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Pricing of Defaultable Coupon Bonds When Firm Values and
Interest Rates are Stochastic
German Finance Association
Sixth Annual Meeting, Aachen, Germany
Mark C. W. Wong
Financial Options Research Centre
Warwick Business School, University of Warwick,
Coventry, CV4 7AL, UK
Off: 44 (024) 7657 2572 ext 24691
Res: 44 (024) 7657 2572 ext 30365
Fax: 44 (024) 7652 4650
Email: [email protected]
Stewart D. Hodges
Financial Options Research Centre
Warwick Business School, University of Warwick,
Coventry, CV4 7AL, UK
Off: 44 (024) 7652 4118
Fax: 44 (024) 7652 4167
Email: [email protected]
June, 17
Title:
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Pricing of defaultable bonds when interest rates and firm values are
stochastic
Mark Wong, Stewart D. Hodges
Warwick Business School, University of Warwick, CV4 7AL, UK
(Off) 44 (0)1203 572572 ext 24465, (Res) 44 (0)1203 572572 ext 30365
44 (0)1203 524650
[email protected]
ABSTRACT
This paper develops a flexible binomial approach to valuing risky debt when interest
rates and firm values are stochastic. By using a hypothetical asset as a numeraire, it can be
shown that not only does the use of this numeraire significantly simplify the analytic valuation of
risky discount bonds, but also gives an implication that the computations of two-factor model can
be implemented easily.
By extending the method employed by Ho, Stapleton and
Subrahmanyam (1995), we suggest an efficient computation algorithm for the pricing of general
risky coupon bonds. We investigate the properties of defaultable bonds in an economy of
stochastic interest rates. Interactions of market and credit risk are also discussed in this paper.
This approach has a number of interesting implications, including: Firstly, the recovery
rate of a defaulted bond is generated endogenously depending on the remaining values of the
firm.
Secondly, our model integrates market and credit risk together to allow for a more
complete picture of the underlying risks, as Longstaff and Schwartz (1995) provides strong
evidence that both default risk and interest rate risk are necessary components for a valuation
model for corporate debt. We show that when the firm value is low, its credit spread is more
sensitive to the changes in interest rates. This confirms our intuition that firms with low credit
quality should have more market risk than firms with high credit quality.
Thirdly, while
providing conceptual insights into default behaviour, the flexibility of our method allows for
efficient pricing of bond options, credit risk put options on a general defaultable coupon bond
and floating rate notes. Fourthly, this structural approach also paves the way for a further
analysis of more complicated debt structures.
Keywords:
Defaultable Coupon Bonds, Stochastic Interest Rates, Stochastic Firm Values,
Numeraire, Binomial Method, Market and Credit Risk.
1. INTRODUCTION
In a seminal paper Black and Scholes (1973) show how to price simple European call
and put options on a non-dividend paying stock. They suggest that almost all corporate liablities
can be viewed as combinations of options, and hence their analysis can be used to value
corporate liabilities such as shares, corporate loans and warrants. The literature on the valuation
of defaultable term-structure started with Merton (1974). In this paper Merton adopt the Black
and Scholes option pricing model to the pricing of risky discount bond. Under the assumption of
a constant interest rates economy, Merton’s model yields important insight into the determinants
of the risk structure, and shows how the default risk premium is affected by changes in the firm’s
business risk, debt maturity and the prevailing interest rate. As a generalization to Merton’s
model of defaultable discount bond, Geske (1977) applies the technique for valuing compound
options to the problem of risky coupon bonds. He derives an analytic formula, which consists of
multivariate normal integrals with dimensions up to the total number of contractual payments. It
is shown that with a special auto-correlation structure, an application of an integral reduction
may simplify the numerical computations.
Subsequently, in a further paper Geske (1979)
develop a general theory for pricing compound options in terms of multivariate normal integrals.
In practice, a wide variety of important problems have turned out to be very closely related to the
valuation of compound options.
In 1983, Selby in his Ph.D. thesis generalizes Geske’s (1977) work on pricing risky
discrete coupon bonds in three ways. First, a continuous dividend, as a known proportion of the
firm value, is paid to the equity holders. Second, he derives a general valuation formula for
valuing individual risky discrete coupon bonds, or tranches of such bonds with equal seniority.
Third, he derives general formulae for valuing senior and subordinate bonds with discrete
coupons when two alternative default clauses, liquidation and reorganization in the event of a
default, are considered. By using the preference-independent approach first suggested by Cox
and Ross (1976), the derivations become much simpler than in the alternative backwards
recursion techniques employed by Geske (1977).
Shimko, Tjima and Van Deventer (1993) generalizes Merton’s risky debt pricing model
to allow for stochastic interest rates as in Vasicek (1977). The method of analysis is based on
Merton’s (1973) earlier work on valuation of options with stochastic interest rates and timevarying volatility. They obtain a risky discount bond pricing formula, which yields comparative
static results in consistence with Merton’s (1974). They also show that the combined effect of
term structure of interest rates and credit variables are significantly important for risky bond
pricing.
Each of the valuation formulae for risky coupon bonds developed has a number of
common features. From the viewpoint of economic and computational implementation, one
common point of paramount importance is that each valuation formula gives rise to a sum of
multinormal distribution functions. Moreover, the multinormal distributions are nested, in the
sense that the integration region at any stage is dependent of those at the stage of higher
dimension.
Selby and Hodges (1987) prove a general identity relating sums of nested
multinormal distributions. By reducing the number of integrals to be evaluated, the application
of the identity significantly improves the computational aspects of both the Geske and Johnson’s
(1984) analytical American put and the Roll’s (1977) formula for American call option.
However, all these results are more of theoretical interest than practical use.
Although
mathematicians have come up with a lot of more efficient algorithms, the computation of high
dimensional normal integrals has yet been a very challenging problem.
On the numerical
solution of Geske’s (1977) formula, the computation is not as onerous as would first appear. In
the one-factor case where a firm’s value is modeled as a geometric Brownian motion and interest
rate is taken as a constant, the prices of a defaultable coupon bond can be computed efficiently
and accurately by building a binomial tree for the firm value.
Instead of modeling a recovery rate as endogenously given, Longstaff and Schwartz
(1995) adopt a new approach to valuing risky debt by extending Merton’s (1973) and Black and
Cox’s (1976) models in two ways. Firstly, their model incorporates both default risk and interest
rate risk. Secondly, they derive the model in such a way that allows for deviations from strict
absolute priority. Given the recovery rate as a constant, analytic formulae for fixed rate debt and
floating rate debt are derived. By construction the ratio of the firm value to the face amount of
the debt is a sufficient statistic for default risk in this model, in order to value a coupon bond it is
not necessary to condition on the pattern of cash payments to be made before the maturity. As a
consequence, coupon bonds can be valued as simple portfolios of discount bonds. Though this is
a result that provides much of the tractability of the model, it is highly questionable whether or
not it is reasonable to retain the linear properties in cash flows for the modeling of defaultable
coupon bonds. Furthermore, the implication that a firm has a constant value upon default in the
typical diffusion approach is problematic. On the one hand, this approach emphasizes the central
role of firm value in the determination of default. On the other hand, the approach can not allow
the variation in the recovery rate of a risky bond to depend on the firm’s remaining value at
default.
In addition, the empirical results suggest that the implications of this valuation model are
consistent with the properties of credit spreads implicit in Moody’s corporate bond yield
averages, which in turn provide strong evidence that both default risk and interest rate risk are
necessary components for a valuation model for corporate debt.
This paper is structured as follows. Section 2 presents a set of assumptions under which
risky discount bonds are priced. By using a hypothetical asset as a numeraire, we show that not
only does the use of this numeraire significantly simplify the analytic valuation of risky discount
bonds, but also gives an implication that the computations of two-factor model can be
implemented easily. Section 3 derives some basic properties of two underlying processes. In
section 4, we propose an algorithm for constructing the binomial processes in section 3 by
extending a method suggested by Ho, Stapleton and Subrahmanyam (1995). The method is to
approximate a bivariate lognormal distribution by a bivariate binomial process.
Based on
Geske’s idea, we can price a defaultable coupon bond efficiently by using a three dimensional
lattice. A brief discussion on Geske’s method of evaluating defaultable coupon bonds is given in
section 5. Section 6 presents an efficient computation algorithm for the pricing of general risky
coupon bonds. We generalize Geske’s (1977) and Selby’s (1983) valuation models of risky
coupon bonds to allow for stochastic interest rates as in Vasicek (1977). Section 7 illustrates the
efficiency of our computation algorithm. We investigate the properties of defaultable bonds in
an economy of stochastic interest rates. Discussions on interaction of market and credit risk are
given in this section. Section 8 is devoted to further applications of the framework. Section 9
concludes.
2. VALUATION OF RISKY DISCOUNT BONDS
2.1 Assumptions and Notation
In this section we shall consider a firm which has one discount bond and has no other
form of loan. We make the standard assumptions as follows:
(A1): Frictionless Markets

Trading in assets takes place continuously in time,

There are no taxes, bankruptcy, agency or transaction costs, nor are there
problems of indivisibility of assets,

Every individual acts as through the market price is independent of the
amount bought or sold,

Borrowing and lending are at the same cost,

Short sales are permitted, as is full use of the proceeds.
(A2): The Term Structure

The term structure of interest rates, r , is assumed to follow Vasicek’s
interest rate model.
(A3): Firm Value Process
The dynamics for the value of the firm, V , follows a Geometric Brownian
Motion with instantaneous standard derivation, V , where  is nonstochastic and is known. The firm value is assumed to be a traded asset, and
the correlation coefficient between the interest rate process and the firm
value process is a constant denoted by .
(A4): Maturity Payment

Maturity payment, , is financed by rights issues only taken up by existing
shareholders.
(A5): Dividends

Shareholders are entitled to receive a continuous dividend which is a
constant proportion of the value of the assets of the firm.
2.2 Models
Merton (1974) first derives the value of a pure discount corporate bond by employing
assumption (A3).
Following Merton’s (1973) derivation of Black and Scholes model, he
considers forming a three-securities portfolio containing the firm, the corporate bond and a
riskless debt such that the total investment in the portfolio is zero. By usual no-arbitrage
arguments, a parabolic partial differential equation for the corporate bond is derived. Having
observed that the differential equation is identical to Black and Scholes equation (1973, p.643,
(7)) for a European call option on a non-dividend paying stock, where firm value in his equation
corresponds to stock price and maturity payment corresponds to exercise price, the isomorphic
price relationship between equity of the firm and a call option immediately allows him to write
down the solution to his equation directly. An alternative to Merton’s (1974) method of analysis
is using a probabilistic approach to pricing corporate bonds, we discuss this approach as follows.
By assumption (A2) and (A3), we can express the stochastic processes of interest rates and the
firm values in the following forms
drt  (  rt )dt  dBtr
(1)
dVt  ( rt  a )Vt dt  Vt dBt
(2)
where Btr , Bt 1are two standard Brownian motions under an equivalent martingale measure Q 2,
and satisfy cor ( Btr , Bt )   .  is a constant depending on the market price of risk of interest
rate risk   3.
t

Let t  exp( rudu) be the money market account at time t by starting 1 unit of cash at
0
time 0. If D(Vt , rt , t ) 4represents the price of the risky discount bond at time t, then the relative
price of D(Vt , rt , t ) to  t is a Q -martingale. We write
D(Vs , rs , s)
s
 EQ[
D(Vt , rt , t )
t
s ]
where the expectation is under Q conditional on the information set s , for any t  s  0 .
1
Mathematically Bt 
r
Bt  1   2 Bt' ,
where
Bt , Bt' are two independent standard Brownian
motions.
2
A measure that is equivalent to the original objective measure. It is also known as a risk neutral measure.
The existence of this measure is guaranteed by no-arbitrage arguments. See Harrison and Kreps (1979),
and Harrison and Pliska (1981).
3
If   rt is the drift term under the original objective measure, then      where    0 is the
market price of risk of interest rates. In general two market prices of risk are needed in our setting,
however, by the assumption that the firm value is a traded asset, only the market price of interest rate risk
appears.
4
D(Vt , rt , t ) must be a C 2 function with respect to the first and second coordinates.
If the risky discount bond matures at time T , then the price of the bond at time 0 is given by
D(V0 , r0 ,0)  E Q [
Min (VT , K )
T
0 ] .
This means that the probabilistic approach to pricing a corporate bond is essentially a problem of
computing an expectation under Q , which is in turn a problem of computing multivariate normal
integrals. The practicality of this approach lies in whether we can express VT and T 5 in form
of simple expressions. In the case of Merton (1974) where interest rate is assumed to be
constant, the expectation is a univariate normal integral, the computation of which is trivial. In
our case of two-factor model, the problem becomes handling of bivariate normal integrals.
Though it is known that when the integration regions are of some particular forms high
dimensional integrals can be reduced to ones with lower dimensions, introduction of stochastic
interest rates necessarily complicates the computation issues.
2.3 Change of Numeraire
Most of the bond pricing models we mentioned before are based on the approach of
using money market account as a numeraire, associated with which we have the risk-neutral
measure Q . While this convenient choice of numeraire leads to an intuitive idea of risk-neutral
pricing for asset pricing problems, it is by no means necessary. For example, Geman, Karoui and
Rochet (1995) shows that many other probability measures can be defined in a similar way to
solve option pricing problems.
In this section, we shall discuss the use of an alternative
numeraire which provides computational convenience for bond pricing problems.
By assumption (A5) we assume that continuous dividends are paid at a constant rate of
a per unit of the firm value. The solution of equation (2) is expressed as follows,
1
Vt  V0  t exp( at   2t  Bt ) .
2
Because of the presence of a which is in general non-zero, it is not difficult to see that
the relative price of Vt to the money market account is not a Q -martingale. However, by
assuming the existence of a hypothetical asset Vt  Vt exp(at ) 6, we see that
H
Note that VT and T are log-normally distributed under Q .
It can be regarded as the value of an identical firm that pays no dividends to its shareholders. This
assumption is in fact not important, for we merely want to find out a stochastic process, which has nice
5
6
Vt
H
1
 exp(   2t  Bt )
t
2
is a Q -martingale. Now let us quote without proof a well-known result in probability theory
which is central to the analysis of changes of measure7.
Let  
Given two equivalent measures P1 and P2 .
LEMMA 1
dP2
be the RadondP1
Nikody`m derivative of P2 relative to P1 . Then for any random variable X integrable with
respect to (, P2 ) , the following abstraction of Bayes theorem holds
E P1 [ X ]  E P1 [ ]E P2 [ X ] .
This lemma suggests that under some regularity conditions if there exists an equivalent
measure QV such that the following holds
EQ[
D(Vt , rt , t )
t
V
s ]  E Q [ t
for any t  s  0 , then the relative price
H
t
s ] E Q [
V
D(Vt , rt , t )
s ] ,
H
Vt
D(Vt , rt , t )
is a QV -martingale. The pricing problem
H
Vt
is now turned into finding the new measure QV .
The following theorem guarantees the
existence of such a measure, which can be obtained by a simple transformation of standard
Brownian motions.
THEOREM 2 Under the above assumptions on the processes of the interest rates and the firm
value,
there
exists
a
measure
transformation BtV  t  Bt ,
Proof
t
Vt
H

QV equivalent
to
Q
such
that
under
the
t
V
H is a Q -martingale.
Vt
1
1
exp(  2t  Bt ) . By Ito lemma we have
V0
2
mathematical properties such that the relative price of the bond to this process is a martingale under another
measure.
7
See Musiela and Rutkowski (1997), p458.
d
t
Vt
H
 
t
Vt
H
( dt  dBt ) .
Then since  is a constant, then by virtue of Girsanov’s theorem there exists an equivalent
martingale
d
t
Vt
H
 
QV
measure
t
Vt
H
such
BtV  t  Bt
that
V
dBt is driftless, this implies that
is
a
QV -martingale.
Since
t
H is an exponential martingale. Hence
Vt
result follows.
2.4 Valuation of Risky Discount Bonds
With the results in the last section, the valuation of the risky discount bond becomes
fairly straightforward. By applying LEMMA 1, we have
V
EQ [
V 
D(Vt , rt , t )
D(Vt , rt , t )
s ]  E Q [ Ht s ]E Q [
s ] .
H
t
Vt
Vt
1
2
T

Define a normal random variable ZT   rudu   2T  BTV . Since by THEOREM 2
0
a QV -martingale and the relative price of D(Vt , rt , t ) to  t is a Q -martingale,
t
H is
Vt
D(Vt , rt , t )
is a
H
Vt
QV -martingale, which implies that
D(V0 , r0 ,0)  V0 E Q [
V
V
Min (VT , K )
K
0 ]  V0 E Q [ Min (exp( aT ), exp( ZT )) 0 ]
H
V0
VT
(3)
Suppose that ZT has mean T and variance  T2 , then the price of the defaultable discount bond
at time T is given by8
D(V0 , r0 ,0)  V0e aT N [
Log (V0 / K )  aT  T
T
]  KP(T ) N [
Log (V0 / K )  aT  T
T
  T ] (4)
where N [.] is the cumulative normal distribution function. If P(T ) is the price of a default-free
zero coupon bond with the face amount of unity, then
P(T )  exp[
R ( ) 
8
1  e  T

( R()  r0 )  TR() 
2
(1  e  T )2 ] ,
3
4
 1 2
,

 2 2
The computations are essentially the same as those required for a single factor model.
aT  
bT2 
9
1

(1  e  T )[ r0 
 T
 e T  1
(e  1)]  (
T)



2
2
2 2
 2 T
T

(
1

e
)

(1  e  T )
2
3
3

2

1 2
 1  e  T
T  aT   T 
(
T)
2


10
 T2  bT2   2T  2
 1  e  T
(
T)


We can consider  T2 as the integrated instantaneous variance of ZT over the life of the
risky debt. It is interesting to note that  T2 is a quadratic function of  . For typical values of the
parameters  and T , this function has a positive coefficient in the leading term. This implies
that  T2 is an increasing function of  when  is positive, and  T2 attains its minimum value at
some positive value of  when  is negative. For small values of  11, an important implication
on the bond prices is that a decreasing trend is expected when the firm value has a positive
correlation with the interest rates. On the contrary when the firm value has a negative correlation
with the interest rates, an increasing trend in bond prices is expected.
Let B(T )  KP(T ) be the price of a risk-free discount bond, which pays K at maturity
time T . Then we can rewrite (3) as follows:
D(V0 , r0 ,0)  V0e  aT N [ h1T ]  B(T ) N [h2T ]
Log (
where
h1T 
V0
1
)  aT   T2
B(T )
2
T
(5)
,
h2T  h1T   T .
Note that (5) resembles the solution derived by Merton (1974). The firm will be able to make its
promised payment, K , if and only if its value at the bond maturity is at least K . The first term in
(5) represents the expected present value of the firm if default happens at maturity, and the
V
The mean and variance are evaluated under the equivalent martingale measure Q .
10
See APPENDIX 1 for a proof.
11
To avoid a high probability that the interest rates go negative, we have to choose small values of
use.
9
 to
second term is the expected present value of the promised payment if default does not occur at
maturity.
3. BASIC PROPERTIES OF rt and Z t .
In the last section, we derived the pricing formula for a defaultable discount bond. The
method is appealing in its simplicity by assuming Z t as the only source of variability. Before
going further into the pricing of general coupon bonds, it is tempting to conjecture that such nice
properties of Z t is preserved in multi-period case so that analytic simplicity can be followed in
the same line as the single period case. However, it turns out that this conjecture is wrong. The
reason is as follows:

t
1
2
Given the definition Z t   ru du   2t  BtV , we can easily see that Z t satisfies the
0
following stochastic differential equation
1
dZ t  ( rt   2 )dt  dBtV .
2
This implies that Z t is a Markovian process of two state variable rt and Z t . THEOREM 3
gives some more properties of Z t .
THEOREM 3 For any time 0  s  t , under the martingale measure12 QV we have the
following results:
E[rt | s ]  as,t  e  ( t s ) rs ,
  ( t s )
E[Zt | s ]  bs,t  Z s  1e 
(6)
rs ,
where as,t   
(1  e  ( t s ) )

bs,t   12  2 (t  s)   
(t  s  1e 

  ( t s )
) are deterministic functions.
Furthermore, if we express the above two equations in the following forms:
rt  as,t  e  ( t s ) rs   s,t ,r ,
  ( t s )
Zt  bs,t  Zs  1e 
12
rs   s,t ,Z ,
(7)
(8)
From now on, we implicitly assume that all expectations and variances are computed under the martingale
measure
QV .
where  s ,t ,r and  s ,t ,Z are
two
normal
random
variables
independent
of
s satisfying
E[ s ,t ,r | s ]  0 and E[ s ,t ,Z | s ]  0 , then
Var[rt ]  e2  ( t s )Var[rs ]   s2,t ,r
  ( t s )
Var[Zt ]  Var[Z s ]  ( 1e 
  ( t s )
)2Var[rs ]  2( 1e 
cov[ rt , Zt ]  e  ( t s ) cov[ rs , Zs ]  e  ( t s ) ( 1e 
  ( t s )
) cov( rs , Z s )   s2,t ,Z
)Var[rs ]   s,t ,r ,Z
where we assume the conditional variances Var[ s ,t ,r | s ]   s2,t ,r , Var[ s,t ,Z | s ]   s2,t ,Z .
Proof: See APPENDIX 2.
This theorem shows that the value of Z t at time t can be predicted by using the
knowledge of rs and Z s at time s . Although Z t is not a Markovian process on its own, its
simple decomposition into rs , Z s suggests that it is possible to solve the problem of pricing
defaultable coupon bonds by using binomial trees. In the following section, we shall extend the
method employed by Ho, Stapleton and Subrahmanyam (1995) to build binomial trees for rs and
Zs .
4. A METHOD FOR CONSTRUCTING BINOMIAL PROCESSES
4.1 One-Period Case
Ho, Stapleton and Subrahmanyam (1995) shows how to construct a multivariatebinomial approximation to a joint lognormal distribution of variables with a recombining
binomial lattice. Each variable is lognormally distributed and Markovian on its own. In the
present case, we need to modify the procedure, allowing the value of Z t to depend on the values
of rs and Z s . As interest rates are assumed to follow normal distributions, the construction of
the interest rate tree is the same as they propose. For simplicity, we first consider a one-period
case. For detailed construction of two-period case, see APPENDIX 3. Let X s  exp( rs ) . Our
method involves the construction of one binomial distribution X s .
Exhibit 1
X s ,0
X s ,1
X s,2
A discrete process for
X s ,3
X0
Xs .
X s,4
X s ,5
X s ,6
X s ,7
0
s
We approximate X s by a vector of ns  1 numbers:
X s,i  X 0usi d sns i ,
for i  0,..., ns , where
ds 
2( E[ X s ] / X 0 )1/ ns
,
1  exp( 2 0,s ,r / ns )
us  2( E[ X s ] / X 0 )1 / ns  d s .
On the time-interval [0, s ] , we choose a transition probability qs of an up-movement at the initial
node such that property (6) holds, that is
r0  ns (qs Log[us ]  (1  qs ) Log[d s ])  a0,s  e s r0 ,
hence,
qs 
a0,s  e  s r0  r0  ns Log[d s ]
.
ns ( Log[us ]  Log[d s ])
Given the above interest rate tree, we are now ready to construct a second tree for
Ys  exp( Z s ) conditional on rs . At time s , a tree of similar structure Ys is approximated by a
vector of (ms  1) numbers
Ys,i  usi d sms i ,
for i  0,..., ms , where
ds 
2( E[Ys ])1/ ms
,
1  exp( 2 0,s ,Z |rs / ms )
us  2( E[Ys ])1 / ms  d s ,
where  0,s ,Z |rs is the volatility of Z s conditional on rs .
In order to incorporate the correlation structure into the trees, we relate the noise terms in
property (7) and (8) through their correlation 0 s as follows
 0,s ,Z

 0,s 0,s ,r  1  02,s  0,s ,
 0,s , Z
 0,s ,r
where  0,s is a normal standard random variable which is independent of  0 , s,r .
By property (8) we have
E[ Z s | rˆs ]  b0,s  1e r0  0,s
 s
 0,s ,Z
ˆ .
 0, s ,r 0, s ,r
We choose a transition probability p s of an up-movement at initial node such that the above
equation holds, that is
b0,s 
ps 
1  e s

r0  0,s
 0, s , Z
ˆ  ms Log[d s ]
 0, s ,r 0, s ,r
ms ( Log[us ]  Log[d s ])
.
4.2 Multiple-period case
In general, both J -period trees can be constructed similarly. Suppose that there are
J future dates t1 , t2 ,..., tJ on which we are interested in the asset prices. We are interested in the
joint distributions of X t and Yt on these dates. Over each of the time interval [t j 1 , t j ] , we
assume that there are exogenously given number of binomial steps nt j and mt j for X t and Yt
respectively. The construction of both trees are summarized as follows:
For j  1,2,..., J , we approximate X t j by a vector of N j  1 numbers
N j i
X t j ,i  X 0u ij d j
,
for i  0,..., N j , where
1/ N j
dj 
2( E [ X t j ] / X 0 )
1  exp( 2 t j 1 ,t j ,r / nt j )
u j  2( E [ X t j ] / X 0 )
1/ N j
,
dj,
j
N j   ntl .
l 1
On the time interval [t j 1 , t j ] , we choose a transition probability q j of an up-movement at node
i at time t j 1 such that property (6) holds,
qj 
at j 1 ,t j  e
  ( t j t j 1 )
rˆt j 1  r0  iLog [u j ]  ( N j 1  i ) Log[d j ]  nt j Log[d j ]
nt j ( Log[u j ]  Log[d j ])
.
At time t j , Yt j is approximated by a vector of M j  1 numbers
M j i
Yt j ,i  u ji d j
,
for i  0,..., M j , where
dj 
2( E[Yt j ])
1/ M j
1  exp( 2 t j 1 ,t j ,Z |rt / mt j )
,
j
u j  2( E[Yt j ])
1/ M j
dj ,
j
M j   mtl
l 1
where  t j 1 ,t j ,Z |rt is the volatility of Z t j conditional on t j 1 and rt j . We choose a transition
j
probability p j of an up-movement at node i at time t j 1 as follows
bt j 1 ,t j
1 e
 Zˆ t j 1 
pj 
  ( t j t j 1 )

rˆt j 1  t j 1 ,t j
t
j 1 ,t j , Z
t
j 1 ,t j ,r
ˆt
j 1 ,t j ,r
 iLog [u j ]  ( M j 1  i ) Log[d j ]  mt j Log[d j ]
mt j ( Log[u j ]  Log[d j ])
The following theorem shows the basic properties of the constructed r̂t and Ẑ t
processes that the estimated means, variances and covariances converge to their true values in the
limit.
THEOREM 4 Suppose that the X t and Yt trees are constructed as above.
j  1,2,..., J , we have
Then for
E[rˆt j ]  E[rt j ]
(9)
E[ Zˆ t j ]  E[ Z t j ]
(10)
ˆ t2
(11)
j 1 ,t j ,r
  t2j 1 ,t j ,r as nt j   ,
ˆ t2
j 1 ,t j , Z |rt j
ˆ t2
j 1 ,t j , Z
ˆ t
 (1  t2j 1 ,t j ) t2j 1 ,t j ,Z as mt j   ,
  t2j 1 ,t j ,Z as mt j , nt j   ,13
j 1 ,t j ,r , Z
(12)
(13)
  t j 1 ,t j ,r ,Z as mt j , nt j   ,
(14)
Furthermore,
E[ Xˆ t j ]  E[ X t j ] as nt j ,..., nt1   ,
(15)
E [Yˆt j ]  E [Yt j ] as mt j , nt j ,..., mt1 , nt1   .
(16)
Proof: We can prove (9) by induction. Note that by the choice of the probabilities q j , we have
E[rˆt j | t j 1 ]  at j 1 ,t j  e
  ( t j t j 1 )
rˆt j 1 .
This implies that
E[rˆt j ]  at j 1 ,t j  e
  ( t j t j 1 )
E[rˆt j 1 ] .
It is easy to check that
E[rˆt1 ]  E[ rt1 ] .
By induction, result (9) follows.
Similarly we can prove (10).
For (11), we only consider the case when j  1 . The proof for a general j is similar.
At time t1 , a realization of rt1 can be estimated by
rˆt1  r0  iLog [u1 ]  (nt1  i ) Log[d1 ] ,
where i is a binomial random variable with parameters ( nt1 , q1 ) . Hence,
13
Here the order of operations is important. The limits are taken over
mt j , nt j respectively.
ˆ 02,t ,r  nt q1 (1  q1 )( Log[u1 ]  Log[d1 ]) 2
1
Since q1 
1
1
as nt1   , we have ˆ 02,t1 ,r   02,t1 ,r as nt j   . Result (11) obtains.
2
Similarly we can prove that
ˆ t2
j 1 ,t j , Z |rt j
  t2j 1 ,t j ,Z |rt as mt j   .
j
By property (8), we have
Zt j  bt j 1 ,t j  Zt j 1  1e
where  t j 1 ,t j ,Z  t j 1 ,t j
t
t
j 1 ,t j , Z
t
j 1 ,t j , r
  ( t j  t j 1 )

 1   t2
j 1 , t j
t
rs   t j 1 ,t j ,Z ,
j 1 ,t j , Z
t
j 1 ,t j
, for a standard normal random
j 1 ,t j , r
variable  t j 1 ,t j that is independent of  t j 1 ,t j ,r . This implies that  t2j 1 ,t j ,Z |rt  (1  t2j 1 ,t j ) t2j 1 ,t j ,Z .
j
Hence (13) holds.
By construction, the choice of p j satisfies
E[ Zˆ t j | t j 1 , rˆt j ]  bt j 1 ,t j  Zˆ t j 1  1e
  ( t j  t j 1 )

rˆt j 1  t j  ,t j
t
j 1 ,t j , Z
t
j 1 ,t j , r
ˆt
j 1 ,t j , r
,
which implies that
Var[ E[ Zˆt j | t j 1 , rˆt j ] | t j 1 ]  t2j 1 ,t j
 t2
j 1 ,t j , Z

2
t j 1 ,t j ,r
ˆ t2
j 1 ,t j ,r
 t2j 1 ,t j t2j 1 ,t j ,Z
as nt j   by (11). As ˆ t2j 1 ,t j ,Z  Var[ E[ Zˆ t j | t j 1 , rˆt j ] | t j 1 ]  E[Var[ Zˆ t j | t j 1 , rˆt j ] | t j 1 ] ,
result (13) follows.
Proof of (14) is similar.
To justify (15), it suffices to prove by induction on j that for any real number 
E[exp( rˆt j )]  E[exp( rt j )] as nt j ,..., nt1   .
For j  1 ,
E[exp( rˆt j )]  X 0 [q1ut1  (1  q1 )dt1 ] t1
n
1
nt1 

 X 0 exp[ (a0,t1  e  t1 r0  r0 )   2 02,t1 ,r ]
2
1
 exp[ ( a0,t1  e  t1 r0 )   2 02,t1 ]
2
 E[exp( rt1 )]
Suppose that the proposition is true for some j . For any real number  , conditional on the
node i at time t j we consider
E[exp( rˆt j 1 ) | t j ]  X 0 (utj 1 )i (d tj 1 )

t j 1


 X 0 exp[ (at j ,t j 1  e
n
N j i
  ( t j 1 t j )
[q j 1utj 1  (1  q j 1 )d tj 1 ]
nt j 1
1
rt j  r0 )   2 t2j ,t j 1 ,r ]
2
1
  ( t t )
 exp[ ' rt j ] exp[at j ,t j 1   2 t2j ,t j 1 ,r ] , where  '  e j 1 j
2
By assumption, we have
nt j 1 ,..., nt1 
1 2
1
E[exp( rˆt j 1 )] 
  exp[ ' E[rt j ]   ' Var[rt j ]] exp[at j ,t j 1   2 t2j ,t j 1 ,r ]
2
2
By THEOREM 3, right hand side of the above expression is of the form
1
exp[E[rt j 1 ]   2Var[ rt j 1 ]] .
2
Hence the proposition holds for j  1 .
To prove (16), it suffices to prove an alternative proposition by induction on j for any
real number  ,
E[exp( Zt j  rˆt j )]  E[exp( Zt j  rt j )] as mt j , nt j ,..., mt1 , nt1   .
The method is similar to the proof of (15).
COROLLARY 5
Suppose that the X t and Yt trees are constructed as above. Then for
j  1,2,..., J , convergence in distribution is guaranteed
rˆt j | t j 1  rt j | t j 1 as nt j   .
Zˆ t j | ( t j 1 , rˆt j )  Z t j | (t j 1 , rˆt j ) as mt j   .
Proof. The proof is followed by the constructions of q j and p j , and the properties of
conditional variances.
5. GESKE’S METHOD OF EVALUATING DEFAULTABLE COUPON BONDS
For ease of exposition, we first give a review on the methods of analysis given by Geske
(1977) and Selby (1983). Under the assumption of constant interest rate, Geske demonstrated
that risky securities could be valued as compound options. Selby generalized Geske’s pricing
model by incorporating continuous dividends as a constant proportion of the total value of the
firm assets. He used the “preference independent” approach first suggested by Cox and Ross
(1976) to simplify the mathematical derivation of the model. An important assumption that
underlay the analysis by them is that the firm finances each coupon payment with equity only
taken up by shareholders, and that bankruptcy occurs when the firm fails to make a coupon
payment because it is unable to raise enough money to fund the payment. Black and Cox (1976)
argue that this will happen whenever the value of the equity, after payment is made, is less than
the value of the payment. Black and Cox’s (1976) argument is intuitive, in that the firm will find
no takers for its stock if they know the stock will become less valuable than the total value that
they need to contribute to the promised payment. Suppose that the firm has an obligation to meet
a coupon payment C t at time t . Let S t  be the value of the stock immediately after time t .
Then by Black and Cox’s (1976) argument, the firm will be able to finance the coupon payment
by rights issues if
St   Ct
(17)
By the Modigliani and Miller theorem, the value of the firm is independent of its capital
structure. Therefore the above inequality can be rewritten as
Vt  Ct  Dt  ,
(18)
where Vt  Vt   Vt  14 is the firm value at time t . This implies that the value of the firm at time
t should be greater than the total value of the coupon to be honored and the debt immediately
after the coupon date. Looking from another viewpoint, the right hand side of inequality (18) is
the value of the debt at time t . Hence by the Modigliani and Miller theorem again, if the firm is
able to honor its obligation on interest payment by issuing new equity, then the stock has a
14
The money raised by rights issues is used to finance the coupon payment. Firm value remains unchanged
before and after the payment.
positive value. Inequality (18) is of particular importance when we are working on numerical
computations of defaultable bonds.
In the case of Geske’s (1977) model, it is fairly straightforward to imply from inequality
(17) the existence of a critical value of the firm V ( below which a default happens ). The
reason is that when the interest rate is assumed to be constant, there is only one stochastic
variable Vt in the function of the stock price. Moreover, the monotonic increasing property of
the stock prices on the firm value always guarantees the existence of such a point V .
Mathematically we write
St  (V )  Ct .
(19)
Assuming the existence of such critical values of the firm allows us to price risky coupon bonds
analytically. With the “risk-neutral technique” of Cox and Ross (1976), the pricing problems
simply become finding expected present value of a stream of promised payments on a series of
events on coupon dates that determine whether or not a default has happened 15. The derivation
becomes much simpler than the alternative backward recursion techniques employed by Geske
(1977). A formula containing multi-variate normal integrals can be derived. However, the result
is more of theoretical interest than practical use. Even with the aid of fast computers in existence
today, the computation of high dimensional normal integrals is still a very challenging problem16.
Alternatively the Geske’s (1977) formula can be computed numerically. In the one-factor case
where a firm’s value is modeled as a geometric Brownian motion and interest rate is taken as a
constant, the prices of a defaultable coupon bond can be computed efficiently and accurately by
building a binomial tree for the firm value.
In a multi-factor framework, the method of solution for the pricing of defaultable coupon
bonds is in general more complicated. One complication is the introduction of the stochastic
interest rates that makes the determination of an analytic solution for coupon bond prices
difficult, if not impossible. In our two-factor case, the stock price is a function of two variables,
namely the firm value and the interest rate, and so the solution of equation (18) is a function of
future interest rates on a coupon date, which are unknown at the time of issue. In other words,
V is a moving default boundary. In the following section, instead of looking for an analytic
15
16
See Cox and Ross (1976), and Selby (1983).
See Papageorgiou and Traub (1997), Paskov and Traub (1997).
solution, we shall propose an efficient numerical technique to price risky coupon bonds by
building two binomial trees.
6. APPLICATIONS TO PRICING OF DEFAULTABLE BONDS
In section 2.4, we derived an analytic formula for a defaultable discount bond. Using the
hypothetical asset Vt H as a numeraire, we showed that by combining two sources of variability
together to form a single variable Z t the analytic valuation is significantly simplified. In section
3, we discussed the properties of Z t , which implies that the analytic simplicity in valuation of
defaultable bonds can not be retained in multi-period case. We have to resort to compute coupon
bond prices numerically. In principle computation is independent of the choice of a numeraire,
however, appropriate choice of it does provide much of the computational simplicity. It is
important and interesting to investigate the differences in computational efficiency between
using the money market account  t and the hypothetical asset Vt H as numeraires in bond pricing
problems. In the case of a defaultable discount bond with maturity T , we consider the following
cases. Suppose we use VTH as a numeraire to price the bond, the initial price is estimated by
taking average17 of the following expression
V0
Min (VT , K )
VTH
 Min (V0 exp( aT ), K exp( ZT ))
With the results in section 4, the bond price can be approximated by building two binomial trees
rT and ZT . On the other hand, if the money market account T is used as a numeraire, the bond
price is estimated by taking average18 of
Min (VT , K )
T
T
1
 Min (V0 exp( aT   2T  BT ), K exp(   ru du ))
0
2
T

Note that exp(  ru du) is a Markovian process on its own and the interest rate. By using the
0
T

similar techniques as in section 4, exp(  ru du) can be mimicked by two binomial trees. This
0
17
Under
QV .
result implies that with this choice of numeraire the construction of three binomial trees is
necessary to price the bond. Therefore the use of the numeraire VTH is supported by the
computational convenience by saving one binomial tree. The following theorem establishes the
theoretical justification for numerical computation of discount bonds.
THEOREM 6 With the construction of two one-period binomial trees rT and ZT with binomial
steps n and m respectively, the price of a defaultable discount bond of maturity T can be
estimated by the following two steps:
(i) E[ Min (V0 exp( aT ), K exp( ZˆT )) | rˆT ]  E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ]
as m   .
(ii) E[ E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ]]  E[ Min (V0 exp( aT ), K exp( ZT ))]
as n   , where the limit is the price of the defaultable discount bond.
Proof
(i)
By
COROLLARY
5,
ZˆT | rˆT
converges
to
ZT | rˆT
in
distribution.
Since Min (V0 exp( aT ), K exp( ZT )) | rˆT is continuous at ZT , Billingsley(1995, p.334, corollary
1) implies that
Min (V0 exp( aT ), K exp( ZˆT )) | rˆT  Min (V0 exp( aT ), K exp( ZT )) | rˆT ,
as m   .
Note that Min (V0 exp( aT ), K exp( ZT )) | rˆT is uniformly integrable as it is
bounded above by V0 . By Billingsley(1995, p.338, Theorem 25.12), (i) follows.
(ii) Since r̂T converges to rT in distribution and E[ Min (V0 exp( aT ), K exp( ZT )) | rT ]
is continuous at rT , by Billingsley(1995, p.334, corollary 1) again
E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ]  E[ Min (V0 exp( aT ), K exp( ZT )) | rT ] ,
as n   .
E[ Min (V0 exp( aT ), K exp( ZT )) | rˆT ] is uniformly integrable, and so by
Billingsley(1995, p.338, Theorem 25.12) again,
E[ E[ Min (V0 exp( aT ), K exp( ZˆT )) | rˆT ]]  E[ E[ Min (V0 exp( aT ), K exp( ZT ))] | rT ]
as n   . Result follows.
18
Under Q .
This theorem shows that computations of a discount bond price consist of three main
steps. Firstly, we compute Min (V0 exp( aT ), K exp( ZˆT )) at each node Ẑ T of the ZT tree.
Secondly, given each r̂T , we find the mean of the above expression. Thirdly, the bond price is
estimated by taking the mean of the results in step 2 over all r̂T . Similarly we can estimate the
price of a defaultable coupon bond by binomial methods. With the same notation used as before
we first consider a two-period case where t1 and t2 are the coupon and maturity dates
respectively. As similar to the proof of THEOREM 6, we can prove that
(i) E[ Min (V0 exp( at2 ), K (1  c) exp( Zˆt2 )) | t1 , rˆt2 ]
 E[ Min (V0 exp( at2 ), K (1  c) exp( Z t2 )) | t1 , rˆt2 ] ,
as m2   , and
(ii) E[ E[ Min (V0 exp( at2 ), K (1  c) exp( Z t2 )) | t1 , rˆt2 ] | t1 ]
 E[ Min (V0 exp( at2 ), K (1  c) exp( Z t2 )) | t1 ] ,
as n2   .
Let
B( rt1 , Z t1 )  E[ Min (V0 exp( at2 ), K (1  c) exp( Z t2 )) | t1 ] .
Then
Vt1H B(rt1 , Zt1 ) / V0 is the bond price at time t1 . In other words, the bond price at time t1 is
approximated by (i) and (ii) THEOREM 6. By (18), the firm will be able to finance the coupon
payment by a rights issue if Vt1  Vt1H B(rt1 , Zt1 ) / V0  Kc , and so the bond price immediately
before the coupon date is
Min (Vt1 ,Vt1H B(rt1 , Zt1 ) / V0  Kc) .
By the martingale property, its initial price is given by
V0 E[ Min (Vt1 ,Vt1H B(rt1 , Zt1 ) / V0  Kc)]
 E[ Min (V0 exp( at1 ), B( rt1 , Z t1 )  Kc exp( Z t1 ))] .
As similar to THEOREM 6, we can estimate this bond price by the following two steps:
(iii) E[ Min (V0 exp( at1 ), B( rˆt1 , Zˆt1 )  Kc exp( Zˆt1 )) | rˆt1 ]
 E[ Min (V0 exp( at1 ), B( rˆt1 , Z t1 )  Kc exp( Z t1 )) | rˆt1 ]
as m1   , and
(iv) E[ E[ Min (V0 exp( at1 ), B( rˆt1 , Z t1 )  Kc exp( Z t1 )) | rˆt1 ]]
 E[ Min (V0 exp( at1 ), B( rt1 , Z t1 )  Kc exp( Z t1 ))]
as n1   .
With one- and two-period cases, we have depicted the essence of numerical techniques
for computations of general defaultable coupon bonds. For j  1,2,..., J , on each interval
[t j 1 , t j ] , the relative price of the bond to the hypothetical asset Vt H agrees with martingale
properties, and computations consist of three similar steps as described above. Firstly, we work
out the default barrier and the payoff immediately before time t j in accordance with Geske’s
idea. Secondly, we find the relative value of the payoff to Vt Hj and take the mean of it given each
r̂t j . Thirdly, we take the average of the results in the second step over all r̂t j .
7. NUMERICAL COMPUTATIONS
7.1 Prices of Defaultable Bonds
For a J -period model, our method of constructing binomial trees rt and Z t requires the
choice of J -binomial steps nt1 , nt2 ,..., ntJ and mt1 , mt2 ,..., mtJ respectively. As discussed before,
the order of operations is important. To guarantee the convergence of the estimated bond prices
to the true value, the limits should be taken in the order of mtJ , ntJ ,..., mt1 , nt1 by letting them tend
to infinity.
In general, the estimation of the true value by a choice of realizations of
mtJ , ntJ ,..., mt1 , nt1 is a difficult task, and depends on the essence of the problem. Improper choice
of binomial steps may lead to a possibility that the sequence of approximated bond prices does
not converge to the true limit. For J  1 , we compute the prices of one-year discount bonds
with par value 70, 100 and 130 analytically.
Exhibit 2
Example of bond prices with par value 70, 100 and
130. The prices are computed analytically using
parameter values r =0.04,  =0.06,  =1,  =0.031,
 =-0.25,  =0.2, V0 =100, a =0.12.
K =70
one-year discount 66.2571
K =100
84.314
K =130
88.3116
Exhibit 3 shows bond price convergence graphs for a one-year discount bond with par
value K =70, 100 and 130. It shows that the prices of the discount bonds converge to the true
values in Exhibit 2. The rate of convergence is fairly high, and suggests that it is not necessary
to compute with many binomial steps.
Binomial steps with n =5 and m =5 appear to be
sufficiently large to give good estimates of bond prices. Exhibit 4 shows convergence graphs for
the same instruments with one difference in the use of binomial steps that n =2 and m =2,3,…,20.
However, it exhibits the same pattern as indicated in Exhibit 3. Exhibit 5 gives further details on
the estimated prices of a discount bond with par value K =70. It shows that the bond prices are
insensitive to the changes of interest rate depicted by the first tree rt , and the binomial steps m
plays a more influential role in the precision of estimates. This can be explained by the fact that
the variable Z t already captures most of the variability in rt , and so the increases in binomial
steps n have merely negligible influence on the generated bond prices. Higher values of m tend
to give a more accurate approximation of the bond prices. For this reason we shall keep using
small values of n in the subsequent computations.
Exhibit 3
Bond price convergence graphs against binomialsteps
for a one-year discount bond. The bond prices are
computed numerically assuming stochastic interest
rate using parameter values r =0.04,  =0.06,  =1,
 =0.031,  =-0.25,  =0.2, V0 =100, a =0.12,
n  m =2,3,…,20
(i) K =70(___),
(ii) K =100(- - -),
(iii) K =70(---).
Insert Exhibit 3 here
Exhibit 4
Bond price convergence graphs against binomial
steps for a one-year discount bond. The bond prices
are computed numerically assuming stochastic
r =0.04,
interest rate using parameter values
 =0.06,  =1,  =0.031,  =-0.25,  =0.2,
V0 =100, a =0.12, n =2, m =2,3,…,20.
(i) K =70(___),
(ii) K =100(- - -),
(iii) K =70(---).
Insert Exhibit 4 here
Exhibit 5
Example of discount bond prices with par value 70
for the cases (i) n =2, m =2,3,…,20, (ii)
The prices are computed
n  m =2,3,…,20.
numerically using parameter values r =0.04,
 =0.06,  =1,  =0.031,  =-0.25,  =0.2,
V0 =100, a =0.12.
Case (i)
Case (ii)
m=2
66.7393
66.7393
m=3
66.3665
66.3665
m=4
66.2381
66.2381
m=5
66.3429
66.3429
m=10
66.257
66.2568
m=11
66.3263
66.3261
m=12
66.2616
66.2614
m=13
66.2727
66.2726
m=18
66.2678
66.2676
m=19
66.2886
66.2884
m=20
66.256
66.2558
It is worthwhile to note that the estimated discount bond prices converge to the true limit
at a fairly fast fashion. Because of the similarity of the binomial construction of coupon bonds to
discount ones, high rate of convergence is also expected in the computations of coupon bond
prices over each coupon period as long as the binomial steps m is chosen to be sufficiently
large. Therefore to resolve the problem we discussed above, in the computations of coupon
bonds we choose mt1  mt2  ...  mt J .
In the following table, we illustrate the efficiency of our method by showing the time
taken to compute a one-year discount bond, a one-year 8% coupon bond and a two-year 8%
coupon bond. It shows that the computation time increases approximately linearly with the
binomial step m .
Exhibit 6
Example of bond prices and their computation times
with par value 70 for n =2, m =2,4,…,10. The
prices are computed numerically using parameter
values r =0.04,  =0.06,  =1,  =0.031,  =0.25,
 =0.2, V0 =100, a =0.12.
one-year
one-year 8%
two-year 8%
discount
m=2
66.7393 (0.16s)
71.395 (1.75s)
70.765 (13.79s)
m=4
66.2381 (0.21s)
71.4713 (2.75s)
70.576 (24.02s)
m=6
66.358 (0.27s)
71.3807 (3.9s)
70.6929 (35.51s)
m=8
66.3159 (0.34s)
71.3782 (5.05s)
70.6299 (47.56s)
m=10
66.257 (0.38s)
71.4299 (6.04s)
70.6068 (57.77s)
Exhibit 7
Two-year 8% risky coupon bond prices with
stochastic interest rates: r0 =0.04,  =0.06,  =1,
 =0.031,  =-0.25,  =0.2, V0 =100, K =70,
a =0.12, c =0.04, unless otherwise stated.
In Exhibit 7.1:- _____: K =70, - - -: K =100, ----:
K =130.
In Exhibit 7.3:- _____:  =-0.25, - - -:  =0.25.
Insert Exhibit 7 here
7.2 Credit Spreads of Defaultable Bonds
Having worked out the prices of a coupon bond, we can solve for the credit spread.
Credit spread is defined as a spread level over the yield of a default-free bond with the same
promised payments and maturity. In other words, a defaultable discount bond can be expressed
as a default-free discount bond with an upward adjustment of a credit spread in its yield. In the
case of a coupon bond, a credit spread can be solved numerically by assuming that a defaultable
coupon bond takes a simple form of an associated default-free coupon bond where each term is
of the same upward adjustment of a credit spread. The plots of credit spreads of a 2-year 8%
coupon bond against different parameters are shown in Exhibit 8.
Exhibit 8
Credit spreads of a two-year 8% risky coupon bond
with stochastic interest rates: r0 =0.04,  =0.06,
 =1,  =0.031,  =-0.25,  =0.2, V0 =100,
K =70, a =0.12, c =0.04 and n =2, m =2,4,…,10,
unless otherwise stated.
In Exhibit 8.1:- _____: K =70, - - -: K =100, ----:
K =130.
In Exhibit 8.3:- _____:  =-0.25, - - -:  =0.25.
Insert Exhibit 8 here
As expected, in most of the plots credit spreads move in a direction opposite to the bond prices.
However, in Exhibit 7.4 and 8.4 bond prices and credit spreads are both decreasing functions of
the interest rates. The reason for this phenomenon is attributable to the fact that in the risk
neutral world the firm value tends to increase in the response to the increasing interest rates.
Bankruptcy is less likely to happen as a result of higher stock prices, therefore credit spreads
decrease with interest rates.
This result agrees with the empirical findings in Longstaff and
Schwartz (1995).
7.3 Interaction of Market and Credit Risk
Market risk refers to changes in bond prices as a result of changes in interest rates.
Credit risk refers to the risk that the issuer of a bond may default. As market events have shown,
there is an important interplay between both concepts. Our model integrates market and credit
risk together to allow for a more complete picture of the underlying risk. Exhibit 9, 10 show
bond price sensitivity and credit spread sensitivity to parameters.
Exhibit 9
Two-year
r0 =0.04,
8%
coupon
bond
price
sensitivity:
 =0.06,  =1,  =0.031,  =-0.25,
 =0.2, V0 =100, K =70, a =0.12, c =0.04, unless
otherwise stated.
Insert Exhibit 9 here
Exhibit 10
Credit spread sensitivity of a two-year 8% coupon
bond: r0 =0.04,  =0.06,  =1,  =0.031,  =0.25,
 =0.2, V0 =100, K =70, a =0.12, c =0.04,
unless otherwise stated.
Insert Exhibit 10 here
Exhibit 10.1 shows that when the firm value is low, credit spread is more sensitive to the
changes in interest rates. This confirms our intuition that firms with low credit quality should
have more market risk than firms with high credit quality. On the contrary, firms with high
credit quality are those which we expect have only a base level of interest rate exposure.
Furthermore, it is clear from 10.2 and 10.3 that when dividend and coupon rates are higher, credit
spread is more sensitive to firm value volatility. This implies that under the assumption that
coupons are financed by right issues, the bond is of higher default risk as coupon rate increases.
A similar trend holds for the case where shareholders are entitled to receive higher dividend rate
as a proportion of firm value.
Exhibit 11
Credit spread sensitivity of a one-year 8% coupon
bond: r0 =0.04,  =0.06,  =1,  =0.2, V0 =100,
K =70, a =0.12, c =0.04, unless otherwise stated.
Insert Exhibit 11 here
Exhibit 11 plots the relation of credit spreads with respect to interest rate volatility 
and correlation  .
As shown, the effect of correlation can be very significant.
When
correlation is high, credit spread appears to be sensitive to the changes in interest rate volatility.
When correlation is small, credit spread decreases slightly over a wide range of interest rate
volatility. As in Longstaff and Schwartz (1995), these results are consistent with empirical
evidence that credit spreads for bonds of equal rating vary across sections.
8. POSSIBLE EXTENSIONS
In addition to providing conceptual insights into default behaviour, the flexibility of our
method allows for efficient pricing of some other financial instruments. These include bond
options, credit risk put options on a general defaultable coupon bond and floating rate notes.
Many papers in financial literature have addressed the important topic of bond option valuation.
Of these papers, Jamshidian (1989) and Longstaff (1993) provide an analytic formula for the
value of an option on a coupon bond with stochastic interest rates 19. However, they derive the
formulae under the assumption that the underlying coupon bonds are non-defaultable. In this
section, we shall show that the pricing algorithm for risky coupon bonds can be modified to price
options on a defaultable bond.
Consider a  -year European option on a T -year defaultable coupon bond D(Vt , rt , t )
with a fixed exercise price, where 0  t  T   . Here we are focusing on options on bonds with
final maturity date   T . Since the bond option is a derivative of the defaultable coupon bond
D(Vt , rt , t ) , which is a function of Vt , rt and t , we can express its value at time s as
20
C (Vs , rs , s) , where 0  s   . Note that
that
C (Vs , rs , s )
s
D (Vs , rs , s )
s
is a Q -martingale, Ito’s Lemma implies
is also a Q -martingale. By LEMMA 1 and THEOREM 2 in section 2, it is easy
19
They assume that interest rates follow a Vasicek process and Cox, Ingeroll and Ross (1985) process
respectively.
20
2
It has to be a C function with respect to the first and second coordinates.
to prove that under the transformation BsV  s  Bs ,
C (Vs , rs , s )
is a QV -martingale. Now
H
Vs
we state this result in the following lemma.
LEMMA 7
The relative price of C (Vs , rs , s) to VsH is a QV -martingale for 0  s   . The
initial price of the bond option is given by
C (V0 , r0 ,0)  V0 E Q [
V
C (V , r , )
0 ] ,
H
V
where C (V , r , ) is the payoff of the option at its maturity.
With this lemma, the numerical valuation of European bond options becomes
straightforward. This is due to the fact that the relative price of bond option C (Vs , rs , s) to the
same numeraire VsH (as what we employed for the pricing of defaultable coupon bonds) evolves
in the same way as that of the underlying asset, and so immediately lends the numerical
algorithm developed to the efficient pricing of bond options. The numerical valuation algorithm
for defaultable coupon bonds can be easily modified to price European and American bond calls
and bond puts. Exhibit 12 and 13 show bond option prices for different levels of initial interest
rates. In Exhibit 12, the first six columns represent call and put option prices on a risky coupon
bond, and the last three columns are the prices of a call option on a default-free coupon bond
with the same payment schedule as the risky one. The corresponding put option prices are not
shown in Exhibit 12 because of their extremely small values. This can be explained by the fact
that when the underlying bond is non-defaultable, its price at maturity is likely to be greater than
the exercise level, and so the put option becomes deep out-of-the money. As shown, the prices of
the call option on a non-defaultable bond are uniformly decreasing with the interest rate levels,
and increasing with the exercise prices. This result agrees with Longstaff’s (1993) findings on
the valuation of options on default-free coupon bonds. Not surprisingly, when the underlying
bond is defaultable, the corresponding calls become less valuable because of lower values of the
underlying. More interestingly, the risky call prices move in a direction opposite to the riskless
call prices. The reason for this is similar to the explanation we gave at the end of section 7.2.
The gradually increasing feature of the bond call prices can be explained by the fact that, in the
risk-neutral world, the value of the firm increases in the response to the increasing interest rates.
As a result, the bond calls become more valuable as interest rates increase.
Exhibit 12
Examples of prices for 2-year call and put options on a 5-year 8% coupon
bond with face value 70 for different initial interest rate levels r, and
exercise price E. The option values are computed using parameters
 =0.06,  =1,  =0.031,  =-0.25,  =0.2, V0 =100, K =70, a =0.12,
c =0.04, and binomial steps n =2, m =5.
Call Prices (Risky)
Put Prices (Risky)
Call Prices (Riskless)
r
E=60
E=65
E=70
E=60
E=65
E=70
E=60
E=65
E=70
0.01
3.411
3.472
3.354
3.586
3.623
3.659
3.697
3.735
3.774
3.813
3.852
3.894
3.935
3.981
4.027
1.340
1.368
1.400
1.424
1.441
1.458
1.475
1.491
1.514
1.537
1.560
1.583
1.606
1.629
1.657
0.317
0.325
0.332
0.337
0.339
0.340
0.344
0.347
0.352
0.357
0.363
0.368
0.374
0.380
0.385
2.716
2.590
2.468
2.359
2.256
2.154
2.055
1.961
1.867
1.775
1.686
1.600
1.515
1.436
1.358
5.281
5.081
4.887
4.714
4.551
4.392
4.234
4.079
3.933
3.788
3.645
3.504
3.364
3.227
3.095
8.893
8.633
8.376
8.142
7.926
7.713
7.504
7.298
7.095
6.900
6.699
6.503
6.310
6.119
5.929
15.01
14.79
14.57
14.36
14.15
13.94
13.74
13.54
13.34
13.13
12.94
12.75
12.56
12.38
12.19
10.37
10.20
10.02
9.846
9.674
9.505
9.338
9.174
9.011
8.851
8.694
8.538
8.384
8.233
8.084
5.739
5.601
5.464
5.329
5.197
5.066
4.938
4.812
4.687
4.565
4.444
4.326
4.209
4.093
3.980
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
Exhibit 13
Examples of prices for  -year call and put options on a 5-year 8% risky
coupon bond with face value 70 for different initial interest rate levels r.
The option values are computed using parameters  =0.06,  =1,
 =0.031,  =-0.25,  =0.2, V0 =100, K =70, a =0.12, c =0.04, E =70
and binomial steps n =2, m =5.
Call Prices
r
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14

=1
0.295
0.282
0.269
0.257
0.245
0.233
0.222
0.211
0.200
0.189
0.179
0.166
0.150
0.138
Put Prices

=2
0.317
0.325
0.332
0.337
0.339
0.340
0.344
0.347
0.352
0.357
0.363
0.368
0.374
0.380

=5
0.280
0.289
0.299
0.308
0.315
0.322
0.330
0.338
0.347
0.356
0.365
0.374
0.384
0.394

=1
6.067
5.994
5.922
5.850
5.778
5.707
5.635
5.563
5.492
5.420
5.348
5.297
5.268
5.242

=2
8.893
8.633
8.376
8.142
7.926
7.713
7.504
7.298
7.095
6.900
6.699
6.503
6.310
6.119

=5
13.96
13.61
13.25
12.91
12.59
12.28
11.96
11.66
11.36
11.06
10.77
10.48
10.19
9.913
0.15
0.127
0.385
0.405
5.216
5.929
9.637
This shows that default risk has a significant effect on the pricing of bond options on a
defaultable asset.
Another interesting feature of defaultable coupon bond option prices is
their relationship with the option maturity  . Here we are focusing on the underlying bonds
with maturity  +5 years. In Exhibit 13, it is easy to see that both call and put prices are
decreasing functions of initial interest rate when  =1. This is because when  is small, the
option values are close to their intrinsic value, and so move in the same direction as the
underlying bond does. However, when  becomes large, option features play a more important
role in call and put prices. Call prices are uniformly increasing, and put prices are uniformly
decreasing with interest rates. This result agrees with those in Exhibit 12.
Another possibility of generalization is through the introduction of bankruptcy costs.
Jones, Mason, and Rosenfeld’s (1984) attempt to implement the Merton model on U.S. corporate
bonds proves disappointing.
The models do not fit very well and tend to systematically
underestimate observed yields when plausible values of asset volatility were employed. To
improve the levels of credit spreads, we suggest that bankruptcy results in costly liquidation, and
so the creditor will receive the assets of the firm net of any liquidation costs. We assume that
liquidation costs take the following form
C  (1   )Vt ,
where C represents a fixed initial liquidation cost,  is a constant between 0 and 1, and t is the
time at which liquidation happens. Thus at the liquidation barrier the value of the bond is given
by
Max (Vt  C,0) .
The flexibility of our computational method allows for easy incorporation of liquidation costs in
our bond pricing model. With a slight modification of the default payoffs, recovery rates now
become a consequence of the combined effects of the endogenously generated mechanism and
the exogenously given structure as above. In Exhibit 14, we show that the deployment of the
simple functional form of liquidation costs improves the levels of credit spreads generated by our
framework, and hence provides a more pragmatic approach to the solution of defaultable bonds.
Exhibit 14
Term structure of credit spreads for a 8% coupon
bond: r0 =0.04,  =0.06,  =1,  =0.2, V0 =100,
a =0.12, c =0.04, unless otherwise stated.
In Exhibit 14.1: K =50, _____: without liquidation
costs ; - - -: with liquidation costs  =0.05 and C =2
In Exhibit 14.2: K =70, _____: without liquidation
costs ; - - -: with liquidation costs  =0.05 and C =2
In Exhibit 14.3: K =90, _____: without liquidation
costs ; - - -: with liquidation costs  =0.05 and C =2
Insert Exhibit 14 here
9. CONCLUSION
In this paper we have generalized, in computational aspects, Geske’s (1977) and Selby’s
(1983) valuation models of risky coupon bonds to allow for stochastic interest rates proposed by
Vasicek (1977). By using the hypothetical asset Vt H as a numeraire, it has been shown that not
only does the use of this numeraire significantly simplify the analytic valuation of risky discount
bonds, but also gives an implication that the two-factor model can be implemented easily. We
have discussed computational efficiency when the hypothetical asset Vt H is used as a numeraire,
and showed that this is an appropriate choice of numeraire. In addition, we have suggested an
efficient computation algorithm for the pricing of general risky coupon bonds by generalizing the
models proposed by Ho, Stapleton and Subrahmanyam (1995). Much of the simplicity of this
method lies in the fact that two sources of variability, namely interest rate risk and asset value
risk, are combined together to form a single stochastic process Z t . This approach to pricing
general risky debt is particularly appealing when the underlying assets of a derivative instrument
can be managed to be driven by Z t .
This analysis can be extended in several ways. The Vasicek model suffers from its
implicit assumption that interest rates can become negative with a positive probability at any
given time. While it has been shown that this probability can usually be made small by properly
adjusting the process parameters, the weakness of negative interest rates is perhaps offset by Hull
and White’s (1990) observation that the extended Vasicek model can be used to fit any
observable term structure. Moreover, the fact that the Vasicek process can be embedded in a
framework of HJM’s (1992) model leaves us with an implication that the methods developed in
this paper would be readily generalizable to incorporate a more general term structure consistent
interest rate process. Despite this, it still retains much of the computational tractability. For
example in the extended Vasicek model, apart from being consistent with initial term structure,
any volatility term that is a deterministic function of time can readily be fitted into our
framework. Our method of construction of binomial trees can be easily generalized to cope with
more general Markovian processes dependent on several stochastic variables. This should be
useful in pricing of some more complicated instruments, for example currency swaps.
Another drawback is with the assumption that asset values are log-normally distributed.
The traditional Merton’s (1973) approach to pricing risky debt has been criticized as being
incapable of generating credit spreads consistent with those observed in corporate debt markets.
One of the reasons is that the model lacks the fat-tailness properties that we normally observe in
asset returns. The introduction of jump processes to the bond pricing model would be able to
resolve part of the issues, though the mathematics for dealing with point processes is more
involved. Another reason is the absence of a mechanism that allows for costly liquidation in the
event of bankruptcy. We have shown that the pragmatic deployment of a simple functional form
of liquidation costs improves the levels of credit spreads generated by our framework.
Finally, we observe that while this traditional approach to modeling risky debt does not
provide practical tools for valuing realistic types of corporate securities, it has been an
indispensable tool for discussing the distribution of the firm’s value between shareholders and
bondholders. In addition to providing conceptual insights into default behaviour, the flexibility
of our method allows for efficient pricing of bond options, credit risk put options on a general
defaultable coupon bond and floating rate notes. This structural approach also paves the way for
a further analysis of more complicated debt structures. Incorporation of bankruptcy costs in the
model is an important avenue that can be explored in our framework. Efficient numerical
valuation of general risky debts when interest rates and firm’s values are stochastic should be a
crucial step forward in understanding the full complexity of credit analysis.
APPENDIX 1
Proof
Since rt follows a Vasicek process, it is not difficult to prove that under the equivalent
measure Q ,
T
  rsds  aT 
0
where aT  
1

(1  e  T )[ r0 
  T T s r  r
e  e dBs  BT
0


 T
 e T  1
(e  1)]  (
 T ) and Bsr  Bs  1   2 Bs' . By



THEOREM 2, under the equivalent martingale measure QV
and the transformation
BtV  t  Bt we have
T
  rsds  aT 
0
  T T s

 1  e  T
e  e d [ BsV  1   2 Bs' ]  [ BTV  1   2 BT' ] 
[
T]
0




which is a normal random variable.
T

1
2
By definition ZT   rudu   2T  BTV ,
0
 T2  Var[ ZT ]  bT2   2T  2
 1  e  T
(
T)


2
2
2 2
 2 T
where b  2 T 
(1  e
)  3 (1  e  T ) .
3

2

2
T
APPENDIX 2
Under that assumption that rt follows a Vasicek process as described by (1), we have
rt  r0e  t 
t

(1  e  t )  e t  e s dBsr ,
0

where Btr  Bt  1   2 Bt' . By THEOREM 2, under the equivalent martingale measure
QV and the transformation BtV  t  Bt , the above equation can be expressed as
rt  r0e  t 
t
  
(1  e  t )  e  t  e s d [ BsV  1   2 Bs' ] .
0

(*)
We can prove that
E[rt | s ]  r0e  t 
s
  
(1  e  t )  e  t  e u d [ BuV  1   2 Bu' ] .(**)
0

Result follows after eliminating the integrals in (*), (**).

t
1
2
By definition Z t   ru du   2t  BtV , we have
0
t
1
E[ Z t | s ]  Z s   2 (t  s )  E[  ru du | s ] .
s
2
After integrating (*) and substituting the result and (*) into the above equation, result follows.
The proofs for the rest of results are trivial.
APPENDIX 3
In the text, we have constructed the one-period trees for X s , Ys . Now we consider the
construction of the trees at a second time period t , with t  s , by following the similar fashion
to approximate X t by a vector of ns  nt  1 numbers:
X t ,i  X 0uti dtns nt i ,
for i  0,..., ns  nt , where
dt 
2( E[ X t ] / X 0 )1/(ns nt )
,
1  exp( 2 s ,t ,r / nt )
ut  2( E[ X t ] / X 0 )1 /(ns nt )  d t .
On the time-interval [ s, t ] , we choose a transition probability qt of an up-movement at node i
at time s such that property (6) holds, that is
r0  iLog[ut ]  (ns  i ) Log[dt ]  nt (qt Log[ut ]  (1  qt ) Log[dt ])  as,t  e  ( t s ) rˆs ,
hence,
qt 
as ,t  e   ( t  s ) rˆs  r0  iLog [ut ]  ( ns  i ) Log[d t ]  nt Log[d t ]
.
nt ( Log[ut ]  Log[d t ])
Given the above interest rate tree, we are now ready to construct a second tree for Yt
conditional on rt . At time t , we create a vector of (ms  mt  1) numbers
Yt ,i  uti dtms mt i ,
for i  0,1,..., (ms  mt ) , where
2( E[Yt ])1/(ms mt )
,
dt 
1  exp( 2 s ,t ,Z |rt / mt )
ut  2( E[Yt ])1 /(ms mt )  d t ,
where  s ,t ,Z |rt is the volatility of Z t conditional on s and rt . A transition probability pt of an
up-movement at node i at time s is chosen such that the following property holds
 (ts )
E[ Z t | s , rt ]  bs ,t  Z s  1e 
rs   s ,t
 s ,t ,Z

 s ,t ,r s ,t ,r
that is
iLog [ut ]  (ms  i ) Log[d t ]  mt [ pt Log[ut ]  (1  pt ) Log[d t ]]
 bs ,t  Zˆ s 
1 e
bs ,t  Zˆ s 
pt 
  ( t s )

rˆs   s ,t
1  e  (t s )

rˆs   s ,t
 s ,t ,Z
ˆs ,t ,r
 s ,t ,r
 s ,t ,Z
ˆ  iLog[ut ]  (ms  i ) Log[d t ]  mt Log[d t ]
 s ,t ,r s ,t ,r
mt ( Log[ut ]  Log[d t ])
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