Valuation of the Defaultable Bonds
Bi Yusheng1, Wang Xiaoli2
(School of Economics and Management, Tongji University, Shanghai, 200092)
(Email: [email protected])
Abstract： Default occurs when the assets' value of a firm reaches a lower threshold or at the first jump time of a
Cox process related to the firm. By changing numeraire the dimension of PDE to be solved can be reduced, and
closed form solutions are derived for pricing defaultable bonds when spot interest rate evolves as a diffusion of
Vasicek model and hazard rate is governed by a CIR-model process. Thus the classical structural and reduced form
approaches are integrated here. Given the closed form solutions, credit spreads are analyzed.
Keywords： defaultable bonds
credit spread
partial differential equation
可违约公司债定价研究
毕玉升，王效俐
(同济大学 经济与管理学院，上海 200092)
摘要：考虑了两种公司违约方式：公司资产值碰到破产边界以及该公司某Cox过程发生首次跳。在瞬时利
率服从Vasicek扩散模型和危险率服从CIR扩散模型的情况下，通过计价单位变换，将所求的偏微分方程降
维，得到公司债券价格的闭合解。因此经典的结构模型和约化模型在这里得到结合。由闭合解出发，本文
还分析了信用价差。
关键字：公司债
信用价差 偏微分方程
1 Introduction
To describe one firm's default, there are generally two existing approaches: structural and
reduced form models. The first is pioneered by Merton (1974) and extended by Longstaff and
Schwartz (1995), where default is defined as occurring at maturity or when the assets' value
diffuses to a prespecified lower threshold for the first time. The structural approach is conceptually
important because it offers perceptions on default behavior and on the implications for debt
pricing of changes in firm variables such as capital structure. By reason of the complete
information of the firm, default will never occur by surprise. However, this is not necessarily the
case in real-life markets.
The latter approach has been considered by Jarrow and Turnbull (1995), Madan and Unal
(1998). In the literature, an exogenous model is given to describe default. Unlike structural models,
reduced form models assume modelers have incomplete knowledge of the firm's condition. As a
result, this imperfect knowledge leads to an unpredictable default time. Such a sudden default may
be caused by lots of surprise including unexpected devaluations, sudden breach of contract by
suppliers or buyers, and so on.
This paper seeks to provide a combination of first passage time method and stochastic hazard
rate model (Cox process or Doubly stochastic Poisson process) in closed form. A key assumption
of this paper is that stochastic hazard rate is a stochastic process governed by CIR model.
Realdon (2003), Cathcart and El-Jahel (2003) have already dealt with default in both
expectedly and unexpectedly manner. However, they assume hazard rate either a constant or a
linear function of spot interest rate. This can likely be improved by relating the hazard rate to
1毕玉升（1982.4－）
，男，山东莱芜人，同济大学经济管理学院
06 级博士，研究方向：风险管理。
2王效俐：
（1960—）
，男，山西万荣人，同济大学经管学院教授，博士生导师，研究方向：系统工程
some firm's specific information variables such as market value of cash assets, which is proposed
by Madan and Unal (2000). Another defect in these literatures is the independence of spot interest
rate and the firm's assets' value. We release these restrictions in this work: interest rate and the
value of a firm's assets are dependent while hazard rate is assumed to be a stochastic process
uncorrelated with them.
2 The Basic Assumptions
All the work is done in an equivalent martingale measure. Traded in an economy are two classes
of zero-coupon bonds with the same maturity: default-free and defaultable, and their spot prices at
time t are denoted by D(t; T ) and P (t ; T ) .
Assumption 1 The market is perfectly frictionless and all the concerned bonds and securities are
continuously traded.
Assumption 2 The short-term interest rate follows a diffusion of Vasicek model, i.e. the dynamics
are:
drt  (a  brt )dt   r dWr
(2.1)
where a , b ,  r are constants and Wr is a standard Brownian motion.
Under this assumption, the price of a default-free discount bond can be derived from the
affine structure solution of a PDE, and the solution is:
(2.2)
D(t , r; T )  exp{ A(T  t )  B(T  t )r}
where
A( ) 
1  exp{b }
1
1 2  r B 2 ( )
, B ( ) 
(
B
(

)


)(
ab

r ) 
2
b
b
2
4b
Assumption 3 The value of the underlying firm's assets denoted Vt evolves as a Geometric
Browmian motion:
dVt
 rdt   V dWV
Vt
(2.3)
where  V are constants and WV is another standard Brownian motion which is correlated with
Wr
Cov(dWr ,dWV )  dt
( -1 < < 1 )
Assumption 4 Here we assume default occurs when Vt hits the prespecified default
boundary  D (t ; T ) . Then the default time is defined as  '  inf{t  0 : Vt   D(t ; T )} .Obviously,
 ' is a predictable stopping time. There is another situation, in which default time  " is a
stopping time generated by a Cox process, whose intensity (also called hazard rate) t is a
stochastic process. Note that in this situation  " is an unpredictable stopping time. Thus the
time when the concerned firm defaults is    '   " .
Assumption 5 We suppose the hazard rate t  X t and its dynamics are of the CIR type:
dX t  k (l  X t )dt   X X t dWX
(2.4)
where k , l ,  X are constants ( 2kl   X ) and standard Brownian motion WX is uncorrelated
2
with both Wr and WV .
Assumption 6 If the firm defaults during the life of the concerned bond, the bond-holders
can receive  times the value of the risk-free discount bond.
3 A Closed-form Solution to the Valuation Model
This section presents a closed form solution for defaultable discount bond whose par value is
1. At the maturity T, the bondholders receive 1 if default has not occurred or  otherwise,
i.e. the bondholder receives I {  T }  wI {  T } , where I {} is the indicator function.
Employing standard valuation arguments, we know the price of the corporate discount
V
bond P(t , r , Y , X ; T ) (define Y  ) must satisfy the following partial differential equation

(PDE):
1 2
1 2 2

 Pt  (a  br ) Pr  rYPY  2  r Prr   r V rYPrV  2  V Y PYY  k (l  X ) PX

1

+  X 2 XPXX  X ( D  P)  rP  0

2

P
(
t
,
r
,
D
(
t
,
r
;
T
),
X
;
T
)


D
(
t
,
r
;
T
)

 P(T , r , Y , X ; T )  1
Y>D(T,r;T)=1

(3.1)
Where P and P , denote the first and second order partial derivatives w.r.t t , r , Y , or X .
To derive the solutions of the equation (3.1), we use risk-free discount bond D (t , r ; T ) as
the numeraire, i.e. let F (t , z , X ; T )  P(t , r , Y , X ; T ) / D(t , r ; T ) and z  Y / D(t , r; T ) . After this
transformation, define   T  t and we get a PDE of F (t , z, X ; T ) :
1 2
1 2

2
 F  2 W ( ) z Fzz  2  X XFXX  k (l  X ) FX  X (  F )  0

 F ( ,1, X )  
 F (0, z, X )  1 (z>1)


(3.2)
where W 2 ( )   r 2 B 2 ( )  2  r V B( )   V 2 . What we will do now is to change the general
boundary condition of (3.1) to a homogeneous condition. Let G ( , z, X )  F ( , z, X )   , we
1 
have:
1 2
1 2

2
G  2 W ( ) z Gzz  2  X XGXX  k (l  X ) g X  XG  0

(3.3)
G ( ,1, X )  0
G (0, z, X )  1 (z>1)


Note that F  G   (1  G ) and G ( , z, X ) is exactly the survival probability of the firm,
i.e. the probability of the events that neither does the value of the firm's assets reach the lower
threshold, nor does default occur unexpectedly between the time t and T.
To solve equation (3.3), we follow the methods of Cathcart and El-Jahel (2003) and
let G( , z, X )  f ( , z ) g ( , X ) , where the functions f ( , z ) and g ( , X ) depend only on
 , z and  , X respectively. Substitute this to (3.3), we get:
LX g Lz f

 ( )
g
f
(3.4)
where  ( ) denotes some function of  , and partial differential operators LX g , Lz f are
1
1
defined by: LX g   g   X 2 Xg XX  k (l  X ) g X  Xg , Lz f  f  W 2 ( ) z 2 f zz
2
2
To find the solution of equation (3.3), we just need to solve (3.4) in a special case of
 ( ) due to the uniqueness of PDE (3.3). Without losing generalness, suppose  ( )  0 and
evaluate the appropriate boundary conditions, two PDEs are received:
 Lz f  0
(a) 
 f (0, z )  1 (z>1)
 f ( ,1)  0

(b)
 LX g  0

 g (0, X )  1
(3.5)
Equation (3.5a) is a heat equation on a semi-infinite space [0, T ]  (1, ) and we can easily

find its solution by letting   W 2 ( s )ds and x  ln z . Equation (3.5b) happens to be the

0
same as the PDE followed by default-free zero-coupon bond when spot interest rate is of CIR
model, if we change X , g in (3:5b) to r,D respectively. Its solution is of affine structure
form, which is given in the following. The two PDEs' solutions are given by:
l nz  ˆw( )
 l nz ˆ w ( )
f ( , z ) N (
) z N (
)
2wˆ ( )
w
2ˆ  ( )
（3.6）
g ( , X )  eM1 ( ) M 2 ( ) X
（3.7）
where
1
2
N ( x) 
M1 ( ) 
2kl
X
ln
2
x
 exp{


s2
1
}ds , wˆ ( )   ( r 2 B 2 ( s )  2  r V B( s)   V 2 )ds ,
20
2
2 e( k  ) / 2
2(e  1)
,
,   k 2  2 X 2
M
(

)

2
(k   )(e  1)  2
(k   )(e  1)  2
Summarizing, we have the following result:
Theorem 3.1 The price of defaultable discount bond at time t maturing at T is:


V
P(t , r ,V , X ; T )  D(t , r; T )   (1   ) g (T  t , X ) f (T  t ,
)
 D(t , r; T ) 

(3.8)
As shown in formula (3.8), P consists of three parts: discount factor, the payoffs when
issuer defaults its obligations and when it still survives at the maturity. Formula (3.8) reflects
the fact that the price of a defaultable bond is the discounted average of the payoffs at the
maturity under the martingale measure.
Theorem 3.1 shows that the value of the defaultable discount bond depends on V and 
only through their ratio Y . An important implication of this is that defaultable bonds can be
priced without separating V and  . This feature presents simple applications for our model
because the value of the firm's assets is often invisible and we have avoided direct
observations on this value. From (3.8), the price P is an explicit function of t, r, V , X and this
closed form solution is very convenient to be applied.
4 Numerical Results and Analysis on Credit Spreads
One important question on credit risk is: How much is the spread between rates on corporate
bonds and government bonds? This question is very fundamental for pricing both bonds and
credit derivatives and it leads people to investigate credit spreads. Credit spread is defined as
the difference between the yields of defaultable and default free treasury bonds. Given the
explicit solution for risky debt in this paper, credit spread is given by the expressions:
l nD (t ,r ;T ) l P
n t ( r, V, X, T; )
(4.1)
cs 
T t
To quantify the credit spreads, we give the base case parameters: t  0 , T  10 (annum),
a  0.005 , b  0.01 ,  r  0.01 , k  0.005 , l  0.003 ,  X  0.2 ,   0.6 ,   0.55 ,
  0.25 ,  V  0.4 , r  0.03 , V  0.9 , X  0.003 .
Fig.1 credit spreads varing omega
Fig.2 credit spreads varing Delta
Figure 1 plots the relations between the bond prices, the term structure of credit spreads
and default intensity X. As X increases from 0.003 to 0.05, the maximum credit spread
increases from approximately 0.0095 to approximately 0.0115. Note that an increase in X
leads to a decrease in bond price, because of the increase of the unexpected-type default
possibility.
Figure 2 gives the changes of bond prices and credit spreads when the level of interest
rate changes. As discussed in other literature, increasing r tends to reduce the default
probability owing to the effect on the drift of risk-neutral process for V . Thus, an increase
in r results in a decrease in the credit spread.
Fig.3 credit spreads varing sigma
Fig.4 credit spreads varing rho
The relation between credit spreads and the firm's assets' value is shown in Figure 3.
The better the firm's financial condition is, the smaller credit spreads are.
Figure 4 graphs the term structure of credit spreads for varying values of  . Credit
spread is an increasing function of  . However, the term structures can take on different
shapes for different  , because the maximum credit spread occurs at different time for
different maturities. As shown in Longstaff and Schwartz (1995), differences in credit spreads
can be viewed as the term structure of priority, which is the bond-holders' grades to recover
their claims in bankruptcy reorganizations.
Fig.5 credit spreads varing interest rate
Fig.6 credit spreads varing X
Figure 5 shows the credit spreads given different values of the firm's assets' volatility  V .
When  V increases, the firm's assets' value fluctuates more seriously and there are more
risks of default. As a result, the bond issued by the firm will depreciate, which leads to an
increase of credit spread.
These relations between credit spreads and the variables and parameters are fundamental
and crucial to corporate bonds' research. To estimate the parameters in equation (3.15), we
should take stock of adequate information about the concerned firm's bonds with different
maturities. From the empirical statistics, we can find that some parameters influence
significantly the credit spreads such as the assets' volatility  V , correlation coefficient  ,
and recovery rate  . This phenomenon originates from the fact that these parameters
represent the risk level and a movement in the parameters results in a significant change in the
possibility of default as well as credit spreads.
5 Conclusions
This paper presents a combination of structural and reduced form model for pricing risky
corporate bonds and investigating credit spreads, and gives a closed form expression. In
addition, this model allows for the assumption of stochastic hazard rate and dependence of the
firm's assets and risk-free interest rates.
We show that the credit spreads are tightly related to the interest rate, the the assets'
volatility, the correlation and the recovery rate. The results in this paper suggest that short
term credit spreads are usually estimated high, while long term credit spreads are lower for
some speculative companies. Finally, the idea of changing numeraire to reduce the dimension
of PDE is basic but important to our model.
References
[1] Merton R., On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance, 1974,
29:449-470.
[2] Longstaff F., Schwartz E., A Simple Approach to Valuing Risky Fixed and Floating Rate Debt, Journal of
Finance, 1995, 50:789-819.
[3] Jarrow R., Turnbull S., Pricing Derivatives on Financial Securities Subject to Credit risk, Journal of Finance,
1995, 50:53-85.
[4] Realdon M., Corporate Bond Valuation With Both Expected and Unexpected Default, working paper,
Department of Economics and Related Studies, University of York, 2003.
[5] Cathcart L., El-Jahel L., Semi-Analytical Pricing of Defaultable Bonds In a Signaling Jump-Default Model,
Journal of Computational Finance, 2003, 6: n.3.
[6] Madan D. and H. Unal, Pricing the Risks of Default, Review of Derivatives Research, 1998, 2, 121-160.