general properties of solutions to inhomogeneous black

Report No: KISU-MATH-E-R-024: Version 2

General Properties of Solutions to Inhomogeneous Black-Scholes

Equations with Discontinuous Maturity Payoffs and Application

Hyong-Chol O 1 and Ji-Sok Kim

2

1, 2Faculty of Mathematics, Kim Il Sung University, Pyongyang, D. P. R. Korea

2 Corresponding Author E-mail address: [email protected]

Abstract: We provide representations of solutions to terminal value problems

of inhomogeneous Black-Scholes equations and studied such general

properties as min-max estimates, gradient estimates, monotonicity and

convexity of the solutions with respect to the stock price variable, which

are important for financial security pricing. In particular, we focus on

finding representation of the gradient (with respect to the stock price

variable) of solutions to the terminal value problems with discontinuous

terminal payoffs or inhomogeneous terms. Such terminal value problems

are often encountered in pricing problems of compound-like options such

as Bermudan options or defaultable bonds with discrete default barrier,

default intensity and endogenous default recovery. Our results are applied

in pricing defaultable discrete coupon bonds.

Keywords : Black-Scholes equation, inhomogeneous, representation of solution,

min- max estimates, gradient estimate, monotonicity, convexity,

defaultable coupon bond

2010 MSC : 35C15, 35K15, 35B50, 35Q91

First version written on 12 Aug., 2013, this version revised on 27 Sep., 2013

1. Introduction

Since the development of the famous Black-Scholes formula for option pricing in [3], the

Black-Scholes equations have been widely studied and became one of the most important

tools in financial mathematics, in particular, in pricing options and many other financial

securities.

In this paper, we study general properties of the solutions to terminal value problems of

inhomogeneous Black-Scholes partial differential equations. We provide the solution -

representations, min-max estimates of the solutions, gradient estimates with respect the stock

price variable, monotonicity and convexity of the solutions.

mailto:[email protected]

Hyong-Chol O and Ji-Sok Kim

2

It is well known that standard European call and put options have pricing formulae

(solution representation of a special terminal value problems of Black-Scholes equations) and

their price functions have various monotonicity (with respect to stock price, exercise price,

short rate, dividend rate, and volatility) and convexity (with respect to stock price and

exercise price) [10, 13].

In 1973, Fischer Black and Myron Scholes [3] gave the call option pricing formula, so

called Black-Scholes formula. More general formulae of the solutions to Black-Scholes

equations with any terminal payoffs are described in [13, 17] and etc.

Merton [14] and Jagannathan [9] show that a call option’s price is an increasing and

convex function of the stock price under the assumption that the risk-neutralized stock price

follows a proportional stochastic process (under which the pricing model becomes a simple

Black - Scholes equation with zero short rate and dividend rate). Cox et al. [5] generalize this

result and show that, under the same proportionality assumption of the stock price process,

the price function of any European contingent claim, not just a call option, inherits qualitative

properties of the claim’s contractual payoff function [2]. (This means that the solution to

Black-Scholes equation inherits such qualitative properties as monotonicity or convexity of

the maturity payoff function.)

Bergman et al. [2] generalized the results of [5, 9, 14] into the case when the stock price

follows a diffusion whose volatility not only depends on time but also is a differentiable

function of and current stock price. They used PDE method to study gradient estimate and

convexity preservation. In particular, the property of convexity preservation is related to the

monotonicity of option price with respect to the volatility and widely studied. El Karoui et al.

[6] obtained the same results as [2] with stochastic approach. Hobson [8] simplified the result

of [6] using stochastic coupling. Janson et al. [11] generalized the result of [2] by replacing

the differentiability of volatility with respect to current stock price into local Hölder

continuity and improved the results of [8]. Ekström et al [7] provided a counter example of

the solution not preserving convexity when the stock price follows a jump-diffusion process

and studied the condition to preserve convexity. All the above mentioned works dealt with

homogenous Black-Scholes (integro-) differential equations while Janson et al [12] studied

preservation of convexity of solutions to general (inhomogeneous) parabolic equations under

a regularity assumption of coefficients.

On the other hand, in [16] the authors provided an estimate for gradient of the solution to

Black – Scholes equation with respect to stock price (which is stricter than the gradient

estimate of [2], theorem 1. (i)) and applied it to comprehensive pricing of some exotic option

with several expiry times. They used such a stricter estimate to prove existence of unique

roots of nonlinear equations related to option prices and such roots were used as exercise

prices in the next calculations. In [1] studying defaultable coupon bonds, Agliardi used the

root V* of a nonlinear equation EN(V, TN1) = CN1 at page 751 without any mention about the

existence and uniqueness which are justified by the lemma 1 of [16, at page 253].

The purpose of this article is to study general properties of the solutions to terminal value

General Properties of Solutions to Inhomogeneous Black-Scholes Equations …

3

problems of inhomogeneous Black-Scholes equations (with discontinuous data). We provide

representations of solutions to terminal value problems of inhomogeneous Black-Scholes

equations and study such general properties as min-max estimates, gradient estimates,

monotonicity and convexity of the solutions with respect to the stock price variable. In

particular, we focus on finding representation of the gradient (with respect to the stock price

variable) of solutions to the terminal value problems with discontinuous terminal payoffs or

inhomogeneous terms.

As Black and Sholes mentioned in [3], almost all corporate liabilities can be viewed as

combinations of options. In particular we often encounter Black-Scholes equations with

discontinuous maturity payoffs or inhomogeneous terms in pricing defaultable corporate

bonds. See the cases with such recovery rate that 0 <1 or taxes on coupons in [1] and

(4.4) in [17] when Re Ru for Black-Scholes equations with discontinuous maturity payoffs.

See the problems (4.11) in [17] and (4.5) in [18] for inhomogeneous Black-Scholes equations

with discontinuous maturity payoffs. It is not appropriate to assume that these equations have

smooth coefficients. This fact shows the necessity of studying general properties of

inhomogeneous Black-Scholes equations with discontinuous maturity payoffs.

Our aim is not to establish a general theory generalizing the results of [2, 6, 8 or 11] but

to develop the theory enough to cover the circumstance of inhomogeneous Black-Scholes

equations with discontinuous maturity payoffs arising in the pricing problems on defaultable

bonds considered in [1, 17 and 18] (or further study of this direction).

The remainder of the article is organized as follows. In section 2 we provide

representations of solutions to the terminal value problems of inhomogeneous Black-Scholes

equations and min-max estimates. In section 3 we study the derivatives with respect to the

stock price variable in the case with continuous terminal payoffs and inhomogeneous terms

and the solutions’ monotonicity and convexity. In section 4 we study the derivatives with

respect to the stock price variable in the case with discontinuous terminal payoffs or

inhomogeneous terms, gradient estimate and the solutions’ monotonicity. In section 5 we

give an application to the pricing of a defaultable discrete coupon bond using our gradient

estimate covering discontinuous data.

2. Representation of Solutions and Min-Max Estimates for Terminal Value

Problems of Inhomogeneous Black-Scholes Equations

In this section we study representations of solutions to the terminal value problems of

inhomogeneous Black-Scholes partial differential equations and estimate the minimum and

maximum values of the solutions.

The terminal value problem of an inhomogeneous Black-Scholes partial differential

equation with the risk free short rate )(tr , the dividend rate )(tq of the underlying asset, the

volatility )(t of the underlying asset’s price process, maturity payoff f(x) and inhomogeneous


4

term g(x) is as follows:

2 22

2

( )( ( ) ( )) ( ) ( ) 0, 0 , 0

2

V t V Vx r t q t x r t V g x t T x

t x x

, (2.1)

)(),( xfTxV . (2.2)

Throughout the whole paper, we assume that )(tr , )(tq , )(t , f(x) and g(x) are all

piecewise continuous on their domains. Let denote

2 2( , ) ( ) , ( , ) ( ) , ( , ) ( ) , 0T T T

t t tr t T r s ds q t T q s ds t T s ds t T . (2.3)

.),(2

1),(),()/ln(),(),,/( 2

12

TtTtqTtrKxTtTtKxd (2.4)

Theorem 1.(Representation of Solutions to inhomogeneous Black-Scholes equations) Assume

that there exist nonnegative constants A, B, and such that

|)(| xf xAx ln , xBxxg ln|)(| , a.e. 0x . (2.5)

Then the solution of (2.1) and (2.2) is provided as follows:

.0,)(

2

)(

2

)()();,(

02

,,2

1

),(

02

,,2

1

),(

0

,,2

1

),(

0

,,2

1

),(

),(),(

),(2),(2

22

22

22

22

Ttddzzg

z

eedzzf

z

eex

ddzzg

z

eedzzf

z

eeTtxV

T

t

tz

xd

tq

Ttz

xd

Ttq

T

t

tz

xd

tr

Ttz

xd

Ttr

tTt

tTt

(2.6)

Proof. The solution of (2.1) and (2.2) is provided by the sum of the solution of the

corresponding homogeneous equation of (2.1) with terminal value f(x) and the solution of

inhomogeneous equation with terminal value 0. By the lemma 1 of [17, at page 4] the

solution of the corresponding homogeneous equation with terminal value f(x) is given by the

first term of (2.6) under the condition (2.5) on f(x). Now consider the inhomogeneous

equation (2.1) with terminal value 0. We can use the Duhamel's principle to solve it.

Fix ),0( T and let );,( txW be the solution to the following terminal value problem:

.0),(),(

,0,0,0)())()((2

)( 22

xxgxW

xtWtrxWtqtrWxt

W xxxt

This is just the homogeneous Black-Scholes equation with terminal value g(x). Thus using

the lemma 1 of [17] again, under the condition (2.5) on g(x) its solution is given as follows:

.)(,,2

1exp

11);,(

0

2

),(

),(22

dzzgtz

xd

zetxW

t

tr


5

According to Duhamel's principle, )0,0();,( xTtdtxWT

t is the solution to of the

inhomogeneous Black-Scholes equation with terminal value 0. Thus we have (2.6).(QED)

Remark 1. The theorem 1 generalizes the lemma 1 of [17, at page 4] to the case of the

inhomogeneous Black-Scholes equation.

Corollary 1. (Min-Max estimate for inhomogeneous Black-Scholes equation) Let f(x) and g(x)

be bounded and let )(sup)(),(inf)(),(sup)(),(inf)( xggMxggmxffMxffm xxxx . Let V(x,

t) be the solution of (2.1) and (2.2). Then we have the following min-max estimates:

i) .0,)()(),()()( ),(),(),(),( TtdegMefMtxVdegmefmT

t

trTtrT

t

trTtr (2.7)

In particular if r(t) r is constant, then we have

.0,0),)(()(),())(()(

,0),1()(

)(),()1()(

)( )()()()(

TtrtTgMfMtxVtTgmfm

rer

gMefMtxVe

r

gmefm tTrtTrtTrtTr

(2.7)’

ii) Let )}()(:{ gmzfzE f , )}()(:{ fMzfzF f . If 0fE or 0gE , then we have the

stricter estimate at the time before the maturity:

TttxVdegmefmT

t

trTtr 0),,()()( ),(),( . (2.8)

Similarly if 0fF or 0gF , then we have

TtdegMefMtxx

V T

t

trTtr

0,)()(),( ),(),( . (2.9)

Proof. In the two improper integrals of (2.6), if we use the substitutions ),,/( Ttzxdy

and ),,/( tzxdy , respectively, then we have

T

t

ytr

yTtr

ddytycxgee

dyTtycxfee

txV

)),,(()),,((),( 2/),(

2/),(

22

22. (2.10)

Here

.),(2

1),(),(),(e x p),,( 22

TtTtqTtrTtyTtyc (2.11)

From (2.10), if we consider 1)( 2/1 2

2

dye y , then the results are easily proved. (QED)

Remark 2. Maximum value estimates for Black-Scholes equations might not have been

mentioned before. The reason seems that call option price itself is unbounded and such a

maximum estimate is direct result of the solution representation like [13, 17]. But in the study

of compound-like options such as Bermudan options or defaultable bonds with discrete

default information we encounter such terminal value problems of (inhomogeneous)

Black-Scholes equations that terminal values are solutions to other (inhomogeneous)

Black-Scholes equations and we need to solve linear equation related to the solutions to such


6

terminal value problems. In that case we need such a min-max estimate.

Corollary 2. Let f(x) and g(x) be such functions that )0(),0( gf and )(),( gf exist. Let

V(x, t) be the solution of (2.1) and (2.2). Then ),0( tV and ),( tV are given as follows:

.)()(),(

,)0()0(),0(

),(),(

),(),(

T

t

trTtr

T

t

trTtr

degeftV

degeftV

(2.12)

Corollary 3. Assume that 0 f(x) x and 0 g(x) x. Let V(x, t) be the solution of

(2.1) and (2.2). Then we have

.0,),(0 ),(),( TtdeextxVT

t

tqTtq

In particular if q(t) q is constant, then we have

.0,0)],([),(0

,0,)1(),(0 )()(

TtqtTxtxV

qeq

extxV tTqtTq

(2.13)

Furthermore let }.)(:{~

},)(:{},0)(:{ xxhxFxxhxFxhxE hhh If 0|| fE and 0|| fF

(or 0|| gE and 0|~

| gF ), then we can replace “” into “<” in the above estimates.

Proof. These estimates are the direct conclusion from the second formula of (2.6) and the

assumption of the lemma. (QED)

3. The Preservation of Monotonicity and Convexity in the Case with

Continuous Terminal Payoffs and Inhomogeneous Terms

In this section we study the derivatives (with respect to the stock price variable) of

solutions to the terminal value problems of inhomogeneous Black-Scholes equations with

continuous terminal payoffs and inhomogeneous terms and the solutions’ monotonicity and

convexity.

Theorem 2.(x-gradient of Solution of inhomogeneous Black-Scholes equation) Let f(x) and g(x)

be continuous and piecewise differentiable and satisfy

,|)(),(),(),(| ln xAxxgxgxfxf a.e. 0x (A, > 0).

Let V(x, t) be the solution of (2.1) and (2.2). Then we have the following conclusions:


7

i)

dyTtyxcfee

txx

V TtyTtq

)),,((),(

22 ),(

2

1),(

2

,)),,((

2

2 ),(2

1),(

2

T

t

tytq

dydtyxcgee

Tt 0 . (3.1)

Here ),,( Ttyc is as in (2.11).

ii) If f(x) and g(x) are increasing (or decreasing), then V(x, t) is x-increasing (or

decreasing), too.

iii) If )0(),0( gf and )(),( gf exist, then ),(),,0( tVtV xx are given as follows:

.)()(),(

,)0()0(),0(

),(),(

),(),(

T

t

tqTtq

T

t

tqTtq

degeftx

V

degeftx

V

(3.2)

iv) If )(xf and )(xg are bounded, then we have

T

t

tqTtqT

t

tqTtq degMefMx

Vdegmefm ),(),(),(),( )()()()( . (3.3)

Furthermore if 0' fE or 0' gE , then we have the stricter estimate on x-gradient:

Tttxx

Vdegmefm

T

t

tqTtq

0),,()()( ),(),( . (3.4)

Similarly if 0' fF or 0' gF , then

TtdegMefMtxx

V T

t

tqTtq

0,)()(),( ),(),( . (3.5)

Here ),(),( fMfm 'fE , 'fF are same as in the corollary 1 of the theorem 1.

Proof. If we differentiate the both side of (2.10) on x, then we obtain the results of i). The

results of ii), iii) and iv) are evident from (3.1). (QED)

Remark 3. The theorem 2, iv) is a generalization of the lemma 1 of [16, at page 253] which

considered homogeneous Black-Sholes equations with constant coefficients. Bergman et al.

[2] (theorem 1, (i)) obtained a gradient estimate for homogeneous Black-Sholes equation with

stock price and time dependent coefficients but our estimate is stricter than it.

Theorem 3.(Convexity of Solutions of inhomogeneous Black-Scholes equation) Let f(x) and

g(x) be the same with those in the theorem 1 and V(x, t) the solution of (2.1) and (2.2). If both

f(x) and g(x) are convex upward (or downward), then V(x, t) is convex upward (or

downward) on x , too.


8

Proof. Let assume f(x) and g(x) are convex upward. Then for any (0, 1), we have

f(x1 + (x2 x1)) f(x1) + (f(x2) f(x1)), g(x1 + (x2 x1)) g(x1) + (g(x2) g(x1)). (3.6)

By the formula (2.10), we have

.)),,()](([

)),,()](([);),((

1212/

),(

1212/

),(

121

2

2

2

2

T

t

ytr

yTtr

ddytycxxxgee

dyTtycxxxfee

TtxxxV

Here we use (3.6) and then use (2.10) again, then we obtain

V(x1 + (x2 x1),t ; T) V(x1,t ; T) + (V(x2,t ; T)) V(x1, t ; T)).

The downward convexity is similarly proved. (QED)

Remark 4. The theorem 3 generalizes the well-known convexity preservation results of standard

Black-Scholes equations to inhomogenous cases. Janson et al. [12] studied preservation of

convexity of solutions to general inhomogeneous parabolic equations but their result didn’t

cover ours because of regularity assumption of coefficients.

4. Estimates of x-Derivatives in the Case with Discontinuous Data

In this section we study the derivatives (with respect to the stock price variable) of

solutions to the terminal value problems of Black-Scholes equations with discontinuous

terminal payoffs or inhomogeneous terms and the solutions’ monotonicity. Such terminal

value problems are often encountered in pricing problems of defaultable bonds with discrete

default barrier, discrete default intensity and endogenous default recovery. See the cases with

such recovery rate that 0 <1 or the case with taxes on coupons in [1] and (4.4) in [17, at

page 17] when Re Ru for homogeneous Black-Scholes equations with discontinuous

terminal payoffs. See the (4.5) in [18, at page 21] and (4.11) in [17, at page 18] for

inhomogeneous Black-Scholes equations with discontinuous terminal payoffs.

Theorem 4.(x-gradient of Solution of Black-Scholes equation with discontinuous maturity payoff

and inhomogeneous term) Let f(x) and g(x) be piecewise differentiable and satisfy

,|)(),(),(),(| ln xAxxgxgxfxf a.e. 0x (A, > 0).

Let V(x, t) be the solution of (2.1) and (2.2). Assume that f(x) and g(x) have the only jump

discontinuities Kf and Kg, respectively. Then we have the following conclusions:

i)

)(

),(

)),,((2

),(2

),,/(2

1

),(2

1),(

22

2

f

f

TtKxdTtyTtq

Kf

Tt

edyTtycxfe

etx

x

V

K

f

T

t

g

g

tKxdtytq

dKg

t

edytycxge

e

K

g

)(

),(

)),,((2 2

),,/(2

1

),(2

1),(

22

2

, Tt 0 . (4.1)


9

Here )(xh = )0()0( xhxh is the jump of h at x and ),,( Ttyc is as in (2.11).

ii) If f(x) and g(x) are increasing (or decreasing), then V(x, t) is x-increasing (or

decreasing), too.

iii) If )0(),0( gf and )(),( gf exist, then ),(),,0( tVtV xx are given as in (3.2).

iv) If )(xf and )(xg are bounded, then we have

),(2

)]([)()(),(

,

),(2

)]([

),(2

)]([

)()(),(

2

),(),(),(

2

),(

2

),(

),(),(

Tt

e

K

KfdegMefMtxV

d

t

e

K

Kg

Tt

e

K

Kf

degmefmtxV

Ttq

f

fT

t

tqTtqx

T

t

tq

g

gTtq

f

f

T

t

tqTtqx

T

t

tq

g

gd

t

e

K

Kg

),(2

)]([

2

),(

, 0 t < T. (4.2)

Furthermore if 0|| ' fE or 0|| ' gE , we have the stricter minimum estimate on x-gradient

.0,

),(2

)]([

),(2

)]([

)()(),(

2

),(

2

),(

),(),(

Ttd

t

e

K

Kg

Tt

e

K

Kf

degmefmtxV

T

t

tq

g

gTtq

f

f

T

t

tqTtqx

(4.3)

Similarly if 0' fF or 0' gF , then we have the stricter maximum estimate on x-gradient

.0,

),(2

)]([

),(2

)]([

)()(),(

2

),(

2

),(

),(),(

Ttd

t

e

K

Kg

Tt

e

K

Kf

degMefMtxV

T

t

tq

g

gTtq

f

f

T

t

tqTtqx

(4.4)

Here ),(),( fMfm 'fE , 'fF are same as in the corollary 1 of the theorem 1 and [x]+

=

max{x, 0}, [x]

= min{x, 0}.

Proof. i) In (2.10), let denote ),(),(),( 21 txUtxUtxV , where

.)),,((),(,)),,((),( 2/),(

22/

),(

1

22

22

T

t

ytr

yTtr

dydtycxgee

txUdyTtycxfee

txU

Now Calculate ),(1 txUx . Note that Kf is the only jump discontinuity of f(x). Therefore the

function h(y) = )),,(( Ttycxf has the only jump discontinuity 0y such that fKTtyxc ),,( 0 . We

find 0y using (2.11) and use (2.4) to get


10

1

200 ),()(,,,)(

TtxxyTt

K

xdxy

f

. (4.5)

Using this 0y , the integral of ),(1 txU is divided into the integrals of continuous functions:

0

20 2

)),,(()),,((),( 2/2/),(

12 y

yy

yTtr

dyTtycxfedyTtycxfee

txU

.

Using the theorem on differentiation of integral with parameter in elementary analysis, (4.5)

and the fact that c(y, t, T) is y-decreasing, we have

)].0()0([

),(2

)),,((2

)()),,0((),,()),,((

)()),,0(()),,()),,((2

),(

2

2/),(),(2

1),(

002/2/

002/2/

),(1

20

2

2

20

0

2

20

0 2

ff

yTtrTtyTtq

y

y

y

yy

yTtr

KfKf

Tt

edyTtycxfe

e

xyTtycxfedyTtycTtycxfe

xyTtycxfedyTtycTtycxfee

txx

U

x

If we use (4.5) and (2.4), then we have

),(

)()),,((

2),(

2

),,/(2

1

),(2

1),(

1

22

2

TtK

eKfdyTtycxfe

etx

x

U

f

TtKxd

fTtyTtq

f

.

Similarly we obtain ),(2 txUx :

T

t g

tKxd

gtytq

d

tK

eKgdytycxge

etx

x

Ug

),(

)()),,((

2),(

2

),,/(2

1

),(2

1),(

2

22

2

.

Adding these two equalities, we obtain (4.1).

The results of ii), iii) and iv) are evident from (4.1). (QED)

Corollary Under the assumptions of theorem 4, assume that f(x) and g(x) have the finite

jump discontinuities ),,1( niyi and ),,1( mkzk , respectively. Then we have

.0,

),(

)()),,((

2

),(

)()),,((

2),(

1 2

),,/(2

1

),(2

1),(

1 2

),,/(2

1

),(2

1),(

22

2

22

2

T

t

m

k

tzxd

k

ktytq

n

i

Ttyxd

i

iTtyTtq

Ttd

t

e

z

zgdytycxge

e

Tt

e

y

yfdyTtycxfe

etx

x

V

k

i

(4.6)

Furthermore, the conclusions of ii) and iii) of theorem 4 hold and the analogues of the results

of iv) of theorem 4 hold, too.


11

Remark 5. The theorem 4 and its corollary generalize the theorem 2. In particular such stricter

estimates as (4.3) and (4.4) (and (2.8), (2.9), (3.4), (3.5)) are important in application (see the

section 3 of [16] and the following section 5). Such gradient estimates for Black-Scholes

equation with discontinuous maturity payoff or inhomogeneous term can be used in

calculation of default boundary in pricing real defaultable bonds with discrete coupon dates.

Theorem 5. (second order derivative representation of Solutions to inhomogeneous Black-Scholes

equations) Let f(x) and g(x) be piecewise twice differentiable and satisfy

,|)(),(),(),(),(),(| ln xAxxgxgxgxfxfxf a.e. 0x (A, > 0).

Let V(x, t) be the solution of (2.1) and (2.2). Assume that )(xf and )(xf have the only jump

discontinuity fK and g(x) and )(xg has the only jump discontinuity gK . Then we have the

following second order derivative representation:

TtK

xde

TtK

Kf

Tt

e

K

KfdyTtycxfe

e

Ttxx

V

f

TtTtK

xd

f

f

TtTtK

xd

f

fTtyTtTtqTtr

f

f

,,

),(

1)(

),(

)()),,((

2

);,(

2

2

2

22

22

),(,,2

1

22

2

),(,,2

1

),(22

1),(),(2),(

2

2

)7.4(.,,

),(

1)(

),(

)()),,((

2

2

2

2

22

22

),(),,/(2

1

22

2

),(,,2

1

),(2

1),(),(2),(

dtK

xde

tK

Kg

t

e

K

Kgdytycxge

e

g

ttKxd

g

g

T

t

ttK

xd

g

gtyttqtr

g

g

Proof. From the assumption and the formula (4.1), we have for Tt 0 ,

.

),(

)()),,((

2

),(

)()),,((

2),(

4321

),,/(2

1

2

),(2

1),(

),,/(2

1

2

),(2

1),(

2

2

2

2

2

2

UUUU

de

t

Kgdytycxge

e

e

Tt

KfdyTtycxfe

etx

x

V

T

t

tKxd

g

gtytq

TtKxd

f

fTtyTtq

g

f

K

K

From the assumption, )),,(()( Ttycxfyh has the only jump discontinuity )(0 xy such that

fKTtyxc ),,( 0 and )),,(();( tycxgyH has the only jump discontinuity );(1 xy such

that gKtyxc ),,( 1 . From (2.11) and (2.4), we have


12

.),()(,,,)(;),()(,,,)(

1

211

1

200

txxyt

K

xdxyTtxxyTt

K

xdxy

gf

Considering this fact, differentiating ),( txVx with respect to x, we have

)]0()0([

),(2

)),,((2

)())],,0(()),,0(([2

),,()),,((2

)),,((2

2

2

22

2

20

2

2

2

2

),,/(2

1

2

),(

),(22

1),(),(2),(

000

),(2

1),(

),(2

1),(

),(2

1),(

1

ff

TtKxdTtq

TtyTtTtqTtr

TtyTtq

TtyTtq

TtyTtq

KfKfe

Ttx

e

dyTtycxfee

xyTtycxfTtycxfee

dyTtycTtycxfee

dyTtycxfex

e

x

U

f

),(

)()),,((

2 2

),(,,2

1

),(22

1),(),(2),(

2

22

22

TtK

eKfdyTtycxfe

e

f

TtTtK

xd

fTtyTtTtqTtr f

.

Similarly we have

T

t g

ttK

xd

gtyttqtr

T

t

tytq

d

tK

eKgdytycxge

e

ddytycxgex

e

x

U

g

.

),(

)()),,((

2

)),,((2

2

),(,,2

1

),(2

1),(),(2),(

),(2

1),(

3

2

22

22

2

2

On the other hand

.,,

)(

2),(

),(

1,,

)(

),(2

)(

),(2

2

22

2

2

),(,,/2

1

22

),(),(2),(

2

),,/(2

1

2

),(

),,/(2

1

2

),(2

TtK

xde

K

Kf

Tt

e

Tt

TtK

xde

K

Kf

Tt

e

exK

Kf

Tt

e

x

U

f

TtTtKxd

f

fTtTtqTtr

f

TtKxd

f

fTtq

TtKxd

f

fTtq

f

f

f

x


13

Similarly

.,,

2),(

)(

),(2

)(

2

22

2

),(),,/(2

1

2

),(),(2),(

2

),,/(2

1

2

),(4

T

t g

ttKxdttqtr

g

g

T

t

tKxdtq

g

g

dtK

xde

t

e

K

Kg

dex

t

e

K

Kg

x

U

g

g

Thus we have proved (4.7). (QED)

5. Application to the Pricing of Defaultable Discrete Coupon Bonds

5.1 Assumptions

1) The short rate follows the Vasicek model

)()( 121 tdWsdtraadr rt (5.1)

under the risk neutral martingale measure and a standard Wiener process 1W . Under this

assumption the price Z(r, t ; T) of default free zero coupon bond is the solution to the problem

22

1 22

1( ) 0,

2

( , ) 1.

r

Z Z Zs a a r rZ

t rr

Z r T

(5.2)

The solution is given by

rTtBTtAeTtrZ ),(),();,( . (5.3)

Here A(t, T) and B(t, T) are given as follows [19]:

.),(2

1),(),(,

1),( 22

22

)(2

T

t

r

tTa

duTuBsTuBaTtAa

eTtB (5.4)

2) The firm value )(tV follows a geometric Brown motion

dttVbrtdV t )()()( )()( 2 tdWtVsV

under the risk neutral martingale measure and a standard Wiener process 2W and ,( 1dWE

dtdW )2 . The firm continuously pays out dividend in rate b 0 (constant) for a unit of

firm value.

3) A firm issues a corporate bond (debt) with maturity T and face value F (unit of

currency). TTTTT NN 1100 and Ti (i = 1,…, N) are the discrete coupon dates. At

time Ti (i = 1,…, N1) the bond holder receives the coupon of quantity CiZ(r,Ti ;T) (unit of

currency) from the firm, at time TN = T the bond holder receives the face value F and the last

coupon CN (unit of currency) and Ck 0. This means that the time T-value of the sum of the

face value and the coupons of the bond is kNk CF 1 .

4) The expected default occurs only at time Ti (i =1,…, N) when the firm value is not


14

enough to pay debt and coupon. If the expected default occurs, the bond holder receives V

as default recovery. Here is called a fractional recovery rate of firm value at default.

5) The unexpected default can occur at any time. The unexpected default probability in

the time interval ],[ ttt is t . Here the default intensity 0 is a constant. If the

unexpected default occurs at time t ),( 1 ii TT , the bond holder receives min{Vt, (F+

kN

ik C1 )Z(r, t ; T)} as default recovery. (Here the reason why the expected default recovery

and unexpected recovery are given in different forms is to avoid the possibility of paying at

time t ),( 1 ii TT more than the current price of risk free zero coupon bond with the face

value kN

ik CF 1 as a default recovery when the unexpected default event occurs. See [17]

and [18].)

6) The price of our corporate bond is given by a sufficiently smooth function ),,( trVBi

in the subinterval ),( 1ii TT , ( 1,,0 Ni ).

5.2 The Pricing Model for Defaultable Discrete Coupon Bond

According to [19], under the above assumptions 1) and 2), the price B of defaultable

bond with a constant default intensity and unexpected default recovery Rud satisfies the

following PDE:

.0)()(22

12

22

2

2

222

udrrrVV RBr

r

Ba

V

BVbr

r

Bs

rV

BVss

V

BVs

t

B

Therefore under the above assumptions 3), 5) and 6) we can know that our bond price

),,( trVBi ( 1,,0 Ni ) satisfies the following PDE in the subinterval ),( 1ii TT :

.0);,()(,min)(

)()(22

1

1

212

22

2

2

222

TtrZCFVBr

r

Braa

V

BVbr

r

Bs

rV

BVss

V

BVs

t

B

kN

iki

iiir

irV

iV

i

(5.5)

Now let derive the terminal value conditions from the conditions 3) and 4). At time T = TN,

the expected default occurs and the bond holder receives VT if VT < F + CN. Otherwise the

bond holder receives F + CN. Thus we have

}{1}{1)(),,(1 NNNNN CFVVCFVCFTrVB . (5.6)

When i{1,…, N1}, assume that the bond price ),,( trVBi in the subinterval ],( 1ii TT are

already calculated. The ),,( trVBi is just the price of our bond (the quantity of debt) in the

subinterval ],( 1ii TT under the condition that the default didn’t occur before or at the time Ti .

So the fact that the expected default did not occur at time Ti means that firm value must not be

less than ),,( ii TrVB after paying coupon at time Ti, that is, VTi ),,( ii TrVB + CiZ(r,Ti ;T). Thus

we have

)];,(),,([),,( 11111 TTrZCTrVBTrVB iiiiii 1{V > ),,( 11 ii TrVB + Ci+1 Z(r, Ti+1 ; T)}+

V 1{V ),,( 11 ii TrVB + Ci+1 Z(r, Ti+1 ; T)}, i = 0,…, N2. (5.7)

The problem (5.5), (5.6), (5.7) is just the pricing model of our defaultable discrete coupon


15

bond.

Remark 6. 1) As shown in the above in our model the consideration of unexpected default risk

and dividend of firm value is added to the model on defaultable discrete coupon bond of [1].

Another difference from [1]’s approach is that the bond is considered as a derivative of risk

free rate r (but not a derivative of default free zero coupon bond) and the coupon prior to the

maturity is provided as a discounted value of the maturity (predetermined) value.

2) The equation (5.5) is just the same type with (3.7) in [17]. In (5.6) the default barrier is

explicitly shown. But in (5.7) the default conditions don’t show the default barrier explicitly.

So in this stage, it is still not clear to find the similarity of this problem with the problem of

[17] but through more careful consideration we can use the method of [17] to get the pricing

formula of our defaultable discrete coupon bond (see the following subsection 5.3). As

shown in bellow, solving this problem is an implementation of the method of [17].

We introduce the following notation for simplicity of pricing formulae.

.1,,0,

;;

;1,,1,;

1

1

NiTTT

CFKCF

NiCcCFc

iii

NNkN

iki

iiNN

(5.8)

That is, ic is the time TN -value of the payoff to bondholders at time Ti (i =1,…,N) and is

the time TN -value of the sum of the face value and all coupons of the bond. KN denotes the

default barrier at time TN as in [17].

5.3 The Pricing Formulae of the Defaultable Discrete Coupon Bond

Theorem 6. The solution of (5.5), (5.6) and (5.7) is provided as follows:

1),),;,(/();,(),,( iiNiNi TtTtTtrZVuTtrZtrVB , 1,,0 Ni . (5.9)

Here 1),,( iii TtTtxu , 1,,0 Ni are the solutions to the following problems

.2,0,}),({1}),({1]),([),(

},{1}{1),(

,1,,0,,0,0,min)(2

1

1111111111

1

12

222

NicTxuxxcTxuxcTxuTxu

KxxKxcTxu

NiTtTxxux

ubx

x

uxt

t

u

iiiiiiiiiii

NNNNN

iiiiiii

Here )(t is given as follows when B(t, T) is as (5.4):

0),(),(2)( 222 TtBsTtBssst rrVV . (5.10)

1),,( iii TtTtxu , 1,,0 Ni are provided as combinations of higher order binaries and

their integrals.

(i) If every nonlinear equation

x iii cTxu ),( , x 0 ( 1,,1 Ni ) (5.11)

has unique root )1,,1( NiKi , respectively, then ),( txui , 1,,0 Ni are provided as

follows:

1

11111 ),,,;,(),,;,(),(1111

N

im

mmiKKKmiKKmi TTTtxATTtxBctxummimi


16

.0,,);,();,(

),,,;,(),,,;,(

1//

1

1

1

1

1

1

1

1

xTtTdtxAtxB

dTTtxATTtxB

ii

T

t

i

miKK

N

im

T

T

miKK

m

i

ii

mmi

m

m

mmi

(5.12)

Here ),,;,( 11

mKKTTtxB

m

and

mm KKK

A11

),,,;,( 11 mm TTTtx are the price of m-th order

bond and asset binaries with risk free rate , dividend rate +b and volatility )(t (the

theorem 1 of [17, pages 5~6]). In particular the initial price of the bond is given by

.)],,,;0,/(),,,;0,/([

),,,;0,/(),,;0,/(

)0,,(

1

11

1111

100/100/

1

0

1100110010

0000

m

m

mmmm

mmm

T

T

mKKmKKmm

N

m

mmKKKmKKm

dTTZVATTZVB

TTTZVATTZVBcZ

rVBB

(5.13)

Here );0,( 00 TrZZ .

(ii) If some of the equations x 111 ),( iii cTxu ( }2,,0{ Ni ) have several roots

Ki+1(j), j=1, …, Mi , then the corresponding ),( txui is provided as combinations of higher

order binaries (related to Ki+1(j), j = 1, … , M ) and their integrals.

Remark 7. Generally, the nonlinear equations (5.11) have an odd number of roots. See the proof.

For example In the case when = 1, all the nonlinear equations (5.11) have the unique roots Ki,

respectively. When 0 < 1, if the volatility )(t of the relative price of the firm value x =

V/Z is enough large, that is, if there exist a sequence 1 NN dd 11 d such that

),1,,1(0if)(2)1(),(1

112

NkbddTT kkkk

,0if

12

)1(),(

)()(1

)(

12

b

eb

edd

eTT

kk

k

TbTbkk

Tb

kk

(5.14)

then all the nonlinear equations (5.11) have the unique roots Ki, respectively, too. In the case

when (5.14) do not satisfy, if the coupons at time Ti (i=1,…, N1) is not too large, then all the

nonlinear equations (5.11) have the unique roots Ki. (See Lemmas 1 and 2 in the proof.) The

case when some of the equations (5.11) have several roots seems not appropriate in financial

meaning. (See the remark 9 in the proof.)

Proof: Now we solve the problem (5.5), (5.6), (5.7). For i=N1, the problem (5.5) and

(5.6) becomes as follows:


17

.0},{1}{1),,(

0,,0);,(,min)()(

)()()()(2)(2

1

1

1111

21

1

2

12

212

2

12

221

xKVVKVcTrVB

xTtTTtrZVBrr

Braa

V

BVbr

r

Bts

rV

BVtsts

V

BVt

t

B

NNNNN

NNNNN

NNr

NrV

NN

Here if we let 1 Ni and use the change of numeraire

1),;,(/),,(),(),;,(/ iiii TtTTtrZtrVBtxuTtrZVx (5.15)

and consider (5.2), then we have

.0},{1}{1),(

,0,,0,min)(2

1

1

1111

2

12

221

xKxxKxcTxu

xTtTxux

ubx

x

uxt

t

u

NNNNN

NNNNNNN

This problem is just the same problem with the (2.1) and (2.2) with the coefficients

)(,)(,)( tbtqtr and the inhomogeneous term

})/{1}/{1(,min)( 1111 NNNN xxxxxg .

Furthermore the terminal value

}{1}{1)( NNN KxxKxcxf

is a discontinuous function if < 1.

Using the theorem 1, we have

.})/{1}/{1(

}){1}{1(

2

);,(

0

1112

),(2

),(2

1)(ln

)(

02

),(2

),(2

1)(ln

)(1

),(2

),(

2

22

2

22

N

N

NN

N

T

t

NNN

t

ttbz

x

t

NNN

N

Tt

TttTbz

x

tTNN

ddzzzze

e

dzKzzKzce

eTtxu

tz

Ttz

From the definition of bond and asset binaries [4, 16, 17] we can know that the above two

terms are just the representation of linear combinations of bond and asset binaries and the

integrals of them. Thus we have

.0,,]);,();,([

);,();,(),(

1//1

1

11

xTtTdtxAtxB

TtxATtxBctxu

NN

T

tN

NKNKNN

N

NN

NN

(5.16)

Here );,( TtxBK and );,( TtxAK

are the prices of bond and asset binary options with the coeffic

ients )(,)(,)( tbtqtr and maturity T, which are provided by the theorem 1 of [17, at

page 5~6]. Returning to original variables using (5.15), we obtain BN1 :

),);,(/();,(),,( 11 tTtrZVuTtrZtrVB NN , )0,( 1 xTtT NN .

For the next step of study we investigate ),( 11 NN Txu and ),( 11 NNx Txu . Note that

1)(,)(,0)()( NNN gMKcfMgmfm , f and g are increasing, f(x) x and g(x)


18

x. From the corollary 1 and corollary 3 of the theorem 1, ii) of the theorem 4, we know

that ),( 11 NN Txu is bounded and increasing. Furthermore we have

}.),,(min{),(0),0( 111111 xTuTxuTu NNNNNN (5.17)

Here 111 )1(),( 11

NT

NT

NNNN eKeTu

. (See the figure 1.)

Figure 1. Convex and concave intervals of ),( 11 NN Txu that is bellow y = x.

)(xf has the jump NN KKf )1()( at KN and )()( gmfm ,0 NdfM )( , )(gM

. Thus using iv) of the theorem 4, we have

)18.5(.0if

),(2

1),(0

,0if

),(2

)1()1(),(0

12

11

12

)()()(

11

1

11

b

TT

dTxu

b

TT

ee

bedTxu

NN

NNNx

NN

TbTbTb

NNNx

N

NN

Therefore if = 1, then

1),(0 11 NNx Txu . (5.19)

If < 1 and the condition (5.14) with k = N1 holds, then we have

1),(0 111 NNNx dTxu (5.20)

That is, (5.14) is a sufficient condition for (5.19) to be true.

Now let study the convexity of ),( 11 NN Txu . Note that f has jump at x = KN, g has

jump N1 at x = N1/ and )()( xgxf 0 , a.e. x > 0. Using the theorem 5, we have

2

12

112

1),(,,

2

1

12

),()2(

12

12

),(2

),(

NNNNN

NNNTTTT

K

xd

NNN

TTTb

NN e

TTK

eTx

x

u

.,,

),(2

)1(

),(2

2

12

112

1

2

12

11

12

1

),(,,2

1

1

12

),()2(

),(,,2

1

12

),()(2(

1

2 )

NNNNN

NNN

NNN

NN

TTTTK

xd

NNN

NNN

TTTb

T

t

TTx

d

N

TTb

N

eTTK

xd

TTK

e

de

T

e


19

From this, we can know that if =1 then 0),( 112 NNx Txu , that is, ),( 11 NN Txu is x-upward

convex and if 0 ≤ δ < 1 then there exists unique point x0 such that

0),( 1012 NNx Txu ; 0),( 11

2 NNx Txu , x < x0 ; 0),( 112 NNx Txu , x > x0.

(See the figure 2.) That is, if x < x0, then ),( 11 NN Txu is downward convex and if x > x0, then

),( 11 NN Txu is upward convex. (See the figure 1.)

Figure 2. A graph of ),( 112

NNx Txu , ,3,2,05.0,1.0,8.0 1 NN TTb

15.0,5.0,077.0,379.0,6.1,3.1 21 VrNN ssaK

Now since we have enough information about ),( 11 NN Txu , let consider the case when

i=N2 in the problem (5.5) and (5.7).

)}.,,(),,({1

)},,(),,({1)],,(),,([

),,(

,,0);,(,min)(),(

)()()()(2)(2

1

1111

11111111

12

1222222

2

2

22

222

2

22

222

NNNNN

NNNNNNNNNN

NN

NNNNNNN

r

NNr

NrV

NV

N

TTrZcTrVBVV

TTrZcTrVBVTTrZcTrVB

TrVB

TtTTtrZVBrr

Btra

V

BVbr

r

Bts

rV

BVtsts

V

BVts

t

B

In this problem , we use (5.8) with i=N2 to get

,0,min)(2

122

2

2

22

222

xux

ubx

x

uxtS

t

uNN

NNX

N (5.21)

}),({1}),({1]),([),( 11111111112 NNNNNNNNNNN cTxuxxcTxuxcTxuTxu . (5.22)

The equation (5.21) is just the same type with the Black-Scholes equation for 1Nu but the

inhomogenous term is

xxg N ,min)( 2


20

and the terminal condition (5.22) is not a standard binary type. To make it into a standard

binary type, we now consider the equation 111 ),( NNN cTxux . If (5.19) holds, then the

equation x 111 ),( NNN cTxu has unique root 01 NK and we have

111 ),( NNN cTxux 1 NKx . Note that 11 0 NN Kc 0 . If (5.19) does not hold,

then there exists a x1 such that 1),( 111 NNx Txu and we can know that the equation

111 ),( NNN cTxux can have one root 1NK or three roots )2(),1( 11 NN KK and )3(1NK such

that )2()1( 11 NN KK )3(1 NK .(See the figure 3.)

Figure 3. roots of 111 ),( NNN cTxux .

The figure 3 allows us to provide a necessary and sufficient condition for the equation

111 ),( NNN cTxux to have unique root.

Lemma 1. The equation 111 ),( NNN cTxux has unique root if and only if one of

the three conditions satisfies.

(i) If 1x is the maximum root of 1),( 111 NNx Txu , then

),( 11111 NNN Txuxc . (5.23)

(ii) If 0x is the minimum root of 1),( 101 NNx Txu , then

),( 10101 NNN Txuxc . (5.24)

Remark 8. The lemma 1 provides a theoretical necessary and sufficient condition for the

equation 111 ),( NNN cTxux to have unique root. For example (5.24) seems not

appropriate in financial meaning. But (5.23) seems to provide an appropriate (in financial

meaning) upper bound for the coupon.

If the 111 ),( NNN cTxux has one root 1NK then the terminal value (5.22) can be

written as follows:

).(}{1}{1]),([),( 1111112 xfKxxKxcTxuTxu NNNNNNN (5.25)

If the equation 111 ),( NNN cTxux have three roots )2()1( 11 NN KK )3(1NK , then the

terminal value (5.22) can be written as follows (See the figure 4.):

).()}]3()2({1)}1({1[

)}]3({1)}2()1({1[]),([),(

111

11111112

xfKxKKxx

KxKxKcTxuTxu

NNN

NNNNNNNN

(5.26)

The problem (5.21) with (5.25) or (5.26) is just same type with the problem for 1Nu . Its


21

terminal value f(x) is just the binary type. So we can get the solution-representation of the

problem by the method of [17].

Figure 4. )(),( 12 xfTxu NN

Remark 9. As shown in (5.25) and (5.26), 1NK or )2(),1( 11 NN KK and )3(1NK give the

default barrier at time TN1. In (5.26), )2()1( 11 NN KxK is a survival interval but

)2(1NK )3(1 NKx is a default interval for x = V/Z . It seems not compatible with

financial reality. Thus we will not consider (5.26) in what follows.

Now we postpone finding the representation of ),(2 txuN for a while and first estimate

2Nu and 2 Nxu . From (5.25) and (5.17), we have )( fm 0, )1()( 11 NN T

NT

eKefM

1 Nc , f is increasing and f(x) x. Using again the corollary 1 and corollary 3 of the

theorem 1, ii) of the theorem 4, ),( 22 NN Txu is bounded, increasing and ),(0 22 NN Txu

.x (Note that when f(x) is given by (5.26), f(x) is not increasing and thus ),( 22 NN Txu is not

increasing either but other all conclusions on ),( 22 NN Txu still hold.)

On the other hand, from (5.25), )(xf has the jump 11 )1()( NN KKf at KN1 and from

(5.18), 0)( fm , )( fM . Remind 0)( gm and )(gM . Thus using iv) of the

theorem 4, we have

)27.5().0(,

),(2

1)(),(0

)0(,

),(2

)1()1()(),(0

122

22

222

)()()(

22

2

22

b

TT

fMTxu

b

TT

ee

befMTxu

NN

NNx

NN

TbTbTb

NNx

N

NN

If =1, then 1)( fM , )(gM from (5.25) and (5.19). Thus from (5.27) we have

.1),(0 22 NNx Txu (5.28)

If < 1 and the condition (5.14) holds, then from (5.20) we have 1)( NdfM and thus from

(5.27) we have

1),(0 222 NNNx dTxu . (5.29)

(Again note that (5.14) is a sufficient condition for (5.28) to be true.) Using the theorem 5, we

can obtain ),( 222

NNx Txu and finite number of convex or concave intervals of ),( 22 NN Txu

as the case when i=N1 but here we will omit the detail.


22

Now let find the representation of ),(2 txuN (the solution to the problem (5.21) and

(5.25)). From (5.16), we can rewrite (5.25) as

}.{1}{1

}]{1);,(}{1);,([

}{1);,(}{1);,(),(

111

11/11/2

111112

122

NNN

T

TNNNNN

NNNKNNNKNNN

KxxKxc

dKxTxAKxTxB

KxTTxAKxTTxBcTxu

N

NNN

NN

Using the theorem 1, we have:

.})/{1}/{1(

),(

2

),(

1 2

22

12

2

12

1

1

0

2222

),(2

),(2

1)(ln

)(

0

12

12

),(2

),(2

1)(ln

)(2

),(2

),(

N

N

NN

N

T

t

NNN

t

ttbz

x

t

NN

N

Tt

TttTbz

x

tTN

ddzzzze

e

dzTzue

etxu

tz

Ttz

From the definition of second order binaries [4, 16, 17], the lemma 2 of [17] on the integrals

of binaries, we have

.0,,);,();,(

);,();,(

),;,(),;,(

),;,(),;,(),(

12//2

111

1/1/1

112

1

22

11

11111

11

xTtTdtxAtxB

TtxATtxBc

dTtxATtxB

TTtxATTtxBctxu

NN

T

tN

NKNKN

T

TNKNKN

NNKKNNKKNN

N

NN

NN

N

NNNNN

NNNN

(5.30)

Here ),;,( 2121TTtxB aa

and ),;,( 2121

TTtxAaa

are the prices of second order bond and asset

binary options with the coefficients )(,)(,)( tbtqtr , two expiry dates T1 and T2, two

exercise prices a1 and a2, which are provided by the theorem 1 of [17, at page 5~6]. Returning

to original variables using (5.15), we obtain the representation of BN-2 :

),);,(/();,(),,( 22 tTtrZVuTtrZtrVB NN , )0,( 12 xTtT NN

Now let consider the case when i=N3. In (5.5) and (5.7), we use (5.15) with i=N3 and

the notation (5.8) to get

,0,,0,min)(2

12333

3

2

32

223

xTtTxux

ubx

x

uxt

t

uNNNN

NNN (5.31)

.0},),({1

}),({1]),([),(

222

22222223

xcTxuxx

cTxuxcTxuTxu

NNN

NNNNNNNN

(5.32)

The equation for 3Nu is just the same type of Black-Scholes equation for 1Nu or 2Nu but the

terminal condition (5.32) is not a standard binary type. To make it into a standard binary type

as the above, we consider the equation ),( 22 NN Txux 2 Nc . If (5.28) holds, then the


23

equation has unique root 2NK 0 and we have x 222 ),( NNN cTxu 2 NKx . Even

though (5.28) does not hold, from the above considered properties of ),( 22 NN Txu the set S

= {x > 0: 1),( 22 NNx Txu } is a finite set and we can prove the following lemma:

Lemma 2. The equation ),( 22 NN Txux 2 Nc has unique root if and only if one of

the two conditions satisfies.

(i) 12212 ),( xcTxu NNN for every .1 Sx

(ii) 12212 ),( xcTxu NNN for every .1 Sx

Similarly as noted in the remark 8, in this lemma, (ii) seems not appropriate in financial

meaning. But (i) seems to provide an appropriate (in financial meaning) upper bound for the coupon.

Thus, if 2Nc is in an appropriate range, then the equation has unique root 2NK 0 and

we have x 222 ),( NNN cTxu 2 NKx , too. Therefore the terminal value of the

problem (5.32) can be written as follows:

}.{1}{1]),([),( 2222223 NNNNNNN KxxKxcTxuTxu (5.33)

This gives the default barrier 2NK at time TN2.

The problem (5.31) and (5.33) is just the same problem for the inhomogenous

Black-Scholes equation with binary type of terminal value. Thus using the theorem 1 in the

section 2, the definition of third order binaries [16, 17] and the lemma 2 of [17] on the

integrals of binaries, we have

).0,(,)];,();,([

)],;,(),;,([

)],,;,(),,;,([

);,();,(

),;,(),;,(

),,;,(),,;,(),(

23//3

2/2/2

12/12/1

222

12121

12123

2

33

1

2

2222

1

112112

22

1212

1212

xTtTdtxAtxB

dTtxATtxB

dTTtxATTtxB

TtxATtxBc

TTtxATTtxBc

TTTtxATTTtxBctxu

NN

T

t

N

T

T

NKNKN

T

T

NNKKNNKKN

NKNKN

NNKKNNKKN

NNNKKKNNNKKKNN

N

NN

N

N

NNNN

N

N

NNNNNN

NN

NNNN

NNNNNN

Returning to original variables using (5.15), we obtain the representation of BN3 :

),);,(/();,(),,( 33 tTtrZVuTtrZtrVB NN , )0,( 23 xTtT NN .

Similarly (by induction), if we assume that we have already ),,(1 trVBi ( 21 ii TtT ),

we can prove the following lemma:

Lemma 3. Under the assumption of the theorem 6, if all the equations ),( 11 ii Txux

)2,,0(1 Nici have unique roots 01 iK , then 111 ),( iii cTxux 1 iKx and

furthermore the solution of (5.5) and (5.7) is provided as follows:

1),),;,(/();,(),,( iiNiNi TtTtTtrZVuTtrZtrVB .


24

Here

.0,,);,();,(

),,,;,(),,,;,(

),,,;,(),,;,(),(

1//

1

1

1

1

1

11111

1

1

1

1

1111

xTtTdtxAtxB

dTTtxATTtxB

TTTtxATTtxBctxu

ii

T

t

i

miKK

N

im

T

T

miKK

m

N

im

mmiKKKmiKKmi

i

ii

mmi

m

m

mmi

mmimi

),,,;,(),,,;,( 11111111

mmKKKmKK

TTTtxATTtxBmmm

are the price of m-th order bond and

asset binaries with risk free rate , dividend rate +b and volatility )(t (the theorem 1 of

[17, pages 5~6]).

Thus the theorem 6 has been proved. (QED)

Let denote the k-th coupon rate by ck = Ck/F (k = 1,…, N). Then we have the following

representation of the initial price of the our defaultable discrete coupon bond in terms of the

debt, coupon rates, default recovery rate, default intensity, initial prices of the default free

zero coupon bond and firm value and multi-variate normal distribution functions.

Corollary 1. Under the assumption of theorem 1, the initial price of the bond can be

represented as follows:

)34.5(.))](~

);,(~

,,,()1(

))(~

);,(~

,,,([

);,,,(

);,,();,,,()1(

),;,;,,,(

111101

1

0

1111)(

0

1

0

1111)(

0

2

0

11111110

00100

1

1

1

dAdddNeZcF

AdddNeV

AdddNeV

AddNecAdddNecFZ

VZccFBB

mmmmkNk

N

m

T

T

mmmmb

N

m

mmmmTb

N

m

mmmT

mNNNNT

N

N

m

m

m

mN

Here );,,( 1 AaaN mm is the cumulative distribution function of m-variate normal distribution with

zero mean vector and a covariance matrix 1A (the theorem 1 of [17]),

;)(2

1

)1(ln)(

;1,,1,)(2

1ln)(

,)2

1exp(det

)2(

1);,,(

0

2

0

02/1

0

2

0

2

0

02/1

0

2

1

1

NN

ii

m

T

NN

T

N

T

ii

T

i

a a

mmm

dttbTcFZ

Vdttd

NidttbTKZ

Vdttd

dyAyyAAaaN

,


25

.,,1;,)(2

1

)1(ln)(),(

~1

0

2

10

02/1

0

2 NiTTdttbcFZ

Vdttd ii

kNk

i

The matrix mjijim rA 1,

1 )()( is given by

jir ),,1,(,,)()(0

2

0

2 mjijirrdttdtt jiij

TT ji

,

mjijim rA 1,

1 ))(~())(~

( is the matrix whose m-th row and column are given by

)(~ mir )1,,1(),(~)(~,)()(0

2

0

2 mimirrdttdtt miim

Ti

and other elements coincide with those of 1)( mA . The matrices m

jijim rA 1,1 )()(

and 1))(~

( mA

mjijir 1,))(~(

have such m-th rows and columns that

mimi rr

, imim rr

; )(~)(~ mimi rr

, )(~)(~ imim rr

, )1,,1( mimi

and other elements coincide with those of 1)( mA and 1))(

~( mA , respectively .

Remark 10. If b = 0 and = 0, then our pricing formula (5.34) nearly coincides with the

formula (5) of [1, at page 752] and the only difference comes from the fact that k-th coupon

is provided as a discounted value of the maturity-value in our model. If b = 0, Ck = 0(i =

1,…, N) and = 0, then the formula (5.34) coincides with the formula (5) of [1, at page 752]

when Ck = 0 (i = 1,…, N) which is a known formula for defaultable zero coupon bond that

generalizes Merton (1974) [15].

References

[1] Agliardi, R., A Comprehensive Structural Model for Defaultable Fixed-income Bonds,

Quantitative Finance, 11:5, 2011, 749-762, DOI:10.1080/14697680903222451

[2] Bergman, Y.Z., Grundy B.D. and Wiener Z., General Properties of Option Prices, The Journal of

Finance, 51, 1996, 1573 – 1610, DOI:10.1111/j.1540-6261.1996.tb05218.x

[3] Black, F. and Scholes, M., The pricing of options and corporate liabilities. J. polit. Econ., 81,

1973, 637-654.

[4] Buchen, P.; The Pricing of dual-expiry exotics, Quantitative Finance, 4 (2004) 101-108, DOI:

10.1088/1469-7688/4/1/009

[5] Cox, J.C. and Ross S.A., A Survey of Some New Result in Financial Option Pricing Theory, The

Journal of Finance, 31, 1976, 383 – 402, DOI: 10.1111/j.1540-6261.1976.tb01893.x

[6] El Karoui, N., Jeanblance-Picqué, M. and Shreve, S. E., Robustness of the Black and Scholes

formula, Mathematical Finance, 8, 1998, 93-126, MR1609962, DOI:10.1111/1467-9965.00047

http://dx.doi.org/10.1080/14697680903222451

http://dx.doi.org/10.1111/j.1540-6261.1996.tb05218.x

http://dx.doi.org/10.1088/1469-7688/4/1/009

http://dx.doi.org/10.1111/j.1540-6261.1976.tb01893.x

http://www.ams.org/mathscinet-getitem?mr=1609962

http://dx.doi.org/10.1111/1467-9965.00047


26

[7] Ekström, E. and J. Tysk, Properties of option prices in models with jumps, arXiv preprint: pp.1-14,

2005, arXiv:math/0509232v2 [math.AP]

[8] Hobson, D., Volatility misspecification, option pricing and superreplication via coupling, The

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[9] Jagannathan, R., Call Options and the Risk of Underlying Securities, Journal of Financial

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Probability, Vol. 13, no.3, 2003, 890 – 913, DOI:10.1214/aoap/1060202830

[12] Janson, S. and Tysk, J., Preservation of convexity of solutions to parabolic equations, J.

Differential Equations 206 (2004) 182-226. DOI: 10.1016/j.jde.2004.07.016

[13] Kwok,Y.K., Mathematical models of Financial Derivatives, Springer-verlag, Berlin, 1999

[14] Merton, R.C., Theory of rational option pricing, The Bell Journal of Economics and

Management Science, 4, 1973, 141-183. Cross-ref

[15] Merton, R.C.; On the pricing of corporate debt: the risk structure of interest rates, Journal of

Finance, 29, 1974, pp. 449 - 470

[16] O, Hyong-chol and Kim, Mun-chol; Higher Order Binary Options and Multiple Expiry Exotics,

Electronic Journal of Mathematical Analysis and Applications, Vol. 1(2) July 2013, pp. 247-259,

Cross-ref, arXiv preprint: arXiv:1302.3319v8[q-fin.PR]

[17] O, Hyong-chol; Kim, Yong-Gon and Kim, Dong-Hyok; Higher Order Binaries with Time

Dependent Coefficients and Two Factors - Model for Defaultable Bond with Discrete Default

Information, arXiv preprint, pp 1-20, arXiv: 1305.6868v4[q-fin.PR]

[18] O, Hyong-chol, Kim, Dong-Hyok; Ri, Song-Hun and Jo, Jong-Jun: Integrals of Higher Binary

Options and Defaultable Bonds with Discrete Default Information, arXiv preprint, pp 1-27,

arXiv: 1305.6988v4[q-fin.PR]

[19] Wilmott, P., Derivatives: the Theory and Practice of Financial Engineering, John Wiley

& Sons Inc., 1998

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http://dx.doi.org/10.1214%2faoap%2f1060202830

http://dx.doi.org/10.1016/j.jde.2004.07.016

http://www.jstor.org/discover/10.2307/3003144?uid=3737800&uid=2134&uid=2&uid=70&uid=4&sid=21102526440141

http://ejmaa.6te.net/Vol2_Papers/Volume2_Paper12_Abstract_New.htm

http://arxiv.org/abs/1302.3319v8



general properties of solutions to inhomogeneous black

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