第36卷 第5期
2007年1 0月
上海师范大学学报(自然科学版)
Journal of Sharlghai Normal University(Natural Sciences)
Vo1．36．No．5
2 0 0 7 ，Oct．
Quality spread differentials of fixed rate
risky debts in jump diffusion models
ZHANG Chao，ZHANG Ji—zhou
(College of Mathematics and Sciences，Shanghai Normal University，Shanghai 200234，China)
Abstract：Assume that the firm value follows the jump—diffused process and the interest rate follows the Hull—
White mode1．We derive the closed form formula for the defau1t premium for fixed rate debts．Then，we study the de—
fault premium for fixed rate debts when interest rate follows the jump—diffused process，and spread the price formulas
of default premium equation in the ecumenical jump—diffused process．
Key words：Hull—Wh ite model；risky debts；jump diffusion process；quality spread differential
CLC number：0175．2 Docum ent code：A Article ID：1000—5137(2007)05-0010-05
1 Introduction
A corporation may raise capital by issuing either fixed rate or floating rate debts．In fixed rate debts，the
par value of the debt paid by the issuer at maturity is fixed．A floating rate debt is similar to a money market
account，where the par account paid by the issuer at maturity is the sum of principal and accrued interests．
The amount of accrued interests depends on the realization of the stochastic interest rate process over the life of
the floating rate debt．It is a common practice for corporations to issue both fixed rate and floating rate debts．
The preference of fixed rate over floating rate may signal the management view on possible rise of interest rate．
In determining the appropriate proportion of debts into either fixed rate type or floating rate type，corporate ex—
ecutives consider financial attribute like balance sheet duration，current interest rate environment． and peer
group practices．Assuming jump diffusion process for the firm value and Vasicek mean reversion process for
the interest rate，the quality spread differential is studied in[1]．In this paper，we assume that the interest rate
follows the Hull—White mode1．We derive the closed form form ula for the default premium for fixed rate debt
．
The firm value of the issuer follows the Geometric Brownian motion with random jumps．The governing
stochastic differential equation for is given by
Received date：2007-05-08
Foundation item：Supported by the special Funds for major specialties(05DZ10)and Development Funds of Shanghai
Higher Education(T0401)of Shanghai Education Committee．
Biography：ZHANG Chao(1977一)，male，master，College of Mathematics and Sciences，Shanghai Normal University；
ZHANG Ji—zhou(1958一)，male，professor，College of Mathematics and Sciences，Shanghai Normal University．
维普资讯
第5期 张 超，张 寄 洲：跳扩散模型下的固定利率债券的套利成本差 1l
By intergration，we obtain
dV,=(ot一 u) d￡+or dW,+ —UdN
In V 一In =( —hv一了or)(T一￡)+ (W 一 )+∑ In(1+ )，
．_ J 0
where ot is the instantaneous expected return of the firm value and h is the mean number of Poisson arrivals per
unit time．The jump amptitude 1+U is assumed to be lognormally distributed with mean In(1+ )一 ／2 and
variance ，and independent of{ }l≥0 and{Ⅳl}I≥0．The jump component is the sum of Poisson distributed
(with Poisson arrival rate h)Nr一ⅣI iid random shocks．All parameters are assumed to be constant．
We adopt the form ulation of Merton s risky debt mode1．A simple capital structure of the firm is assumed
i’n the Merton model，where the liabilities of the firm only consist of a single debt．Default can be triggered on—
ly at maturity and this OCCUrs when the firm value cannot meet its obligation．Accordingly，the payoff to the
bondholders at maturity T is given by
Br=min(Vr，Fr)=Vr—max(Vr—Fr，0)，
where Fr is the par amount paid at maturity．The potential loss due to the default of the issuer can be visualized
as the value of a put option on the firm value with strike price Fr．
2 Default premium of fixed rate debts when interest rate follows continuous model
Th e interest rate rI follows the Hull—White model
dr =(ot(t)一 (t)r1)dt+or(t)dZ ，
where dW,and dZ are correlated with dW,dZf P dt，ot(t)， (t)and or(t)are nonrandom functions oft．
尸(t，T)denote the price of the default free zero coupon．It can be shown that
where
弓 =r d 一 ( ， )dz ，
r
orp(￡， )=or(￡)exp I K(￡)}J．exp{一K(y)}dy，
K(￡)=
Consider a fixed rate debt whose fixed par value at maturity T is F．Y．The payoff to the bondholders is given by
V7 一max(V7．一FY，0)．Let B denote the price of the risky fixed debt at the current time t．By the Black—
Scholes— Merton option pricing theory， the debt price is given by the discounted expectation of the
terminal payoff
E fy， (e-P[V7．一ma (Vr一
．
o)])
= 一 尸(￡，T)Erd；’ {max(V 一F ，O))．
In order to derive B ，we only calculate the second term ．We regard it as the value a call option
Let ( ，P，t；F )denotes the price of a call option with maturity Tand F is the strike price．By the re。
suits in[2]，we obmin
d
、，
／
r ● ●●J 0
维普资讯
12 上海师范大学学报(自然科学版)
韭
O t
v O B 1
- p er erp + P2 +
h E[ (V(1+U)，P，t)一 (V，P，t)]=0，
=max(Vr—Fx，0)．
2007证
(1)
Theorem 1 The solution of the equation(1)is
)= f I(1+ xp{-hv t ，驯 ，(2)
where H(V，P，t；Fx)is the option value with stochastic interest rate proposed by Merton and E is an expected
operator on
Proof
Then
Let
In order to prove
= 0．
this conclusion，we only need to verify that(2)satisfies(1)
P ( )=旦 P— —= ， = f
。：
I
。
(1+Ui)exp{-hv t}
磐= ㈤E u O H．·，VzO Z B= ㈤E U20 2 H．，
a B
a P ，筝=
=
弘㈤E ( O2H"0 l， n= o』
E (U a0 2
a
HI)’-
喾=一^ +^耋 H)一^t， ( O H)+n壹=O ) O H)=
嚣+ ㈤ O H⋯n =O㈩ ， ， ，
where m = n— 1．So
喾 骞+ 塑o v2-p er ere + 1 2 P 睾=
( U 2a02_ _H_H+p Or Orp P + 1 2 P oZH一 O H卜^t， 韭O V+
膪+^t， 嚣一^ ) (H( ， ， )=
E ( +p o'orp U,,P丽02 H+ 1 2 oZH一 )m —
Since H(V，P，t，Fx)satisfies
÷( +2p⋯ 02 H．)一 O H_0，
we have for any n，
1( U 2
a
02
_ _ ff _H+2p o'o'
p
U．P + ； oZH．)
一
O H
=。，
(3)
(4)
、
+
／ n
／
n P
∑
l1
／
∑
／
∑
／
∑
、
F
P
+ m
／ H
／
m
、
／
m
P
∑
^
维普资讯
第5期 张 超，张 寄 洲：跳扩散模型下的固定利率债券的套利成本差 13
and
El+v[ ( (1+ )，P，t，F )]=E +u[∑P (￡)E (日( (1+ )，P，t，F ))]=
n 0
∑P (￡)E (日(U州，P，t，F ))． (5)
n ：U
Finally，substituting(4)and(5)into(3)，we can obtain(1)．
In the following，we will prove the(2)satisfies the marginal condition
E [H( ，1，T，F )]=E [max(U 一F )]≤ E (U )=V(1+t，) ．
Then
⋯
lim ㈤ 删 )]≤
im S exp(一ht)exP((1+v)ht)一1)=0·
( ，1， ， )= im
，
13(v，P )= 尸0(￡)E( ，P )=max( 一Fx,0)
Therefore，
B = 一 ( ，P，t，F )=
一 e卫 I FI(1+ )exp}-hv t}’P )]．
Last，we substitute the result above into the formula of the default premium
仃 =专·n FX ， 了m “— 一，
and we can obtain the default premiuln when interest rate follows the Hull—White mode1．
3 Default premium of fixed rate debts when interest rate process is a jump—diffu—
slon process
LetP(t，T)denote the price of the default free zero coupon and follow
!j ：( ，一 )d + ，dz + ，dⅣ ，
where／xe is the instantaneous expected return of the firm value， P is the volatility of the debt
， 1+X is the
jump amplitude and kP=E(X)．All parameters are assumed to be constant．
Similarly by the results in[2]，we can obtain
喾 v 0a 13 -h ke P嚣+ 1 +p o'o'p + 1 2 P +
^E[13(V(1+ )，P(1+ )，￡)一13(v，P，￡)]：0 ’ (6)
Bl：r=max(Vr—F ，0)
Theorem 2 The solution of the equation(6)is
( ，P，t，F )=
n =O
n
(1+ exp 川，吨(1+ exp(-hke t},t
维普资讯
14 上海师范大学学报(自然科学版) 2007钲
The proof of Theorem 2 is similar to that of Theorem 1．
Then
B，Y= —B(V，P，t，Fx)=
E 刖虞(1+ exp 川，P虞(1+ xp{
Last，we substitute the above result into the formula of the default premium
仃 =了1 ln fX
， 仃-Y 了m n—彳一，
and we can obtain the default premium when interest rate process is a jump—diffused process．
References：
[1]
[2]
[3]
[4]
[5]
WONG H Y，KWOK Y K，Jump diffusion models for risky debts：Quality spread differentials[J]．International Journal of
Theoretical and Applied Finance，2003，6：655—662．
CHEN CHAO，option pricing model when interest rate process is a jump—diffusion process[J]．Institute of quantitative
technical economics，2000，11：110—114．
FIMA C KLEBANER．Introduction to stochastic calculus with applications[M]．London：Imperial College Press，20o4．
KWOK Y K，Mathematical models of Financial derivatives[M]．Singapore：Springer—Verlag Singapore，1998，
JIANG L SH．Mathematical modeling and methods of option pricing[M]．Beijing：Higher Education Press，2003．
跳扩散模型下的固定利率债券的套利成本差
张 超，张 寄 洲
(上海师范大学 数理信息学院，上海 200234)
摘要：首先假设公司资产价值服从跳扩散模型，利率服从 Hull—w}Iite模型，得到固定利率债券的套利成本差的闭式解
其次在假设利率服从跳扩散模型时推导固定利率债券的套利成本差，推广了一般跳扩散模型下的套利成本差定价公式
关键词 ：Huu—White模型；风险债券；跳扩散过程；套利成本差
(责任编辑：冯珍珍)
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