:
Parameters: r 0 = 0.1  = 0.05 c = 0.5 r = 0.1
 = 0.2  = 0.9 V = 150
v
0
1
T
T
T
T
0.9
Probability density of V
T
0.8
=
=
=
=
5
10
15
20
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 100 200
400
600
800 1000 1200
Value of firm V T
1400
1600
1800
2000
34
Figure 2c
Probability Density of VT for Vasicek with σr = 0.1, ρ = - 0.9
Model: Vasicek:
Parameters: r 0 = 0.1  = 0.05 c = 0.5 r = 0.1
 = 0.2  = -0.9 V = 150
v
0
1
T
T
T
T
0.9
Probability density of V
T
0.8
=
=
=
=
5
10
15
20
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 100 200
400
600
800 1000 1200
Value of firm V T
1400
1600
1800
2000
35
Figure 3
The Behavior of Default Risky Spot Rates and Spreads Dependent upon the
Distribution of Vt : Probability of Default Increasing with Maturity
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
600
T
E(V )
500
Expectation of V
T
400
300
200
100
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3a
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
120
110
90
80
Volatility of V
T
T
vol(V )
100
70
60
50
40
30
20
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3b
36
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.1
0.09
0.08
d
P (K,D,T)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3c
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.035
0.03
0.02
d
Derivative of P (T) vs. T
0.025
0.015
0.01
0.005
0
-0.005
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3d
37
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
Rdf(T)
0.14
Rd(T)
0.13
Spot Rate
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3e
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.02
0.018
0.016
df
0.012
d
S(T) = P (T) - P (T)
0.014
0.01
0.008
0.006
0.004
0.002
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 3f
38
Figure 4
The Behavior of Default Risky Spot Rates and Spreads Dependent upon the
Distribution of Vt : Probability of Default Humped with Respect to Maturity
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
600
T
E(V )
500
Expectation of V
T
400
300
200
100
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4a
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
120
110
90
80
Volatility of V
T
T
vol(V )
100
70
60
50
40
30
20
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4b
39
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.1
0.09
0.08
0.06
d
P (K,D,T)
0.07
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4c
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.035
0.03
0.02
d
Derivative of P (T) vs. T
0.025
0.015
0.01
0.005
0
-0.005
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4d
40
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
Rdf(T)
0.14
Rd(T)
0.13
Spot Rate
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4e
Vasicek: r 0 = 0.06 c = 0.5 r = 0.05
 = 0.2  = 0 V = 150 K = 100 D = 75
v
0
0.02
0.018
0.016
df
0.012
d
S(T) = P (T) - P (T)
0.014
0.01
0.008
0.006
0.004
0.002
0
0
1
2
3
4
5
6
Maturity (T)
7
8
9
10
Figure 4f
41