imaman Credit Linked note pricing

8/11/2019 imaman Credit Linked note pricing

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IMA Journal of Management Mathematics(2014) Page 1 of 20

doi:10.1093/imaman/dpt028

Explicit formulas for pricing credit-linked notes with counterparty risk under

reduced-form framework

Lei Ge

Center for Financial Engineering, Soochow University,

Suzhou 215006, P.R. China

[email protected]

and

Xiaosong Qian and Xingye YueSchool of Mathematical Sciences and Center for Financial Engineering, Soochow University, Suzhou

215006, P.R. China

Corresponding author: [email protected] [email protected]

[Received on 22 April 2013; accepted on 14 December 2013]

A credit-linked note (CLN) is a type of credit derivative, constructed with a bond and an embedded

credit default swap, which allows the issuer to transfer a specific credit risk to credit investors. In this

paper, we model CLNs with and without counterparty risk in the reduced-form framework. For CLNs

with counterparty risk, we consider two different scenarios, i.e. the issuer of CLNs and reference assets

have either positive correlation or negative correlation. Assuming the interest rate follows the Cox

IngersollRoss (CIR) model(Coxet al.,1985) and the default events mainly depend on the interest rate,

we model the two different correlations. Explicit formulas for value functions are obtained through a

partial differential equation approach. In addition, counterparty valuation adjustment and the dependence

on related parameters are also investigated.

Keywords: credit-linked note; reduced-form; CVA.

1. Introduction

Since the financial crisis in 2008, investors have realized that it is possible even for the largest banks and

institutes to go bankrupt. The wide-spread credit risk has profound influence on quantitative methods

in different markets, such as fixed income market (Patrizia et al., 2012). But credit market and credit

derivatives are subject to more public scrutiny. Various credit instruments are designed to separate or

transfer credit risk of the underlying assets, such as credit default swap (CDS), credit-linked note (CLN),

collateralized debt obligation (CDO) and total return swap.

A CLN is a contract ensuring protection against certain default. A standard CLN involves three

entities: a protection buyer (CLN issuer), a protection seller (CLN investor) and a reference firm. More

precisely, a standard CLN is structured as follows (see Fig. 1): the investor of the CLN buys the note

from the issuer at the beginning and pays the face value. If there is no credit event before maturity,

the noteholder receives interest payments which are usually higher than the payment in the case of

comparable bonds. Upon maturity, the principal of the CLN is repaid at the nominal value. In the case

of a credit event, the CLN is repaid partially.

To model default and value credit derivatives, there are two main approaches: structuralmodel and

reduced-formmodel. The structural model, first introduced by Merton(1976) and further developed by

cThe authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Management Mathematics Advance Access published January 20, 2014


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Fig. 1. The cash flow of standard CLN.

Black & Cox(1976), uses the assets and liabilities of the firm as the underlying of the credit derivatives

and the default event is defined as the asset value of reference firm touching some deterministic or

stochastic default barrier (see, e.g. Hui et al., 2007). Recently, some other theories like optimal stochastic

control are incorporated into the study of structural models (see, e.g. Liang & Jiang, 2012). Hui &

Lo(2002) used structural model to describe the default premiums of CLNs and got the closed-form

solutions as functions of firm values and the short-term interest rate.Wu(2010,2011) gave the default

premium of basket CLNs using the copula method to denote the correlation of reference assets and

discussed effects of the issuer default risk on default premium. Although the structural models canbe understood intuitively, they have a limited usage in practice because of the incompleteness of the

publicly available information in regard to the reference firms.

The reduced-form model was first introduced by Pye (1974) and Litterman & Iben (1991) and further

developed byJarrow & Turnbull(1995) andMadan & Unal(2000). The reduced-form models describe

default as an exogenous stochastic event. Hazard rate, also known as default intensity, is usually used

to describe the default probability of risky assets. In practice, we often calibrate the intensity according

to market data, such as CDS quotes or risky bond prices. So, under the reduced-model, the valuation of

credit derivatives is compatible with the market data. Wang & Chang(2007) have obtained an explicit

formula for the value of CLNs using the reduced-form model. However,Wang & Chang(2007) consid-

ered only the special situation where the default intensity had a positive linear relationship with short

rate, which is not always the case in practice. The term zero coupon bond in the explicit formula also

made the calibration much more complicated. In their numerical implementation, the authors dealt with

this problem by assuming a special form for zero coupon bond.

In this paper, we use the partial differential equation (PDE) approach to value CLNs with fixed

coupon rates under reduced-form model. We study the valuation of CLNs with and without counter-

party risk. For the CLNs with counterparty risk, the default of issuer and reference assets are usually

positively correlated, and the default risk mainly comes from wrong way risk. Wrong way risk is defined

by the International Swaps and Derivatives Association as the risk that occurs when exposure to a coun-

terparty is adversely correlated with the credit quality of that counterparty. By assuming that both of

their default intensities depend on the short rate that follows CoxIngersollRoss (CIR) process, we

model the positive correlation between the issuer and reference assets. We model the intensity into two

different scenarios: they are both in proportion to the interest rate and they are in inverse proportion to

the interest rate. By analysing cash flows of CLNs, we obtain the pricing PDE for CLNs and solve it by

introducing a transformation. Analytic solutions are obtained. We also study the counterparty valuation

adjustment (CVA) of CLNs. CVA is defined as the difference between the value of the contract without

counterparty credit risk and the value with counterparty credit risk. That is, the market value of counter-

party credit risk. The ignorance of counterparty risk made people pay a high price in the financial crisis

in 2008. Using the model, we also investigate the dependence of the CLN value on related parameters.

The rest of the paper is organized as follows. In Section 2, we analyse the cash flows of CLNs with

and without counterparty risk in the reduced-form framework. In Section 3, we consider the default


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Before the default time1of the reference firm, the default time 2 of the issuer and maturityT, theinvestor obtains premium with ratekcontinuously.

If the reference firm defaults before2 andT, the investor gets a constant recovery payoffR1 fromthe issuer.

If the issuer defaults before1 andT, the investor gets a recovery payoff which is proportional tothe market value of the CLN contract. We assume a constant recovery rateR2here.

If there is no default during the life of CLN, the issuer pays the face value 1 back to the investor.On the base of the former analysis, we have the following definition.

Definition2.2 The value of a CLN with counterparty risk at timetbefore maturityT is

U(t) =E T

t

kI1>sI2>se s

t rd ds+It1TI2>1R1e

1t

rd

+ It2R2U(2) e2

t rd +I1>TI2>Te

Tt

rdGt

. (2.2)

Remark2.3 Usually, people tend to describe the recovery value as R2V(2) at the default time 2 of

counterpart, where V(t) is the value of contract without counterparty risk. However, this traditional

method which does not take counterparty risk into account leads underestimation of risk. Here, we use

R2U(2)to describe the recovery value at counterpart default time 2, which is similar to the CDS con-

tracts with counterparty risk studied byCrpeyet al.(2010) andHuet al.(2012). While it is impossible

to get any explicit solutions for CDS situations, we can obtain the explicit pricing formulas for CLNs

with counterparty risk here, due to the special structure of CLN contracts.

Then we have the CVA of CLNs with counterparty risk at timet, i.e.

CVA(t) = V(t)U(t), 0 t T. (2.3)

2.2 Reduced-form model

In the section, we consider the valuation of CLNs under the framework of reduced-form models. In this

framework, the default probability of reference firm and CLNs issuer can be described by hazard rates

1t and2t, which are non-negative Ft-progressively measurable stochastic processes. The default times

1 and2 can be constructed through the canonical method discussed inBielecki & Rutkowski(2001),

satisfying P(1> t|Ft) = e t

0 1d and P(2> t|Ft) = e t

0 2d.First, we introduce two lemmas, which can be found inBielecki & Rutkowski(2001).

Lemma2.4 For any integral random variableYG,

E[I1>tY|G1t] =I1>tet

01dE[I1>tY|Ft]. (2.4)


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EXPLICIT FORMULAS FOR PRICING CLNS WITH COUNTERPARTY RISK 5 of 20

Lemma2.5 If{Zt}t0is a bounded, Ft-predictable process, then for anyt< T,

E[IttE T

t

Zuue u

t 1d du |Ft

. (2.5)

Using Lemmas 2.4 and 2.5, we obtain the following result.

Proposition 2.6 The value of a CLN without counterparty risk at time t before maturity T can be

represented as

V(t) =I1>tE T

t

(k+R11s ) es

t(r+1) d ds+e

Tt

(r+1) dFt

. (2.6)

To deal with the CLN with counterparty risk, we assume that 1 and 2are conditional independent

with respect to the filtration F, i.e. for anyt> 0,

E[I1>tI2>t|Ft] =E[I1>t|Ft]E[I2>t|Ft] = et

0(1+2) d. (2.7)

Remark 2.7 The conditional independence assumption means that the probability of the reference

firm and issuer of the CLN is affected by some common exogenous variables. These variables are

usually assumed to be some macro factors in business cycle, such as interest rates. Conditionally, on the

evolution of the variables, defaults are independent of each other. This assumption is commonly used in

valuation of credit derivatives such as defaultable bonds, CDO and CDS (see Duffie & Garleanu,2001;

Herbertssonet al.,2011;Jarrowet al.,2005).

We introduce the following two results based on the conditional independence assumption, whichcould also be found inBielecki & Rutkowski(2001).

Lemma2.8 For any integral random variableYG,

E[I1>tI2>tY|Gt] =I1>tI2>tet

0(1+2) dE[I1>tI2>tY|Ft]. (2.8)

Lemma2.9 If{Z1t}t0,{Z2t}t0are bounded, Ft-adapted processes, then for any t< T,

E[I1>tI2>t(Z11

I1tE T

t

(Z1u 1u+Z2u 2u) e

ut (1+2) d

Ft

. (2.9)

Using Lemmas 2.8 and 2.9, we obtain the following result for the CLN with counterpartyrisk.

Proposition2.10 The value of a CLN with counterparty risk at timet( T)can be represented as

U(t) =I1>tI2>tE T

t

(k+R11s+R2U(s)2s ) es

t(r+1+2) d ds+e

Tt

(r+1+2) dFt

. (2.10)


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3. PDE of CLNs

In this section, we derive the pricing PDE for CLNs. We assume that spot interest rate rtsatisfies CIR

process

drt=

(

rt) dt

+

rtdWt, (3.1)

where , , are positive constant parameters and satisfy the Feller condition

2 > 2. (3.2)

LetF be the filtration generated by the process rt and the intensity processes 1t,

2tare both smooth

functions ofrt. By the strong Markov property ofrt, we obtain the following probabilistic representation

ofV(t)and U(t)in (2.6) and(2.10), on{1> t}and{1> t, 2> t}, respectively,

V(t) =E T

t

(k+R11s ) es

t(r+1) d ds+e

Tt

(r+1) d rt

, (3.3)

U(t) =E T

t(k+R11s+R2U(s)2s ) e

s

t(r+1

+2

) d ds+e T

t (r+1

+2

) d rt . (3.4)

We can denoteV(t) = v(t, rt)and U(t) = u(t, rt), and call them the pre-default value of CLNs.Usually, the CLN issuer may hold some risky assets of the reference firm and could suffer some

loss when the reference firm defaults. So, the default probabilities of a CLN issuer and reference firm

could increase simultaneously in this scenario. It is also possible that a CLN issuer wants to speculate

in some reference credit asset and he could get some profit from the default of the reference firm. Here,

we suppose that the default intensities of the reference firm and the CLN issuer both depend on the

observable interest rate rt, but in two different types, i.e. positive correlation or negative correlation

between reference asset and CLN issuer. Let us consider the simplest cases as follows:

Positive correlation: 1t= 1rt+1, 2t= 2rt+2 (3.5)or

1t=1

rt+1, 2t=

2

rt+2. (3.6)

Negative correlation:1t= 1rt+1, 2t=

2

rt+2 (3.7)

or

1t=1

rt+1, 2t= 2rt+2, (3.8)

where1, 2, 1, 2, 1, 2, 1, 2are all positive constants.

Remark3.1 When they are positively correlated withrt, the default intensities satisfy the CIR process.

When they are negatively correlated withrt, the default intensities satisfy so-called inverse CIR (ICIR)

process, which was first proposed byAhn & Gao(1999), and intensively studied byHurd & Kuznetsov

(2008).Qianet al.(2012) used the ICIR process to describe the prepayment rate and obtained explicit

pricing formulas for Mortgage-backed securities. Based on the ideas presented in Hurd & Kuznetsov

(2008) andQianet al.(2012), we use the ICIR model to study the pricing of CLNs.


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3.1 Positive correlation

3.1.1 CIR model of hazard rate. When the hazard rates satisfy the CIR model in (3.5), using the

FeynmanKac formula, we can easily obtain from (3.3) and(3.4) that the CLN pre-default valuesv(t, r)

andu(t, r)satisfy the following PDEs:

v

t+ 1

22r

2v

r2+ (r) v

r[(1+1)r+1]v

+ k+R1(1r+1) = 0 (0 < r< , 0 t< T),v(T, r) = 1 (0 < r< ),

(3.9)

u

t+ 1

22r

2u

r2+ (r) u

r[(1+1+(1R2)2)r+1

+ (1R2)2]u+k+R1(1r+1) = 0 (0 < r< , 0 t< T),u(T, r) = 1 (0 < r< ).

(3.10)

To solve equation (3.9), we first introduce the following transformation:

P(t, r) = ea1ra2tvt

, (3.11)

where

a1=

2 +22(1+1)

2 , a2= 1 a1. (3.12)

Then we have

v(t, r) = 1+ T

t

ea1r+a2sP(s, r) ds.

An easy calculation will show that the function P(t, r)satisfies the following PDE:

P

t+ 1

22r

2P

r2+ (r) P

r= 0 (0 < r< , 0 t< T),

P(T, r) = ea1ra2T[((R11)11)r+(R11)1+k] (0 < r< ),(3.13)

where

= a12, =

a12. (3.14)

Remark3.2 It is easy to check

P(T, r) = ea1ra2tvt

t=T

= ea1ra2t

(r) vr

+ 12

2r2v

r2((1+1)r+1)v+k+R1(1r+1)

t=T

.

Noticingv(T, r) = 1 in(3.9), we can obtain the terminal condition in (3.13).To solve equation (3.13), we introduce the following lemma (seeFeller,1951).
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Lemma3.3 The fundamental solution to the following equation:

P

t+ 1

22r

2P

r2+ (r) P

r= 0 (0 < r< , 0 t< T),

P(T, r;y) = (ry) (0 < r< )(3.15)

isG(t, r;y), where ()is Dirac delta function and the function G is defined by

G(t, r;y) = c epq

q

p

z/2Iz(2(pq)

1/2), (3.16)

with

c = 2

2(1e (Tt)) , z =2

2 1,

p = cre (Tt), q = cy,(3.17)

andIz()is the modified Bessel function of the first kind of order z,

Iz(2(pq)1/2) = (pq)z/2

k=0

(pq)k

k! (z+k+1) . (3.18)

Then we can deduce the solution of (3.13)

P(t, r) =

0

[((R11)11)y+(R1)1+k] ea1ya2TG(t, r;y) dy

= ((R11)11) ea2T

0

y ea1yG(t, r;y) dy

+((R11)1+k) ea2T

0

ea1yG(t, r;y) dy. (3.19)

It is easy to verify that

0

y ea1yG(t, r;y) dy =

0

y ea1yc epq

q

p

z/2Iz(2(pq)

1/2) dy

= ep

c

0

qz+1 e(a1/c+1)q

k=0

(pq)k

k! (z+k+1) dq

=ep

c

k=0

(pq)k

k! (z+k+1)

0qz+1 e(a1/c+1)q dq

= ep

c

c

a1+c

z+2 k=0

(z+k+2)k! (z+k+1)

cp

a1+c

k

= cz+1(a1+c)z3(cp+(z+1)(a1+c)) ea1p/(a1+c), (3.20)

where (x) =

0 tx1 et dtis Gamma function.


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Similarly, we have

0

ea1yG(t, r;y) dy =

0

ea1yc epq

q

p

z/2Iz(2(pq)

1/2) dy

=

c

a1+c

z+1

ea1p/(a1+c). (3.21)

Substituting (3.20) and (3.21) into (3.19), we obtain the following explicit expression for the

functionP(t, r):

P(t, r) =

((R11)11)(cp+(z+1)(a1+c))(a1+c)2

+(R11)1+k

c

a1

+c

z+1e(a1p/(a1+c))a2T. (3.22)

Then we can conclude by the following theorem.

Theorem 3.4 The pre-default value of the CLN without counterparty risk at time t( T) can be

expressed as

v(t, r) = 1+ T

t

ea1r+a2sP(s, r) ds, (3.23)

wherea1, a2 and P(t, r)are defined in (3.12) and(3.22).

A similar calculation leads to the following result.

Theorem 3.5 The pre-default value of CLN with counterparty risk and default intensities satisfying(3.5) at timet( T)can be expressed as

u(t, r) = 1+ T

t

eb1r+b2sQ(s, r) ds, (3.24)

whereQ(t, r)is defined by

Q(t, r) =

((R11)1+(R21)21)(cp+(z+1)(b1+c))(b1+c)2

+(R11)1

+(R21)2+k cb1+cz+1

e(b

1p/(b

1+c))

b

2T

, (3.25)

with

b1=

2 +22(1+1+(1R2)2)

2 , b2= 1+(1R2)2 a1, (3.26)

andc,p,zdefined in (3.17).


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3.1.2 ICIR model of hazard rate. When the hazard rates follow the ICIR model in (3.6), using the

FeynmanKac formula, we can obtain that the CLN pre-default values v(t, r) and u(t, r) satisfy the

following PDEs:

v t

+ 12

2r

2

vr2

+ (r) vr

r+ 1r

+1 v+k (0 < r< , 0 t< T),+R1

1

r+1

= 0

v(T, r) = 1 (0 < r< ),

(3.27)

u

t+ 1

22r

2u

r2+ (r) u

rr+ 1

r+(1R2)

2

r+1+(1R2)2

u+k+R1

1

r+1

= 0 (0 < r< , 0 t< T),

u(T, r) = 1 (0 < r< ).

(3.28)

To solve equation (3.27), we first introduce the following transformation:

P(t, r) = ra3 ea1ra2tvt

, (3.29)

where

a1=

2 +222

, a2= b1a1a1a32 +1,

a3=(1/2)2

+(

(1/2)2)2

+212

2 .

(3.30)

Then we haveu(t, r) = 1+T

t ra3 ea1r+a2sP(s, r) ds, andP(t, r)satisfies the following PDE:

P

t+ 1

22r

2P

r2+ (r) P

r= 0 (0 < r< , 0 t< T),

P(T, r) = ra3 ea1ra2T

(R11)1

rr+(R11)1+k

(0 < r< ),

(3.31)

where

= a12

, =2b1

+

a12 . (3.32)Then we can deduce the solution of (3.31) by using Lemma 3.3,

P(t, r) =

0

(R11)

1

yy+(R11)1+k

ya3 ea1ya2TG(t, r;y) dy

= (R11)1ea2T

0

ya31 ea1yG(t, r;y) dy


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+((R11)1+k) ea2T

0


ea2T

0

ya3+1 ea1yG(t, r;y) dy. (3.33)

It is easy to verify that

0


=

0

ya31 ea1yc epq

q

p

z/2Iz(2(pq)

1/2) dy

=

0

ca3+1qa31 e(a1/c)q epq

q

p

z/2(pq)z/2

k=0

(pq)k

k! (z+k+1) dq

= ca3+1 ep

k=0

pk

k! (z+k+1)

0

qz+ka31 e(a1/c+1)q dq

= ca3+1 ep

c

a1+c

za3 k=0

(z+ka3)k! (z+k+1)

cp

a1+c

k

= ca3+1 ep

c

a1+c

za3 (za3) (z+1) M

za3,z+1,

cp

a1+c

, (3.34)

where M(, ,z) =k=0(()k/k!()k)zk is the confluent hypergeometric function with ()k definedby()k

=(

+1)

(

+k

1)for k> 0,()0

=1.

Similarly, we have

0


= ca3 ep

c

a1+c

za3+1 (za3+1) (z+1) M

za3+1,z+1,

cp

a1+c

(3.35)

and

0

ya3+1 ea1yG(t, r;y) dy

= ca31 ep

c

a1+c

za3+2 (za3+2) (z+1) M

za3+2,z+1,

cp

a1+c

. (3.36)

Substituting(3.343.36) into(3.33), we have

P(t, r) = (R11)1ca3+1 ea2Tp

c

a1+c

za3M

za3,z+1,

cp

a1+c

(za3) (z+1)


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ca31 ea2Tp

c

a1+c

za3+2M

za3+2,z+1,

cp

a1+c

(za3+2)

(z+1)

+((R1

1)1

+k)ca3 ea2Tp

c

a1+cza3+1

Mza3

+1,z

+1,

cp

a1+c (za3+1)

(z+1) . (3.37)

Then we conclude by the following theorem.

Theorem 3.6 The pre-default value of the CLN without counterparty risk at time t( T) can be

expressed as

v(t, r) = 1+ T

t

ra3 ea1r+a2sP(s, r) ds, (3.38)

whereP(t, r)is defined by(3.37), witha1, a2, a3 defined by(3.30).

A similar calculation leads to the following result.

Theorem 3.7 The pre-default value of the CLN with counterparty risk and default intensities satisfying

(3.6) at timet( T)can be expressed as

u(t, r) = 1+ T

t

rb3 eb1r+b2sQ(s, r) ds, (3.39)


Q(t, r) = cb31 eb2Tp

c

b1+

c

zb3+2M

zb3+2,z+1,

cp

b1+

c

(zb3+2)

(z+

1)

+((R11)1+(R21)2)cb3+1 eb2Tp

c

b1+c

zb3 (zb3) (z+1)

M

zb3,z+1,cp

b1+c

+((R11)1+(R21)2+k)cb3 eb2Tp

c

b1+c

zb3+1 (zb3+1) (z+1)

M

zb3+1,z+1,cp

b1+c

, (3.40)

with

b1=

2 +222

, b2= b3 b1b1b32 +1+(1R2)2,

b3=(1/2)2 +

((1/2)2)2 +2(1+(1R2)2)2

2 ,

(3.41)

and , , c,p,zare defined in (3.17) and (3.32).


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3.2 Negative correlation

In this section, we give only the results for CLN values with counterparty risk in the negative correlation

case. The proof procedures are similar to previous theorems.

Theorem 3.8 The pre-default value of the CLN with counterparty risk and default intensities satisfying(3.7) at timet( T)can be expressed as

u(t, r) = 1+ T

t



Q(t, r) = (R1111)cb31 eb2Tp

c

b1+c

zb3+2M

zb3+2,z+1,

cp

b1+c

(zb3+2)

(z+1)

+(R2

1)2c

b3+1 eb2Tp cb1+c

zb3 (zb3) (z+1)

M

zb3,z+1,cp

b1+c

+((R11)1+(R21)2+k)cb3 eb2Tp

c

b1+c

zb3+1 (zb3+1) (z+1)

M

zb3+1,z+1,cp

b1+c

, (3.43)

with

b1=

2 +22(1+1)2

, b2= b3b1b1b32 +1+(1R2)2,

b3=(1/2)2 +

((1/2)2)2 +2(1R2)22

2 ,

(3.44)


Theorem 3.9 The pre-default value of the CLN with counterparty risk and default intensities satisfying

(3.8) at timet( T)can be expressed as

u(t, r)=

1+

T

t



Q(t, r) = (R2221)cb31 eb2Tp

c

b1+c

zb3+2M

zb3+2,z+1,

cp

b1+c

(zb3+2)

(z+1)

+(R11)1cb3+1 eb2Tp

c

b1+c

zb3 (zb3) (z+1)


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Fig. 2. The values of CLNs with the CIR model.

M

zb3,z+1,cp

b1+c

+((R11)1+(R21)2+k)cb3 eb2Tp

c

b1+c

zb3+1 (zb3+1) (z+1)

M

zb3+1,z+1,cp

b1+

c

, (3.46)

with

b1=

2 +22(1+2R22)

2 , b2= b3 b1b1b32 +1+(1R2)2,

b3=(1/2)2 +

((1/2)2)2 +212

2 ,

(3.47)


4. Numerical analysis

In this section, numerical results are presented to show how the parameter values affect the price of the

CLN and the CVA.

We use the optimization procedure discussed in Nocedal & Wright (2006) to calibrate the parameters

of the CIR model using market data on 19 May 2011, which consists of the quotes of Libor, interest rate

futures and swaps. We obtain the following estimated parameters:

= 0.0904, = 0.0594, = 0.0295.


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Fig. 3. The values of CVA with the CIR model.

Fig. 4. The dependence of CLNs on 1 with the CIR model.

The parameters of hazard rates are assumed to be

CIR: 1= 2, 1= 0.16, 2= 1.5, 2= 0.1,ICIR : 1= 0.002, 1= 0.16, 2= 0.0015, 2= 0.1,

(4.1)

and other parameters are shown as follows: T= 1.5,k= 0.1,R1= 0.4,R2= 0.4.


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Fig. 5. The dependence of CLNs on 1 with the CIR model.

Fig. 6. The values of CLNs with the ICIR model.

When the default intensities of reference firm and CLN issuer are positively correlated, we consider

the situations where hazard rates satisfy CIR models as shown in Figs 25.The results for ICIR cases

are shown in Figs69.

Figures2and6 show the values of CLNs with and without counterparty risk. As we know, the

holder of CLNs with counterparty risk faces the default risk of issuer, so the values of these CLNs are

always higher than values of CLNs without counterparty risk. When the hazard rates follow the CIR

model, with the increase in interest rate, the discount factor decided by interest rate goes down and


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Fig. 7. The values of CVA with the ICIR model.

Fig. 8. The dependence of CLNs on1 with the ICIR model.

hazard rate goes up, so the values of CLNs decrease. When the hazard rates follow the ICIR model,

at first the values of CLNs increase with interest rate because of the decrease in hazard rate. But when

the interest rate becomes large enough, the discount factor plays a more important role, and the values

decrease, as shown in Fig.6.

Figures3and7show the CVA values of CLNs. In case of CIR model, when the interest rate is

small, the CVA of CLNs is positively correlated with it. Usually, high interest rate will have negative

impacts on market, and the investor will be subjected to relatively severe counterparty risk. However, if


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Fig. 9. The dependence of CLNs on1 with the ICIR model.

Fig. 10. The values of CLNs with default intensities satisfying (3.7).

the hazard rate of counterpart satisfies the ICIR model, high interest rate means that the issuer of CLNs

is less likely to default. So, the CVA of CLNs is negatively correlated with interest rate.

In Figs4,5,8 and 9,we investigate the dependence of CLNs values without counterparty risk on

parameters of hazard rates in both CIR and ICIR models. Higher hazard rate makes the reference asset

more likely to default, so values of CLNs decrease when the interest rate ascends. For CLNs values with

counterparty risk, the dependence structures are similar.


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Fig. 11. The values of CLNs with default intensities satisfying(3.8).

In Figs10and11,we investigate the situations where the default intensities of a reference firm and

CLN issuer are negatively correlated. We consider the CLNs with or without counterparty risk. The

dependence relations of parameters with negative correlation are similar with the cases with positive

correlation.

5. Conclusions

In the reduced-form framework, we obtain the analytic solutions of CLN values without and with coun-terparty risk. The formulas will help us to value the CLN contracts and to estimate the counterparty risk

much more convenient, compared with Monte Carlo or finite difference methods.

Acknowledgements

The authors thank Prof. Lishang Jiang, Prof. Srdjan Stojanovic and Dr Wei Wei for their valuable dis-

cussions. They are very grateful to the referees for their careful reading of the manuscript and several

constructive suggestions.

Funding

This work is supported in part by NSF of China under the grant 11271281 and 11371274.

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